Caio V Z Cipro
email: caiovzc@gmail.com

Yves Cherel

Florence Caurant

Pierre Miramand

Paula Méndez- Fernandez

Paco Bustamante

Trace elements in tissues of white-chinned-petrels (Procellaria aequinoctialis) from Kerguelen waters, Southern Indian Ocean

Keywords: Heavy metals, seabirds, Procellariiformes, Southern Ocean, subantarctic islands

The use of seabirds to assess marine contamination by trace elements in areas remote from pollutant emission points has already been done at various latitudes. Nevertheless, little information is available concerning the Southern Indian Ocean. Determining the contaminants levels there appears necessary not only due to several deleterious effects reported in literature, but also as previous studies have highlighted elevated concentrations of cadmium (Cd) and mercury (Hg) in mollusks, crustaceans and fish. Within this context, the white-chinned-petrel appears as a key species due to its lifespan, diet and trophic position. Thirty three accidentally killed (collision with lights/bycatch in longline vessels) individuals collected in Kerguelen waters were analysed for Cd, copper (Cu), Hg, selenium (Se) and zinc (Zn) in liver, kidney, pectoral muscle, feathers and for mature males, testis. Elevated Hg concentrations (average: 58.4 g g -1 dw in liver) are likely due to the presence of mesopelagic prey in the diet of P. aequinoctialis. Cd concentrations (average of 65.7 g g -1 dw in kidney) can be attributed to a high level of fisheries offal consumption, as well as crustacean and squid ingestion.

Correlation of Hg with Se indicates its detoxification by co-precipitation, and correlation of Cd with Zn suggests its displacement by Cd on metallothioneins binding sites. This work also indirectly confirms ecological data (range and diet composition) from the wintering period of the species, which is rather scarce. Seasonal diet change and moulting accounted more for the obtained results than sex of the birds.

INTRODUCTION

Contamination of the marine environment is a problem of concern at the worldwide scale. Oceanic ecosystems are often considered less impacted than coastal ones because they are not directly submitted to river influxes, runoff, and point-source pollution from urban, industrial and agricultural emissions [START_REF] Mailman | Heavy metals[END_REF]. Nonetheless, many of these contaminants, e.g. persistent organic pollutants or mercury (Hg) are transported by the atmosphere all over the world and they deposit into oceanic environments [START_REF] Fitzgerald | The case for atmospheric mercury contamination in remote areas[END_REF]. In the aquatic environment, contaminants such as trace elements are more mobile than in terrestrial ones [START_REF] Selin | Global Biogeochemical Cycling of Mercury: A Review[END_REF], and species foraging in these environments could, therefore, be considered more vulnerable.

Trace elements encompass essential elements involved in physiological and biochemical processes [START_REF] Abdulla | New aspects on the distribution and metabolism of essential trace elements after dietary exposure to toxic metals[END_REF]) that can however be toxic when exposure and assimilation are excessive. Others, such as cadmium (Cd) and Hg, are nonessential elements that are potentially toxic even at relatively low concentrations. Exposure of marine organisms is governed by various factors, including foraging location and trophic position [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF]. Among trace elements, Cd and Hg are known to bioaccumulate in food webs, resulting to high concentrations in long-lived top predator species such as marine birds. This process is particularly exacerbated at high latitudes [START_REF] Dietz | Have arctic marine mammals adapted to high cadmium levels?[END_REF][START_REF] Elliott | Heavy metal and metallothionein concentrations in seabirds from the Pacific coast of Canada[END_REF], where the baseline concentrations are probably higher than at more temperate latitudes [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF]. Food constitutes the main pathway of exposure of seabirds to contaminants (e.g., [START_REF] Ochoa-Acuña | Mercury in feathers from Chilean birds: influence of location, feeding strategy, and taxonomic affiliation[END_REF].

Nevertheless, numerous factors such as phylogeny, moult pattern, sex, life span and diet composition are likely to influence trace element bioaccumulation in seabird tissues, with Cd and Hg concentrations showing the largest variation among seabird species [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF][START_REF] Walsh | The use of seabirds as monitors of heavy metals in the marine environment[END_REF]. Because seabirds are long-lived species situated at the top of the food web, they are also submitted to the process of biomagnification of contaminants [START_REF] Furness | Seabirds as monitors of the marine environment[END_REF]. For these reasons, marine birds are considered to be useful bioindicators of environmental contamination [START_REF] Furness | Seabirds as monitors of the marine environment[END_REF][START_REF] Monteiro | Seabirds as monitors of mercury in the marine environment[END_REF]. They seem to be particularly appropriate to survey the health status of marine ecosystems as they generally display lower coefficients of variation of contaminant levels than fish or marine mammals [START_REF] Gilbertson | Seabirds as indicators of marine pollution[END_REF]. Furthermore, they are reflecting large areas integrating contaminants even from remote places that would be otherwise difficult to sample [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF][START_REF] Carravieri | Penguins as bioindicators of mercury contamination in the Southern Ocean: Birds from the Kerguelen Islands as a case study[END_REF]. The use of seabirds to monitor marine contamination by trace elements in areas remote from pollutant emission points has already been done at various latitudes [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF][START_REF] Fitzgerald | The case for atmospheric mercury contamination in remote areas[END_REF][START_REF] Kojadinovic | Trace elements in three marine birds breeding on Reunion Island (Western Indian ocean): part 1-factors influencing their bioaccumulation[END_REF]. However, information concerning the Southern Indian Ocean is still scarce. Only one study reported trace element concentrations in liver, kidney and muscle for 5 small petrels, prions and diving-petrels from the Kerguelen Islands, all of them mainly feeding on zooplankton [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF]. The study highlighted that the diet was a key factor driving the exposure and trace element levels in the tissues of these birds. More recently, Hg concentrations were determined in feathers from a large number of species, revealing a strong influence of the feeding behaviour on Hg concentrations [START_REF] Blévin | Wide range of mercury contamination in chicks of southern ocean seabirds[END_REF][START_REF] Carravieri | Penguins as bioindicators of mercury contamination in the Southern Ocean: Birds from the Kerguelen Islands as a case study[END_REF]. Identification of high levels of Cd and Hg in mollusks, crustaceans and fish from the Kerguelen waters [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF][START_REF] Bustamante | Distribution of trace elements in the tissues of benthic and pelagic fish from the Kerguelen Islands[END_REF]Bustamante et al., , 1998) ) would result in elevated exposure of seabirds. Moreover, several deletery effects of such elements have already been reported in birds, for instance: decrease in egg laying, egg hatchability and increase in embryonic mortality; and besides neurotoxicity, there is also evidence for immunotoxic effects, such as decreased immune response and histological changes [START_REF] Scheuhammer | Effects of environmental methylmercury on the health of wild birds, mammals, and fish[END_REF]Goutte et al. in press) and several endocrine effects as well (e.g. [START_REF] Tan | The endocrine effects of mercury in humans and wildlife[END_REF][START_REF] Tartu | To breed or not to breed: endocrine response to mercury contamination by an arctic seabird[END_REF]. In this context, the monitoring of the contaminants in the Southern Indian Ocean appears therefore a matter of importance.

