FORMALIZED MATHEMATICS Volume 13, Number 4, Pages 435–441 University of Białystok, 2005

Set Sequences and Monotone Class

Bo Zhang Shinshu University

Nagano, Japan Hiroshi Yamazaki Shinshu University Nagano, Japan Yatsuka Nakamura Shinshu University Nagano, Japan

Summary. In this paper we first defined the partial-union sequence, the partial-intersection sequence, and the partial-difference-union sequence of given sequence of subsets, and then proved the additive theorem of infinite sequences and sub-additive theorem of finite sequences for probability. Further, we defined the monotone class of families of subsets, and discussed the relations between the monotone class and the σ-field which are generated by the field of subsets of a given set.

MML identifier: PROB 3, version: 7.5.01 4.39.921

The articles [4], [3], [2], [20], [23], [19], [9], [21], [22], [18], [16], [6], [1], [13], [11], [24], [7], [8], [15], [14], [10], [12], [26], [25], [17], and [5] provide the notation and terminology for this paper.

For simplicity, we adopt the following rules: n, m, k are natural numbers, g is a real number, x, X, Y, Z are sets, A_{1} is a sequence of subsets of X, F_{1} is a finite sequence of elements of 2^{X}, R_{1} is a finite sequence of elements of ℝ, S_{1} is a σ-field of subsets of X, O_{1} is a non empty set, S_{2} is a σ-field of subsets of O_{1}, A_{2}, B_{1} are sequences of subsets of S_{2}, and P is a probability on S_{2}.

One can prove the following propositions:

(1) For every finite sequence f holds 0 ∉ dom f.

(2) For every finite sequence f holds n ∈ dom f iff n ≠ 0 and n ≤ len f.

(3) Let f be a sequence of real numbers. Given k such that let given n. If k ≤ n, then f(n) = g. Then f is convergent and lim f = g.

(4) (P · A_{2})(n) ≥ 0.

(5) If A_{2}(n) ⊆ B_{1}(n), then (P · A_{2})(n) ≤ (P · B_{1})(n).

(6) If A_{2} is non-decreasing, then P · A_{2} is non-decreasing.

(7) If A_{2} is non-increasing, then P · A_{2} is non-increasing.

Let A_{1} be a function. The partial intersections of A_{1} constitute a function defined by the conditions (Def. 1).

(Def. 1)(i) dom (the partial intersections of A_{1}) = ℕ, (ii) (the partial intersections of A_{1})(0) = A_{1}(0), and (iii) for every natural number n holds (the partial intersections of A_{1})(n +

1) = (the partial intersections of A_{1})(n) ∩ A_{1}(n + 1). Let X be a set and let A_{1} be a sequence of subsets of X. Then the partial

intersections of A_{1} is a sequence of subsets of X.

Let A_{1} be a function. The partial unions of A_{1} constitute a function defined by the conditions (Def. 2).

(Def. 2)(i) dom (the partial unions of A_{1}) = ℕ, (ii) (the partial unions of A_{1})(0) = A_{1}(0), and (iii) for every natural number n holds (the partial unions of A_{1})(n+1) = (the

partial unions of A_{1})(n) ∪ A_{1}(n + 1). Let X be a set and let A_{1} be a sequence of subsets of X. Then the partial

unions of A_{1} is a sequence of subsets of X. The following propositions are true:

(8) (The partial intersections of A_{1})(n) ⊆ A_{1}(n).

(9) A_{1}(n) ⊆ (the partial unions of A_{1})(n).

(10) The partial intersections of A_{1} are non-increasing.

(11) The partial unions of A_{1} are non-decreasing.

(12) x ∈ (the partial intersections of A_{1})(n) iff for every k such that k ≤ n holds x ∈ A_{1}(k).

(13) x ∈ (the partial unions of A_{1})(n) iff there exists k such that k ≤ n and x ∈ A_{1}(k).

(14) Intersection (the partial intersections of A_{1}) = Intersection A_{1}.

(15) ⋃ (the partial unions of A_{1}) = ⋃ A_{1}.

Let A_{1} be a function. The partial diff-unions of A_{1} constitute a function defined by the conditions (Def. 3).

(Def. 3)(i) dom (the partial diff-unions of A_{1}) = ℕ, (ii) (the partial diff-unions of A_{1})(0) = A_{1}(0), and (iii) for every natural number n holds (the partial diff-unions of A_{1})(n+1) =

A_{1}(n + 1) \ (the partial unions of A_{1})(n). Let X be a set and let A_{1} be a sequence of subsets of X. Then the partial

diff-unions of A_{1} is a sequence of subsets of X.

One can prove the following propositions:

(16) x ∈ (the partial diff-unions of A_{1})(n) iff x ∈ A_{1}(n) and for every k such that k < n holds x ∉ A_{1}(k).

(17) (The partial diff-unions of A_{1})(n) ⊆ A_{1}(n).

(18) (The partial diff-unions of A_{1})(n) ⊆ (the partial unions of A_{1})(n).

(19) The partial unions of the partial diff-unions of A_{1} = the partial unions of A_{1}.

(20) ⋃ (the partial diff-unions of A_{1}) = ⋃ A_{1}.

Let us consider X, A_{1}. Let us observe that A_{1} is disjoint valued if and only if:

(Def. 4) For all m, n such that m ≠ n holds A_{1}(m) misses A_{1}(n).

We now state the proposition

(21) The partial diff-unions of A_{1} are disjoint valued.

Let X be a set, let S_{1} be a σ-field of subsets of X, and let X_{1} be a sequence of subsets of S_{1}. Then the partial intersections of X_{1} is a sequence of subsets of S_{1}.

Let X be a set, let S_{1} be a σ-field of subsets of X, and let X_{1} be a sequence of subsets of S_{1}. Then the partial unions of X_{1} is a sequence of subsets of S_{1}.

Let X be a set, let S_{1} be a σ-field of subsets of X, and let X_{1} be a sequence of subsets of S_{1}. Then the partial diff-unions of X_{1} is a sequence of subsets of S_{1}.

Next we state a number of propositions:

(22) P · the partial unions of A_{2} is non-decreasing.

(23) P · the partial intersections of A_{2} is non-increasing.

(24) (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ} is non-decreasing.

(25) (P · the partial unions of A_{2})(0) = (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}(0). (26)(i) P · the partial unions of A_{2} is convergent, (ii) lim(P · the partial unions of A_{2}) = sup(P · the partial unions of A_{2}),

and (iii) lim(P · the partial unions of A_{2}) = P (⋃ A_{2}).

(27) If A_{2} is disjoint valued, then for all n, m such that n < m holds (the partial unions of A_{2})(n) misses A_{2}(m).

(28) If A_{2} is disjoint valued, then (P ·the partial unions of A_{2})(n) = (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}(n).

(29) If A_{2} is disjoint valued, then P · the partial unions of A_{2} = (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}.

(30) If A_{2} is disjoint valued, then (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ} is convergent and lim((∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}) = sup((∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}) and lim((∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}) = P (⋃ A_{2}).

(31) If A_{2} is disjoint valued, then P (⋃ A_{2}) = ∑(P · A_{2}).

Let us consider X, F_{1}, n. Then F_{1}(n) is a subset of X.

One can prove the following two propositions:

(32) There exists a finite sequence F_{1} of elements of 2^{X} such that for every k such that k ∈ dom F_{1} holds F_{1}(k) = X.

(33) For every finite sequence F_{1} of elements of 2^{X} holds ⋃ rng F_{1} is a subset

of X. Let X be a set and let F_{1} be a finite sequence of elements of 2^{X}. Then ⋃ F_{1}

is a subset of X.

We now state the proposition

(34) x ∈ ⋃ F_{1} iff there exists k such that k ∈ dom F_{1} and x ∈ F_{1}(k). Let us consider X, F_{1}. The functor Complement F_{1} yields a finite sequence

of elements of 2^{X} and is defined by: (Def. 5) len Complement F_{1} = len F_{1} and for every n such that n ∈

dom Complement F_{1} holds (Complement F_{1})(n) = F_{1}(n)^{c}. Let us consider X, F_{1}. The functor Intersection F_{1} yields a subset of X and

is defined by:

(Def. 6) Intersection F_{1} = \{ \begin{matrix}(⋃ Complement F_{1})^{c}, if F_{1} ≠ ∅, \\ ∅, otherwise. \end{matrix}

Next we state several propositions:

(35) dom Complement F_{1} = dom F_{1}.

(36) If F_{1} ≠ ∅, then x ∈ Intersection F_{1} iff for every k such that k ∈ dom F_{1} holds x ∈ F_{1}(k).

(37) If F_{1} ≠ ∅, then x ∈ ⋂ rng F_{1} iff for every n such that n ∈ dom F_{1} holds x ∈ F_{1}(n).

(38) Intersection F_{1} = ⋂ rng F_{1}.

(39) Let F_{1} be a finite sequence of elements of 2^{X}. Then there exists a sequence A_{1} of subsets of X such that for every k such that k ∈ dom F_{1} holds A_{1}(k) = F_{1}(k) and for every k such that k ∉ dom F_{1} holds A_{1}(k) = ∅.

(40) Let F_{1} be a finite sequence of elements of 2^{X} and A_{1} be a sequence of subsets of X. Suppose for every k such that k ∈ dom F_{1} holds A_{1}(k) = F_{1}(k) and for every k such that k ∉ dom F_{1} holds A_{1}(k) = ∅. Then A_{1}(0) = ∅ and ⋃ A_{1} = ⋃ F_{1}.

Let X be a set and let S_{1} be a σ-field of subsets of X. A finite sequence of elements of 2^{X} is said to be a finite sequence of elements of S_{1} if:

(Def. 7) For every k such that k ∈ dom it holds it(k) ∈ S_{1}.

Let X be a set, let S_{1} be a σ-field of subsets of X, let F_{2} be a finite sequence of elements of S_{1}, and let us consider n. Then F_{2}(n) is an event of S_{1}.

We now state two propositions:

(41) Let F_{2} be a finite sequence of elements of S_{1}. Then there exists a sequence A_{2} of subsets of S_{1} such that for every k such that k ∈ dom F_{2} holds A_{2}(k) = F_{2}(k) and for every k such that k ∉ dom F_{2} holds A_{2}(k) = ∅.

(42) For every finite sequence F_{2} of elements of S_{1} holds ⋃ F_{2} ∈ S_{1}. Let X be a set, let S be a σ-field of subsets of X, and let F be a finite

sequence of elements of S. The functor F^{c} yielding a finite sequence of elements of S is defined as follows:

(Def. 8) F^{c} = Complement F.

We now state the proposition

(43) For every finite sequence F_{2} of elements of S_{1} holds Intersection F_{2} ∈ S_{1}. In the sequel F_{3} denotes a finite sequence of elements of S_{2}. The following two propositions are true:

(44) dom(P · F_{3}) = dom F_{3}.

(45) P · F_{3} is a finite sequence of elements of ℝ.

Let us consider O_{1}, S_{2}, F_{3}, P. Then P · F_{3} is a finite sequence of elements of ℝ.

Next we state several propositions:

(46) len(P · F_{3}) = len F_{3}.

(47) If len R_{1} = 0, then ∑ R_{1} = 0.

(48) Suppose len R_{1} ≥ 1. Then there exists a sequence f of real numbers such that f(1) = R_{1}(1) and for every n such that 0 ≠ n and n < len R_{1} holds f(n + 1) = f(n) + R_{1}(n + 1) and ∑ R_{1} = f(len R_{1}).

(49) Let F_{3} be a finite sequence of elements of S_{2} and A_{2} be a sequence of subsets of S_{2}. Suppose for every k such that k ∈ dom F_{3} holds A_{2}(k) = F_{3}(k) and for every k such that k ∉ dom F_{3} holds A_{2}(k) = ∅. Then (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ} is convergent and ∑(P · A_{2}) = (∑_{α=0}^{κ}(P · A_{2})(α))_{κ∈ℕ}(len F_{3}) and P (⋃ A_{2}) ≤ ∑(P · A_{2}) and ∑(P · F_{3}) = ∑(P · A_{2}).

(50) P (⋃ F_{3}) ≤ ∑(P · F_{3}) and if F_{3} is disjoint valued, then P (⋃ F_{3}) =

∑(P · F_{3}). Let us consider X and let I_{1} be a family of subsets of X. We say that I_{1} is

non-decreasing-union-closed if and only if: (Def. 9) For every sequence A_{1} of subsets of X such that A_{1} is non-decreasing

and for every n holds A_{1}(n) ∈ I_{1} holds ⋃ A_{1} ∈ I_{1}. We say that I_{1} is non-increasing-intersection-closed if and only if:

(Def. 10) For every sequence A_{1} of subsets of X such that A_{1} is non-increasing

and for every n holds A_{1}(n) ∈ I_{1} holds Intersection A_{1} ∈ I_{1}. We now state three propositions:

(51) Let I_{1} be a family of subsets of X. Then I_{1} is non-decreasing-union-closed if and only if for every sequence A_{1} of subsets of X such that A_{1} is non-decreasing and for every n holds A_{1}(n) ∈ I_{1} holds lim A_{1} ∈ I_{1}.

(52) Let I_{1} be a family of subsets of X. Then I_{1} is non-increasing-intersection-closed if and only if for every sequence A_{1} of subsets of X such that A_{1} is

non-increasing and for every n holds A_{1}(n) ∈ I_{1} holds lim A_{1} ∈ I_{1}.

(53) 2^{X} is non-decreasing-union-closed and 2^{X} is non-increasing-intersection-

closed. Let us consider X. A family of subsets of X is said to be a monotone class

of X if: (Def. 11) It is non-decreasing-union-closed and it is non-increasing-intersection-

closed. Next we state four propositions:

(54) Z is a monotone class of X if and only if the following conditions are

satisfied:

(i) Z ⊆ 2^{X}, and (ii) for every sequence A_{1} of subsets of X such that A_{1} is monotone and

for every n holds A_{1}(n) ∈ Z holds lim A_{1} ∈ Z.

(55) Let F be a field of subsets of X. Then F is a σ-field of subsets of X if and only if F is a monotone class of X.

(56) 2^{O_{1}} is a monotone class of O_{1}.

(57) Let X be a family of subsets of O_{1}. Then there exists a monotone class Y of O_{1} such that X ⊆ Y and for every Z such that X ⊆ Z and Z is a monotone class of O_{1} holds Y ⊆ Z.

Let us consider O_{1} and let X be a family of subsets of O_{1}. The functor monotone-class(X) yielding a monotone class of O_{1} is defined as follows:

(Def. 12) X ⊆ monotone-class(X) and for every Z such that X ⊆ Z and Z is a

monotone class of O_{1} holds monotone-class(X) ⊆ Z. We now state two propositions:

(58) For every field Z of subsets of O_{1} holds monotone-class(Z) is a field of subsets of O_{1}.

(59) For every field Z of subsets of O_{1} holds σ(Z) = monotone-class(Z).

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442 bo zhang et al.

A Theory of Sequential Files

Hirofumi Fukura Shinshu University Nagano, Japan Yatsuka Nakamura Shinshu University Nagano, Japan

Summary. This article is a continuation of [6]. We present the notion of files and records. These are two finite sequences. One is a record and another is a separator for the carriage return and/or line feed. So, we define the record. The sequential text file contains records and separators. Generally, a record and a separator are paired in the file. And in a special situation, the separator does not exist in the file, for that the record is only one record or record is nothing. And the record does not exist in the file, for that some separator is in the file. In this article, we present a theory for files and records.

MML identifier: FILEREC1, version: 7.5.01 4.39.921

The terminology and notation used here are introduced in the following articles: [11], [12], [7], [1], [10], [13], [8], [2], [3], [4], [9], [5], and [6].

In this paper a, b, c denote sets. The following propositions are true:

(1) Let D be a non empty set and p, q, r, s be finite sequences of elements of D. Then p^{⌢} q^{⌢} r^{⌢} s = p^{⌢} (q^{⌢} r)^{⌢} s and (p^{⌢} q^{⌢} r)^{⌢} s = p^{⌢} q^{⌢} (r^{⌢} s) and (p^{⌢} (q^{⌢} r))^{⌢} s = p^{⌢} q^{⌢} (r^{⌢} s).

(2) For every set D and for every finite sequence f of elements of D holds f↾ len f = f.

(3) For every non empty set D and for all finite sequences p, q of elements of D such that len p = 0 holds q = p^{⌢} q.

(4) Let D be a non empty set, f be a finite sequence of elements of D, and n, m be natural numbers. If n ≤ m, then len(f_{⇂m}) ≤ len(f_{⇂n}).

(5) For every non empty set D and for all finite sequences f, g of elements of D such that len g ≥ 1 holds mid(f ^{⌢} g, len f + 1, len f + len g) = g.

(6) Let D be a non empty set, f, g be finite sequences of elements of D, and i, j be natural numbers. If 1 ≤ i and i ≤ j and j ≤ len f, then mid(f ^{⌢} g, i, j) = mid(f, i, j).

(7) Let D be a non empty set, f be a finite sequence of elements of D, and i, j, n be natural numbers. If 1 ≤ i and i ≤ j and i ≤ len(f↾n) and j ≤ len(f↾n), then mid(f, i, j) = mid(f↾n, i, j).

(8) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a⟩ holds a ∈ D.

(9) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a, b⟩ holds a ∈ D and b ∈ D.

(10) Let D be a non empty set and f be a finite sequence of elements of D. If f = ⟨a, b, c⟩, then a ∈ D and b ∈ D and c ∈ D.

(11) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a⟩ holds f↾1 = ⟨a⟩.

(12) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a, b⟩ holds f_{⇂1} = ⟨b⟩.

(13) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a, b, c⟩ holds f↾1 = ⟨a⟩.

(14) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a, b, c⟩ holds f↾2 = ⟨a, b⟩.

(15) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a, b, c⟩ holds f_{⇂1} = ⟨b, c⟩.

(16) For every non empty set D and for every finite sequence f of elements of D such that f = ⟨a, b, c⟩ holds f_{⇂2} = ⟨c⟩.

(17) For every non empty set D and for every finite sequence f of elements of D such that len f = 0 holds Rev(f) = f.

(18) Let D be a non empty set, r be a finite sequence of elements of D, and i be a natural number. If i ≤ len r, then Rev(r_{⇂i}) = Rev(r)↾(len r −′ i).

(19) Let D be a non empty set and f, C_{1} be finite sequences of elements of D. If C_{1} is not a substring of f and C_{1} separates uniquely, then instr(1, f ^{⌢} C_{1}) = len f + 1.

(20) For every non empty set D and for every finite sequence f of elements of D holds every finite sequence f, g of elements of D is a preposition of (f ^{⌢} g)_{⇂len f}.

(21) Let D be a non empty set and f, C_{1} be finite sequences of elements of D. Suppose C_{1} is not a substring of f and C_{1} separates uniquely. Then f ^{⌢} C_{1} is terminated by C_{1}.

Let D be a set. We introduce file of D as a synonym of finite sequence of elements of D.

Let D be a non empty set and let r, f, C_{1} be files of D. We say that r is a record of f and C_{1} if and only if:

(Def. 1) C_{1} ^{⌢} r is a substring of addcr(f, C_{1}) or r is a preposition of addcr(f, C_{1})

but r is terminated by C_{1}. The following propositions are true:

(22) For every non empty set D and for every finite sequence r of elements of D holds ovlpart(ε_{D}, r) = ε_{D} and ovlpart(r, ε_{D}) = ε_{D}.

(23) For every non empty set D holds every finite sequence C_{1} of elements of D is a record of ε_{D} and C_{1}.

(24) Let D be a non empty set, a, b be sets, and f, r, C_{1} be files of D. Suppose a ≠ b and D = {a, b} and C_{1} = ⟨b⟩ and f = ⟨b, a, b⟩ and r = ⟨a, b⟩. Then C_{1} is a record of f and C_{1} and r is a record of f and C_{1}.

(25) For every non empty set D and for all files f, C_{1} of D holds f is a preposition of f ^{⌢} C_{1}.

(26) For every non empty set D and for all files f, C_{1} of D holds f is a preposition of addcr(f, C_{1}).

