@MISC{Formalized10thesum, author = {Formalized and Vol}, title = {The Sum and Product of Finite Sequences of Complex Numbers}, year = {2010} }

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Abstract

Summary. This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11]. MML identifier: RVSUM 2, version: 7.11.07 4.156.1112 The notation and terminology used in this paper have been introduced in the following papers: [5], [7], [6], [4], [8], [13], [9], [2], [3], [15], [10], [12], and [14]. Auxiliary Theorems Let F be a complex-valued binary relation. Then rng F is a subset of C. Let D be a non empty set, let F be a function from C into D, and let F 1 be a complex-valued finite sequence. Note that F · F 1 is finite sequence-like. For simplicity, we adopt the following rules: i, j denote natural numbers, x, x 1 denote elements of C, c denotes a complex number, F , F 1 , F 2 denote complex-valued finite sequences, and R, R 1 denote i-element finite sequences of elements of C. The unary operation sqrcomplex on C is defined as follows: (Def. 1) For every c holds (sqrcomplex)(c) = c 2 . Next we state two propositions: Let us observe that the functor F 1 + F 2 is commutative. Let us consider i, R 1 , R 2 . Then R 1 + R 2 is an element of C i . The following propositions are true: Let us consider F . Then −F is a finite sequence of elements of C and it can be characterized by the condition: Let us consider i, R. Then −R is an element of C i . The following propositions are true: Let us consider F 1 , F 2 . Then F 1 − F 2 is a finite sequence of elements of C and it can be characterized by the condition: The following propositions are true: Let us consider F , c. We introduce c · F as a synonym of c F. The sum and product of finite sequences of . . . 109 Let us consider F , c. Then c · F is a finite sequence of elements of C and it can be characterized by the condition: One can prove the following four propositions: is a finite sequence of elements of C and it can be characterized by the condition: Let us note that the functor Next we state four propositions: Finite Sum of Finite Sequence of Complex Numbers One can prove the following propositions: 110 keiichi miyajima and takahiro kato The Product of Finite Sequences of Complex Numbers One can prove the following propositions: Modified Part of [1] We now state several propositions: (50) For every complex-valued finite sequence x holds len(−x) = len x. (51) For all complex-valued finite sequences x 1 , x 2 such that len x 1 = len x 2 holds len(x 1 + x 2 ) = len x 1 . (52) For all complex-valued finite sequences x 1 , x 2 such that len x 1 = len x 2 holds len(x 1 − x 2 ) = len x 1 . (53) For every real number a and for every complex-valued finite sequence x holds len(a · x) = len x. (54) For all complex-valued finite sequences x, y, z such that len x = len y = len z holds (x + y) References [1] Kanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of n-dimensional topological space. Formalized Mathematics, 11