Procellaria aequinoctialis, even though one of the most abundant seabirds in the Southern Ocean is classified as Vulnerable in the IUCN red list since literature presents historical declines of populations within the Southern Ocean, namely at Crozet [START_REF] Barbraud | Demographic response of a population of white-chinned petrels Procellaria aequinoctialis to climate and longline fishery bycatch[END_REF]) and South Georgia [START_REF] Berrow | Using acoustics to monitor burrownesting white-chinned petrels Procellaria aecquinoctialis at Bird Island, South Georgia[END_REF], but not conclusively in Kerguelen Islands [START_REF] Barbraud | Estimates of population size of white-chinned petrels and grey petrels at Kerguelen Islands and sensitivity to fisheries[END_REF]). Very high rates of incidental mortality in longline fisheries have been recorded in recent years (Birdlife International, 2013). The probability that these circumstances will continue and its susceptibility to predation and the degradation of breeding habitat indicate that a rapid and on-going population decline is likely (Birdlife International, 2013). Specifically for the Kerguelen waters, the mortality caused by fisheries constitutes a serious threat for P. aequinoctialis at least at the regional scale of the Southern Indian Ocean [START_REF] Barbraud | Estimates of population size of white-chinned petrels and grey petrels at Kerguelen Islands and sensitivity to fisheries[END_REF]. A similar conclusion was taken for the Crozet Islands population, even though estimates of the number of birds killed by the longline fishing vessels remained within the threshold levels (estimated around 8000 individuals per annum in this specific case) after 2003 [START_REF] Barbraud | Demographic response of a population of white-chinned petrels Procellaria aequinoctialis to climate and longline fishery bycatch[END_REF]. Association to commercial fisheries is due whether because P. aequinoctalis target the same areas of high productivity than fishermen and/or because fisheries appear to represent an additional potential food source for these opportunistic seabirds in the form of offal and discards of non-target species [START_REF] Delord | At-sea distribution and diet of an endangered top predator: relationship between white-chinned petrels and commercial longline fisheries[END_REF]. In their wintering grounds, i.e., the eastern boundary upwelling system of the Benguela current [START_REF] Péron | Seasonal variation in oceanographic habitat and behaviour of white-chinned petrels Procellaria aequinoctialis from Kerguelen Island[END_REF], P. aequinoctalis population's from the Kerguelen waters mainly feed on offal, i.e., fish heads and guts scavenged from commercial trawlers [START_REF] Jackson | Diets of the white-chinned petrel and sooty shearwater in the southern Benguela region, South Africa[END_REF].

Nevertheless, their diet also includes "natural" pelagic fish prey, crustaceans and cephalopods, in this order. Prey given to chicks during the breeding season is mainly composed of fish, followed by cephalopods and then crustaceans [START_REF] Delord | At-sea distribution and diet of an endangered top predator: relationship between white-chinned petrels and commercial longline fisheries[END_REF]). In the Crozet Islands (3.2 degrees north from Kerguelen), the diet of P. aequinoctalis is somewhat similar with fish prevailing, especially mesopelagic species [START_REF] Catard | Exploitation of distant Antarctic waters and close shelf-break waters by white-chinned petrels rearing chicks[END_REF][START_REF] Connan | Trophic relationships of whitechinned petrels from Crozet Islands: combined stomach oil and conventional dietary analyses[END_REF][START_REF] Ridoux | The diets and dietary segregation of seabirds at the subantarctic Crozet Islands (part 1)[END_REF]. However, crustaceans and cephalopods are important prey as well.

In this study, we examine the levels of Cd, copper (Cu), Hg, selenium (Se), and zinc (Zn) concentrations in the feathers, liver, kidney and muscle tissues (and also testis in a smaller extent) of the white-chinned-petrel P. aequinoctialis from Kerguelen waters using only incidentally killed animals. The second objective of this work was to make the relationship between trace element levels in internal tissues, reflecting the risk of toxicity for the birds, and their levels in the feathers that, if related to the internal organs, could provide a wider non-destructive monitoring basis [START_REF] Agusa | Body distribution of trace elements in blacktailed gulls from Rishiri Island, Japan: age-dependent accumulation and transfer to feather and eggs[END_REF].

MATERIALS AND METHODS

Sampling

Sample collection occurred from October/1998 to March/2000 at Kerguelen Archipelago, Southern Indian Ocean. Thirty three white-chinned petrels were collected as bycatch of longline fishing vessels or found recently dead on land killed following collision with the Scientific Station lights. Only intact or almost intact birds were considered for this study. The petrels were weighed in the field, immediately frozen and kept at -20°C until analyses. Birds were thawed overnight and dissected for liver, kidney, muscle (pectoral), testis (only on three mature males) and body feather removal, as body feathers (mainly pectoral ones, avoiding brood patch areas) give the best measure of the mercury contamination of a bird [START_REF] Furness | Seabirds as monitors of the marine environment[END_REF]. They were also sexed by visual observation of the gonads: 6 individuals could be identified as females and 12 as males.

Trace element analyses

Analyses were performed as described by [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF]: for Cd, Cu and Zn determination, two aliquots of approximately 200-300 mg of each homogenised dried sample were digested in 5 ml of suprapure 14 N nitric acid at 60°C on a hot plate until the solution was clear. After evaporation, the residues were dissolved in 10 ml of 0.3 N suprapure nitric acid. Trace elements were determined by flame Atomic Absorption Spectrophotometry (AAS) with a Varian spectrophotometer Spectra 250 Plus with a deuterium background correction (high Cd concentrations, Cu and Zn) and by graphite furnace AAS (Hitachi Z-5000) with Zeeman background correction (low Cd concentrations and Se). Feather analyses included a previous cleaning step in order to remove surface lipids and contaminants, by using a 2:1 chloroform:methanol solution followed by two successive methanol rinses. After cleaning, body feathers were oven dried for 48 h at 50 °C.

Mercury analysis was carried out with an Automatic Mercury Analyser spectrophotometer, ALTEC AMA 254, which does not require an acid-digestion of the samples. Aliquots ranging from 10 to 50 mg of dried sample were directly analysed after being inserted in the oven of the apparatus. After drying, the samples were heated under an oxygen atmosphere for 3 min, and the Hg liberated and subsequently amalgamated on an Aunet. The net was then heated to liberate the collected Hg, which was measured by AAS.

Accuracy and reproducibility of the methods were tested using dogfish liver (DOLT-2) and muscle (DORM-2) and lobster hepatopancreas (TORT-2) (National Research Council, Canada) reference standards. Standard and blanks were analysed along with each set of samples and recoveries of the certified values and recoveries of the metals ranged from 92 to 105%. Measurements were also validated by the IAEA Intercalibration Exercise [START_REF] Coquery | The analytical performance study for the Medpol region: determination of trace elements and methylmercury in estuarine sediment sample IAEA-405[END_REF]. Concentrations are expressed in dry weight in order to compensate eventual moisture loss during freezing and to facilitate comparison between tissues and with other studies. Blanks were analysed at the beginning of each set of samples and the detection limit of the method was 0.005 μg g -1 dry mass.

Statistical analyses.

Tests were performed using Microsoft Excel and Statsoft Statistica. Before analyses, data was checked for normality of distribution and homogeneity of variances using Shapiro-Wilk and Brown-Forsythe tests, respectively. Parametrical (Tukey HSD/ANOVA) and nonparametrical tests (Kruskal-Wallis/ANOVA) followed accordingly. Comparison between sexes was performed by means of Wilcoxon-Mann-Whitney test. Statistically significant results were set at =0.05.

RESULTS

Cd, Cu, Hg, Se and Zn concentrations in the tissues of P. aequinoctialis are presented in Table 1. No significant correlations (Pearson's product-moment) were detected between tissue concentrations (both in wet weight and dry weight basis) and total body weight of the birds. The results overall showed that liver and kidney are the main sites for bioaccumulation of non essential trace elements, with mean values of 58.4 ± 33.4 g g -1 dw of Hg in the liver and 65.7 ± 22.2 g g -1 dw of Cd in the kidney. In these tissues, Cd and Hg concentrations were one order of magnitude higher than those of muscle and feathers, which were the only tissues that did not present statistically significant difference for both contaminants. In contrast, Cd concentrations in the testis were as high as in the kidney.

Cu presented similar concentrations and ranges in liver, kidney, muscle and testis.

Nevertheless, Cu variability was always lower than 30% in the 3 former tissues whereas it was above 70% in the testis, mostly due to the small sampling number (n=3) that precluded further statistical analyses. Feathers showed the lowest concentrations in average, but with a relatively high variability (coefficient of variation of 43%). Se concentrations were the lowest in the liver and highest in the kidney, with average concentrations of 6.2 ± 2.0 and 16.1 ± 4.8 g g -1 dw, respectively. Nevertheless, the variation of Se concentrations remained always < 35%. For Zn, the concentrations were the lowest in muscles and highest in feathers in which the variability was relatively high (coefficient of variation of 73%). Nevertheless, this difference was statistically significant. Only muscle and liver, and liver and kidney (paired) did not present such differences. The other tissues, liver, kidney and testis, showed similar concentrations that were intermediate between those of muscle and feathers.

Males and females have shown significant contaminant differences in only one case:

Cd in muscle was higher in males. All other comparisons were not statistically significant.

