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In order to allow the use of axioms in a second‐order system of extracting programs from proofs, we define constant terms, a form of Curry‐Howard terms, whose types are intended to correspond to those axioms. We also define new reduction rules for these new terms so that all consequences of the axioms can be represented. We finally show that the extended Curry‐Howard terms are strongly normalizable.  相似文献   
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ABSTRACT

In this study, the evaluation of the salt content of canned sardines in brine using near-infrared (NIR) spectroscopy with a Fourier transform NIR spectrometer in diffuse reflectance mode over a range of 12,500–4000?cm?1 was investigated. The reference salt content of the samples was measured using an autotitrator. The optimum partial least squares regression model was developed using vector normalization spectra in the ranges of 9403.8–5446.3?cm?1 and 4605.4–4242.9?cm?1 with seven factors. The model provided a coefficient of determination for a prediction of 0.82, a root mean square error of prediction of 0.069%, a bias of ?0.019%, and a ratio of prediction deviation of 2.5. NIR spectroscopy has great potential as an alternative method for measuring the salt content of canned sardines in brine, and the calibration model is useful for the off-line quality control of canned sardines in brine processing. In addition, the oxygen–hydrogen bond stretch and deformation and the carbon–hydrogen bond stretch and deformation had obvious effects on the prediction of the salt content of canned sardines in brine.  相似文献   
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We write for the cardinality of the set of finite sequences of a set which is of cardinality . With the Axiom of Choice (), for every infinite cardinal where is the cardinality of the permutations on a set which is of cardinality . In this paper, we show that “ for every cardinal ”  is provable in and this is the best possible result in the absence of . Similar results are also obtained for : the cardinality of the set of finite sequences without repetition of a set which is of cardinality .  相似文献   
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