(27) For every non empty set D and for all files r, C_{1} of D such that C_{1} is a postposition of r holds 0 ≤ len r − len C_{1}.

(28) For every non empty set D and for all files C_{1}, r of D such that C_{1} is a postposition of r holds r = addcr(r, C_{1}).

(29) For every non empty set D and for all files C_{1}, r of D such that r is terminated by C_{1} holds r = addcr(r, C_{1}).

(30) For every non empty set D and for all files f, g of D such that f is terminated by g holds len g ≤ len f.

(31) For every non empty set D and for all files f, C_{1} of D holds len addcr(f, C_{1}) ≥ len f and len addcr(f, C_{1}) ≥ len C_{1}.

(32) For every non empty set D and for all finite sequences f, g of elements of D holds g = (ovlpart(f, g))^{⌢} ovlrdiff(f, g).

(33) For every non empty set D and for all finite sequences f, g of elements of D holds ovlcon(f, g) = (ovlldiff(f, g))^{⌢} g.

(34) For every non empty set D and for all files C_{1}, r of D holds addcr(r, C_{1}) = (ovlldiff(r, C_{1}))^{⌢} C_{1}.

(35) Let D be a non empty set and r_{1}, r_{2}, f be files of D. If f = r_{1} ^{⌢} r_{2}, then r_{1} is a substring of f and r_{2} is a substring of f.

(36) Let D be a non empty set and r_{1}, r_{2}, r_{3}, f be files of D. Suppose f = r_{1} ^{⌢} r_{2} ^{⌢} r_{3}. Then r_{1} is a substring of f and r_{2} is a substring of f and r_{3} is a substring of f.

(37) Let D be a non empty set and C_{1}, r_{1}, r_{2} be files of D. Suppose r_{1} is terminated by C_{1} and r_{2} is terminated by C_{1}. Then C_{1} ^{⌢} r_{2} is a substring 446 hirofumi fukura and yatsuka nakamura

of addcr(r_{1} ^{⌢} r_{2}, C_{1}).

(38) Let D be a non empty set, f, g be files of D, and n be a natural number. If 0 < n and g = ∅, then instr(n, f) = n.

(39) Let D be a non empty set, f, g be files of D, and n be a natural number. If 0 < n and n ≤ len f, then instr(n, f) ≤ len f.

(40) For every non empty set D and for every file f of D holds every file f, C_{1} of D is a substring of ovlcon(f, C_{1}).

(41) For every non empty set D and for every file f of D holds every file f, C_{1} of D is a substring of addcr(f, C_{1}).

(42) Let D be a non empty set, f, g be finite sequences of elements of D, and n be a natural number. If g is a substring of f↾n and len g > 0 and len g ≤ n, then g is a substring of f.

(43) For every non empty set D and for all files f, C_{1} of D holds there exists a file of D which is a record of f and C_{1}.

(44) For every non empty set D and for all files f, C_{1}, r of D such that r is a record of f and C_{1} holds r is a record of r and C_{1}.

(45) Let D be a non empty set and C_{1}, r_{1}, r_{2}, f be files of D. Suppose r_{1} is terminated by C_{1} and r_{2} is terminated by C_{1} and f = r_{1} ^{⌢} r_{2}. Then r_{1} is a record of f and C_{1} and r_{2} is a record of f and C_{1}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.

[6] Hirofumi Fukura and Yatsuka Nakamura. Concatenation of finite sequences reducing overlapping part and an argument of separators of sequential files. Formalized Mathematics, 12(2):219–224, 2004.

[7] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[8] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275–278, 1992.

[9] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Formalized Mathematics, 6(2):255–263, 1997.

[10] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[13] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Circled Sets, Circled Hull, and Circled Family

Fahui Zhai Qingdao University of Science and Technology China

Jianbing Cao Qingdao University of Science and Technology China

Xiquan Liang Qingdao University of Science and Technology China

Summary. In this article, we prove some basic properties of the circled sets. We also define the circled hull, and give the definition of a circled family.

      MML identifier: CIRCLED1, version: 7.5.01 4.39.921

The articles [15], [19], [14], [3], [4], [12], [5], [11], [13], [18], [9], [8], [2], [17], [16], [6], [1], [7], and [10] provide the terminology and notation for this paper.

1. Circled Sets

One can prove the following proposition

(1) For every real linear space V and for all circled subsets A, B of V holds A − B is circled.

Let V be a real linear space and let M, N be circled subsets of V. Note that M − N is circled.

Next we state the proposition

(2) Let V be a non empty RLS structure and M be a subset of V. Then M is circled if and only if for every vector u of V and for every real number r such that |r| ≤ 1 and u ∈ M holds r · u ∈ M.

Let V be a non empty RLS structure and let M be a subset of V. Let us observe that M is circled if and only if:

(Def. 1) For every vector u of V and for every real number r such that |r| ≤ 1

and u ∈ M holds r · u ∈ M. The following propositions are true:

(3) Let V be a real linear space, M be a subset of V, and r be a real number. If M is circled, then r · M is circled.

(4) Let V be a real linear space, M_{1}, M_{2} be subsets of V, and r_{1}, r_{2} be real numbers. If M_{1} is circled and M_{2} is circled, then r_{1} ·M_{1} +r_{2} ·M_{2} is circled.

(5) Let V be a real linear space, M_{1}, M_{2}, M_{3} be subsets of V, and r_{1}, r_{2}, r_{3} be real numbers. Suppose M_{1} is circled and M_{2} is circled and M_{3} is circled. Then r_{1} · M_{1} + r_{2} · M_{2} + r_{3} · M_{3} is circled.

(6) For every real linear space V holds Up(0_{V}) is circled.

(7) For every real linear space V holds Up(Ω_{V}) is circled.

(8) For every real linear space V and for all circled subsets M, N of V holds M ∩ N is circled.

(9) For every real linear space V and for all circled subsets M, N of V holds M ∪ N is circled.

2. Circled Hull and Circled Family

Let V be a non empty RLS structure and let M be a subset of V. The functor Circled-Family M yields a family of subsets of V and is defined as follows:

(Def. 2) For every subset N of V holds N ∈ Circled-Family M iff N is circled

and M ⊆ N. Let V be a real linear space and let M be a subset of V. The functor Cir M

yielding a circled subset of V is defined by: (Def. 3) Cir M = ⋂ Circled-Family M.

Let V be a real linear space and let M be a subset of V. Note that Circled-Family M is non empty.

We now state several propositions:

(10) For every real linear space V and for all subsets M_{1}, M_{2} of V such that M_{1} ⊆ M_{2} holds Circled-Family M_{2} ⊆ Circled-Family M_{1}.

(11) For every real linear space V and for all subsets M_{1}, M_{2} of V such that M_{1} ⊆ M_{2} holds Cir M_{1} ⊆ Cir M_{2}.

(12) For every real linear space V and for every subset M of V holds M ⊆ Cir M.

(13) Let V be a real linear space, M be a subset of V, and N be a circled subset of V. If M ⊆ N, then Cir M ⊆ N.

(14) For every real linear space V and for every circled subset M of V holds Cir M = M.

(15) For every real linear space V holds Cir(∅_{V}) = ∅.

(16) For every real linear space V and for every subset M of V and for every real number r holds r · Cir M = Cir(r · M).

3. Basic Properties of Combination

Let V be a real linear space and let L be a linear combination of V. We say that L is circled if and only if the condition (Def. 4) is satisfied.

(Def. 4) There exists a finite sequence F of elements of the carrier of V such that (i) F is one-to-one, (ii) rng F = the support of L, and (iii) there exists a finite sequence f of elements of ℝ such that len f = len F

and ∑ f = 1 and for every natural number n such that n ∈ dom f holds f(n) = L(F (n)) and f(n) ≥ 0.

The following propositions are true:

(17) Let V be a real linear space and L be a linear combination of V. If L is circled, then the support of L ≠ ∅.

(18) Let V be a real linear space, L be a linear combination of V, and v be a vector of V. If L is circled and L(v) ≤ 0, then v ∉ the support of L.

(19) For every real linear space V and for every linear combination L of V such that L is circled holds L ≠ 0_{LC_{V}} .

(20) For every real linear space V holds there exists a linear combination of

V which is circled. Let V be a real linear space. One can check that there exists a linear

combination of V which is circled.

Let V be a real linear space. A circled combination of V is a circled linear combination of V.

We now state the proposition

(21) For every real linear space V and for every non empty subset M of V

holds there exists a linear combination of M which is circled. Let V be a real linear space and let M be a non empty subset of V. Note

that there exists a linear combination of M which is circled.

Let V be a real linear space and let M be a non empty subset of V. A circled combination of M is a circled linear combination of M.

Let V be a real linear space. The functor circledComb V is defined as follows: (Def. 5) For every set L holds L ∈ circledComb V iff L is a circled combination

of V.

Let V be a real linear space and let M be a non empty subset of V. The functor circledComb M is defined by:

(Def. 6) For every set L holds L ∈ circledComb M iff L is a circled combination

of M. The following propositions are true:

(22) Let V be a real linear space and v be a vector of V. Then there exists a circled combination L of V such that ∑ L = v and for every non empty subset A of V such that v ∈ A holds L is a circled combination of A.

(23) Let V be a real linear space and v_{1}, v_{2} be vectors of V. Suppose v_{1} ≠ v_{2}. Then there exists a circled combination L of V such that for every non empty subset A of V if {v_{1}, v_{2}} ⊆ A, then L is a circled combination of A.

(24) Let V be a real linear space, L_{1}, L_{2} be circled combinations of V, and a, b be real numbers. Suppose a·b > 0. Then the support of a·L_{1}+b·L_{2} = (the support of a · L_{1}) ∪ (the support of b · L_{2}).

(25) Let V be a real linear space, v be a vector of V, and L be a linear combination of V. If L is circled and the support of L = {v}, then L(v) = 1 and ∑ L = L(v) · v.

(26) Let V be a real linear space, v_{1}, v_{2} be vectors of V, and L be a linear combination of V. Suppose L is circled and the support of L = {v_{1}, v_{2}} and v_{1} ≠ v_{2}. Then L(v_{1}) + L(v_{2}) = 1 and L(v_{1}) ≥ 0 and L(v_{2}) ≥ 0 and ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2}.

(27) Let V be a real linear space, v be a vector of V, and L be a linear combination of {v}. If L is circled, then L(v) = 1 and ∑ L = L(v) · v.

(28) Let V be a real linear space, v_{1}, v_{2} be vectors of V, and L be a linear combination of {v_{1}, v_{2}}. Suppose v_{1} ≠ v_{2} and L is circled. Then L(v_{1}) + L(v_{2}) = 1 and L(v_{1}) ≥ 0 and L(v_{2}) ≥ 0 and ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2}.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[7] Czesław Byliński. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99–107, 2005.

[8] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53–58, 2003.

[9] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23–28, 2003.

circled sets, circled hull, … 451

[10] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Topology of real unitary space. Formalized Mathematics, 11(1):33–38, 2003.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[12] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[13] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[14] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[15] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[16] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.

[17] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

[18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291– 296, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

452 fahui zhai and jianbing cao and xiquan liang

On the Borel Families of Subsets of Topological Spaces^{1}

Adam Grabowski Institute of Mathematics University of Białystok Akademicka 2, 15-267 Białystok, Poland

Summary. This is the next Mizar article in a series aiming at complete formalization of “General Topology” [14] by Engelking. We cover the second part of Section 1.3.

MML identifier: TOPGEN 4, version: 7.5.01 4.39.921

The papers [27], [30], [31], [9], [1], [2], [26], [3], [28], [10], [12], [21], [29], [22], [5], [16], [6], [23], [32], [11], [20], [17], [18], [19], [7], [13], [25], [24], [15], [4], and [8] provide the terminology and notation for this paper.

1. Preliminaries

Let T be a 1-sorted structure. The functor TotFam T yielding a family of subsets of T is defined by:

(Def. 1) TotFam T = 2^{the carrier of T} . The following proposition is true

(1) For every set T and for every family F of subsets of T holds F is countable iff F^{c} is countable.

Let us note that ℚ is countable.

The scheme FraenCoun11 concerns a unary predicate P, and states that: {{n}; n ranges over elements of ℚ: P[n]} is countable for all values of the parameters.

1This work has been partially supported by the KBN grant 4 T11C 039 24 and the FP6 IST grant TYPES No. 510996.

One can prove the following proposition

(2) For every non empty topological space T and for every subset A of T holds Der A = \{x; x ranges over points of T: x ∈ \overline{A \ \{x\}}\}.

Let us note that every topological structure which is finite is also secondcountable.

One can verify that ℝ is non countable. One can verify the following observations:

∗ every set which is non countable is also non finite,

∗ every set which is non finite is also non trivial, and

∗ there exists a set which is non countable and non empty. We adopt the following rules: T is a non empty topological space, A, B are

subsets of T, and F, G are families of subsets of T. One can prove the following propositions:

(3) A is closed iff Der A ⊆ A.

(4) Let T be a non empty topological structure, B be a basis of T, and V be a subset of T. Suppose V is open and V ≠ ∅. Then there exists a subset W of T such that W ∈ B and W ⊆ V and W ≠ ∅.

2. Regular Formalization: Separable Spaces

The following propositions are true:

(5) density T ≤ weight T.

(6) T is separable iff there exists a subset of T which is dense and countable.

(7) If T is second-countable, then T is separable. One can check that every non empty topological space which is second-

countable is also separable.

The following four propositions are true:

(8) Let T be a non empty topological space and A, B be subsets of T. If A and B are separated, then Fr(A ∪ B) = Fr A ∪ Fr B.

(9) If F is locally finite, then Fr ⋃ F ⊆ ⋃ Fr F.

(10) For every discrete non empty topological space T holds T is separable iff \overline{\overline{Ω_{T}}} ≤ ℵ_{0}.

(11) For every discrete non empty topological space T holds T is separable iff T is countable.

3. Families of Subsets Closed for Countable Unions and Complement

Let us consider T, F. We say that F is all-open-containing if and only if: (Def. 2) For every subset A of T such that A is open holds A ∈ F.

Let us consider T, F. We say that F is all-closed-containing if and only if: (Def. 3) For every subset A of T such that A is closed holds A ∈ F.

Let T be a set and let F be a family of subsets of T. We say that F is closed for countable unions if and only if:

(Def. 4) For every countable family G of subsets of T such that G ⊆ F holds

⋃ G ∈ F. Let T be a set. Note that every σ-field of subsets of T is closed for countable

unions.

One can prove the following proposition

(12) For every set T and for every family F of subsets of T such that F is

closed for countable unions holds ∅ ∈ F. Let T be a set. One can verify that every family of subsets of T which is

closed for countable unions is also non empty.

Next we state the proposition

(13) Let T be a set and F be a family of subsets of T. Then F is a σ-field of subsets of T if and only if F is closed for complement operator and closed for countable unions.

Let T be a set and let F be a family of subsets of T. We say that F is closed for countable meets if and only if:

(Def. 5) For every countable family G of subsets of T such that G ⊆ F holds

⋂ G ∈ F. Next we state four propositions:

(14) Let F be a family of subsets of T. Then the following statements are

equivalent

(i) F is all-closed-containing and closed for complement operator, (ii) F is all-open-containing and closed for complement operator.

(15) For every set T and for every family F of subsets of T such that F is closed for complement operator holds F = F^{c}.

(16) Let T be a set and F, G be families of subsets of T. If F ⊆ G and G is closed for complement operator, then F^{c} ⊆ G.

(17) Let T be a set and F be a family of subsets of T. Then the following

statements are equivalent

(i) F is closed for countable meets and closed for complement operator, (ii) F is closed for countable unions and closed for complement operator.

Let us consider T. One can verify that every family of subsets of T which is all-open-containing, closed for complement operator, and closed for countable unions is also all-closed-containing and closed for countable meets and every family of subsets of T which is all-closed-containing, closed for complement operator, and closed for countable meets is also all-open-containing and closed for countable unions.

4. On the Families of Subsets

Let T be a set and let F be a countable family of subsets of T. Note that F^{c} is countable.

Let us consider T. Note that every family of subsets of T which is empty is also open and closed.

Let us consider T. One can check that there exists a family of subsets of T which is countable, open, and closed.

We now state the proposition

(18) For every set T holds ∅ is an empty family of subsets of T.

Let us observe that every set which is empty is also countable.

5. Collective Properties of Families

One can prove the following two propositions:

(19) If F = {A}, then A is open iff F is open.

(20) If F = {A}, then A is closed iff F is closed.

Let T be a set and let F, G be families of subsets of T. Then F ⋒ G is a family of subsets of T. Then F ⋓ G is a family of subsets of T.

Next we state a number of propositions:

(21) If F is closed and G is closed, then F ⋒ G is closed.

(22) If F is closed and G is closed, then F ⋓ G is closed.

(23) If F is open and G is open, then F ⋒ G is open.

(24) If F is open and G is open, then F ⋓ G is open.

(25) For every set T and for all families F, G of subsets of T holds \overline{\overline{F ⋒ G}} ≤ \overline{\overline{[: F, G:]}}.

(26) For every set T and for all families F, G of subsets of T holds \overline{\overline{F ⋓ G}} ≤ \overline{\overline{[: F, G:]}}.

(27) For all sets F, G holds ⋃(F ⋓ G) ⊆ ⋃ F ∪ ⋃ G.

(28) For all sets F, G such that F ≠ ∅ and G ≠ ∅ holds ⋃ F ∪⋃ G = ⋃(F ⋓G).

(29) For every set F holds ∅ ⋓ F = ∅.

(30) For all sets F, G such that F ⋓ G = ∅ holds F = ∅ or G = ∅.

(31) For all sets F, G such that F ⋒ G = ∅ holds F = ∅ or G = ∅.

(32) For all sets F, G holds ⋂(F ⋓ G) ⊆ ⋂ F ∪ ⋂ G.

(33) For all sets F, G such that F ≠ ∅ and G ≠ ∅ holds ⋂(F ⋓G) = ⋂ F ∪⋂ G.

(34) For all sets F, G such that F ≠ ∅ and G ≠ ∅ holds ⋂ F ∩⋂ G = ⋂(F ⋒G).

6. F_{σ} and G_{δ} Types of Subsets

Let us consider T, A. We say that A is F_{σ} if and only if: (Def. 6) There exists a closed countable family F of subsets of T such that A =

⋃ F. Let us consider T, A. We say that A is G_{δ} if and only if:

(Def. 7) There exists an open countable family F of subsets of T such that A =

⋂ F. The following propositions are true:

(35) ∅_{T} is F_{σ}.

(36) ∅_{T} is G_{δ}.

Let us consider T. Note that ∅_{T} is F_{σ} and G_{δ}.

Next we state two propositions:

(37) Ω_{T} is F_{σ}.

(38) Ω_{T} is G_{δ}.

Let us consider T. One can verify that Ω_{T} is F_{σ} and G_{δ}.

One can prove the following propositions:

(39) If A is F_{σ}, then A^{c} is G_{δ}.

(40) If A is G_{δ}, then A^{c} is F_{σ}.

(41) If A is F_{σ} and B is F_{σ}, then A ∩ B is F_{σ}.

(42) If A is F_{σ} and B is F_{σ}, then A ∪ B is F_{σ}.

(43) If A is G_{δ} and B is G_{δ}, then A ∪ B is G_{δ}.

(44) If A is G_{δ} and B is G_{δ}, then A ∩ B is G_{δ}.

(45) For every subset A of T such that A is closed holds A is F_{σ}.

(46) For every subset A of T such that A is open holds A is G_{δ}.

(47) For every subset A of ℝ^{1} such that A = ℚ holds A is F_{σ}.

7. T_{1/2} Topological Spaces

Let T be a topological space. We say that T is T_{1/2} if and only if: (Def. 8) For every subset A of T holds Der A is closed.

We now state three propositions:

(48) For every topological space T such that T is T_{1} holds T is T_{1/2}.

(49) For every non empty topological space T such that T is T_{1/2} holds T is T_{0}.