Inter and intra-tissue correlations are shown in Table 2. Significant intra-tissue correlations appeared mostly within liver and kidney and to a lesser extent in muscle, whereas no correlation was found in feathers. In liver, Cd was positively correlated to Hg, Se and Zn, Cu was positively correlated to Zn, and Se to Hg. In kidney, Se was positively correlated to Cd, Cu and Zn and Zn was positively correlated to Cd, Cu, and Se. In muscle, only Cu was negatively correlated to Zn and Se positively correlated to Hg.

Almost no significant inter-tissue correlations were detected for Cu and Zn, with only concentrations in the liver being negatively correlated to those in the feathers. Significant inter-tissue correlations appear mostly between the liver and kidney for non essential elements but also for Se. Thus, Cd concentrations in the liver correlated positively with those in the kidney but negatively with those in the feathers. Concerning Hg, concentrations in the liver positively correlated to those in kidney and in muscle. Similar correlations were found for Se, as well as a significant correlation between liver and muscle Se concentrations.

In order to better understand detoxification processes, other correlations were also investigated for non essential elements (Fig. 1 & 2). Figure 1 presents the liver Hg concentrations as a function of kidney Cd concentrations. The relationship between both non essential elements in their main tissues of bioaccumulation was however not significant.

Figure 2 presents the molar relationship between Se and Hg. It highlighted a significant increase of the number of atoms of Se with those of Hg in the liver, but not in the kidney.

DISCUSSION

Limited information is currently available for trace element concentrations in the tissues of the P. aequinoctalis with, to the best of our knowledge, only three studies reported such levels for feathers [START_REF] Blévin | Wide range of mercury contamination in chicks of southern ocean seabirds[END_REF], for soft tissues [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF] and for both [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF]. The concentrations of trace elements in liver and kidney of P. aequinoctialis from Kerguelen waters were overall in the same order of magnitude than those reported for P. aequinoctialis collected in 1995 and 1996 as bycatch of tuna fisheries around New Zealand [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF]. Moreover, it presents the same distribution pattern between liver and kidney tissues for all elements (except Se which was not analysed by these authors). However, Hg concentrations in the feathers of adult birds from the present study were obviously higher (4.2 times) than those in chicks from the same location as a result of longer term exposure in adults [START_REF] Blévin | Wide range of mercury contamination in chicks of southern ocean seabirds[END_REF].

In birds from Kerguelen waters, Cu concentrations in feathers were half those of P. aequinoctialis collected in Bird Island, South Georgia, during the 2001/02 austral summer [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF]. In contrast, Zn values from the present study were 1.5 times higher when compared to Bird Island seabirds. Even though trace elements contamination is known to have several deleterious effects in birds testis [START_REF] Scheuhammer | The chronic toxicity of aluminium, cadmium, mercury, and lead in birds: a review[END_REF], data for analyses of such tissue is scarce. [START_REF] Kim | Distribution of 14 elements in tissues and organs of oceanic seabirds[END_REF] present data with similar distribution and, in a general way, within the same orders of magnitude as the present work. Elevated Cu variation in this tissue might be linked to body condition, as suggested by [START_REF] Malinga | Age, sex and spatial dependent variations in heavy metals levels in the Glaucous Gulls (Larus hyperboreus) from the Bjørnøya and Jan Mayen, Arctic[END_REF]. These authors also suggested that the variation of Cu concentrations could be due to different stages of development of the gonads, which was visually confirmed in the present study. More generally, sex might play a role in trace element distribution in some seabird species [START_REF] Kim | Distribution of 14 elements in tissues and organs of oceanic seabirds[END_REF]. Sexual size dimorphism or different age composition in each gender (see [START_REF] Bugoni | Age composition and sexual size dimorphism of albatrosses and petrels off Brazil[END_REF]) could affect contaminant levels, but the main driving sex-related factor is the excretion of potentially toxic materials into the egg that reduce the contaminant burdens of breeding females [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF]. Combined to the different turnover rates of the different tissues, it is most likely that the significantly higher Cd muscle concentrations in males than in females of white-chinned petrels from the Kerguelen waters could be explained by the muscle turnover possibly matching the egg laying period, since it is long known that this tissue has a slower turnover rate than liver [START_REF] Hobson | Assessing avian diets using stable isotopes I: turnover of 13 C in tissues[END_REF]. Moreover, no dilution with growth or emaciation effect was detected, as previously stated. So, further discussion considers the whole grouped dataset, with no sex separation. [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF] classified procellariiform seabirds in function of their pattern of accumulation of Hg in the liver and Cd in the kidney. White-chinned petrels from Kerguelen fall into the same category as Procellaria petrels (including the white-chinned petrel) from New Zealand that are characterized by moderately high liver Hg and high kidney Cd concentrations. No significant relationship between liver Hg and kidney Cd was found, which can be attributed to the distinctive biochemistry of the two metals [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF].

Influence of trophic ecology on trace element concentrations

Elevated Hg concentrations in the tissues of P. aequinoctialis are likely due to the presence of mesopelagic prey in its diet that is mainly composed of fish and cephalopods in Kerguelen waters (66 and 17% by mass, respectively; [START_REF] Delord | At-sea distribution and diet of an endangered top predator: relationship between white-chinned petrels and commercial longline fisheries[END_REF]. In marine predators, trophic position is not always the main factor in determining Hg concentrations, because mesopelagic prey was shown to increase disproportionately their Hg burdens [START_REF] Ochoa-Acuña | Mercury in feathers from Chilean birds: influence of location, feeding strategy, and taxonomic affiliation[END_REF][START_REF] Thompson | Seabirds as biomonitors of mercury inputs to epipelagic and mesopelagic marine food chains[END_REF]. Indeed, mesopelagic fish have elevated Hg concentrations as the result of enhanced Hg methylation in deep waters [START_REF] Chouvelon | Enhanced bioaccumulation of mercury in deep-sea fauna from the Bay of Biscay (north-east Atlantic) in relation to trophic positions identified by analysis of carbon and nitrogen stable isotopes[END_REF][START_REF] Monteiro | Mercury concentrations in prey fish indicate enhanced bioaccumulation in mesopelagic environments[END_REF]. Cephalopods can also be considered as an important predator Hg source since they mainly contain Hg under organic forms, which is highly bioavailable for the upper trophic levels (Bustamante et al., 2006). Long-term Hg exposure of P. aequinoctalis population from Kerguelen waters is also influenced by the species feeding habits outside the breeding period.

