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This paper contains new proofs of the following two theorems of the untyped extensional l-calculus: the Curry theorem stating that any l-term has a bh-normal form if and only if it has a b-normal form and the normalization theorem for bh-reduction. The proposed approach is based on the following well-known results: the postponement theorem of h-reduction and the strong normalization property of h-reduction that allow one to naturally extend some propositions from the usual l-calculus to the extensional case.Keywords: untyped extensional l-calculus, postponement of h-reduction, theorem on bh-normal form, normalization theorem for bh-reduction.This article gives new simple proofs of the following two important results of the untyped extensional l-calculus: the Curry theorem stating that an arbitrary l-term has a bh-normal form if and only if it has a b-normal form and the normalization theorem for bh-reduction. The approach being used is based on the following two well-known results: the postponement theorem of h-reduction and the obvious strong normalization property of h-reduction that, as will be seen in what follows, allow one to naturally extend some propositions from the usual l-calculus to the extensional case. In this case, the postponement theorem of h-reduction allows for an elementary proof without using any special knowledge on b-reduction and h-reduction, which is also presented in this work.Note that works on the l-calculus [1-3] consider rather long and technically complicated proofs of the mentioned results, but their advantage is their constructivity. In the present article, some shorter and logically more transparent proofs are constructed and, at the same time, only elementary methods of l-calculus are used.In what follows, we will use the terminology from [1]. In considering l-terms, we adhere to the convention on the use of variables. Taking into account the general orientation of this work and tending to make its content self-sufficient from the viewpoint of the extensional l-calculus, we will give necessary definitions and also formulations and proofs of results pertinent to bh-reduction and bh-conversion and being used in obtaining the main results; as to b-reduction and b-conversion, we will restrict ourselves to references to the corresponding literature.Hereafter, we denote l-terms by the symbols t , p, q, u, u, and w, variables by the symbols x and y , redexes by the symbol D , and the collection of all l-terms and the set of all variables by the symbols L and X , respectively. Definition 1. Let R 0 and R 1 be arbitrary binary relations defined on some set A. We will say that R 0 is postponed with respect to R 1 if the following diagram takes place:(1)i.e., if R R R R, where the symbol o denotes the usual composition of binary relations. 529Note that if the relations R 0 and R 1 are functional (i.e., are unary operations on A), then R 0 is postponed with respect to R 1 if and only if R R R R; however, in the general case, the fact that R 0 is postponed with respect to R 1 does not imply ...

Research on automated reasoning systems based on a number of paradigms that support human activity in formalized text processing began in the late 1950s -early 1960s, when computer performance and memory space became sufficient for programming of complex intelligent processes. The so-called evidential paradigm was among them and it can be viewed as a way for integrating all reasonable paradigms oriented to the development of computer languages for representing formalized texts in the form most suitable for a user, formalization and development of the evidence of a computer-made proof step, creation of the information environment having influence on a current evidence of a machine proof step, and an active human-machine interaction. This work contains a brief description of the evidential paradigm and its implementation in the form of intelligent systems intended for the symbolic and deductive processing of mathematical texts focusing main attention on their features and peculiarities.