(50) For every non empty topological space T holds every point p of T is

isolated in Ω_{T} or an accumulation point of Ω_{T}. Let us note that every topological space which is T_{1/2} is also T_{0} and every

topological space which is T_{1} is also T_{1/2}.

8. Condensation Points

Let us consider T, A and let x be a point of T. We say that x is a condensation point of A if and only if:

(Def. 9) For every neighbourhood N of x holds N ∩ A is not countable. In the sequel x denotes a point of T. One can prove the following proposition

(51) If x is a condensation point of A and A ⊆ B, then x is a condensation

point of B. Let us consider T, A. The functor A^{0} yielding a subset of T is defined as

follows: (Def. 10) For every point x of T holds x ∈ A^{0} iff x is a condensation point of A.

The following propositions are true:

(52) For every point p of T such that p is a condensation point of A holds p is an accumulation point of A.

(53) A^{0} ⊆ Der A.

(54) A^{0} = \overline{A^{0}}.

(55) If A ⊆ B, then A^{0} ⊆ B^{0}.

(56) If x is a condensation point of A ∪ B, then x is a condensation point of A or a condensation point of B.

(57) A ∪ B^{0} = A^{0} ∪ B^{0}.

(58) If A is countable, then there exists no point of T which is a condensation point of A.

(59) If A is countable, then A^{0} = ∅.

Let us consider T and let A be a countable subset of T. Note that A^{0} is empty.

The following proposition is true

(60) If T is second-countable, then there exists a basis of T which is countable.

Let us mention that there exists a topological space which is second-countable and non empty.

9. Borel Families of Subsets

Let us consider T. Observe that TotFam T is non empty, all-open-containing, closed for complement operator, and closed for countable unions.

We now state four propositions:

(61) For every set T and for every sequence A of subsets of T holds rng A is a countable non empty family of subsets of T.

(62) Let T, F be sets. Then F is a σ-field of subsets of T if and only if F is a closed for complement operator σ-field of subsets-like non empty family of subsets of T.

(63) For all families F, G of subsets of T such that F is all-open-containing and F ⊆ G holds G is all-open-containing.

(64) Let F, G be families of subsets of T. Suppose F is all-closed-containing and F ⊆ G. Then G is all-closed-containing.

Let T be a 1-sorted structure. A σ-field of subsets of T is a σ-field of subsets of the carrier of T.

Let T be a non empty topological space. Note that there exists a family of subsets of T which is closed for complement operator, closed for countable unions, closed for countable meets, all-closed-containing, and all-open-containing.

We now state the proposition

(65) σ(TotFam T) is all-open-containing, closed for complement operator, and closed for countable unions.

Let us consider T. One can verify that σ(TotFam T) is all-open-containing, closed for complement operator, and closed for countable unions.

Let T be a non empty 1-sorted structure. Note that there exists a family of subsets of T which is σ-field of subsets-like, closed for complement operator, closed for countable unions, and non empty.

Let T be a non empty topological space. One can verify that every σ-field of subsets of T is closed for countable unions.

We now state the proposition (66) Let T be a non empty topological space and F be a family of subsets of

T. Suppose F is closed for complement operator and closed for countable unions. Then F is a σ-field of subsets of T.

Let T be a non empty topological space. Note that there exists a σ-field of subsets of T which is all-open-containing.

Let T be a non empty topological space. Note that Topology(T) is open and all-open-containing.

We now state the proposition

(67) Let X be a family of subsets of T. Then there exists an all-open-containing closed for complement operator closed for countable unions family Y of subsets of T such that

(i) X ⊆ Y, and (ii) for every all-open-containing closed for complement operator closed for

countable unions family Z of subsets of T such that X ⊆ Z holds Y ⊆ Z. Let us consider T. The functor BorelSets T yields an all-open-containing

closed for complement operator closed for countable unions family of subsets of T and is defined by the condition (Def. 11).

(Def. 11) Let G be an all-open-containing closed for complement operator closed

for countable unions family of subsets of T. Then BorelSets T ⊆ G. Next we state three propositions:

(68) For every closed family F of subsets of T holds F ⊆ BorelSets T.

(69) For every open family F of subsets of T holds F ⊆ BorelSets T.

(70) BorelSets T = σ(Topology(T)).

Let us consider T, A. We say that A is Borel if and only if: (Def. 12) A ∈ BorelSets T.

Let us consider T. Note that every subset of T which is F_{σ} is also Borel. Let us consider T. Note that every subset of T which is G_{δ} is also Borel.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[3] Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65–69, 1991.

[4] Grzegorz Bancerek. On constructing topological spaces and Sorgenfrey line. Formalized Mathematics, 13(1):171–179, 2005.

[5] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.

[7] Józef Białas and Yatsuka Nakamura. Dyadic numbers and T_{4} topological spaces. Formalized Mathematics, 5(3):361–366, 1996.

[8] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

on the borel families of subsets … 461

[10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[11] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257–261, 1990.

[12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[13] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces – fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[14] Ryszard Engelking. General Topology, volume 60 of Monografie Matematyczne. PWN – Polish Scientific Publishers, Warsaw, 1977.

[15] Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13(1):139–146, 2005.

[16] Jolanta Kamieńska. Representation theorem for Heyting lattices. Formalized Mathematics, 4(1):41–45, 1993.

[17] Zbigniew Karno. The lattice of domains of an extremally disconnected space. Formalized Mathematics, 3(2):143–149, 1992.

[18] Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285–294, 1998.

[19] Andrzej Nȩdzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

[20] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

[21] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[22] Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.

[23] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[24] Marta Pruszyńska and Marek Dudzicz. On the isomorphism between finite chains. Formalized Mathematics, 9(2):429–430, 2001.

[25] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233–236, 1996.

[26] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[27] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[28] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[29] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[30] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[31] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[32] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990.

462 adam grabowski

Linearity of Lebesgue Integral of Simple Valued Function^{1}

Noboru Endou

Gifu National College of Technology Gifu, Japan Yasunari Shidama Shinshu University Nagano, Japan

Summary. In this article the authors prove linearity of the Lebesgue integral of simple valued function.

MML identifier: MESFUNC4, version: 7.5.01 4.39.921

The notation and terminology used here are introduced in the following papers: [16], [17], [1], [15], [2], [18], [7], [9], [8], [3], [4], [5], [6], [10], [11], [12], [14], and [13].

One can prove the following propositions:

(1) Let F, G, H be finite sequences of elements of \overline{ℝ}. Suppose that

(i) for every natural number i such that i ∈ dom F holds 0\overline{ℝ} ≤ F (i), (ii) for every natural number i such that i ∈ dom G holds 0\overline{ℝ} ≤ G(i), (iii) dom F = dom G, and (iv) H = F + G.

Then ∑ H = ∑ F + ∑ G.

(2) Let X be a non empty set, S be a σ-field of subsets of X, M be a σmeasure on S, n be a natural number, f be a partial function from X to \overline{ℝ}, F be a finite sequence of separated subsets of S, and a, x be finite sequences of elements of \overline{ℝ}. Suppose that f is simple function in S and dom f ≠ ∅ and for every set x such that x ∈ dom f holds 0\overline{ℝ} ≤ f(x) and F and a are representation of f and dom x = dom F and for every natural number i such that i ∈ dom x holds x(i) = a(i) · (M · F)(i) and len F = n.

Then ∫_{X} f dM = ∑ x.

1This work has been partially supported by the MEXT grant Grant-in-Aid for Young Scientists (B)16700156.

(3) Let X be a non empty set, S be a σ-field of subsets of X, f be a partial function from X to \overline{ℝ}, M be a σ-measure on S, F be a finite sequence of separated subsets of S, and a, x be finite sequences of elements of \overline{ℝ}. Suppose that

(i) f is simple function in S, (ii) dom f ≠ ∅, (iii) for every set x such that x ∈ dom f holds 0\overline{ℝ} ≤ f(x), (iv) F and a are representation of f,

(v) dom x = dom F, and (vi) for every natural number n such that n ∈ dom x holds x(n) = a(n) ·

(M · F)(n).

Then ∫_{X} f dM = ∑ x.

(4) Let X be a non empty set, S be a σ-field of subsets of X, f be a partial function from X to \overline{ℝ}, and M be a σ-measure on S. Suppose f is simple function in S and dom f ≠ ∅ and for every set x such that x ∈ dom f holds 0\overline{ℝ} ≤ f(x). Then there exists a finite sequence F of separated subsets of S and there exist finite sequences a, x of elements of \overline{ℝ} such that

(i) F and a are representation of f, (ii) dom x = dom F, (iii) for every natural number n such that n ∈ dom x holds x(n) = a(n) ·

(M · F)(n), and

(iv) ∫_{X} f dM = ∑ x.

(5) Let X be a non empty set, S be a σ-field of subsets of X, M be a σ-measure on S, and f, g be partial functions from X to \overline{ℝ}. Suppose that

(i) f is simple function in S, (ii) dom f ≠ ∅, (iii) for every set x such that x ∈ dom f holds 0\overline{ℝ} ≤ f(x), (iv) g is simple function in S,

(v) dom g = dom f, and (vi) for every set x such that x ∈ dom g holds 0\overline{ℝ} ≤ g(x).

Then (vii) f + g is simple function in S, (viii) dom(f + g) ≠ ∅, (ix) for every set x such that x ∈ dom(f + g) holds 0\overline{ℝ} ≤ (f + g)(x), and

(x) ∫_{X} f + g dM = ∫_{X} f dM + ∫_{X} g dM.

(6) Let X be a non empty set, S be a σ-field of subsets of X, M be a σmeasure on S, f, g be partial functions from X to \overline{ℝ}, and c be an extended real number. Suppose that f is simple function in S and dom f ≠ ∅ and for every set x such that x ∈ dom f holds 0\overline{ℝ} ≤ f(x) and 0\overline{ℝ} ≤ c and

linearity of lebesgue integral … 465

c < +∞ and dom g = dom f and for every set x such that x ∈ dom g

holds g(x) = c · f(x). Then ∫_{X} g dM = c · ∫_{X} f dM.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[5] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.

[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[9] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[10] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491–494, 2001.

[11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495–500, 2001.

[12] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525–529, 2001.

[13] Grigory E. Ivanov. Definition of convex function and Jensen’s inequality. Formalized Mathematics, 11(4):349–354, 2003.

[14] Yasunari Shidama and Noboru Endou. Lebesgue integral of simple valued function. Formalized Mathematics, 13(1):67–71, 2005.

[15] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

466 noboru endou and yasunari shidama

The Fashoda Meet Theorem for Continuous Mappings

Yatsuka Nakamura Shinshu University Nagano, Japan

Andrzej Trybulec Institute of Computer Science University of Białystok Sosnowa 64, 15-887 Białystok Poland

Artur Korniłowicz Institute of Computer Science University of Białystok Sosnowa 64, 15-887 Białystok Poland

MML identifier: JGRAPH 8, version: 7.5.01 4.39.921

The articles [21], [25], [2], [20], [26], [5], [27], [6], [3], [1], [24], [10], [18], [16], [9], [4], [13], [11], [19], [23], [17], [7], [8], [22], [12], [15], and [14] provide the terminology and notation for this paper.

We use the following convention: n is a natural number, p_{1}, p_{2} are points of E_{T}^{n}, and a, b, c, d are real numbers.

Let us consider a, b, c, d. One can verify that ClosedInsideOfRectangle(a, b, c,

d) is convex.

Let us consider a, b, c, d. Observe that Trectangle(a, b, c, d) is convex.

The following propositions are true:

(1) Let e be a positive real number and g be a continuous map from 𝕀 into E_{T}^{n}. Then there exists a finite sequence h of elements of ℝ such that

(i) h(1) = 0, (ii) h(len h) = 1, (iii) 5 ≤ len h, (iv) rng h ⊆ the carrier of 𝕀,

(v) h is increasing, and (vi) for every natural number i and for every subset Q of 𝕀 and for every

subset W of E^{n} such that 1 ≤ i and i < len h and Q = [h_{i}, h_{i+1}] and W = g^{∘}Q holds ∅W < e.

(2) For every subset P of E_{T}^{n} such that P ⊆ L(p_{1}, p_{2}) and p_{1} ∈ P and p_{2} ∈ P and P is connected holds P = L(p_{1}, p_{2}).

(3) For every path g from p_{1} to p_{2} such that rng g ⊆ L(p_{1}, p_{2}) holds rng g = L(p_{1}, p_{2}).

(4) Let P, Q be non empty subsets of E_{T}^{2}, p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}, f be a path from p_{1} to p_{2}, and g be a path from q_{1} to q_{2}. Suppose that

(i) rng f = P, (ii) rng g = Q, (iii) for every point p of E_{T}^{2} such that p ∈ P holds (p_{1})_{1} ≤ p_{1} and p_{1} ≤ (p_{2})_{1}, (iv) for every point p of E_{T}^{2} such that p ∈ Q holds (p_{1})_{1} ≤ p_{1} and p_{1} ≤ (p_{2})_{1},

(v) for every point p of E_{T}^{2} such that p ∈ P holds (q_{1})_{2} ≤ p_{2} and p_{2} ≤ (q_{2})_{2}, and

(vi) for every point p of E_{T}^{2} such that p ∈ Q holds (q_{1})_{2} ≤ p_{2} and p_{2} ≤ (q_{2})_{2}.

Then P meets Q.

(5) Let f, g be continuous maps from 𝕀 into E_{T}^{2} and O, I be points of 𝕀. Suppose that O = 0 and I = 1 and f(O)_{1} = a and f(I)_{1} = b and g(O)_{2} = c and g(I)_{2} = d and for every point r of 𝕀 holds a ≤ f(r)_{1} and f(r)_{1} ≤ b and a ≤ g(r)_{1} and g(r)_{1} ≤ b and c ≤ f(r)_{2} and f(r)_{2} ≤ d and c ≤ g(r)_{2} and g(r)_{2} ≤ d. Then rng f meets rng g.

(6) Let a_{1}, b_{1}, c_{1}, d_{1} be points of Trectangle(a, b, c, d), h be a path from a_{1} to b_{1}, v be a path from d_{1} to c_{1}, and A_{1}, B_{1}, C_{1}, D_{1} be points of E_{T}^{2}. Suppose (A_{1})_{1} = a and (B_{1})_{1} = b and (C_{1})_{2} = c and (D_{1})_{2} = d and a_{1} = A_{1} and b_{1} = B_{1} and c_{1} = C_{1} and d_{1} = D_{1}. Then there exist points s, t of 𝕀 such that h(s) = v(t).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in E^{2}. Formalized Mathematics, 6(3):427–440, 1997.

[8] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[9] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces – fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

the fashoda meet theorem … 469

[10] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

[11] Alicia de la Cruz. Totally bounded metric spaces. Formalized Mathematics, 2(4):559–562, 1991.

[12] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

[13] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[14] Artur Korniłowicz. The fundamental group of convex subspaces of E_{T}^{n}. Formalized Mathematics, 12(3):295–299, 2004.

[15] Artur Korniłowicz and Yasunari Shidama. Some properties of rectangles on the plane. Formalized Mathematics, 13(1):109–115, 2005.

[16] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board – part I. Formalized Mathematics, 3(1):107–115, 1992.

[17] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

[18] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[19] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[20] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[23] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[24] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

470 yatsuka nakamura et al.

Tietze Extension Theorem

Artur Korniłowicz Institute of Computer Science

University of Białystok Sosnowa 64, 15-887 Białystok

Poland

Grzegorz Bancerek Department of Theoretical Computer Science Białystok Technical University Poland

Adam Naumowicz Institute of Computer Science University of Białystok Akademicka 2, 15-267 Białystok Poland

Summary. In this paper we formalize the Tietze extension theorem using as a basis the proof presented at the PlanetMath web server^{1}.

MML identifier: TIETZE, version: 7.5.01 4.39.921

The articles [24], [26], [1], [2], [23], [11], [4], [21], [27], [5], [28], [7], [6], [17], [16], [22], [18], [20], [19], [25], [9], [10], [13], [14], [8], [12], [3], and [15] provide the notation and terminology for this paper.

We adopt the following rules: r, s denote real numbers, X denotes a set, and f, g, h denote real-yielding functions.

The following propositions are true:

(1) For all real numbers a, b, c such that |a − b| ≤ c holds b − c ≤ a and a ≤ b + c.

(2) If r < s, then ]−∞, r] misses [s, +∞[.

(3) If r ≤ s, then ]−∞, r[ misses ]s, +∞[.

(4) If f ⊆ g, then h − f ⊆ h − g.

(5) If f ⊆ g, then f − h ⊆ g − h.

1http://planetmath.org/encyclopedia/ProofOfTietzeExtensionTheorem2.html

Let f be a real-yielding function, let r be a real number, and let X be a set. We say that f is absolutely bounded by r in X if and only if:

(Def. 1) For every set x such that x ∈ X ∩ dom f holds |f(x)| ≤ r.

Let us mention that there exists a sequence of real numbers which is summable, constant, and convergent.

We now state the proposition

(6) For every empty topological space T_{1} and for every topological space T_{2} holds every map from T_{1} into T_{2} is continuous.

Let T_{1} be a topological space and let T_{2} be a non empty topological space. Observe that there exists a map from T_{1} into T_{2} which is continuous.

We now state several propositions:

(7) For all summable sequences f, g of real numbers such that for every natural number n holds f(n) ≤ g(n) holds ∑ f ≤ ∑ g.

(8) For every sequence f of real numbers such that f is absolutely summable holds |∑ f| ≤ ∑|f|.

(9) Let f be a sequence of real numbers and a, r be positive real numbers. Suppose r < 1 and for every natural number n holds |f(n) − f(n + 1)| ≤ a · r^{n}. Then f is convergent and for every natural number n holds | lim f − f(n)| ≤ \frac{a·r^{n}}{1−r} .

(10) Let f be a sequence of real numbers and a, r be positive real numbers. Suppose r < 1 and for every natural number n holds |f(n) − f(n + 1)| ≤ a · r^{n}. Then lim f ≥ f(0) − \frac{a}{1−r} and lim f ≤ f(0) + \frac{a}{1−r} .

(11) Let X, Z be non empty sets and F be a sequence of partial functions from X into ℝ. Suppose Z is common for elements of F. Let a, r be positive real numbers. Suppose r < 1 and for every natural number n holds F (n) − F (n + 1) is absolutely bounded by a · r^{n} in Z. Then F is uniformconvergent on Z and for every natural number n holds lim_{Z}F − F (n) is absolutely bounded by \frac{a·r^{n}}{1−r} in Z.

(12) Let X, Z be non empty sets and F be a sequence of partial functions from X into ℝ. Suppose Z is common for elements of F. Let a, r be positive real numbers. Suppose r < 1 and for every natural number n holds F (n) − F (n + 1) is absolutely bounded by a · r^{n} in Z. Let z be an element of Z. Then (lim_{Z}F)(z) ≥ F (0)(z) − \frac{a}{1−r} and (lim_{Z}F)(z) ≤ F (0)(z) + \frac{a}{1−r} .

(13) Let X, Z be non empty sets and F be a sequence of partial functions from X into ℝ. Suppose Z is common for elements of F. Let a, r be positive real numbers and f be a function from Z into ℝ. Suppose r < 1 and for every natural number n holds F (n) − f is absolutely bounded by a · r^{n} in Z. Then F is point-convergent on Z and lim_{Z}F = f.

Let S, T be topological structures, let A be an empty subset of S, and let f be a map from S into T. Note that f↾A is empty.

Let T be a topological space and let A be a closed subset of T. Note that T ↾A is closed.

The following propositions are true:

(14) Let X, Y be non empty topological spaces, X_{1}, X_{2} be non empty subspaces of X, f_{1} be a map from X_{1} into Y, and f_{2} be a map from X_{2} into Y. Suppose X_{1} misses X_{2} or f_{1}↾(X_{1} ∩ X_{2}) = f_{2}↾(X_{1} ∩ X_{2}). Let x be a point of X. Then

(i) if x ∈ the carrier of X_{1}, then (f_{1} ∪ f_{2})(x) = f_{1}(x), and (ii) if x ∈ the carrier of X_{2}, then (f_{1} ∪ f_{2})(x) = f_{2}(x).