For instance, P. aequinoctalis winters in the Benguela current, with associated dietary changes [START_REF] Jackson | Diets of the white-chinned petrel and sooty shearwater in the southern Benguela region, South Africa[END_REF]. In this area, fish is still the main prey class (73% by mass), but it includes an important proportion (36%) of offal, mainly fish heads and guts discarded by fishing vessels that keep aboard the commercially interesting part, i.e. the muscle. Muscle is one of the main tissues in fish with the lowest concentrations of trace elements (except Hg and arsenic; [START_REF] Bustamante | Distribution of trace elements in the tissues of benthic and pelagic fish from the Kerguelen Islands[END_REF][START_REF] Kojadinovic | Trace elements in three marine birds breeding on Reunion Island (Western Indian ocean): part 1-factors influencing their bioaccumulation[END_REF][START_REF] Metian | Trace element bioaccumulation in reef fish from New Caledonia: Influence of trophic groups and risk assessment for consumers[END_REF]. Because trace element concentrations in muscle are below concentrations in the whole fish, the consumption of offal could represent a higher intake of trace elements, especially Cd. In Kerguelen fish, liver contains up to 87% of the whole Cd body burden with concentrations >10 µg g -1 dw [START_REF] Bustamante | Distribution of trace elements in the tissues of benthic and pelagic fish from the Kerguelen Islands[END_REF]. Therefore, interactions with fisheries both during and outside the breeding season would lead to an increase intake of Cd in addition to Cd contamination resulting from squid consumption (Bustamante et al., 1998a;[START_REF] Muirhead | Heavy metal concentrations in the tissues of seabirds from Gough Island, South Atlantic Ocean[END_REF]. Indeed, Cd levels in P.aequinoctalis tissues were in the same range as those of species with a higher proportion of cephalopods in their diets, such as albatrosses. For example, the Laysan albatross (Phoebastria immutabilis) presents average renal Cd levels of 59.7 g g -1 dw [START_REF] Honda | Metal concentrations in pelagic seabirds from the North Pacific Ocean[END_REF] and Southern Ocean albatrosses (Diomedea exulans, D. epomorpha, D. cauta, D. bulleri and D. melanophris) show renal Cd levels ranging from 74 to 132 g g -1 dw [START_REF] Stewart | Influence of phylogeny, diet, moult schedule and sex on heavy metal concentrations in New Zealand Procellariiformes[END_REF]. Crustaceans are also likely to contribute to Cd exposure in P. aequinoctalis. For instance, the amphipods Themisto spp. (notably T. gaudichaudii) is naturally enriched in Cd [START_REF] Ritterhoff | Influence of body length, life-history status and sex on trace metal concentrations in selected zooplankton collectives from the Greenland Sea[END_REF][START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF] and may represent a significant source of this element via both direct consumption [START_REF] Catard | Exploitation of distant Antarctic waters and close shelf-break waters by white-chinned petrels rearing chicks[END_REF]Delord et al., 2010) and secondary ingestion, since this crustacean contributes significantly to the diet of several petrel prey [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF]. Assuming that offal have a Cd concentration of 10, "natural prey" fish of 0.1, cephalopods of 25, Themisto spp. of 40 and other crustaceans of 1 g g -1 dw, an estimation of prey contribution in the Benguela waters suggest that cephalopods, offal, crustaceans and natural fish prey would be responsible for approximately 51.8%, 43.5%, 1.2% and 0.4% of the total Cd intake, respectively. Therefore, the hypothesis of offals influencing greatly the birds' Cd levels is likely to be verified. Such calculations were made from diet analysis by mass in the Benguela Region [START_REF] Jackson | Diets of the white-chinned petrel and sooty shearwater in the southern Benguela region, South Africa[END_REF] together with available Cd concentrations of prey from the southern Ocean [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF][START_REF] Bustamante | Distribution of trace elements in the tissues of benthic and pelagic fish from the Kerguelen Islands[END_REF][START_REF] Ritterhoff | Influence of body length, life-history status and sex on trace metal concentrations in selected zooplankton collectives from the Greenland Sea[END_REF].

Diet is the main pathway of incorporation of non-essential but also essential elements in seabirds. Studies on trace elements in prey showed generally higher values for Cu and Zn in cephalopods when compared to mesopelagic fish [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF][START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF]Bustamante et al., 2006a[START_REF] Bustamante | Distribution of trace elements in the tissues of benthic and pelagic fish from the Kerguelen Islands[END_REF]. Moreover, the antagonism between Cd, Cu and Zn could modify absorption, retention and distribution of the other two [START_REF] Underwood | Trace elements in human and animal nutrition[END_REF].

Nevertheless, homeostatic control of internal essential elements plays a major role regulating their absorption in vertebrates, which depends on the individual nutritional requirements [START_REF] Walsh | The use of seabirds as monitors of heavy metals in the marine environment[END_REF]. In P. aequinoctalis from the present study, the correlations of Cu with Zn would reflect their nutritional status. Thus, low intraspecific variation of Cu and Zn concentrations should be expected in the tissues of marine birds in good health. In P. aequinoctialis from Kerguelen waters, Cu and Zn averages were close to the majority of other Kerguelen species (authors' unpublished data), thus highlighting their homeostatic regulation, which is also confirmed by their comparatively low relative standard deviations.

Inter-and intra-tissues relationships between trace elements

Inter-and intra-tissue correlations between elements reflect the different processes of regulation of essential elements and of detoxification of non essential elements.

In the case of intra-tissue analyses, in the liver, Cd positively correlated with Zn, Se and Hg, and Hg positively correlated with Se, which is consistent with previous data for procellariiforms seabirds such as the black-footed albatross Diomedea nigripes from Ryukyu Islands [START_REF] Kim | Distribution of 14 elements in tissues and organs of oceanic seabirds[END_REF]. In regard to Cd and Zn co-accumulation, their correlation is likely to reflect Cd detoxification by metallothioneins (MTs) as already reported for different seabird groups (e.g., [START_REF] Kojadinovic | Trace elements in three marine birds breeding on Reunion Island (Western Indian ocean): part 1-factors influencing their bioaccumulation[END_REF][START_REF] Ninomiya | Metal concentrations in the liver and kidney of aquatic mammals and penguins[END_REF]. Indeed, the great affinity of Cd for MTs leads to the competition with Zn for binding sites and to a reduction of the toxic effects of Cd [START_REF] Klaassen | Metallothionein: an intracellular protein to protect against cadmium toxicity[END_REF], whereas the Zn displacement would induce MT synthesis, increasing the number of binding sites within the cells. In regard to the coaccumulation of Hg with Se, their correlation (the highest coefficient found in the whole study) is indicative of Hg detoxification mechanism involving the biomineralization and storage of non-toxic Se-Hg complex and formation of liver granules when elimination through the feathers cannot match the individual's dietary Hg intake [START_REF] Nigro | Intracellular storage of mercury and selenium in different marine vertebrates[END_REF]. Such a process seems to be operative in P. aequinoctalis from Kerguelen waters in the liver but not in the kidney (Figure 2). Moreover, in the kidney, correlation of Se with Zn could be associated with their interaction in detoxifying processes, even though according to [START_REF] Norheim | Levels and interactions of heavy metals in sea birds from Svalbard and the Antarctic[END_REF] Se plays a more important role. Se correlation with Cu could be inferred from the consumption of some crustacean prey, notably E. superba and T. gaudichaudii (see [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF], since the physico-chemical form in which Se is sequestered by different prey may vary and affect its subsequent bioavailability. Hence, it would appear that intra-specific variation in blood Se concentrations in seabirds most likely reflects individual differences in prey selection (see standard deviation in diet composition in [START_REF] Delord | At-sea distribution and diet of an endangered top predator: relationship between white-chinned petrels and commercial longline fisheries[END_REF].

Besides Hg, Se is involved in detoxification of other heavy metals such as Ag, Cd, Cu and Zn [START_REF] Ikemoto | Detoxification mechanism of heavy metals in marine mammals and seabirds: interaction of selenium with mercury, silver, copper, zinc, and cadmium in liver[END_REF]. In regard to inter-tissue analyses, Se significant positive correlations between liver, muscle and kidney are probably linked to the detoxifying mechanisms it is involved in, which is corroborated by its significant positive correlation with other elements, notably Hg in liver and muscle tissues. Hg significant positive correlation in liver with muscle tissue was already reported for several bird species (e.g. [START_REF] Borgå | Regional and species specific bioaccumulation of major and trace elements in Arctic seabirds[END_REF] and is assumed to result from the bioaccumulation of this element.

The significant positive correlations of Cd and Hg in both the liver and kidney confirm the role of these tissues as the main sites for detoxification and long term storage (e.g. [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF]. The negative correlation between Cd concentrations in liver and Cd concentrations in feather could be related to a different diet during the non breeding period (higher offal proportion) when moulting normally occurs [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF] and in the breeding period, when samples were collected. Indeed, liver has a high protein turnover rate [START_REF] Cherel | Whole-body and tissue protein synthesis during brief and prolonged fasting in the rat[END_REF] that reflects diet a few days before collection. The Zn negative correlation between liver and feather could result from the same processes, which is confirmed by its significant correlation with Cd in the liver. This shift towards liver coaccumulation can happen in periods as short as one or two weeks, as demonstrated by [START_REF] Levengood | Coaccumulation of cadmium and zinc in tissues of sentinel mallards (Anas platyrhynchos) using a former dredge-disposal impoundment[END_REF]. In other words, Cd and Zn are co-accumulated in the liver (emphasised by the significant intra-tissue correlation) and before being released from the liver to the feathers (emphasised by the negative inter-tissue correlation between liver and feathers). Even though, Zn in feathers presented large variation. This could be due to either molecular interactions between Cd, Cu, and Zn or to Zn affinity for liver metallothioneins [START_REF] Bocher | Influence of the diet on the bioaccumulation of heavy metals in zooplankton-eating petrels at Kerguelen archipelago, Southern Indian Ocean[END_REF].