(15) Let X, Y be non empty topological spaces, X_{1}, X_{2} be non empty subspaces of X, f_{1} be a map from X_{1} into Y, and f_{2} be a map from X_{2} into Y. If X_{1} misses X_{2} or f_{1}↾(X_{1} ∩ X_{2}) = f_{2}↾(X_{1} ∩ X_{2}), then rng(f_{1} ∪ f_{2}) ⊆ rng f_{1} ∪ rng f_{2}.

(16) Let X, Y be non empty topological spaces, X_{1}, X_{2} be non empty subspaces of X, f_{1} be a map from X_{1} into Y, and f_{2} be a map from X_{2} into Y. Suppose X_{1} misses X_{2} or f_{1}↾(X_{1} ∩X_{2}) = f_{2}↾(X_{1} ∩X_{2}). Then for every subset A of X_{1} holds (f_{1} ∪ f_{2})^{∘}A = f_{1}^{∘}A and for every subset A of X_{2} holds (f_{1} ∪ f_{2})^{∘}A = f_{2}^{∘}A.

(17) If f ⊆ g and g is absolutely bounded by r in X, then f is absolutely bounded by r in X.

(18) If X ⊆ dom f or dom g ⊆ dom f and if f↾X = g↾X and if f is absolutely

bounded by r in X, then g is absolutely bounded by r in X. In the sequel T is a non empty topological space and A is a closed subset of

T.

One can prove the following propositions:

(19) Suppose r > 0 and T is T_{4}. Let f be a continuous map from T ↾A into ℝ^{1}. Suppose f is absolutely bounded by r in A. Then there exists a continuous map g from T into ℝ^{1} such that g is absolutely bounded by \frac{r}{3} in dom g and f − g is absolutely bounded by \frac{2·r}{3} in A.

(20) Suppose that for all non empty closed subsets A, B of T such that A misses B there exists a continuous map f from T into ℝ^{1} such that f^{∘}A = {0} and f^{∘}B = {1}. Then T is a T_{4} space.

(21) Let f be a map from T into ℝ^{1} and x be a point of T. Then f is continuous at x if and only if for every real number e such that e > 0 there exists a subset H of T such that H is open and x ∈ H and for every point y of T such that y ∈ H holds |f(y) − f(x)| < e.

(22) Let F be a sequence of partial functions from the carrier of T into ℝ.

Suppose that

(i) F is uniform-convergent on the carrier of T, and (ii) for every natural number i holds F (i) is a continuous map from T into 474 artur korniłowicz et al.

ℝ^{1}. Then lim_{the carrier of T} F is a continuous map from T into ℝ^{1}.

(23) Let T be a non empty topological space, f be a map from T into ℝ^{1}, and r be a positive real number. Then f is absolutely bounded by r in the carrier of T if and only if f is a map from T into [−r, r]_{T}.

(24) If f −g is absolutely bounded by r in X, then g−f is absolutely bounded by r in X.

(25) Suppose T is T_{4}. Let given A and f be a map from T ↾A into [−1, 1]_{T}. Suppose f is continuous. Then there exists a continuous map g from T into [−1, 1]_{T} such that g↾A = f.

(26) Suppose that for every non empty closed subset A of T and for every continuous map f from T ↾A into [−1, 1]_{T} there exists a continuous map g from T into [−1, 1]_{T} such that g↾A = f. Then T is T_{4}.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[3] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[8] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in E^{2}. Formalized Mathematics, 6(3):427–440, 1997.

[9] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

[10] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces – fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[12] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[13] Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2(5):665–674, 1991.

[14] Zbigniew Karno. Continuity of mappings over the union of subspaces. Formalized Mathematics, 3(1):1–16, 1992.

[15] Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181–187, 2005.

[16] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[17] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[18] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[19] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[20] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17–21, 1992.

tietze extension theorem 475

[21] Konrad Raczkowski and Andrzej Nȩdzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[22] Konrad Raczkowski and Andrzej Nȩdzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[23] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[24] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[25] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[28] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

476 artur korniłowicz et al.

Homeomorphisms of Jordan Curves

Adam Naumowicz Institute of Computer Science University of Białystok

Akademicka 2, 15-267 Białystok Poland

Grzegorz Bancerek Department of Theoretical Computer Science Białystok Technical University Poland

Summary. In this paper we prove that simple closed curves can be homeomorphically framed into a given rectangle. We also show that homeomorphisms preserve the Jordan property.

      MML identifier: JORDAN24, version: 7.5.01 4.39.921

The notation and terminology used in this paper are introduced in the following articles: [20], [21], [1], [3], [22], [4], [5], [19], [10], [18], [7], [17], [11], [2], [8], [9], [16], [13], [14], [15], [6], [23], and [12].

In this paper p_{1}, p_{2} are points of E_{T}^{2}, C is a simple closed curve, and P is a subset of E_{T}^{2}.

Let n be a natural number, let A be a subset of E_{T}^{n}, and let a, b be points of E_{T}^{n}. We say that a and b realize maximal distance in A if and only if:

(Def. 1) a ∈ A and b ∈ A and for all points x, y of E_{T}^{n} such that x ∈ A and y ∈ A

holds ρ(a, b) ≥ ρ(x, y). Next we state the proposition

(1) There exist p_{1}, p_{2} such that p_{1} and p_{2} realize maximal distance in C. Let M be a non empty metric structure and let f be a map from M_{top} into

M_{top}. We say that f is isometric if and only if: (Def. 2) There exists an isometric map g from M into M such that g = f.

Let M be a non empty metric structure. Note that there exists a map from M_{top} into M_{top} which is isometric.

Let M be a non empty metric space. Observe that every map from M_{top} into M_{top} which is isometric is also continuous.

Let M be a non empty metric space. Note that every map from M_{top} into M_{top} which is isometric is also homeomorphism.

Let a be a real number. The functor Rotate a yields a map from E_{T}^{2} into E_{T}^{2} and is defined as follows:

(Def. 3) For every point p of E_{T}^{2} holds (Rotate a)(p) = [ℜ(p_{1} + p_{2} · i ⟲ a), ℑ(p_{1} +

p_{2} · i ⟲ a)], where a = [r_{1}, 0] and r_{1} = −1. The following propositions are true:

(2) Let a be a real number. Suppose 0 ≤ a and a < 2·π. Let f be a map from (E^{2})_{top} into (E^{2})_{top}. If f = Rotate a, then f is isometric, where a = [r_{1}, 0] and r_{1} = −1.

(3) Let A, B, D be real numbers. Suppose p_{1} and p_{2} realize maximal distance in P. Then (AffineMap(A, B, A, D))(p_{1}) and (AffineMap(A, B, A, D))(p_{2}) realize maximal distance in (AffineMap(A, B, A, D))^{∘}P.

(4) Let A be a real number. Suppose 0 ≤ A and A < 2 · π and p_{1} and p_{2} realize maximal distance in P. Then (Rotate A)(p_{1}) and (Rotate A)(p_{2}) realize maximal distance in (Rotate A)^{∘}P.

(5) For every complex number z and for every real number r holds z ⟲ −r = z ⟲ 2 · π − r.

(6) For every real number r holds Rotate(−r) = Rotate(2 · π − r).

(7) There exists a homeomorphism f of E_{T}^{2} such that [−1, 0] and [1, 0] realize maximal distance in f^{∘}C.

Let T_{1}, T_{2} be topological structures and let f be a map from T_{1} into T_{2}. We say that f is closed if and only if:

(Def. 4) For every subset A of T_{1} such that A is closed holds f^{∘}A is closed. One can prove the following propositions:

(8) Let X, Y be non empty topological spaces and f be a continuous map from X into Y. Suppose f is one-to-one and onto. Then f is a homeomorphism if and only if f is closed.

(9) For every set X and for every subset A of X holds A^{c} = ∅ iff A = X.

(10) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into T_{2}. Suppose f is a homeomorphism. Let A be a subset of T_{1}. If A is connected, then f^{∘}A is connected.

(11) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into T_{2}. Suppose f is a homeomorphism. Let A be a subset of T_{1}. If A is a component of T_{1}, then f^{∘}A is a component of T_{2}.

(12) Let T_{1}, T_{2} be non empty topological spaces, f be a map from T_{1} into T_{2}, and A be a subset of T_{1}. Then f↾A is a map from T_{1}↾A into T_{2}↾f^{∘}A.

(13) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into

homeomorphisms of jordan curves 479

T_{2}. Suppose f is continuous. Let A be a subset of T_{1} and g be a map from T_{1}↾A into T_{2}↾f^{∘}A. If g = f↾A, then g is continuous.

(14) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into T_{2}. Suppose f is a homeomorphism. Let A be a subset of T_{1} and g be a map from T_{1}↾A into T_{2}↾f^{∘}A. If g = f↾A, then g is a homeomorphism.

(15) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into T_{2}. Suppose f is a homeomorphism. Let A, B be subsets of T_{1}. If A is a component of B, then f^{∘}A is a component of f^{∘}B.

(16) For every subset S of E_{T}^{2} and for every homeomorphism f of E_{T}^{2} such that S is Jordan holds f^{∘}S is Jordan.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275–280, 2003.

[7] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257–261, 1990.

[8] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[9] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Simple closed curves. Formalized Mathematics, 2(5):663–664, 1991.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[12] Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics, 7(2):217–225, 1998.

[13] Artur Korniłowicz. Properties of left and right components. Formalized Mathematics, 8(1):163–168, 1999.

[14] Robert Milewski. Real linear-metric space and isometric functions. Formalized Mathematics, 7(2):273–277, 1998.

[15] Yatsuka Nakamura. On Outside Fashoda Meet Theorem. Formalized Mathematics, 9(4):697–704, 2001.

[16] Yatsuka Nakamura and Jarosław Kotowicz. The Jordan’s property for certain subsets of the plane. Formalized Mathematics, 3(2):137–142, 1992.

[17] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

[18] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[19] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[20] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[21] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

480 adam naumowicz and grzegorz bancerek

[23] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle

ratio. Formalized Mathematics, 7(2):255–263, 1998.

Received September 15, 2005

Jordan Curve Theorem

Artur Korniłowicz Institute of Computer Science University of Białystok Sosnowa 64, 15-887 Białystok, Poland

Summary. This paper formalizes the Jordan curve theorem following [42] and [17].

MML identifier: JORDAN, version: 7.5.01 4.39.921

The articles [44], [47], [9], [1], [45], [48], [5], [8], [6], [4], [7], [10], [43], [21], [2], [40], [39], [49], [46], [12], [11], [37], [38], [33], [22], [3], [13], [18], [15], [16], [14], [31], [32], [35], [20], [34], [30], [25], [26], [19], [29], [24], [23], [36], [41], [28], and [27] provide the notation and terminology for this paper.

1. Preliminaries

For simplicity, we adopt the following rules: a, b, c, d, r, s denote real numbers, n denotes a natural number, p, p_{1}, p_{2} denote points of E_{T}^{2}, x, y denote points of E_{T}^{n}, C denotes a simple closed curve, A, B, P denote subsets of E_{T}^{2}, U, V denote subsets of (E_{T}^{2})↾C^{c}, and D denotes a compact middle-intersecting subset of E_{T}^{2}.

Let M be a symmetric triangle Reflexive metric structure and let x, y be points of M. One can verify that ρ(x, y) is non negative.

Let n be a natural number and let x, y be points of E_{T}^{n}. Note that ρ(x, y) is non negative.

Let n be a natural number and let x, y be points of E_{T}^{n}. Observe that |x − y| is non negative.

We now state several propositions:

(1) For all points p_{1}, p_{2} of E_{T}^{n} such that p_{1} ≠ p_{2} holds \frac{1}{2} · (p_{1} + p_{2}) ≠ p_{1}.

(2) If (p_{1})_{2} < (p_{2})_{2}, then (p_{1})_{2} < (\frac{1}{2} · (p_{1} + p_{2}))_{2}.

(3) If (p_{1})_{2} < (p_{2})_{2}, then (\frac{1}{2} · (p_{1} + p_{2}))_{2} < (p_{2})_{2}.

(4) For every vertical subset A of E_{T}^{2} holds A ∩ B is vertical.

(5) For every horizontal subset A of E_{T}^{2} holds A ∩ B is horizontal.

(6) If p ∈ L(p_{1}, p_{2}) and L(p_{1}, p_{2}) is vertical, then L(p, p_{2}) is vertical.

(7) If p ∈ L(p_{1}, p_{2}) and L(p_{1}, p_{2}) is horizontal, then L(p, p_{2}) is horizontal. Let P be a subset of E_{T}^{2}. One can verify the following observations:

∗ L(SW-corner(P), SE-corner(P)) is horizontal,

∗ L(NW-corner(P), SW-corner(P)) is vertical, and

∗ L(NE-corner(P), SE-corner(P)) is vertical.

Let P be a subset of E_{T}^{2}. One can check the following observations:

∗ L(SE-corner(P), SW-corner(P)) is horizontal,

∗ L(SW-corner(P), NW-corner(P)) is vertical, and

∗ L(SE-corner(P), NE-corner(P)) is vertical.

Let us note that every subset of E_{T}^{2} which is vertical, non empty, and compact is also middle-intersecting.

The following propositions are true:

(8) For all non empty compact subsets X, Y of E_{T}^{2} such that X ⊆ Y but W_{min}(Y) ∈ X or W_{max}(Y) ∈ X holds W-bound(X) = W-bound(Y).

(9) For all non empty compact subsets X, Y of E_{T}^{2} such that X ⊆ Y but E_{min}(Y) ∈ X or E_{max}(Y) ∈ X holds E-bound(X) = E-bound(Y).

(10) For all non empty compact subsets X, Y of E_{T}^{2} such that X ⊆ Y but N_{min}(Y) ∈ X or N_{max}(Y) ∈ X holds N-bound(X) = N-bound(Y).

(11) For all non empty compact subsets X, Y of E_{T}^{2} such that X ⊆ Y but S_{min}(Y) ∈ X or S_{max}(Y) ∈ X holds S-bound(X) = S-bound(Y).

(12) W-bound(C) = W-bound(NorthArc(C)).

(13) E-bound(C) = E-bound(NorthArc(C)).

(14) W-bound(C) = W-bound(SouthArc(C)).

(15) E-bound(C) = E-bound(SouthArc(C)).

(16) If (p_{1})_{1} ≤ r and r ≤ (p_{2})_{1}, then L(p_{1}, p_{2}) meets VerticalLine(r).

(17) If (p_{1})_{2} ≤ r and r ≤ (p_{2})_{2}, then L(p_{1}, p_{2}) meets HorizontalLine(r). Let us consider n. One can check that every subset of E_{T}^{n} which is empty is

also Bounded and every subset of E_{T}^{n} which is non Bounded is also non empty.

Let n be a non empty natural number. Note that there exists a subset of E_{T}^{n} which is open, closed, non Bounded, and convex.

Next we state several propositions:

(18) For every compact subset C of E_{T}^{2} holds NorthHalfline UMP C\{UMP C} misses C.

(19) For every compact subset C of E_{T}^{2} holds SouthHalfline LMP C \{LMP C} misses C.

(20) For every compact subset C of E_{T}^{2} holds NorthHalfline UMP C \ {UMP C} ⊆ UBD C.

(21) For every compact subset C of E_{T}^{2} holds SouthHalfline LMP C \ {LMP C} ⊆ UBD C.

(22) If A is an inside component of B, then UBD B misses A.

(23) If A is an outside component of B, then BDD B misses A.

One can prove the following propositions:

(24) For every positive real number r and for every point a of E_{T}^{n} holds a ∈ Ball(a, r).

(25) For every non negative real number r holds every point p of E_{T}^{n} is a point

of Tdisk(p, r). Let r be a positive real number, let n be a non empty natural number, and

let p, q be points of E_{T}^{n}. Observe that \overline{Ball}(p, r) \ \{q\} is non empty.

We now state several propositions:

(26) If r ≤ s, then Ball(x, r) ⊆ Ball(x, s).

(27) \overline{Ball}(x, r) \ Ball(x, r) = Sphere(x, r).

(28) If y ∈ Sphere(x, r), then L(x, y) \ {x, y} ⊆ Ball(x, r).

(29) If r < s, then \overline{Ball}(x, r) ⊆ Ball(x, s).

(30) If r < s, then Sphere(x, r) ⊆ Ball(x, s).

(31) For every non zero real number r holds \overline{Ball(x, r)} = \overline{Ball}(x, r).

(32) For every non zero real number r holds Fr Ball(x, r) = Sphere(x, r). Let n be a non empty natural number. Note that every subset of E_{T}^{n} which

is Bounded is also proper.

Let us consider n. Note that there exists a subset of E_{T}^{n} which is non empty, closed, convex, and Bounded and there exists a subset of E_{T}^{n} which is non empty, open, convex, and Bounded.

Let n be a natural number and let A be a Bounded subset of E_{T}^{n}. Observe that \overline{A} is Bounded.

Let n be a natural number and let A be a Bounded subset of E_{T}^{n}. One can check that Fr A is Bounded.

The following propositions are true:

(33) Let A be a closed subset of E_{T}^{n} and p be a point of E_{T}^{n}. If p ∉ A, then there exists a positive real number r such that Ball(p, r) misses A.

(34) For every Bounded subset A of E_{T}^{n} and for every point a of E_{T}^{n} there exists a positive real number r such that A ⊆ Ball(a, r).

(35) For all topological structures S, T and for every map f from S into T such that f is a homeomorphism holds f is onto.

(36) Let T be a topological space, S be a subspace of T, A be a subset of T,

and B be a subset of S. If A = B, then T ↾A = S↾B. Let T be a non empty T_{2} topological space. Note that every non empty

subspace of T is T_{2}.

Let us consider p, r. Observe that Tdisk(p, r) is closed. Let us consider p, r. Observe that Tdisk(p, r) is compact.

2. Paths

Next we state a number of propositions:

(37) Let T be a non empty topological space, a, b be points of T, and f be a path from a to b. If a, b are connected, then rng f is connected.

(38) Let X be a non empty topological space, Y be a non empty subspace of X, x_{1}, x_{2} be points of X, y_{1}, y_{2} be points of Y, and f be a path from x_{1} to x_{2}. Suppose x_{1} = y_{1} and x_{2} = y_{2} and x_{1}, x_{2} are connected and rng f ⊆ the carrier of Y. Then y_{1}, y_{2} are connected and f is a path from y_{1} to y_{2}.

(39) Let X be an arcwise connected non empty topological space, Y be a non empty subspace of X, x_{1}, x_{2} be points of X, y_{1}, y_{2} be points of Y, and f be a path from x_{1} to x_{2}. Suppose x_{1} = y_{1} and x_{2} = y_{2} and rng f ⊆ the carrier of Y. Then y_{1}, y_{2} are connected and f is a path from y_{1} to y_{2}.

(40) Let T be a non empty topological space, a, b be points of T, and f be a path from a to b. If a, b are connected, then rng f = rng(−f).

(41) Let T be an arcwise connected non empty topological space, a, b be points of T, and f be a path from a to b. Then rng f = rng(−f).

(42) Let T be a non empty topological space, a, b, c be points of T, f be a path from a to b, and g be a path from b to c. If a, b are connected and b, c are connected, then rng f ⊆ rng(f + g).

(43) Let T be an arcwise connected non empty topological space, a, b, c be points of T, f be a path from a to b, and g be a path from b to c. Then rng f ⊆ rng(f + g).

(44) Let T be a non empty topological space, a, b, c be points of T, f be a path from b to c, and g be a path from a to b. If a, b are connected and b, c are connected, then rng f ⊆ rng(g + f).

(45) Let T be an arcwise connected non empty topological space, a, b, c be points of T, f be a path from b to c, and g be a path from a to b. Then rng f ⊆ rng(g + f).

(46) Let T be a non empty topological space, a, b, c be points of T, f be a path from a to b, and g be a path from b to c. If a, b are connected and b, c are connected, then rng(f + g) = rng f ∪ rng g.

(47) Let T be an arcwise connected non empty topological space, a, b, c be points of T, f be a path from a to b, and g be a path from b to c. Then rng(f + g) = rng f ∪ rng g.

(48) Let T be a non empty topological space, a, b, c, d be points of T, f be a path from a to b, g be a path from b to c, and h be a path from c to d. Suppose a, b are connected and b, c are connected and c, d are connected. Then rng(f + g + h) = rng f ∪ rng g ∪ rng h.

(49) Let T be an arcwise connected non empty topological space, a, b, c, d be points of T, f be a path from a to b, g be a path from b to c, and h be a path from c to d. Then rng(f + g + h) = rng f ∪ rng g ∪ rng h.

(50) For every non empty topological space T and for every point a of T holds 𝕀 ⟼ a is a path from a to a.

(51) Let p_{1}, p_{2} be points of E_{T}^{n} and P be a subset of E_{T}^{n}. Suppose P is an arc from p_{1} to p_{2}. Then there exists a path F from p_{1} to p_{2} and there exists a map f from 𝕀 into (E_{T}^{n})↾P such that rng f = P and F = f.

(52) Let p_{1}, p_{2} be points of E_{T}^{n}. Then there exists a path F from p_{1} to p_{2} and there exists a map f from 𝕀 into (E_{T}^{n})↾L(p_{1}, p_{2}) such that rng f = L(p_{1}, p_{2}) and F = f.

(53) Let p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and q_{1} ∈ P and q_{2} ∈ P and q_{1} ≠ p_{1} and q_{1} ≠ p_{2} and q_{2} ≠ p_{1} and q_{2} ≠ p_{2}. Then there exists a path f from q_{1} to q_{2} such that rng f ⊆ P and rng f misses {p_{1}, p_{2}}.

3. Rectangles

Next we state three propositions:

(54) If a ≤ b and c ≤ d, then Rectangle(a, b, c, d) ⊆ ClosedInsideOfRectangle (a, b, c, d).

(55) InsideOfRectangle(a, b, c, d) ⊆ ClosedInsideOfRectangle(a, b, c, d).

(56) ClosedInsideOfRectangle(a, b, c, d) = (OutsideOfRectangle(a, b, c, d))^{c}. Let a, b, c, d be real numbers. Note that ClosedInsideOfRectangle(a, b, c, d)

is closed.

One can prove the following propositions:

(57) ClosedInsideOfRectangle(a, b, c, d) misses OutsideOfRectangle(a, b, c, d).

(58) ClosedInsideOfRectangle(a, b, c, d) ∩ InsideOfRectangle(a, b, c, d) = InsideOfRectangle(a, b, c, d).

(59) If a < b and c < d, then Int ClosedInsideOfRectangle(a, b, c, d) = InsideOfRectangle(a, b, c, d).

(60) If a ≤ b and c ≤ d, then ClosedInsideOfRectangle(a, b, c, d) \ InsideOfRectangle(a, b, c, d) = Rectangle(a, b, c, d).

(61) If a < b and c < d, then Fr ClosedInsideOfRectangle(a, b, c, d) = Rectangle(a, b, c, d).

(62) If a ≤ b and c ≤ d, then W-bound(ClosedInsideOfRectangle(a, b, c, d)) = a.

(63) If a ≤ b and c ≤ d, then S-bound(ClosedInsideOfRectangle(a, b, c, d)) = c.

(64) If a ≤ b and c ≤ d, then E-bound(ClosedInsideOfRectangle(a, b, c, d)) = b.

(65) If a ≤ b and c ≤ d, then N-bound(ClosedInsideOfRectangle(a, b, c, d)) = d.

(66) If a < b and c < d and p_{1} ∈ ClosedInsideOfRectangle(a, b, c, d) and p_{2} ∉ ClosedInsideOfRectangle(a, b, c, d) and P is an arc from p_{1} to p_{2}, then Segment(P, p_{1}, p_{2}, p_{1}, FPoint(P, p_{1}, p_{2}, Rectangle(a, b, c, d))) ⊆ ClosedInsideOfRectangle(a, b, c, d).

4. Some Useful Functions

Let S, T be non empty topological spaces and let x be a point of [: S, T :]. Then x_{1} is an element of S, and x_{2} is an element of T.

Let o be a point of E_{T}^{2}. The functor (□_{2})_{1} − o_{1} yielding a real map of [: E_{T}^{2}, E_{T}^{2} :] is defined as follows:

(Def. 1) For every point x of [: E_{T}^{2}, E_{T}^{2} :] holds ((□_{2})_{1} − o_{1})(x) = (x_{2})_{1} − o_{1}.

The functor (□_{2})_{2} − o_{2} yields a real map of [: E_{T}^{2}, E_{T}^{2} :] and is defined as follows: (Def. 2) For every point x of [: E_{T}^{2}, E_{T}^{2} :] holds ((□_{2})_{2} − o_{2})(x) = (x_{2})_{2} − o_{2}.

The real map (□_{1})_{1} − (□_{2})_{1} of [: E_{T}^{2}, E_{T}^{2} :] is defined as follows: (Def. 3) For every point x of [: E_{T}^{2}, E_{T}^{2} :] holds ((□_{1})_{1} − (□_{2})_{1})(x) = (x_{1})_{1} − (x_{2})_{1}.

The real map (□_{1})_{2} − (□_{2})_{2} of [: E_{T}^{2}, E_{T}^{2} :] is defined as follows: (Def. 4) For every point x of [: E_{T}^{2}, E_{T}^{2} :] holds ((□_{1})_{2} − (□_{2})_{2})(x) = (x_{1})_{2} − (x_{2})_{2}.

The real map (□_{2})_{1} of [: E_{T}^{2}, E_{T}^{2} :] is defined as follows: (Def. 5) For every point x of [: E_{T}^{2}, E_{T}^{2} :] holds (□_{2})_{1}(x) = (x_{2})_{1}.

The real map (□_{2})_{2} of [: E_{T}^{2}, E_{T}^{2} :] is defined by: (Def. 6) For every point x of [: E_{T}^{2}, E_{T}^{2} :] holds (□_{2})_{2}(x) = (x_{2})_{2}.

One can prove the following propositions:

(67) For every point o of E_{T}^{2} holds (□_{2})_{1} − o_{1} is a continuous map from [: E_{T}^{2}, E_{T}^{2} :] into ℝ^{1}.

(68) For every point o of E_{T}^{2} holds (□_{2})_{2} − o_{2} is a continuous map from [: E_{T}^{2}, E_{T}^{2} :] into ℝ^{1}.

(69) (□_{1})_{1} − (□_{2})_{1} is a continuous map from [: E_{T}^{2}, E_{T}^{2} :] into ℝ^{1}.

(70) (□_{1})_{2} − (□_{2})_{2} is a continuous map from [: E_{T}^{2}, E_{T}^{2} :] into ℝ^{1}.

(71) (□_{2})_{1} is a continuous map from [: E_{T}^{2}, E_{T}^{2} :] into ℝ^{1}.

(72) (□_{2})_{2} is a continuous map from [: E_{T}^{2}, E_{T}^{2} :] into ℝ^{1}.

Let o be a point of E_{T}^{2}. One can check that (□_{2})_{1} − o_{1} is continuous and (□_{2})_{2} − o_{2} is continuous.

One can check the following observations:

∗ (□_{1})_{1} − (□_{2})_{1} is continuous,

∗ (□_{1})_{2} − (□_{2})_{2} is continuous,

∗ (□_{2})_{1} is continuous, and

∗ (□_{2})_{2} is continuous. Let n be a non empty natural number, let o, p be points of E_{T}^{n}, and let r be a

positive real number. Let us assume that p is a point of Tdisk(o, r). The functor DiskProj(o, r, p) yielding a map from (E_{T}^{n})↾(\overline{Ball}(o, r) \ \{p\}) into Tcircle(o, r) is defined by:

(Def. 7) For every point x of (E_{T}^{n})↾(\overline{Ball}(o, r) \ \{p\}) there exists a point y of E_{T}^{n}

such that x = y and (DiskProj(o, r, p))(x) = HC(p, y, o, r). The following propositions are true:

(73) Let o, p be points of E_{T}^{2} and r be a positive real number. If p is a point of Tdisk(o, r), then DiskProj(o, r, p) is continuous.

(74) Let n be a non empty natural number, o, p be points of E_{T}^{n}, and r be a positive real number. If p ∈ Ball(o, r), then DiskProj(o, r, p)↾ Sphere(o, r) = id_{Sphere(o,r)}.

Let n be a non empty natural number, let o, p be points of E_{T}^{n}, and let r be a positive real number. Let us assume that p ∈ Ball(o, r). The functor RotateCircle(o, r, p) yields a map from Tcircle(o, r) into Tcircle(o, r) and is defined by:

(Def. 8) For every point x of Tcircle(o, r) there exists a point y of E_{T}^{n} such that

x = y and (RotateCircle(o, r, p))(x) = HC(y, p, o, r). One can prove the following propositions:

(75) For all points o, p of E_{T}^{2} and for every positive real number r such that p ∈ Ball(o, r) holds RotateCircle(o, r, p) is continuous.

(76) Let n be a non empty natural number, o, p be points of E_{T}^{n}, and r be a positive real number. If p ∈ Ball(o, r), then RotateCircle(o, r, p) has no fixpoint.

5. Jordan Curve Theorem

The following propositions are true:

(77) If U = P and U is a component of (E_{T}^{2})↾C^{c} and V is a component of (E_{T}^{2})↾C^{c} and U ≠ V, then \overline{P} misses V.

(78) If U is a component of (E_{T}^{2})↾C^{c}, then (E_{T}^{2})↾C^{c}↾U is arcwise connected.

(79) If U = P and U is a component of (E_{T}^{2})↾C^{c}, then C = Fr P.

One can prove the following propositions:

(80) For every homeomorphism h of E_{T}^{2} holds h^{∘}C satisfies conditions of simple closed curve.

(81) If [−1, 0] and [1, 0] realize maximal distance in P, then P ⊆ ClosedInsideOfRectangle(−1, 1, −3, 3).

(82) If [−1, 0] and [1, 0] realize maximal distance in P, then P misses L([−1, 3], [1, 3]).

(83) If [−1, 0] and [1, 0] realize maximal distance in P, then P misses L([−1, −3], [1, −3]).

(84) If [−1, 0] and [1, 0] realize maximal distance in P, then P ∩ Rectangle(−1, 1, −3, 3) = {[−1, 0], [1, 0]}.

(85) If [−1, 0] and [1, 0] realize maximal distance in P, then W-bound(P) = −1.

(86) If [−1, 0] and [1, 0] realize maximal distance in P, then E-bound(P) = 1.

(87) For every compact subset P of E_{T}^{2} such that [−1, 0] and [1, 0] realize maximal distance in P holds W_{most}(P) = {[−1, 0]}.

(88) For every compact subset P of E_{T}^{2} such that [−1, 0] and [1, 0] realize maximal distance in P holds E_{most}(P) = {[1, 0]}.

(89) Let P be a compact subset of E_{T}^{2}. Suppose [−1, 0] and [1, 0] realize maximal distance in P. Then W_{min}(P) = [−1, 0] and W_{max}(P) = [−1, 0].

(90) Let P be a compact subset of E_{T}^{2}. Suppose [−1, 0] and [1, 0] realize maximal distance in P. Then E_{min}(P) = [1, 0] and E_{max}(P) = [1, 0].

(91) If [−1, 0] and [1, 0] realize maximal distance in P, then L([0, 3], UMP P) is vertical.

(92) If [−1, 0] and [1, 0] realize maximal distance in P, then L(LMP P, [0, −3]) is vertical.

(93) If [−1, 0] and [1, 0] realize maximal distance in P and p ∈ P, then p_{2} < 3.

(94) If [−1, 0] and [1, 0] realize maximal distance in P and p ∈ P, then −3 < p_{2}.

(95) If [−1, 0] and [1, 0] realize maximal distance in D and p ∈ L([0, 3], UMP D), then (UMP D)_{2} ≤ p_{2}.

(96) If [−1, 0] and [1, 0] realize maximal distance in D and p ∈ L(LMP D, [0, −3]), then p_{2} ≤ (LMP D)_{2}.

(97) If [−1, 0] and [1, 0] realize maximal distance in D, then L([0, 3], UMP D) ⊆ NorthHalfline UMP D.

(98) If [−1, 0] and [1, 0] realize maximal distance in D, then L(LMP D, [0, −3]) ⊆ SouthHalfline LMP D.

(99) If [−1, 0] and [1, 0] realize maximal distance in C and P is an inside component of C, then L([0, 3], UMP C) misses P.

(100) If [−1, 0] and [1, 0] realize maximal distance in C and P is an inside component of C, then L(LMP C, [0, −3]) misses P.

(101) If [−1, 0] and [1, 0] realize maximal distance in D, then L([0, 3], UMP D)∩ D = {UMP D}.

(102) If [−1, 0] and [1, 0] realize maximal distance in D, then L([0, −3], LMP D) ∩ D = {LMP D}.

(103) Suppose P is compact and [−1, 0] and [1, 0] realize maximal distance in P and A is an inside component of P. Then A ⊆ ClosedInsideOfRectangle(−1, 1, −3, 3).

(104) If [−1, 0] and [1, 0] realize maximal distance in C, then L([0, 3], [0, −3]) meets C.

(105) Suppose [−1, 0] and [1, 0] realize maximal distance in C. Let J_{1}, J_{2} be compact middle-intersecting subsets of T_{2}. Suppose that J_{1} is an arc from [−1, 0] to [1, 0] and J_{2} is an arc from [−1, 0] to [1, 0] and C = J_{1} ∪ J_{2} and J_{1} ∩ J_{2} = {[−1, 0], [1, 0]} and UMP C ∈ J_{1} and LMP C ∈ J_{2} and W-bound(C) = W-bound(J_{1}) and E-bound(C) = E-bound(J_{1}). Let U_{1} be a subset of E_{T}^{2}. Suppose U_{1} = Component(Down(\frac{1}{2} · (UMP(L(LMP J_{1}, [0, −3]) ∩ J_{2}) + LMP J_{1}), C^{c})). Then U_{1} is an inside component of C and for every subset V of T_{2} such that V is an inside component of C holds V = U_{1}, where T_{2} = E_{T}^{2}.

(106) Suppose [−1, 0] and [1, 0] realize maximal distance in C. Let J_{1}, J_{2} be compact middle-intersecting subsets of T_{2}. Suppose that J_{1} is an arc from [−1, 0] to [1, 0] and J_{2} is an arc from [−1, 0] to [1, 0] and C = J_{1} ∪ J_{2} and J_{1} ∩ J_{2} = {[−1, 0], [1, 0]} and UMP C ∈ J_{1} and LMP C ∈ J_{2} and W-bound(C) = W-bound(J_{1}) and E-bound(C) = E-bound(J_{1}). Then BDD C = Component(Down(\frac{1}{2} · (UMP(L(LMP J_{1}, [0, −3]) ∩ J_{2}) + LMP J_{1}), C^{c})), where T_{2} = E_{T}^{2}.

(107) Let C be a simple closed curve. Then there exist subsets A_{1}, A_{2} of E_{T}^{2}

such that

(i) C^{c} = A_{1} ∪ A_{2}, (ii) A_{1} misses A_{2}, (iii) \overline{A_{1}} \ A_{1} = \overline{A_{2}} \ A_{2}, and 490 artur korniłowicz

(iv) for all subsets C_{1}, C_{2} of (E_{T}^{2})↾C^{c} such that C_{1} = A_{1} and C_{2} = A_{2} holds C_{1} is a component of (E_{T}^{2})↾C^{c} and C_{2} is a component of (E_{T}^{2})↾C^{c}.

(108) Every simple closed curve is Jordan.

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for including me in the team working on the formalization of the Jordan Curve Theorem. Especially, I am very grateful to Professor Nakamura for inviting me to Shinshu University, Nagano, to work on the project together.

I am also thankful to Professor Andrzej Trybulec for his continual help and fruitful discussions during the formalization.

References

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The Inner Product and Conjugate of Matrix of Complex Numbers

Wenpai Chang Shinshu University

Nagano, Japan Hiroshi Yamazaki Shinshu University Nagano, Japan Yatsuka Nakamura Shinshu University Nagano, Japan

Summary. Concepts of the inner product and conjugate of matrix of complex numbers are defined here. Operations such as addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of the conjugate of a matrix of a complex field. Many equations for such operations consist like a case of the conjugate of matrix of a field and some operations on the set of sum of complex numbers are introduced.

      MML identifier: MATRIXC1, version: 7.5.01 4.39.921

The papers [20], [24], [18], [25], [7], [8], [9], [3], [19], [2], [4], [11], [5], [10], [6], [17], [1], [13], [14], [23], [12], [15], [16], [22], and [21] provide the notation and terminology for this paper.

We follow the rules: i, j denote natural numbers, a denotes an element of ℂ, and R_{1}, R_{2} denote elements of ℂ^{i}.

Let M be a matrix over ℂ. The functor \overline{M} yields a matrix over ℂ and is defined by:

(Def. 1) len \overline{M} = len M and width \overline{M} = width M and for all natural numbers i,

j such that ⟨⟨i, j⟩⟩ ∈ the indices of M holds \overline{M} ∘ (i, j) = \overline{M ∘ (i, j)}. One can prove the following propositions:

(1) For every matrix M over ℂ holds ⟨⟨i, j⟩⟩ ∈ the indices of M iff 1 ≤ i and i ≤ len M and 1 ≤ j and j ≤ width M.

(2) For every matrix M over ℂ holds \overline{\overline{M}} = M.

(3) For every complex number a and for every matrix M over ℂ holds len(a· M) = len M and width(a · M) = width M.

(4) Let i, j be natural numbers, a be a complex number, and M be a matrix over ℂ. Suppose len(a · M) = len M and width(a · M) = width M and ⟨⟨i, j⟩⟩ ∈ the indices of M. Then (a · M) ∘ (i, j) = a · (M ∘ (i, j)).

(5) For every complex number a and for every matrix M over ℂ holds \overline{a · M} = \overline{a} · \overline{M} .

(6) For all matrices M_{1}, M_{2} over ℂ holds len(M_{1} + M_{2}) = len M_{1} and width(M_{1} + M_{2}) = width M_{1}.

(7) Let i, j be natural numbers and M_{1}, M_{2} be matrices over ℂ. Suppose len M_{1} = len M_{2} and width M_{1} = width M_{2} and ⟨⟨i, j⟩⟩ ∈ the indices of M_{1}. Then (M_{1} + M_{2}) ∘ (i, j) = (M_{1} ∘ (i, j)) + (M_{2} ∘ (i, j)).

(8) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} holds \overline{M_{1} + M_{2}} = \overline{M_{1}} + \overline{M_{2}} .

(9) For every matrix M over ℂ holds len(−M) = len M and width(−M) = width M.

(10) Let i, j be natural numbers and M be a matrix over ℂ. If len(−M) = len M and width(−M) = width M and ⟨⟨i, j⟩⟩ ∈ the indices of M, then (−M) ∘ (i, j) = −(M ∘ (i, j)).

(11) For every matrix M over ℂ holds (−1) · M = −M.

(12) For every matrix M over ℂ holds \overline{−M} = −\overline{M} .

(13) For all matrices M_{1}, M_{2} over ℂ holds len(M_{1} − M_{2}) = len M_{1} and width(M_{1} − M_{2}) = width M_{1}.

(14) Let i, j be natural numbers and M_{1}, M_{2} be matrices over ℂ. Suppose len M_{1} = len M_{2} and width M_{1} = width M_{2} and ⟨⟨i, j⟩⟩ ∈ the indices of M_{1}. Then (M_{1} − M_{2}) ∘ (i, j) = (M_{1} ∘ (i, j)) − (M_{2} ∘ (i, j)).

(15) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and

width M_{1} = width M_{2} holds \overline{M_{1} − M_{2}} = \overline{M_{1}} − \overline{M_{2}} . Let M be a matrix over ℂ. The functor M^{∗} yields a matrix over ℂ and is

defined by: (Def. 2) M^{∗} = \overline{M^{T}} .