Finally, in regard to the use of feathers as a non destructive monitoring tool, it is noticeable that they presented significant inter-tissue correlation in only two occasions: Zn and Cd, both with liver and both negative as well. No significant correlation to other trace elements was found for none of the tissues, possibly due to two main factors: firstly, the sampling of the birds occurred in different times of the year and not only that, but consequently in different time shifts in regard to the moult. This might result in a degree of fluctuation in the trace element concentration of the internal tissues and, therefore, the static concentration of the feathers may no longer be correlated to those. The second hypothesised factor is that moult occurs normally during the non breeding period [START_REF] Anderson | Element patterns in albatrosses and petrels: influence of trophic position, foraging range, and prey type[END_REF], when diet is different [START_REF] Jackson | Diets of the white-chinned petrel and sooty shearwater in the southern Benguela region, South Africa[END_REF], therefore the variation in internal tissues could be enhanced due to a variation in exposure via diet. A further study with birds sampled in a narrower time interval and with a homogenous set of animals in regard to the last moult could better answer the question whether feathers can be used to monitor trace elements in this species and under what conditions.

Figure 1 -Figure 2 -

 12 Figure 1 -Concentrations of liver Hg versus kidney Cd (µg•g -1 dw) in white-chinned petrels P. aequinoctialis from Kerguelen waters.

Table 1 -

 1 Trace element concentrations (mean ± SD and range; µg•g -1 dw) in tissues of whitechinned petrels P. aequinoctialis from Kerguelen waters. Sampling numbers are given accordingly.

	Tissue	n	Cd	n	Cu	n	Hg	n	Se	n	Zn
	Kidney	29	65.7 ± 22.2	29	15.6 ± 4.4	27	23.6 ± 15.5	24	16.1 ± 4.8	29	131 ± 23
			26.4 -121		11.8 -35.7		8.91 -91.8		7.5 -25.2		96 -200
	Liver	26	20.5 ± 6.5	26	16.2 ± 4.6	26	58.4 ± 33.4	23	6.2 ± 2.0	26	127 ± 31
			7.43 -37.0		10.6 -31.8		17.6 -152		3.4 -10.4		80 -193
	Muscle	32	1.80 ± 0.82	32	12.0 ± 2.2	32	2.86 ± 0.80	31	9.7 ± 3.4	32	80 ± 24
			0.86 -4.04		8.2 -15.7		1.28 -4.54		5.3 -21.0		54 -134
	Feather	22	0.66 ± 0.39	22	6.0 ± 2.6	22	7.63 ± 3.87	-	-	22	196 ± 143
			0.18 -1.53		2.8 -11.8		2.30 -14.2				100 -797
	Testis	3	42.7 ± 14.4	3 19.2 ± 13.7	-	-	-	-	3	145 ± 15
			30.8 -58.8		11.3 -35.0						128 -156

Acknowledgements

The authors thank the fieldworkers who helped with collecting the petrels. The present work was supported financially and logistically by the Agence Nationale de la Recherche (program POLARTOP, O. Chastel), the Institut Polaire Français Paul Emile Victor (IPEV, program no. 109, H. Weimerskirch) and the Terres Australes et Antarctiques Françaises (TAAF). C.V.Z. Cipro was supported financially through a post-doctoral grant from the University of La Rochelle and the CNRS (French acronym for National Council of Scientific Research).


Sana Stojanović

Julien Narboux
email: narboux@unistra.fr

Marc Bezem
email: marc.bezem@ii.uib.no

Predrag Janičić
email: janicic@matf.bg.ac.rs

A Vernacular for Coherent Logic ⋆

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a fragment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many everyday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in L A T E X or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.

Introduction

Mathematics can be done on two different levels. One level is rather informal, based on informal explanations, intuition, diagrams, etc., and typical for everyday mathematical practice. Another level is formal mathematics with proofs rigorously constructed by rules of inference from axioms. A large portion of mathematical logic and interactive theorem proving is aimed at linking these two levels. However, there is still a big gap: mathematicians still don't feel comfortable doing mathematics formally and proof assistants still don't provide enough support for dealing with large mathematical theories, automating technical problems, translating from one formalism to another, etc. We consider the following issue: there are several very mature and popular interactive theorem provers (including Isabelle, Coq, Mizar, HOL-light, see [START_REF] Wiedijk | The Seventeen Provers of the World[END_REF] for an overview), but they still cannot easily share the same mathematical knowledge. This is a significant problem, because there are increasing efforts in building repositories ⋆ The first, second and the fourth author were partly supported by the Serbian-French Technology Co-Operation grant EGIDE/"Pavle Savić" 680-00-132/2012-09/12. The first and the fourth author are partly supported by the grant ON174021 of the Ministry of Science of Serbia. The final publication is available at http://link. springer.com.

of formalized mathematics, but -still developed within specific proof assistants. Building a mechanism for translation between different proof assistants is non-trivial because of many deep specifics of each proof assistant (there are some recent promissing approaches for this task [START_REF] Kaliszyk | Scalable LCF-Style Proof Translation[END_REF]). Instead of developing a translation mechanism, we propose a proof representation and a corresponding XML-based format. The proposed proof representation is light-weight and it does not aim at covering full power of everyday mathematical proofs or full power of first order logic. Still, it can cover a significant portion of many interesting mathematical theories. The underlying logic of our representation is coherent logic, a fragment of first-order logic. Proofs in this format can be generated in an easy way by dedicated, coherent logic provers, but in principle, also by standard theorem provers. The proofs can be translated to a range of proof assistant formats, enabling sharing the same developments. We call our proof representation "coherent logic vernacular". Vernacular is the everyday, ordinary language (in contrast to the official, literary language) of the people of some country or region. A similar term, mathematical vernacular was used in 1980's by de Bruijn within his formalism proposed for trying to put a substantial part of the mathematical vernacular into the formal system [START_REF] Govert De Bruijn | The Mathematical Vernacular, a Language for Mathematics with Typed Sets[END_REF]. Several authors later modified or extended de Bruijn's framework. Wiedijk follows de Bruijn's motivation [START_REF] Wiedijk | Mathematical Vernacular[END_REF], but he also notices:

It turns out that in a significant number of systems ('proof assistants') one encounters languages that look almost the same. Apparently there is a canonical style of presenting mathematics that people discover independently: something like a natural mathematical vernacular. Because this language apparently is something that people arrive at independently, we might call it the mathematical vernacular.

We find that this language is actually closely related to a proof language of coherent logic, which is a basis of our proof representation presented in this paper.

Our proof representation is developed also with readable proofs in mind. Readable proofs (e.g., textbook-like proofs), are very important in mathematical practice. For mathematicians, the main goal is often, not only a trusted, but also a clear and intuitive proof. We believe that coherent logic is very well suited for automated theorem proving with a simple production of readable proofs.

Background

In this section, we give a brief overview of interactive theorem proving and proof assistants, of coherent logic, which is the logical basis for our proof representation, and of XML, which is the technical basis for our proof format.

Interactive Theorem Proving

Interactive theorem proving systems (or proof assistants) support the construction of formal proofs by a human, and verify each proof step with respect to the given underlying logic. The proofs can be written either in a declarative or in a procedural proof style. In the procedural proof style, the proof is described by a sequence of commands which modify the incomplete proof tree. In the declarative proof style the formal document includes the intermediate statements. Both styles are avaible in HOL-Light, Isabelle [START_REF] Wenzel | Isar -A Generic Interpretative Approach to Readable Formal Proof Documents[END_REF] and Coq proof assistants whereas only the declarative style is available in Mizar, see [START_REF] Wiedijk | A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving[END_REF] for a recent discussion. The procedural proof style is more popular in the Coq community.

Formal proofs are typically much longer than "traditional proofs". 4 Progress in the field can be measured by proof scripts becoming shorter and yet contain enough information for the system to construct and verify the full (formal) proof. "Traditional proofs" can often hardly be called proofs, because of the many missing parts, informal arguments, etc. Using interactive theorem proving uncovered many flaws in many published mathematical proofs (including some seminal ones), published in books and journals.