Let x be a finite sequence of elements of ℂ. Let us assume that len x > 0. The functor FinSeq2Matrix x yielding a matrix over ℂ is defined as follows:

(Def. 3) len FinSeq2Matrix x = len x and width FinSeq2Matrix x = 1 and for

every j such that j ∈ Seg len x holds (FinSeq2Matrix x)(j) = ⟨x(j)⟩. Let M be a matrix over ℂ. The functor Matrix2FinSeq M yields a finite

sequence of elements of ℂ and is defined as follows: (Def. 4) Matrix2FinSeq M = M_{□,1}.

Let F_{1}, F_{2} be finite sequences of elements of ℂ. The functor F_{1} • F_{2} yielding a finite sequence of elements of ℂ is defined as follows:

(Def. 5) F_{1} • F_{2} = (·_{ℂ})^{∘}(F_{1}, F_{2}).

Let us observe that the functor F_{1} • F_{2} is commutative.

Let F be a finite sequence of elements of ℂ. The functor ∑ F yields an element of ℂ and is defined as follows:

(Def. 6) ∑ F = +_{ℂ} ⊛ F.

Let M be a matrix over ℂ and let F be a finite sequence of elements of ℂ. The functor M · F yielding a finite sequence of elements of ℂ is defined as follows:

(Def. 7) len(M · F) = len M and for every i such that i ∈ Seg len M holds (M ·

F)(i) = ∑(Line(M, i) • F). We now state the proposition

(16) a · (R_{1} • R_{2}) = a · R_{1} • R_{2}.

Let M be a matrix over ℂ and let a be a complex number. The functor M ·a yielding a matrix over ℂ is defined by:

(Def. 8) M · a = a · M.

We now state three propositions:

(17) For every element a of ℂ and for every matrix M over ℂ holds \overline{M · a} = \overline{a} · \overline{M} .

(18) For all finite sequences x, y of elements of ℂ such that len x = len y holds len(x • y) = len x and len(x • y) = len y.

(19) Let F_{1}, F_{2} be finite sequences of elements of ℂ and i be a natural number.

If i ∈ dom(F_{1} • F_{2}), then (F_{1} • F_{2})(i) = F_{1}(i) · F_{2}(i). Let us consider i, R_{1}, R_{2}. Then R_{1} • R_{2} is an element of ℂ^{i}. We now state a number of propositions:

(20) (R_{1} • R_{2})(j) = R_{1}(j) · R_{2}(j).

(21) For all elements a, b of ℂ holds \overline{+_{ℂ}(a, \overline{b})} = +_{ℂ}(\overline{a}, b).

(22) Let F be a finite sequence of elements of ℂ. Then there exists a function G from ℕ into ℂ such that for every natural number n if 1 ≤ n and n ≤ len F, then G(n) = F (n).

(23) For every finite sequence F of elements of ℂ such that len \overline{F} ≥ 1 holds +_{ℂ} ⊛ \overline{F} = \overline{+_{ℂ} ⊛ F} .

(24) For every finite sequence F of elements of ℂ such that len F ≥ 1 holds ∑ \overline{F} = \overline{∑ F} .

(25) For all finite sequences x, y of elements of ℂ such that len x = len y holds \overline{x • \overline{y}} = y • \overline{x}.

(26) For all finite sequences x, y of elements of ℂ and for every element a of ℂ such that len x = len y holds x • a · y = a · (x • y).

(27) For all finite sequences x, y of elements of ℂ and for every element a of ℂ such that len x = len y holds a · x • y = a · (x • y).

(28) For all finite sequences x, y of elements of ℂ such that len x = len y holds \overline{x • y} = \overline{x} • \overline{y}.

(29) For every finite sequence F of elements of ℂ and for every element a of

ℂ holds ∑(a · F) = a · ∑ F. Let x be a finite sequence of elements of ℝ. The functor FR2FC x yielding

a finite sequence of elements of ℂ is defined as follows: (Def. 9) FR2FC x = x.

Next we state a number of propositions:

(30) Let R be a finite sequence of elements of ℝ and F be a finite sequence of elements of ℂ. If R = F and len R ≥ 1, then +_{ℝ} ⊛ R = +_{ℂ} ⊛ F.

(31) Let x be a finite sequence of elements of ℝ and y be a finite sequence of elements of ℂ. If x = y and len x ≥ 1, then ∑ x = ∑ y.

(32) For all finite sequences F_{1}, F_{2} of elements of ℂ such that len F_{1} = len F_{2} holds ∑(F_{1} − F_{2}) = ∑ F_{1} − ∑ F_{2}.

(33) Let F_{1}, F_{2} be finite sequences of elements of ℂ and i be a natural number. If i ∈ dom(F_{1} + F_{2}), then (F_{1} + F_{2})(i) = F_{1}(i) + F_{2}(i).

(34) Let F_{1}, F_{2} be finite sequences of elements of ℂ and i be a natural number. If i ∈ dom(F_{1} − F_{2}), then (F_{1} − F_{2})(i) = F_{1}(i) − F_{2}(i).

(35) For all finite sequences x, y, z of elements of ℂ such that len x = len y and len y = len z holds (x − y) • z = x • z − y • z.

(36) For all finite sequences x, y, z of elements of ℂ such that len x = len y and len y = len z holds x • (y − z) = x • y − x • z.

(37) For all finite sequences x, y, z of elements of ℂ such that len x = len y and len y = len z holds x • (y + z) = x • y + x • z.

(38) For all finite sequences x, y, z of elements of ℂ such that len x = len y and len y = len z holds (x + y) • z = x • z + y • z.

(39) For all finite sequences F_{1}, F_{2} of elements of ℂ such that len F_{1} = len F_{2} holds ∑(F_{1} + F_{2}) = ∑ F_{1} + ∑ F_{2}.

(40) Let x_{1}, y_{1} be finite sequences of elements of ℂ and x_{2}, y_{2} be finite sequences of elements of ℝ. If x_{1} = x_{2} and y_{1} = y_{2} and len x_{1} = len y_{2}, then (·_{ℂ})^{∘}(x_{1}, y_{1}) = (·_{ℝ})^{∘}(x_{2}, y_{2}).

(41) For all finite sequences x, y of elements of ℝ such that len x = len y holds FR2FC(x • y) = FR2FC x • FR2FC y.

(42) For all finite sequences x, y of elements of ℂ such that len x = len y and len x > 0 holds |(x, y)| = ∑(x • \overline{y}).

(43) For all matrices A, B over ℂ such that len A = len B and width A = width B holds the indices of A = the indices of B.

(44) Let i, j be natural numbers and M_{1}, M_{2} be matrices over ℂ. If len M_{1} = len M_{2} and width M_{1} = width M_{2} and j ∈ Seg len M_{1}, then Line(M_{1} +

M_{2}, j) = Line(M_{1}, j) + Line(M_{2}, j).

(45) For every matrix M over ℂ such that i ∈ Seg len M holds Line(M, i) = \overline{Line(\overline{M} , i)}.

(46) Let F be a finite sequence of elements of ℂ and M be a matrix over ℂ. If len F = width M, then F • \overline{Line(\overline{M} , i)} = \overline{Line(\overline{M} , i) • \overline{F}} .

(47) Let F be a finite sequence of elements of ℂ and M be a matrix over ℂ. If len F = width M and len F ≥ 1, then \overline{M · F} = \overline{M} · \overline{F} .

(48) For all finite sequences F_{1}, F_{2}, F_{3} of elements of ℂ such that len F_{1} = len F_{2} and len F_{2} = len F_{3} holds F_{1} • (F_{2} • F_{3}) = (F_{1} • F_{2}) • F_{3}.

(49) For every finite sequence F of elements of ℂ holds ∑(−F) = −∑ F .

(50) For every element z of ℂ holds ∑⟨z⟩ = z.

(51) For all finite sequences F_{1}, F_{2} of elements of ℂ holds ∑(F_{1} ^{⌢} F_{2}) =

∑ F_{1} + ∑ F_{2}. Let M be a matrix over ℂ. The functor LineSum M yielding a finite sequence

of elements of ℂ is defined as follows: (Def. 10) len LineSum M = len M and for every natural number i such that i ∈

Seg len M holds (LineSum M)(i) = ∑ Line(M, i). Let M be a matrix over ℂ. The functor ColSum M yielding a finite sequence

of elements of ℂ is defined by: (Def. 11) len ColSum M = width M and for every natural number j such that

j ∈ Seg width M holds (ColSum M)(j) = ∑(M_{□,j}). Next we state three propositions:

(52) For every finite sequence F of elements of ℂ such that len F = 1 holds ∑ F = F (1).

(53) Let f, g be finite sequences of elements of ℂ and n be a natural number. If len f = n + 1 and g = f↾n, then ∑ f = ∑ g + f_{len f}.

(54) For every matrix M over ℂ such that len M > 0 holds ∑ LineSum M =

∑ ColSum M. Let M be a matrix over ℂ. The functor SumAll M yielding an element of ℂ

is defined by: (Def. 12) SumAll M = ∑ LineSum M.

Next we state two propositions:

(55) For every matrix M over ℂ holds ColSum M = LineSum(M^{T}).

(56) For every matrix M over ℂ such that len M > 0 holds SumAll M =

SumAll(M^{T}). Let x, y be finite sequences of elements of ℂ and let M be a matrix over

ℂ. Let us assume that len x = len M and len y = width M. The functor QuadraticForm(x, M, y) yielding a matrix over ℂ is defined by the conditions (Def. 13).

(Def. 13)(i) len QuadraticForm(x, M, y) = len x, (ii) width QuadraticForm(x, M, y) = len y, and (iii) for all natural numbers i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of M holds

QuadraticForm(x, M, y) ∘ (i, j) = x(i) · (M ∘ (i, j)) · \overline{y(j)}. The following propositions are true:

(57) Let x, y be finite sequences of elements of ℂ and M be a matrix over ℂ. If len x = len M and len y = width M and len x > 0 and len y > 0, then (QuadraticForm(x, M, y))^{T} = \overline{QuadraticForm(y, M^{∗}, x)}.

(58) Let x, y be finite sequences of elements of ℂ and M be a matrix over ℂ. If len x = len M and len y = width M, then \overline{QuadraticForm(x, M, y)} = QuadraticForm(\overline{x}, \overline{M} , \overline{y}).

(59) For all finite sequences x, y of elements of ℂ such that len x = len y and 0 < len y holds |(x, y)| = \overline{|(y, x)|}.

(60) For all finite sequences x, y of elements of ℂ such that len x = len y and 0 < len y holds \overline{|(x, y)|} = |(\overline{x}, \overline{y})|.

(61) For every matrix M over ℂ such that width M > 0 holds \overline{M^{T}} = \overline{M}^{T}.

(62) Let x, y be finite sequences of elements of ℂ and M be a matrix over ℂ. If len x = width M and len y = len M and len x > 0 and len y > 0, then |(x, M^{∗} · y)| = SumAll QuadraticForm(x, M^{T}, y).

(63) Let x, y be finite sequences of elements of ℂ and M be a matrix over ℂ. If len y = len M and len x = width M and len x > 0 and len y > 0 and len M > 0, then |(M · x, y)| = SumAll QuadraticForm(x, M^{T}, y).

(64) Let x, y be finite sequences of elements of ℂ and M be a matrix over ℂ. If len x = width M and len y = len M and width M > 0 and len M > 0, then |(M · x, y)| = |(x, M^{∗} · y)|.

(65) Let x, y be finite sequences of elements of ℂ and M be a matrix over ℂ. If len x = len M and len y = width M and width M > 0 and len M > 0 and len x > 0, then |(x, M · y)| = |(M^{∗} · x, y)|.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

[5] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164,

the inner product and conjugate of matrix of … 499

[9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[10] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[11] Czesław Byliński and Andrzej Trybulec. Complex spaces. Formalized Mathematics, 2(1):151–158, 1991.

[12] Wenpai Chang, Hiroshi Yamazaki, and Yatsuka Nakamura. The inner product and conjugate of finite sequences of complex numbers. Formalized Mathematics, 13(3):367–373, 2005.

[13] Wenpai Chang, Hiroshi Yamazaki, and Yatsuka Nakamura. A theory of matrices of complex elements. Formalized Mathematics, 13(1):157–162, 2005.

[14] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

[15] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2):265–269, 2001.

[16] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[17] Library Committee of the Association of Mizar Users. Binary operations on numbers. To appear in Formalized Mathematics.

[18] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[19] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[20] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[21] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[22] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291– 296, 1990.

[24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Received October 10, 2005 500 wenpai chang et al.

Partial Sum and Partial Product of Some Series

Jianbing Cao Qingdao University of Science

and Technology China

Fahui Zhai Qingdao University of Science and Technology China

Xiquan Liang Qingdao University of Science and Technology China

Summary. This article contains partial sum and partial product of some series which are often used.

MML identifier: SERIES 4, version: 7.6.01 4.46.926

The notation and terminology used in this paper have been introduced in the following articles: [2], [1], [3], [4], [5], [6], and [7].

We use the following convention: n is a natural number, a, b, c, d are real numbers, and s is a sequence of real numbers.

We now state a number of propositions:

(1) (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 · a · b + 2 · a · c + 2 · b · c.

(2) (a + b)^{3} = a^{3} + 3 · a^{2} · b + 3 · b^{2} · a + b^{3}.

(3) ((a − b) + c)^{2} = (((a^{2} + b^{2} + c^{2}) − 2 · a · b) + 2 · a · c) − 2 · b · c.

(4) (a − b − c)^{2} = ((a^{2} + b^{2} + c^{2}) − 2 · a · b − 2 · a · c) + 2 · b · c.

(5) (a − b)^{3} = ((a^{3} − 3 · a^{2} · b) + 3 · b^{2} · a) − b^{3}.

(6) (a + b)^{4} = a^{4} + 4 · a^{3} · b + 6 · a^{2} · b^{2} + 4 · b^{3} · a + b^{4}.

(7) (a + b + c + d)^{2} = a^{2} + b^{2} + c^{2} + d^{2} + (2 · a · b + 2 · a · c + 2 · a · d) + (2 · b · c + 2 · b · d) + 2 · c · d.

502 jianbing cao and fahui zhai and xiquan liang

(8) (a + b + c)^{3} = a^{3} + b^{3} + c^{3} + (3 · a^{2} · b + 3 · a^{2} · c) + (3 · b^{2} · a + 3 · b^{2} · c) + (3 · c^{2} · a + 3 · c^{2} · b) + 6 · a · b · c.

(9) If a ≠ 0, then ((\frac{1}{a})^{n+1} + a^{n+1})^{2} = (\frac{1}{a})^{2·n+2} + a^{2·n+2} + 2.

(10) If a ≠ 1 and for every n holds s(n) = a^{n}, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1−a^{n+1}}{1−a} .

(11) If a ≠ 1 and a ≠ 0 and for every n holds s(n) = (\frac{1}{a})^{n}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{( \frac{1}{a})^{n}−a}{1−a}.

(12) If for every n holds s(n) = 10^{n} + 2 · n + 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (\frac{10^{n+1}}{9} − \frac{1}{9}) + (n + 1)^{2}.

(13) If for every n holds s(n) = (2 · n − 1) + (\frac{1}{2})^{n}, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (n^{2} + 1) − (\frac{1}{2})^{n}.

(14) If for every n holds s(n) = n · (\frac{1}{2})^{n}, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 2 − (2 + n) · (\frac{1}{2})^{n}.

(15) If for every n holds s(n) = ((\frac{1}{2})^{n} + 2^{n})^{2}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = −\frac{( \frac{1}{4} )^{n}}{3} + \frac{4^{n+1}}{3} + 2 · n + 3.

(16) If for every n holds s(n) = ((\frac{1}{3})^{n} + 3^{n})^{2}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = −\frac{( \frac{1}{9} )^{n}}{8} + \frac{9^{n+1}}{8} + 2 · n + 3.

(17) If for every n holds s(n) = n · 2^{n}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (n · 2^{n+1} − 2^{n+1}) + 2.

(18) If for every n holds s(n) = (2 · n + 1) · 3^{n}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = n · 3^{n+1} + 1.

(19) If a ≠ 1 and for every n holds s(n) = n · a^{n}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{a·(1−a^{n})}{(1−a)^{2}} − \frac{n·a^{n+1}}{1−a} .

(20) If for every n holds s(n) = \frac{1}{(root_{2}(n+1))+(root_{2}(n))}, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = root_{2}(n + 1).

(21) If for every n holds s(n) = 2^{n} + (\frac{1}{2})^{n}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (2^{n+1} − (\frac{1}{2})^{n}) + 1.

(22) If for every n holds s(n) = n! · n + \frac{n}{(n+1)!}, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (n + 1)! − \frac{1}{(n+1)!}.

(23) Suppose a ≠ 1 and for every n such that n ≥ 1 holds s(n) = (\frac{a}{a−1})^{n} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = a·((\frac{a}{a−1})^{n} −1).

(24) If for every n such that n ≥ 1 holds s(n) = 2^{n} · \frac{3·n−1}{4} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 2^{n} · \frac{3·n−4}{2} + 2.

(25) If for every n holds s(n) = \frac{n+1}{n+2}, then (the partial product of s)(n) = \frac{1}{n+2}.

(26) If for every n holds s(n) = \frac{1}{n+1}, then (the partial product of s)(n) = \frac{1}{(n+1)!}.

partial sum and partial product … 503

(27) Suppose that for every n such that n ≥ 1 holds s(n) = n and s(0) = 1. Let given n. If n ≥ 1, then (the partial product of s)(n) = n!.

(28) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{a}{n} and s(0) = 1. Let given n. If n ≥ 1, then (the partial product of s)(n) = \frac{a^{n}}{n!} .

(29) Suppose that for every n such that n ≥ 1 holds s(n) = a and s(0) = 1. Let given n. If n ≥ 1, then (the partial product of s)(n) = a^{n}.

(30) Suppose that for every n such that n ≥ 2 holds s(n) = 1 − \frac{1}{n^{2}} and s(0) = 1 and s(1) = 1. Let given n. If n ≥ 2, then (the partial product of s)(n) = \frac{n+1}{2·n}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Fuguo Ge and Xiquan Liang. On the partial product of series and related basic inequalities. Formalized Mathematics, 13(3):413–416, 2005.

[4] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[5] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[6] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

[7] Konrad Raczkowski and Andrzej Nȩdzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

504 jianbing cao and fahui zhai and xiquan liang

Some Differentiable Formulas of Special Functions

Jianbing Cao Qingdao University of Science and Technology China

Fahui Zhai Qingdao University of Science and Technology China

Xiquan Liang Qingdao University of Science and Technology China

Summary. This article contains some differentiable formulas of special functions.

MML identifier: FDIFF 5, version: 7.6.01 4.46.926

The terminology and notation used in this paper are introduced in the following papers: [13], [15], [16], [2], [4], [10], [12], [3], [1], [6], [9], [7], [8], [11], [17], [5], and [14].

For simplicity, we use the following convention: x, a, b are real numbers, n is a natural number, Z is an open subset of ℝ, and f, f_{1}, f_{2}, g are partial functions from ℝ to ℝ.

Next we state a number of propositions:

(1) Suppose Z ⊆ dom(\frac{f_{1}}{f_{2}} ) and for every x such that x ∈ Z holds f_{1}(x) = a + x and f_{2}(x) = a − x and f_{2}(x) ≠ 0. Then \frac{f_{1}}{f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{f_{1}}{f_{2}} )′_{↾Z}(x) = \frac{2·a}{(a−x)^{2}} .

(2) Suppose Z ⊆ dom(\frac{f_{1}}{f_{2}} ) and for every x such that x ∈ Z holds f_{1}(x) = x − a and f_{2}(x) = x + a and f_{2}(x) ≠ 0. Then \frac{f_{1}}{f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{f_{1}}{f_{2}} )′_{↾Z}(x) = \frac{2·a}{(x+a)^{2}} .

(3) Suppose Z ⊆ dom(\frac{f_{1}}{f_{2}} ) and for every x such that x ∈ Z holds f_{1}(x) = x − a and f_{2}(x) = x − b and f_{2}(x) ≠ 0. Then \frac{f_{1}}{f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{f_{1}}{f_{2}} )′_{↾Z}(x) = \frac{a−b}{(x−b)^{2}} .