Coherent Logic

Coherent logic (CL) was initially defined by Skolem and in recent years it gained new attention [START_REF] Bezem | Automating Coherent Logic[END_REF][START_REF] Fisher | Skolem Machines and Geometric Logic[END_REF][START_REF] Bezem | On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic[END_REF]. It consists of formulae of the following form:

A 1 (x) ∧ . . . ∧ A n (x) ⇒ ∃y(B 1 (x, y) ∨ . . . ∨ B m (x, y)) (1)
which are implicitly universally quantified, and where 0 ≤ n, 0 ≤ m, x denotes a sequence of variables x 1 , x 2 , . . . , x k (0 ≤ k), A i (for 1 ≤ i ≤ n) denotes an atomic formula (involving zero or more of the variables from x), y denotes a sequence of variables y 1 , y 2 , . . . , y l (0 ≤ l), and B j (for 1 ≤ j ≤ m) denotes a conjunction of atomic formulae (involving zero or more of the variables from x and y). For simplicity, we assume that there are no function symbols with arity greater than zero (so, we only consider symbols of constants as ground terms). The definition of CL does not involve negation. For a single atom A, ¬A can be represented in the form A ⇒ ⊥, where ⊥ stands for the empty disjunction, but more general negation must be expressed carefully in coherent logic. In order to reason with negation in general, new predicate symbols are used to abbreviate subformulas. Furthermore, for every predicate symbol R (that appears in negated form), a new symbol R is introduced that stands for ¬R, and the following axioms are postulated (cf. [START_REF] Polonsky | Proofs, Types and Lambda Calculus[END_REF]):

∀x(R(x) ∧ R(x) ⇒ ⊥), ∀x(R(x) ∨ R(x)).
CL allows existential quantifications of the conclusion of a formula, so CL can be considered to be an extension of resolution logic. In contrast to the resolution-based proving, the conjecture being proved is kept unchanged and directly proved (refutation, Skolemization and transformation to clausal form are not used). Hence, proofs in CL are natural and intuitive and reasoning is constructive. Readable proofs (in the style of forward reasoning and a variant of natural deduction) can easily be obtained [START_REF] Bezem | Automating Coherent Logic[END_REF].

A number of theories and theorems can be formulated directly and simply in CL. In CL, constructive provability is the same as classical provability. It can be proved that any first-order formula can be translated into a set of CL formulas (in a different signature) preserving satisfiability [START_REF] Polonsky | Proofs, Types and Lambda Calculus[END_REF] (however, this translation does not always preserve constructive provability).

Coherent logic is semi-decidable and there are several implemented semidecision procedures for it [START_REF] Bezem | Automating Coherent Logic[END_REF]. ArgoCLP [START_REF] Stojanović | A Coherent Logic Based Geometry Theorem Prover Capable of Producing Formal and Readable Proofs[END_REF] is a generic theorem prover for coherent logic, based on a simple proof procedure with forward chaining and with iterative deepening. ArgoCLP can read problems given in TPTP form5 [START_REF] Sutcliffe | The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0[END_REF] and can export proofs in the XML format that we describe in this paper. These proofs are then translated into target languages, for instance, the Isar language or natural language thanks to appropriate XSLT style-sheets.

XML

Extensible Markup Language (XML)6 is a simple, flexible text format, inspired by SGML (ISO 8879), for data structuring using tags and for interchanging information between different computing systems. XML is primarily a "metalanguage"a language for describing other customized markup languages. So, it is not a fixed format like the markup language HTML-in XML the tags indicate the semantic structure of the document, rather than only its layout. XML is a project of the World Wide Web Consortium (W3C) and is a public format. Almost all browsers that are currently in use support XML natively.

There are several schema languages for formaly specifying the structure and content of XML documents of one class. Some of the main schema languages are DTD (Data Type Definition), XML Schema, Relax, etc. [START_REF] Lee | Comparative analysis of six xml schema languages[END_REF]. Specifications in the form of schema languages enable automatic verification ("validation") of whether a specific document meets the given syntactical restrictions.

Extensible style-sheet language transformation (XSLT)7 is a document processing language that is used to transform the input XML documents to output files. An XSLT style-sheet declares a set of rules (templates) for an XSLT processor to use when interpreting the contents of an input XML document. These rules tell to the xslt processor how that data should be presented: as an XML document, as an html document, as plain text, or in some other form.

Proof Representation

The proposed proof representation is very usable and expressive, yet very simple. It uses only a few inference rules, a variant of the rules given in [START_REF] Bezem | On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic[END_REF]. Given a set of coherent axioms AX and a coherent conjecture

A 1 (x) ∧ . . . ∧ A n (x) ⇒ ∃y(B 1 (x, y) ∨ . . . ∨ B m (x, y)
), the goal is to prove, using the rules given below, the following (where a denote a vector of new symbols of constants):

AX, A 1 (a) ∧ . . . ∧ A n (a) ⊢ ∃y(B 1 (a, y) ∨ . . . ∨ B m (a, y))
The rules are applied in a forward manner, so they can be read from bottom to top. In the rules below we assume:

ax ∈ AX is a formula of the form (1) (page 3); a, b, c denote vectors of constants (possibly of length zero); -in the rule mp, b are fresh constants; x and y denote vectors of variables (possibly of length zero); -A i (x) (B i (x, y)) have no free variables other than from x (and y); -A i (a) are ground atomic formulae; -B i (a, b) and B i (c) are ground conjunctions of atomic formulae; -Φ denotes the list of conjuncts in Φ.

Γ, ax, A 1 (a) ∧ . . . ∧ A n (a), B 1 (a, b) ∨ . . . ∨ B m (a, b) ⊢ P Γ, ax, A 1 (a) ∧ . . . ∧ A n (a) ⊢ P mp (modus ponens) Γ, B 1 (c) ⊢ P . . . Γ, B n (c) ⊢ P Γ, B 1 (c) ∨ . . . ∨ B n (c) ⊢ P cs (case split) Γ, B i (a, b) ⊢ ∃y(B 1 (a, y) ∨ . . . ∨ B m (a, y)) as (assumption)
Γ, ⊥ ⊢ P efq (ex falso quodlibet)

None of these rules change the goal P , which helps generating readable proofs as the goal can be kept implicit. Note that the rule mp actually combines universal instantiation, conjunction introduction, modus ponens, and elimination of (zero or more) existential quantifiers. This seems a reasonable granularity for an inference step, albeit probably the maximum for keeping proofs readable. Compared to [START_REF] Rudnicki | Obvious inferences[END_REF] which defines the notion of obvious inference rule by putting constraints on an automated prover, our position is: the obvious inferences are the ones defined by the inference rules above. Compared to the rules given in [START_REF] Bezem | On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic[END_REF], we choose to separate the case split rule (disjunction elimination) and the ex falso quodlibet rule from the single combined rule in [START_REF] Bezem | On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic[END_REF], in order to improve readability. Case distinction (split) is an important way of structuring proofs that deserves to be made explicit. Also, ex falso quodlibet could be seen as a case split with zero cases, but this would be less readable. Any coherent logic proof can be represented in the following simple way (mp is used zero or more time, cs involves at least two other proof objects):

proof ::= mp * (cs(proof ≥2 ) | as | efq)
The proof representation described in Section 3 is used as a basis for our XMLbased proof format. It is developed as an interchange format for automated and interactive theorem provers. Proofs (for Coq and Isabelle/Isar) that are produced from our XML documents are fairly readable. The XML documents themselves can be read by a human, but much better alternative is using translation to human readable proofs in natural language (formatted in L A T E X, for instance). The proof representation is described by a DTD Vernacular.dtd. As an illustration, we show some fragments: The above fragment describes the notion of a theorem and a proof. As said in Section 3, a proof consists of a sequence of applications of the rule modus ponens and closes with one of the remaining proof rules (case split, as, or efq). Within the last three, there is the additional information on whether the proof closes by ⊥ (by detecting a contradiction) or by detecting one of the disjuncts from the goal. This information is generated by the prover and can be used for better readability of the proof but also for some potential proof transformations. Within each proof step there is also the information on indentation. This information, useful for better layout, tells the level of subproofs and as such can be, in principle, computed from the XML representation. Still, for convenience and simplicity of the XSLT style-sheets, it is stored within the XML representation.

We implemented XSL transformations from XML format to Isabelle/Isar (VernacularISAR.xls), Coq (VernacularCoqTactics.xls), and to a natural language (English) in L A T E X form and in HTML form (VernacularTex.xls and VernacularHTML.xls).

The translation from XML to the Isar language is straightforward and each of our proof steps is trivially translated into Isar constructs. 8 Naturally, we use native negation of Isar (and Coq) instead of defined negation in coherent logic.

The translation to Coq has been written in the same spirit as the Isar output despite the fact proofs using tactics are more popular in Coq than declarative proofs. We refer to the assumptions by their statement instead of their name (for example: by cases on (A = B \/ A <> B)). Moreover, when we can, we avoid to refer to the assumptions at all. We did not use the declarative proof mode of Coq because of efficiency issues. We use our own tactics to implement the inference rules of CL to improve readability. Internally, we use an Ltac tactic to get the name of an assumption. The forward reasoning proof steps consist of applications of the assert tactic of Coq. Equality is translated into Leibniz equality.

The translation to L A T E X and HTML includes an additional XSLT stylesheet that optionally defines specific layout for specific relation symbols (so, for instance, (A, B) ∼ = (C, D) can be the layout for cong(A,B,C,D)).

The developed XSLT style-sheets are rather simple and short -each is only around 500 lines long. This shows that transformations for other target languages (other theorem provers, like Mizar and HOL light, L A T E X with other natural languages, MathML, OMDoc or TPTP) can easily be constructed, thus enabling wide access to a single source of mathematical contents.

Our automated theorem prover for coherent logic ArgoCLP exports proofs in the form of the XML files that conforms to this DTD. ArgoCLP reads an input theory and the conjecture given in the TPTP form (assuming the coherent form of all formulae and that there are no function symbols or arity greater than 0). ArgoCLP has built-in support for equality (during the search process, it uses an efficient union-find structure) and the use of equality axioms is implicit in generated proofs. The generated XML documents are simple and consist of three parts: frontpage (providing, for instance, the author of the theorem, the prover used for generating the proof, the date), theory (providing the signature and the axioms) and, organized in chapters, a list of conjectures or theorems with their proofs. This way, some contents (frontpage and theory) can be shared by a number of XML documents. On the other hand, this also enables simple construction of bigger collections of theorems. The following is one example of an XML document generated by ArgoCLP: The overall architecture of the framework is shown in Figure 1. 9

<?xml

Examples

Our XML suite for coherent logic vernacular is used for a number of proofs generated by our prover ArgoCLP. In this section we discuss proofs of theorems from the book Metamathematische Methoden in der Geometrie, by Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski [START_REF] Schwabhäuser | Metamathematische Methoden in der Geometrie[END_REF], one of the twentycentury mathematical classics. The theory is described in terms of first-order logic, it uses only one sort of primitive objects -points, has only two primitive predicates (cong or arity 4 and bet of arity 3, intuitively for congruence and betweenness) and only eleven axioms. The majority of theorems from this book are in coherent logic or can be trivially transformed to belong to coherent logic. After needed transformations, the number of theorems in our development ( 238) is somewhat larger than in the book [START_REF] Stojanović | Synergy Between Interactive and Automated Theorem Proving in Formalization of Mathematical Knowledge: A Case Study of Tarski's Geometry[END_REF].

Here we list a proof of one theorem (4.19) from Tarski's book. The theorem was proved by ArgoCLP (using the list of relevant axioms and theorems produced by a resolution theorem prover), the proof was exported in the XML format, and then transformed to a proof in natural language by appropriate XSL transformation ((A, B) ∼ = (C, D) is an infix notation for cong(A, B, C, D) and it denotes that the pairs of points (A, B) and (C, D) are congruent, bet(A, B, C) denotes that the point B is between the points A and C, col(A, B, C) denotes that the points A, B and C are collinear).

Theorem 1 (th 4 19). Assuming that bet(A, B, C) and AB ∼ = AD and CB ∼ = CD it holds that B = D.

Proof:

1. It holds that bet(B, A, A) (using th 3 1). 2. From the fact(s) bet(A, B, C) it holds that col(C, A, B) (using ax 4 10 3). 3. From the fact(s) AB ∼ = AD it holds that AD ∼ = AB (using th 2 2). In [START_REF] Wiedijk | Mathematical Vernacular[END_REF], Wiedijk proposes a mathematical vernacular that is in a sense the common denominator of the proof languages of Hyperproof, Mizar and Isabelle/Isar. We agree with his conclusion in the last sentence of the quotation in the introduction, but we think that the three proof languages were not discovered independently. Natural deduction has been introduced by the Polish logicians Lukasiewicz and Jaśkowski in the late 1920's, in reaction on the formalisms of Frege, Russell and Hilbert. The term natural deduction seems to have been used first by Gentzen, in German:

Ich wollte zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens". (First of all I wanted to set up a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".)-Gentzen, Untersuchungen über das logische Schließen (Mathematische Zeitschrift 39, pp. 1935) The qualifier natural was of course particularly well-chosen to express that the earlier formalisms were unnatural! As this was indeed the case, natural deduction quickly became the predominant logical system, helped by the seminal work by Gentzen on cut-elimination. (Ironically, this technical work in proof theory is best carried out with proofs represented in sequent calculus, using natural deduction on the meta-level.)

It should thus not come as a surprise that the vernacular we propose also is based on natural deduction. One difference with Wiedijk's vernacular is that ours is based on coherent logic instead of full first-order logic. This choice is motivated in Section 2.2 (easier semi-decision procedure and more readable proofs). Another difference is that Wiedijk allows proofs to be incomplete, whereas we stress complete proof objects. This difference is strongly related to the fact that Wiedijk's vernacular is in the first place an input formalism for proof construction, whereas our vernacular is an output formalism for proof presentation and export of proofs to different proof assistants. As far as we know, the mathematical vernacular proposed by Wiedijk's has not been implemented on its own, although Hyperproof, Mizar and Isabelle/Isar are developed using the same ideas.

A number of authors independently point to this or similar fragments of first-order logic as suitable for expressing significant portions of standard mathematics (or specifically geometry), for instance, Avigad et.al. [START_REF] Avigad | A Formal System for Euclid's Elements. The Review of Symbolic Logic[END_REF] and Givant and Tarski et.al. [START_REF] Tarski | Tarski's system of geometry[END_REF][START_REF] Schwabhäuser | Metamathematische Methoden in der Geometrie[END_REF] in the context of a new axiomatic foundations of geometry. A recent paper by Ganesalingam and Gowers [START_REF] Ganesalingam | A fully automatic problem solver with human-style output[END_REF] is also related to our work. Their goal is comparable to ours: full automation combined with human-style output. They propose inference rules which are very similar to our coherent logic based proof system. For example, their rule splitDisjunctiveHypothesis corresponds to the rule case split, deleteDoneDisjunct corresponds to as, removeTarget corresponds to as (with length of y greater than 0), forwardsReasoning corresponds to the rule mp. Yet, some rules they proposed are not part of our set of rules. The logic they use is full first-order, with a plan to include second-order features (this would also be perfectly possible for coherent logic, which is the first-order fragment of geometric logic, which is in turn a fragment of higher-order logic, see [START_REF] Blass | Topoi and Computation[END_REF]). Upon closer inspection, the paper by Ganesalingam and Gowers seems to stay within the coherent fragment, and proofs by contraposition and contradiction are delegated to future work. We find some support for our approach in the observation by Ganesalingam and Gowers that it will be hard to avoid that such reasoning patterns are applied in "inappropriate contexts". On the other hand, the primary domain of application of their approach is metric space theory so far, with the ambition to attack problems in other domains as well. It would be very interesting to test the two approaches on the same problem sets. One difference is that [START_REF] Ganesalingam | A fully automatic problem solver with human-style output[END_REF] insists on proofs being faithful to the thought processes, whereas we would be happy if the prover finds a short and elegant proof even after a not-so-elegant proof search. Another difference is that we are interested in portability of proofs to other systems. To our knowledge, the prover described in [START_REF] Ganesalingam | A fully automatic problem solver with human-style output[END_REF] is not publicly available.

Compared to OMDoc [START_REF] Kohlhase | An OMDoc primer[END_REF], our proof format is much more specific (as we specify the inference rules we use) and has less features. It can be seen as a specific set of methods elements of the derive element of OMDoc.