(4) Suppose Z ⊆ dom f and for every x such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then \frac{1}{f} is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{f})′_{↾Z}(x) = −\frac{1}{x^{2}} .

(5) Suppose Z ⊆ dom((the function sin) ·\frac{1}{f}) and for every x such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) (the function sin) ·\frac{1}{f} is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((the function sin) ·\frac{1}{f})′_{↾Z}(x) =

−\frac{1}{x^{2}} · (the function cos)(\frac{1}{x}).

(6) Suppose Z ⊆ dom((the function cos) ·\frac{1}{f}) and for every x such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) (the function cos) ·\frac{1}{f} is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((the function cos) ·\frac{1}{f})′_{↾Z}(x) = \frac{1}{x^{2}} ·(the

function sin)(\frac{1}{x}).

(7) Suppose Z ⊆ dom(id_{Z} ((the function sin) ·\frac{1}{f})) and for every x such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) id_{Z} ((the function sin) ·\frac{1}{f}) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (id_{Z} ((the function sin) ·\frac{1}{f}))′_{↾Z}(x) =

(the function sin)(\frac{1}{x}) − \frac{1}{x} · (the function cos)(\frac{1}{x}).

(8) Suppose Z ⊆ dom(id_{Z} ((the function cos) ·\frac{1}{f})) and for every x such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) id_{Z} ((the function cos) ·\frac{1}{f}) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (id_{Z} ((the function cos) ·\frac{1}{f}))′_{↾Z}(x) =

(the function cos)(\frac{1}{x}) + \frac{1}{x} · (the function sin)(\frac{1}{x}).

(9) Suppose Z ⊆ dom(((the function sin) ·\frac{1}{f}) ((the function cos) ·\frac{1}{f})) and for every x such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) ((the function sin) ·\frac{1}{f}) ((the function cos) ·\frac{1}{f}) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (((the function sin) ·\frac{1}{f}) ((the function

cos) ·\frac{1}{f}))′_{↾Z}(x) = \frac{1}{x^{2}} · ((the function sin)(\frac{1}{x})^{2} − (the function cos)(\frac{1}{x})^{2}).

(10) Suppose Z ⊆ dom(((the function sin) ·f) ((_{ℤ}^{n}) · (the function sin))) and

n ≥ 1 and for every x such that x ∈ Z holds f(x) = n · x. Then

(i) ((the function sin) ·f) ((_{ℤ}^{n}) · (the function sin)) is differentiable on Z, and

(ii) for every x such that x ∈ Z holds (((the function sin) ·f) ((_{ℤ}^{n}) · (the function sin)))′_{↾Z}(x) = n · (the function sin)(x)_{ℤ}^{n−1} · (the function sin)((n + 1) · x).

(11) Suppose Z ⊆ dom(((the function cos) ·f) ((_{ℤ}^{n}) · (the function sin))) and

n ≥ 1 and for every x such that x ∈ Z holds f(x) = n · x. Then

(i) ((the function cos) ·f) ((_{ℤ}^{n}) · (the function sin)) is differentiable on Z, and

(ii) for every x such that x ∈ Z holds (((the function cos) ·f) ((_{ℤ}^{n}) · (the function sin)))′_{↾Z}(x) = n · (the function sin)(x)_{ℤ}^{n−1}· (the function cos)((n + 1) · x).

(12) Suppose Z ⊆ dom(((the function cos) ·f) ((_{ℤ}^{n}) · (the function cos))) and

n ≥ 1 and for every x such that x ∈ Z holds f(x) = n · x. Then

(i) ((the function cos) ·f) ((_{ℤ}^{n}) · (the function cos)) is differentiable on Z, and

(ii) for every x such that x ∈ Z holds (((the function cos) ·f) ((_{ℤ}^{n})·(the function cos)))′_{↾Z}(x) = −n · (the function cos)(x)_{ℤ}^{n−1} · (the function sin)((n + 1) ·x).

(13) Suppose Z ⊆ dom(((the function sin) ·f) ((_{ℤ}^{n}) · (the function cos))) and

n ≥ 1 and for every x such that x ∈ Z holds f(x) = n · x. Then

(i) ((the function sin) ·f) ((_{ℤ}^{n}) · (the function cos)) is differentiable on Z, and

(ii) for every x such that x ∈ Z holds (((the function sin) ·f) ((_{ℤ}^{n}) · (the function cos)))′_{↾Z}(x) = n·(the function cos)(x)_{ℤ}^{n−1}·(the function cos)((n+ 1) · x).

(14) Suppose Z ⊆ dom(\frac{1}{f} (the function sin)) and for every x such that x ∈ Z

holds f(x) = x and f(x) ≠ 0. Then

(i) \frac{1}{f} (the function sin) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (\frac{1}{f} (the function sin))′_{↾Z}(x) = \frac{1}{x} · (the

function cos)(x) − \frac{1}{x^{2}} · (the function sin)(x).

(15) Suppose Z ⊆ dom(\frac{1}{f} (the function cos)) and for every x such that x ∈ Z

holds f(x) = x and f(x) ≠ 0. Then

(i) \frac{1}{f} (the function cos) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (\frac{1}{f} (the function cos))′_{↾Z}(x) =

−\frac{1}{x} · (the function sin)(x) − \frac{1}{x^{2}} · (the function cos)(x).

(16) Suppose Z ⊆ dom((the function sin)+(ℝ^{\frac{1}{2}}) · f) and for every x such that

x ∈ Z holds f(x) = x and f(x) > 0. Then

(i) (the function sin)+(ℝ^{\frac{1}{2}}) · f is differentiable on Z, and

(ii) for every x such that x ∈ Z holds ((the function sin)+(ℝ^{\frac{1}{2}}) · f)′_{↾Z}(x) =

(the function cos)(x) + \frac{1}{2} · x_{ℝ}^{− \frac{1}{2}}.

(17) Suppose Z ⊆ dom(g ((the function sin) ·\frac{1}{f})) and g = _{ℤ}^{2} and for every x

such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) g ((the function sin) ·\frac{1}{f}) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (g ((the function sin) ·\frac{1}{f}))′_{↾Z}(x) =

2 · x · (the function sin)(\frac{1}{x}) − (the function cos)(\frac{1}{x}).

(18) Suppose Z ⊆ dom(g ((the function cos) ·\frac{1}{f})) and g = _{ℤ}^{2} and for every x

such that x ∈ Z holds f(x) = x and f(x) ≠ 0. Then

(i) g ((the function cos) ·\frac{1}{f}) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (g ((the function cos) ·\frac{1}{f}))′_{↾Z}(x) =

2 · x · (the function cos)(\frac{1}{x}) + (the function sin)(\frac{1}{x}).

(19) Suppose Z ⊆ dom(log (e) · f) and for every x such that x ∈ Z holds f(x) = x and f(x) > 0. Then log (e) · f is differentiable on Z and for every x such that x ∈ Z holds (log (e) · f)′_{↾Z}(x) = \frac{1}{x}.

(20) Suppose Z ⊆ dom(id_{Z} f) and f = log (e) · f_{1} and for every x such that x ∈ Z holds f_{1}(x) = x and f_{1}(x) > 0. Then id_{Z} f is differentiable on Z and for every x such that x ∈ Z holds (id_{Z} f)′_{↾Z}(x) = 1 + (log (e))(x).

(21) Suppose Z ⊆ dom(g f) and g = _{ℤ}^{2} and f = log (e)·f_{1} and for every x such that x ∈ Z holds f_{1}(x) = x and f_{1}(x) > 0. Then g f is differentiable on Z and for every x such that x ∈ Z holds (g f)′_{↾Z}(x) = x + 2 · x · (log (e))(x).

(22) Suppose Z ⊆ dom(\frac{f_{1}+f_{2}}{f_{1}−f_{2}} ) and for every x such that x ∈ Z holds f_{1}(x) = a and f_{2} = _{ℤ}^{2} and for every x such that x ∈ Z holds (f_{1} − f_{2})(x) > 0. Then \frac{f_{1}+f_{2}}{f_{1}−f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{f_{1}+f_{2}}{f_{1}−f_{2}} )′_{↾Z}(x) = \frac{4·a·x}{(a−x^{2})^{2}} .

(23) Suppose that (i) Z ⊆ dom(log (e) · \frac{f_{1}+f_{2}}{f_{1}−f_{2}} ), (ii) for every x such that x ∈ Z holds f_{1}(x) = a, (iii) f_{2} = _{ℤ}^{2}, (iv) for every x such that x ∈ Z holds (f_{1} − f_{2})(x) > 0, and

(v) for every x such that x ∈ Z holds (f_{1} + f_{2})(x) > 0. Then log (e) · \frac{f_{1}+f_{2}}{f_{1}−f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (log (e) · \frac{f_{1}+f_{2}}{f_{1}−f_{2}} )′_{↾Z}(x) = \frac{4·a·x}{a^{2}−x^{4}} .

(24) Suppose Z ⊆ dom(\frac{1}{f} g) and for every x such that x ∈ Z holds f(x) = x and g = log (e) · f_{1} and for every x such that x ∈ Z holds f_{1}(x) = x and f_{1}(x) > 0. Then \frac{1}{f} g is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{f} g)′_{↾Z}(x) = \frac{1}{x^{2}} · (1 − (log (e))(x)).

(25) Suppose Z ⊆ dom(\frac{1}{f}) and f = log (e) · f_{1} and for every x such that x ∈ Z holds f_{1}(x) = x and f_{1}(x) > 0 and for every x such that x ∈ Z holds f(x) ≠ 0. Then \frac{1}{f} is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{f})′_{↾Z}(x) = −\frac{1}{x·(log (e))(x)^{2}} . some differentiable formulas … 509

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[3] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[4] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[5] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[6] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[7] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

[8] Konrad Raczkowski and Andrzej Nȩdzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[9] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[10] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[11] Yasunari Shidama. The Taylor expansions. Formalized Mathematics, 12(2):195–200, 2004.

[12] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[13] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[14] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[16] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[17] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

510 jianbing cao and fahui zhai and xiquan liang

Formulas and Identities of Hyperbolic Functions

Pacharapokin Chanapat Shinshu University Nagano, Japan Hiroshi Yamazaki Shinshu University Nagano, Japan

Summary. In this article, we proved formulas of hyperbolic sine, hyperbolic cosine and hyperbolic tangent, and their identities.

MML identifier: SIN COS8, version: 7.6.01 4.46.926

The papers [1], [3], [6], [5], [7], [4], and [2] provide the terminology and notation for this paper.

We follow the rules: x, y, z, w are real numbers and n is a natural number. One can prove the following propositions:

(1) tanh x = \frac{sinh x}{cosh x} and tanh 0 = 0.

(2) sinh x = \frac{1}{cosech x} and cosh x = \frac{1}{sech x} and tanh x = \frac{1}{coth x}.

(3) sech x ≤ 1 and 0 < sech x and sech 0 = 1.

(4) If x ≥ 0, then tanh x ≥ 0.

(5) cosh x = \frac{1}{\sqrt{1−(tanh x)^{2}}} and sinh x = \frac{tanh x}{\sqrt{1−(tanh x)^{2}}} .

(6) (cosh x + sinh x)^{n} = cosh(n · x) + sinh(n · x) and (cosh x − sinh x)^{n} = cosh(n · x) − sinh(n · x).

(7)(i) exp x = cosh x + sinh x, (ii) exp(−x) = cosh x − sinh x, (iii) exp x = \frac{cosh( \frac{x}{2} )+sinh( \frac{x}{2} )}{cosh( \frac{x}{2} )−sinh( \frac{x}{2} )},

(iv) exp(−x) = \frac{cosh( \frac{x}{2} )−sinh( \frac{x}{2} )}{cosh( \frac{x}{2} )+sinh( \frac{x}{2} )},

(v) exp x = \frac{1+tanh( \frac{x}{2} )}{1−tanh( \frac{x}{2} )}, and

(vi) exp(−x) = \frac{1−tanh( \frac{x}{2} )}{1+tanh( \frac{x}{2} )}.

(8) If x ≠ 0, then exp x = \frac{coth( \frac{x}{2} )+1}{coth( \frac{x}{2} )−1} and exp(−x) = \frac{coth( \frac{x}{2} )−1}{coth( \frac{x}{2} )+1}.

(9) \frac{cosh x+sinh x}{cosh x−sinh x} = \frac{1+tanh x}{1−tanh x}.

(10) If y ≠ 0, then coth y + tanh z = \frac{cosh(y+z)}{sinh y·cosh z} and coth y − tanh z = \frac{cosh(y−z)}{sinh y·cosh z} .

(11) sinh y · sinh z = \frac{1}{2} · (cosh(y + z) − cosh(y − z)) and sinh y · cosh z = \frac{1}{2} ·(sinh(y+z)+sinh(y−z)) and cosh y·sinh z = \frac{1}{2} ·(sinh(y+z)−sinh(y−z)) and cosh y · cosh z = \frac{1}{2} · (cosh(y + z) + cosh(y − z)).

(12) (sinh y)^{2} − (cosh z)^{2} = sinh(y + z) · sinh(y − z) − 1.

(13) (sinh y − sinh z)^{2} − (cosh y − cosh z)^{2} = 4 · (sinh(\frac{y−z}{2} ))^{2} and (cosh y + cosh z)^{2} − (sinh y + sinh z)^{2} = 4 · (cosh(\frac{y−z}{2} ))^{2}.

(14) \frac{sinh y+sinh z}{sinh y−sinh z} = tanh(\frac{y+z}{2} ) · coth(\frac{y−z}{2} ).

(15) \frac{cosh y+cosh z}{cosh y−cosh z} = coth(\frac{y+z}{2} ) · coth(\frac{y−z}{2} ).

(16) If y − z ≠ 0, then \frac{sinh y+sinh z}{cosh y+cosh z} = \frac{cosh y−cosh z}{sinh y−sinh z} .

(17) If y + z ≠ 0, then \frac{sinh y−sinh z}{cosh y+cosh z} = \frac{cosh y−cosh z}{sinh y+sinh z} .

(18) \frac{sinh y+sinh z}{cosh y+cosh z} = tanh(\frac{y}{2} + \frac{z}{2}) and \frac{sinh y−sinh z}{cosh y+cosh z} = tanh(\frac{y}{2} − \frac{z}{2}).

(19) \frac{tanh y+tanh z}{tanh y−tanh z} = \frac{sinh(y+z)}{sinh(y−z)}.

(20) \frac{sinh(y−z)+sinh y+sinh(y+z)}{cosh(y−z)+cosh y+cosh(y+z)} = tanh y. (21)(i) sinh(y + z + w) = (tanh y + tanh z + tanh w + tanh y · tanh z · tanh w) ·

cosh y · cosh z · cosh w, (ii) cosh(y +z +w) = (1+tanh y ·tanh z +tanh z ·tanh w +tanh w ·tanh y)·

cosh y · cosh z · cosh w, and (iii) tanh(y + z + w) = \frac{tanh y+tanh z+tanh w+tanh y·tanh z·tanh w}{1+tanh z·tanh w+tanh w·tanh y+tanh y·tanh z} .

(22) cosh(2 · y) + cosh(2 · z) + cosh(2 · w) + cosh(2 · (y + z + w)) = 4 · cosh(z + w) · cosh(w + y) · cosh(y + z).

(23) sinh y · sinh z · sinh(z − y) + sinh z · sinh w · sinh(w − z) + sinh w · sinh y · sinh(y − w) + sinh(z − y) · sinh(w − z) · sinh(y − w) = 0.

(24) If x ≥ 0, then sinh(\frac{x}{2}) = \sqrt{\frac{cosh x−1}{2}}.

(25) If x < 0, then sinh(\frac{x}{2}) = −\sqrt{\frac{cosh x−1}{2}}.

(26) sinh(2 · x) = 2 · sinh x · cosh x and cosh(2 · x) = 2 · (cosh x)^{2} − 1 and tanh(2 · x) = \frac{2·tanh x}{1+(tanh x)^{2}} .

(27) sinh(2 · x) = \frac{2·tanh x}{1−(tanh x)^{2}} and sinh(3 · x) = sinh x · (4 · (cosh x)^{2} − 1) and sinh(3 · x) = 3 · sinh x − 2 · sinh x · (1 − cosh(2 · x)) and cosh(2 · x) = 1 + 2 · (sinh x)^{2} and cosh(2·x) = (cosh x)^{2}+(sinh x)^{2} and cosh(2·x) = \frac{1+(tanh x)^{2}}{1−(tanh x)^{2}} and cosh(3·x) = cosh x·(4·(sinh x)^{2}+1) and tanh(3·x) = \frac{3·tanh x+(tanh x)^{3}}{1+3·(tanh x)^{2}} .

(28) \frac{sinh(5·x)+2·sinh(3·x)+sinh x}{sinh(7·x)+2·sinh(5·x)+sinh(3·x)} = \frac{sinh(3·x)}{sinh(5·x)}.

(29) If x ≥ 0, then tanh(\frac{x}{2}) = \sqrt{\frac{cosh x−1}{cosh x+1}}.

(30) If x < 0, then tanh(\frac{x}{2}) = −\sqrt{\frac{cosh x−1}{cosh x+1}}.

(31)(i) (sinh x)^{3} = \frac{sinh(3·x)−3·sinh x}{4}, (ii) (sinh x)^{4} = \frac{(cosh(4·x)−4·cosh(2·x))+3}{8}, (iii) (sinh x)^{5} = \frac{(sinh(5·x)−5·sinh(3·x))+10·sinh x}{16}, (iv) (sinh x)^{6} = \frac{((cosh(6·x)−6·cosh(4·x))+15·cosh(2·x))−10}{32},

(v) (sinh x)^{7} = \frac{((sinh(7·x)−7·sinh(5·x))+21·sinh(3·x))−35·sinh x}{64}, and (vi) (sinh x)^{8} = \frac{(((cosh(8·x)−8·cosh(6·x))+28·cosh(4·x))−56·cosh(2·x))+35}{128}.

(32)(i) (cosh x)^{3} = \frac{cosh(3·x)+3·cosh x}{4}, (ii) (cosh x)^{4} = \frac{cosh(4·x)+4·cosh(2·x)+3}{8}, (iii) (cosh x)^{5} = \frac{cosh(5·x)+5·cosh(3·x)+10·cosh x}{16}, (iv) (cosh x)^{6} = \frac{cosh(6·x)+6·cosh(4·x)+15·cosh(2·x)+10}{32},

(v) (cosh x)^{7} = \frac{cosh(7·x)+7·cosh(5·x)+21·cosh(3·x)+35·cosh x}{64}, and (vi) (cosh x)^{8} = \frac{cosh(8·x)+8·cosh(6·x)+28·cosh(4·x)+56·cosh(2·x)+35}{128}.

(33) cosh(2 · y) + cos(2 · z) = 2 + 2 · ((sinh y)^{2} − (sin z)^{2}) and cosh(2 · y) − cos(2 · z) = 2 · ((sinh y)^{2} + (sin z)^{2}).

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Yuzhong Ding and Xiquan Liang. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(3):243–246, 2004.

[3] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[4] Takashi Mitsuishi and Yuguang Yang. Properties of the trigonometric function. Formalized Mathematics, 8(1):103–106, 1999.

[5] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

[6] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[7] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

514 pacharapokin chanapat and hiroshi yamazaki

Niemytzki Plane - an Example of Tychonoff Space Which Is Not T_{4}

Grzegorz Bancerek Department of Theoretical Computer Science Białystok Technical University Poland

Summary. We continue Mizar formalization of General Topology according to the book [20] by Engelking. Niemytzki plane is defined as halfplane y ≥ 0 with topology introduced by a neighborhood system. Niemytzki plane is not T_{4}. Next, the definition of Tychonoff space is given. The characterization of Tychonoff space by prebasis and the fact that Tychonoff spaces are between T_{3} and T_{4} is proved. The final result is that Niemytzki plane is also a Tychonoff space.