An alternative to using coherent logic provers would be using one of the more powerful automated theorem provers and exploiting existing and ongoing work on proof reconstruction and refactoring (see, for example, [START_REF] Juilf | Robust, Semi-Intelligible Isabelle Proofs from ATP Proofs[END_REF][START_REF] Christian | Redirecting Proofs by Contradiction[END_REF][START_REF] Kaliszyk | PRocH: Proof Reconstruction for HOL Light[END_REF]). This is certainly a viable option. However, reconstructing a proof from the log of a highly optimized prover is difficult. One problematic step is deskolemization, that is, proof reconstruction from a proof of the skolemized version of the problem. (The most efficient provers are based on resolution logic, and clausification including skolemizing is the first step in the solution procedure.) What can be said about this approach in its current stage is that more theorems can be proved, but their proofs can still be prohibitively complicated (or use additional axioms). It has been, however, proved beneficial to use powerful automated theorem provers as preprocessors, to provide hints for ArgoCLP.

The literature contains many results about exchanging proofs between proof assistant using deep or shallow embeddings [START_REF] Obua | Importing HOL into Isabelle/HOL[END_REF][START_REF] Keller | Importing HOL Light into Coq[END_REF]. Boessplug, Cerbonneaux and Hermant propose to use the λΠ-calculus as a universal proof language which can express proof without losing their computational properties [START_REF] Boespflug | The λΠ-calculus Modulo as a Universal Proof Language[END_REF]. To our knowledge, these works do not focus on the readability of proofs.

Conclusions and Further Work

Over the last years a lot of effort has been invested in combining the power of automated and interactive theorem proving: interactive theorem provers are now equipped with trusted support for SAT solving, SMT solving, resolution method, etc [START_REF] Christian Blanchette | Automatic Proof and Disproof in Isabelle/HOL[END_REF][START_REF] Christian Blanchette | Extending Sledgehammer with SMT Solvers[END_REF]. These combinations open new frontiers for applications of theorem proving in software and hardware verification, but also in formalization of mathematics and for helping mathematicians in everyday practice. Exporting proofs in formats such as the presented one opens new possibilities for ex-porting readable mathematical knowledge from automated theorem provers to interactive theorem provers. In the presented approach, the task of generating object-level proofs for proof assistants or proofs expressed in natural language is removed from theorem provers (where it would be hard-coded) and, thanks to the interchange XML format, delegated to simple XSLT style-sheets, which are very flexible and additional XSLT style-sheets (for additional target formats) can be developed without changing the prover. Also, different automated theorem provers can benefit from this suite, as they don't have to deal with specifics of proof assistants.

The presented proof representation is not intended to serve as "the mathematical vernacular". However, it can cover a significant portion of many interesting mathematical theories while it is very simple.

Often, communication between an interactive theorem prover and an external automated theorem prover is supported by a verified, trusted interface which enables direct calling to the prover. On the other hand, our work yields a common format which can be generated by different automated theorem provers and from which proofs for different interactive theorem provers can be generated. The advantage of our approach relies on the fact that the proof which is exported is not just a certificate, it is meant to be human readable.

The current version of the presented XML suite does not support function symbols of arity greated than 0. For the future work, we are planning to add that support to the proof format and to our ArgoCLP prover.

In the current version, for simplicity, the generated Isar and Coq proofs use tactics stronger than necessary. We will try to completely move to basic proofs steps while keeping simplicity of proofs. Beside planning to further improve existing XSLT style-sheets, we are also planning to implement support for additional target languages such as OMDoc.

  -******** Theory **************--> <!ELEMENT theory (theory_name, signature, axiom*) > <!ELEMENT theory_name (#PCDATA)> <!ELEMENT signature (type*, relation_symbol*, constant*) > <!ELEMENT relation_symbol (type*)> <!ATTLIST relation_symbol name CDATA #REQUIRED> <!ELEMENT type (#PCDATA)> <!ELEMENT axiom (cl_formula)> <!ATTLIST axiom name CDATA #REQUIRED> ... The above fragment describes the notion of theory. (Definitions, formalized as pairs of coherent formulae, are used as axioms.) A file describing a theory could be shared among several files with theorems and proofs. ... <!--******** Theorem **************--> <!ELEMENT theorem (theorem_name, cl_formula, proof+)> <!ELEMENT theorem_name (#PCDATA)> <!ELEMENT conjecture (name, cl_formula)> <!--******** Proof **************--> <!ELEMENT proof (proof_step*, proof_closing, proof_name?)> <!ELEMENT proof_name EMPTY> <!ATTLIST proof_name name CDATA #REQUIRED> <!--******** Proof steps **************--> <!ELEMENT proof_step (indentation,modus_ponens)> <!ELEMENT proof_closing (indentation, (case_split|efq|from), (goal_reached_contradiction|goal_reached_thesis))> ...

4 .

 4 It holds that A = B or A = B. 5. Assume that: A = B. 6. From the fact(s) AD ∼ = AB and A = B it holds that AD ∼ = AA. 7. From the fact(s) AD ∼ = AA it holds that A = D (using ax 3). 8. From the fact(s) A = B and A = D it holds that B = D. 9. The conclusion follows from the fact(s) B = D. 10. Assume that: A = B. 11. It holds that A = C or A = C. 12. Assume that: A = C. 13. From the fact(s) bet(A, B, C) and A = C it holds that bet(A, B, A). 14. From the fact(s) bet(A, B, A) and bet(B, A, A) it holds that A = B (using th 3 4). 15. From the fact(s) A = B and A = B we get contradiction. 16. Assume that: A = C. 17. From the fact(s) A = C it holds that C = A. 18. From the fact(s) C = A and col(C, A, B) and CB ∼ = CD and AB ∼ = AD it holds that B = D (using th 4 18). 19. The conclusion follows from the fact(s) B = D. 20. The conclusion follows in all cases. 21. The conclusion follows in all cases. QED Below is the same proof in Isabelle/Isar form: lemma th_4_19 : assumes "bet A B C" and "cong A B A D" and "cong C B C D" shows "(B = D)" proofhave "bet B A A" by (rule th_3_1) from 'bet A B C' have "col C A B" by (rule ax_4_10_3) from 'cong A B A D' have "cong A D A B" by (rule th_2_2) have "A = B ∨ A ~= B" by (subst disj_commute, rule excluded_middle) show ?thesis proof(cases "A = B") case True from 'cong A D A B' and 'A = B' have "cong A D A A" by simp from 'cong A D A A' have "A = D" by (rule ax_3) from 'A = B' and 'A = D' have "B = D" by simp from 'B = D' show ?thesis by assumption next case False have "A = C ∨ A ~= C" by (subst disj_commute, rule excluded_middle) show ?thesis proof(cases "A = C") case True from 'bet A B C' and 'A = C' have "bet A B A" by simp from 'bet A B A' and 'bet B A A' have "A = B" by (rule th_3_4) from 'A ~= B' and 'A = B' have "False" by (rule notE) from this show ?thesis by (rule FalseE) next case False from 'A ~= C' have "C ~= A" by (rule not_sym) from 'C ~= A' and 'col C A B' and 'cong C B C D' and 'cong A B A D' have "B = D" by (rule th_4_18) from 'B = D' show ?thesis by assumption qed qed qed end

The ratio between the length of formal proof script and the length of the informal proof is often called the de Bruijn factor[START_REF] Barendregt | The Challenge of Computer Mathematics[END_REF]. It varies for different parts of mathematics and for different systems, and is currently often around 4. The de Bruijn factor can be below 1 if a lot of automation can be used. It can also be well over 10 when the informal proof is rather sketchy.

http://www.cs.miami.edu/ ~tptp/

http://www.w3.org/XML/

http://www.w3.org/Style/XSL/

The system Isabelle has available a proof method coherent based on a internal theorem prover for coherent logic. Our Isar proofs do not use this proof method.

The whole of our XML suite, along with a collection of theorems is available online from http://argo.matf.bg.ac.rs/downloads/software/clvernacular.zip.

Translating the XML document with 85 proofs by to Isabelle, Coq, HTML, L A T E X (and then to PDF) takes altogether around 20s on a PC with AMD Opteron 6168. The resulting Isabelle document is verified in 30s, and the Coq document in 6s.

Acknowledgements. We are grateful to Filip Marić for his feedback and advices on earlier phases of this work.

From the set of individual theorems (238), the prover ArgoCLP completely automatically proved 85 (36%) of these theorems and generated proofs in the XML format. We created a single XML document that contains all proved theorems and other theorems tagged as conjectures. The whole document matches the original book by Schwabhäuser, Szmielew, and Tarski and can be explored in the L A T E X (or PDF) form, HTML or as Isabelle or Coq development. 10