MML identifier: TOPGEN 5, version: 7.6.01 4.46.926

The notation and terminology used here are introduced in the following papers: [38], [34], [15], [41], [17], [40], [35], [42], [11], [14], [12], [8], [13], [33], [10], [37], [4], [2], [1], [3], [5], [32], [39], [22], [25], [23], [29], [27], [26], [28], [43], [18], [31], [30], [36], [19], [24], [9], [16], [21], [7], and [6].

1. Preliminaries

In this paper x, y are elements of ℝ.

One can prove the following propositions:

(1) For all functions f, g such that f ≈ g and for every set A holds (f+·g)^{−1}(A) = f^{−1}(A) ∪ g^{−1}(A).

(2) For all functions f, g such that dom f misses dom g and for every set A holds (f+·g)^{−1}(A) = f^{−1}(A) ∪ g^{−1}(A).

Let X be a set and let Y be a non empty real-membered set. Note that every relation between X and Y is real-yielding.

Next we state several propositions:

(3) For all sets x, a and for every function f such that a ∈ dom f holds (commute(x⟼ . f))(a) = x⟼ . f(a).

(4) Let b be a set and f be a function. Then b ∈ dom commute(f) if and only if there exists a set a and there exists a function g such that a ∈ dom f and g = f(a) and b ∈ dom g.

(5) Let a, b be sets and f be a function. Then a ∈ dom(commute(f))(b) if and only if there exists a function g such that a ∈ dom f and g = f(a) and b ∈ dom g.

(6) For all sets a, b and for all functions f, g such that a ∈ dom f and g = f(a) and b ∈ dom g holds (commute(f))(b)(a) = g(b).

(7) For every set a and for all functions f, g, h such that h = f ∪ g holds (commute(h))(a) = (commute(f))(a) ∪ (commute(g))(a).

Let us note that every finite subset of ℝ is bounded. The following propositions are true:

(8) For all real numbers a, b, c, d such that a < b and c ≤ d holds ]a, c[ ∩ [b, d] = [b, c[.

(9) For all real numbers a, b, c, d such that a ≥ b and c > d holds ]a, c[ ∩ [b, d] = ]a, d].

(10) For all real numbers a, b, c, d such that a ≤ b and b < c and c ≤ d holds [a, c[∪]b, d] = [a, d].

(11) For all real numbers a, b, c, d such that a ≤ b and b < c and c ≤ d holds [a, c[∩]b, d] = ]b, c[.

(12) For all sets X, Y holds ∏⟨X, Y ⟩ ≈ [: X, Y :] and \overline{\overline{∏⟨X, Y ⟩}} = \overline{\overline{X}} · \overline{\overline{Y}} . In this article we present several logical schemes. The scheme SCH1 deals

with non empty sets A, B, C, two unary functors F and G yielding sets, and a unary predicate P, and states that:

There exists a function f from C into B such that for every element a of A holds

(i) if P[a], then f(a) = F(a), and (ii) if not P[a], then f(a) = G(a)

provided the parameters meet the following conditions:

• C ⊆ A, and

• For every element a of A such that a ∈ C holds if P[a], then F(a) ∈ B and if not P[a], then G(a) ∈ B.

The scheme SCH2 deals with non empty sets A, B, C, three unary functors F, G, and H yielding sets, and two unary predicates P, Q, and states that:

There exists a function f from C into B such that for every element a of A holds

(i) if P[a], then f(a) = F(a), (ii) if not P[a] and Q[a], then f(a) = G(a), and (iii) if not P[a] and not Q[a], then f(a) = H(a)

provided the parameters meet the following conditions: • C ⊆ A, and

• For every element a of A such that a ∈ C holds if P[a], then F(a) ∈ B and if not P[a] and Q[a], then G(a) ∈ B and if not P[a] and not Q[a], then H(a) ∈ B.

The following four propositions are true:

(13) For all real numbers a, b holds |[a, b]|^{2} = a^{2} + b^{2}.

(14) Let X be a topological space, Y be a non empty topological space, A, B be closed subsets of X, f be a continuous function from X↾A into Y, and g be a continuous function from X↾B into Y. If f ≈ g, then f+·g is a continuous function from X↾(A ∪ B) into Y.

(15) Let X be a topological space, Y be a non empty topological space, and A, B be closed subsets of X. Suppose A misses B. Let f be a continuous function from X↾A into Y and g be a continuous function from X↾B into Y. Then f+·g is a continuous function from X↾(A ∪ B) into Y.

(16) Let X be a topological space, Y be a non empty topological space, A be an open closed subset of X, f be a continuous function from X↾A into Y, and g be a continuous function from X↾A^{c} into Y. Then f+·g is a continuous function from X into Y.

2. Niemytzki Plane

One can prove the following proposition

(17) For every natural number n and for every point a of E_{T}^{n} and for every

positive real number r holds a ∈ Ball(a, r). The subset (y = 0)-line of E_{T}^{2} is defined by:

(Def. 1) (y = 0)-line = {[x, 0]}.

The subset (y ≥ 0)-plane of E_{T}^{2} is defined as follows: (Def. 2) (y ≥ 0)-plane = {[x, y] : y ≥ 0}.

We now state several propositions:

(18) For all sets a, b holds ⟨a, b⟩ ∈ (y = 0)-line iff a ∈ ℝ and b = 0.

(19) For all real numbers a, b holds [a, b] ∈ (y = 0)-line iff b = 0.

(20) \overline{\overline{(y = 0)-line}} = 𝔠.

(21) For all sets a, b holds ⟨a, b⟩ ∈ (y ≥ 0)-plane iff a ∈ ℝ and there exists y such that b = y and y ≥ 0.

(22) For all real numbers a, b holds [a, b] ∈ (y ≥ 0)-plane iff b ≥ 0. Let us note that (y = 0)-line is non empty and (y ≥ 0)-plane is non empty. We now state several propositions:

(23) (y = 0)-line ⊆ (y ≥ 0)-plane.

(24) For all real numbers a, b, r such that r > 0 holds Ball([a, b], r) ⊆ (y ≥ 0)-plane iff r ≤ b.

(25) For all real numbers a, b, r such that r > 0 and b ≥ 0 holds Ball([a, b], r) misses (y = 0)-line iff r ≤ b.

(26) Let n be a natural number, a, b be elements of E_{T}^{n}, and r_{1}, r_{2} be positive real numbers. If |a − b| ≤ r_{1} − r_{2}, then Ball(b, r_{2}) ⊆ Ball(a, r_{1}).

(27) For every real number a and for all positive real numbers r_{1}, r_{2} such that r_{1} ≤ r_{2} holds Ball([a, r_{1}], r_{1}) ⊆ Ball([a, r_{2}], r_{2}).

(28) Let T_{1}, T_{2} be non empty topological spaces, B_{1} be a neighborhood system of T_{1}, and B_{2} be a neighborhood system of T_{2}. Suppose B_{1} = B_{2}. Then the topological structure of T_{1} = the topological structure of T_{2}.

In the sequel r is an element of ℝ. Niemytzki plane is a strict non empty topological space and is defined by

the conditions (Def. 3). (Def. 3)(i) The carrier of Niemytzki plane = (y ≥ 0)-plane, and

(ii) there exists a neighborhood system B of Niemytzki plane such that for every x holds B([x, 0]) = {Ball([x, r], r) ∪ {[x, 0]} : r > 0} and for all x, y such that y > 0 holds B([x, y]) = {Ball([x, y], r) ∩ (y ≥ 0)-plane : r > 0}.

The following propositions are true:

(29) (y ≥ 0)-plane \ (y = 0)-line is an open subset of Niemytzki plane.

(30) (y = 0)-line is a closed subset of Niemytzki plane.

(31) Let x be a real number and r be a positive real number. Then Ball([x, r], r) ∪ {[x, 0]} is an open subset of Niemytzki plane.

(32) Let x be a real number and y, r be positive real numbers. Then Ball([x, y], r) ∩ (y ≥ 0)-plane is an open subset of Niemytzki plane.

(33) Let x, y be real numbers and r be a positive real number. If r ≤ y, then Ball([x, y], r) is an open subset of Niemytzki plane.

(34) Let p be a point of Niemytzki plane and r be a positive real number. Then there exists a point a of E_{T}^{2} and there exists an open subset U of Niemytzki plane such that p ∈ U and a ∈ U and for every point b of E_{T}^{2} such that b ∈ U holds |b − a| < r.

(35) Let x, y be real numbers and r be a positive real number. Then there exist rational numbers w, v such that [w, v] ∈ Ball([x, y], r) and [w, v] ≠ [x,

y].

(36) Let A be a subset of Niemytzki plane. If A = ((y ≥ 0)-plane \ (y = 0)-line) ∩ ∏⟨ℚ, ℚ⟩, then for every set x holds \overline{A \ \{x\}} = Ω_{Niemytzki plane}.

(37) Let A be a subset of Niemytzki plane. If A = (y ≥ 0)-plane\(y = 0)-line, then for every set x holds \overline{A \ \{x\}} = Ω_{Niemytzki plane}.

(38) For every subset A of Niemytzki plane such that A = (y ≥ 0)-plane\(y = 0)-line holds \overline{A} = Ω_{Niemytzki plane}.

(39) For every subset A of Niemytzki plane such that A = (y = 0)-line holds \overline{A} = A and Int A = ∅.

(40) ((y ≥ 0)-plane \ (y = 0)-line) ∩ ∏⟨ℚ, ℚ⟩ is a dense subset of Niemytzki plane.

(41) ((y ≥ 0)-plane \ (y = 0)-line) ∩ ∏⟨ℚ, ℚ⟩ is a dense-in-itself subset of Niemytzki plane.

(42) (y ≥ 0)-plane \ (y = 0)-line is a dense subset of Niemytzki plane.

(43) (y ≥ 0)-plane\(y = 0)-line is a dense-in-itself subset of Niemytzki plane.

(44) (y = 0)-line is a nowhere dense subset of Niemytzki plane.

(45) For every subset A of Niemytzki plane such that A = (y = 0)-line holds Der A is empty.

(46) Every subset of (y = 0)-line is a closed subset of Niemytzki plane.

(47) ℚ is a dense subset of Sorgenfrey line.

(48) Sorgenfrey line is separable.

(49) Niemytzki plane is separable.

(50) Niemytzki plane is a T_{1} space.

(51) Niemytzki plane is not T_{4}.

3. Tychonoff Spaces

Let T be a topological space. We say that T is Tychonoff if and only if the conditions (Def. 4) are satisfied.

(Def. 4)(i) T is a T_{1} space, and

(ii) for every closed subset A of T and for every point a of T such that a ∈ A^{c} there exists a continuous function f from T into 𝕀 such that f(a) = 0 and f^{∘}A ⊆ {1}.

Let us observe that every topological space which is Tychonoff is also T_{1} and T_{3} and every non empty topological space which is T_{1} and T_{4} is also Tychonoff.

We now state the proposition

(52) Let X be a T_{1} topological space. Suppose X is Tychonoff. Let B be a prebasis of X, x be a point of X, and V be a subset of X. Suppose x ∈ V and V ∈ B. Then there exists a continuous function f from X into 𝕀 such that f(x) = 0 and f^{∘}V^{c} ⊆ {1}.

Let X be a set and let Y be a non empty real-membered set. Observe that every relation between X and Y is real-yielding.

The following propositions are true:

(53) Let X be a topological space, R be a non empty subspace of ℝ^{1}, f, g be continuous functions from X into R, and A be a subset of X. Suppose that for every point x of X holds x ∈ A iff f(x) ≤ g(x). Then A is closed.

(54) Let X be a topological space, R be a non empty subspace of ℝ^{1}, and f, g be continuous functions from X into R. Then there exists a continuous function h from X into R such that for every point x of X holds h(x) = max(f(x), g(x)).

(55) Let X be a non empty topological space, R be a non empty subspace of ℝ^{1}, A be a finite non empty set, and F be a many sorted function indexed by A. Suppose that for every set a such that a ∈ A holds F (a) is a continuous function from X into R. Then there exists a continuous function f from X into R such that for every point x of X and for every finite non empty subset S of ℝ if S = rng(commute(F))(x), then f(x) = max S.

(56) Let X be a T_{1} non empty topological space and B be a prebasis of X. Suppose that for every point x of X and for every subset V of X such that x ∈ V and V ∈ B there exists a continuous function f from X into 𝕀 such that f(x) = 0 and f^{∘}V^{c} ⊆ {1}. Then X is Tychonoff.

(57) Sorgenfrey line is a T_{1} space.

(58) For every real number x holds ]−∞, x[ is a closed subset of Sorgenfrey line.

(59) For every real number x holds ]−∞, x] is a closed subset of Sorgenfrey line.

(60) For every real number x holds [x, +∞[ is a closed subset of Sorgenfrey line.

(61) For all real numbers x, y holds [x, y[ is a closed subset of Sorgenfrey line.

(62) Let x be a real number and w be a rational number. Suppose x < w. Then there exists a continuous function f from Sorgenfrey line into 𝕀 such that for every point a of Sorgenfrey line holds

(i) if a ∈ [x, w[, then f(a) = 0, and (ii) if a ∉ [x, w[, then f(a) = 1.

(63) Sorgenfrey line is Tychonoff.

4. Niemytzki Plane is Tychonoff Space

Let x be a real number and let r be a positive real number. The functor +(x, r) yielding a function from Niemytzki plane into 𝕀 is defined by the conditions (Def. 5).

(Def. 5)(i) (+(x, r))([x, 0]) = 0, and

(ii) for every real number a and for every non negative real number b holds if a ≠ x or b ≠ 0 and if [a, b] ∉ Ball([x, r], r), then (+(x, r))([a, b]) = 1 and if [a, b] ∈ Ball([x, r], r), then (+(x, r))([a, b]) = \frac{|[x,0]−[a,b]|^{2}}{2·r·b} .

One can prove the following propositions:

(64) Let p be a point of E_{T}^{2}. Suppose p_{2} ≥ 0. Let x be a real number and r be a positive real number. If (+(x, r))(p) = 0, then p = [x, 0].

(65) For all real numbers x, y and for every positive real number r such that x ≠ y holds (+(x, r))([y, 0]) = 1.

(66) Let p be a point of E_{T}^{2}, x be a real number, and a, r be positive real numbers. If a ≤ 1 and |p−[x, r·a]| = r·a and p_{2} ≠ 0, then (+(x, r))(p) = a.

(67) Let p be a point of E_{T}^{2}, x, a be real numbers, and r be a positive real number. If 0 ≤ a and a ≤ 1 and |p−[x, r ·a]| < r ·a, then (+(x, r))(p) < a.

(68) Let p be a point of E_{T}^{2}. Suppose p_{2} ≥ 0. Let x, a be real numbers and r be a positive real number. If 0 ≤ a and a < 1 and |p − [x, r · a]| > r · a, then (+(x, r))(p) > a.

(69) Let p be a point of E_{T}^{2}. Suppose p_{2} ≥ 0. Let x, a, b be real numbers and r be a positive real number. Suppose 0 ≤ a and b ≤ 1 and (+(x, r))(p) ∈ ]a, b[. Then there exists a positive real number r_{1} such that r_{1} ≤ p_{2} and Ball(p, r_{1}) ⊆ (+(x, r))^{−1}(]a, b[).

(70) For every real number x and for all positive real numbers a, r holds Ball([x, r · a], r · a) ⊆ (+(x, r))^{−1}(]0, a[).

(71) For every real number x and for all positive real numbers a, r holds Ball([x, r · a], r · a) ∪ {[x, 0]} ⊆ (+(x, r))^{−1}([0, a[).

(72) Let p be a point of E_{T}^{2}. Suppose p_{2} ≥ 0. Let x, a be real numbers and r be a positive real number. If 0 < (+(x, r))(p) and (+(x, r))(p) < a and a ≤ 1, then p ∈ Ball([x, r · a], r · a).

(73) Let p be a point of E_{T}^{2}. Suppose p_{2} > 0. Let x, a be real numbers and r be a positive real number. Suppose 0 ≤ a and a < (+(x, r))(p). Then there exists a positive real number r_{1} such that r_{1} ≤ p_{2} and Ball(p, r_{1}) ⊆ (+(x, r))^{−1}(]a, 1]).

(74) Let p be a point of E_{T}^{2}. Suppose p_{2} = 0. Let x be a real number and r be a positive real number. Suppose (+(x, r))(p) = 1. Then there exists a positive real number r_{1} such that Ball([p_{1}, r_{1}], r_{1})∪{p} ⊆ (+(x, r))^{−1}({1}).

(75) Let T be a non empty topological space, S be a subspace of T, and B be a basis of T. Then {A ∩ Ω_{S}; A ranges over subsets of T: A ∈ B ∧ A meets Ω_{S}} is a basis of S.

(76) {]a, b[; a ranges over real numbers, b ranges over real numbers: a < b} is a basis of ℝ^{1}.

(77) Let T be a topological space, U, V be subsets of T, and B be a set. If U ∈ B and V ∈ B and B ∪ {U ∪ V } is a basis of T, then B is a basis of T.

(78) {[0, a[; a ranges over real numbers: 0 < a ∧ a ≤ 1} ∪ {]a, 1]; a ranges over real numbers: 0 ≤ a ∧ a < 1} ∪ {]a, b[; a ranges over real numbers, b ranges over real numbers: 0 ≤ a ∧ a < b ∧ b ≤ 1} is a basis of 𝕀.

(79) Let T be a non empty topological space and f be a function from T into 𝕀. Then f is continuous if and only if for all real numbers a, b such that 0 ≤ a and a < 1 and 0 < b and b ≤ 1 holds f^{−1}([0, b[) is open and f^{−1}(]a, 1]) is open.

Let x be a real number and let r be a positive real number. Note that +(x, r) is continuous.

We now state the proposition

(80) Let U be a subset of Niemytzki plane and given x, r. Suppose U = Ball([x, r], r) ∪ {[x, 0]}. Then there exists a continuous function f from Niemytzki plane into 𝕀 such that

(i) f([x, 0]) = 0, and (ii) for all real numbers a, b holds if [a, b] ∈ U^{c}, then f([a, b]) = 1 and if [a,

b] ∈ U \ \{[x, 0]\}, then f([a, b]) = \frac{|[x,0]−[a,b]|^{2}}{2·r·b} . Let x, y be real numbers and let r be a positive real number. The functor

+(x, y, r) yields a function from Niemytzki plane into 𝕀 and is defined by the condition (Def. 6).

(Def. 6) Let a be a real number and b be a non negative real number. Then (i) if [a, b] ∉ Ball([x, y], r), then (+(x, y, r))([a, b]) = 1, and (ii) if [a, b] ∈ Ball([x, y], r), then (+(x, y, r))([a, b]) = \frac{|[x,y]−[a,b]|}{r}. The following propositions are true:

(81) Let p be a point of E_{T}^{2}. Suppose p_{2} ≥ 0. Let x be a real number, y be a non negative real number, and r be a positive real number. Then (+(x, y, r))(p) = 0 if and only if p = [x, y].

(82) Let x be a real number, y be a non negative real number, and r, a be positive real numbers. If a ≤ 1, then (+(x, y, r))^{−1}([0, a[) = Ball([x, y], r · a) ∩ (y ≥ 0)-plane.

(83) Let p be a point of E_{T}^{2}. Suppose p_{2} > 0. Let x be a real number, a be a non negative real number, and y, r be positive real numbers. If (+(x, y, r))(p) > a, then |[x, y]−p| > r·a and Ball(p, |[x, y]−p|−r·a)∩(y ≥ 0)-plane ⊆ (+(x, y, r))^{−1}(]a, 1]).

niemytzki plane - an example of tychonoff … 523

(84) Let p be a point of E_{T}^{2}. Suppose p_{2} = 0. Let x be a real number, a be a non negative real number, and y, r be positive real numbers. Suppose (+(x, y, r))(p) > a. Then |[x, y] − p| > r · a and there exists a positive real number r_{1} such that r_{1} = \frac{|[x,y]−p|−r·a}{2} and Ball([p_{1}, r_{1}], r_{1}) ∪ \{p\} ⊆ (+(x, y, r))^{−1}(]a, 1]).

Let x be a real number and let y, r be positive real numbers. One can verify that +(x, y, r) is continuous.

We now state three propositions:

(85) Let U be a subset of Niemytzki plane and given x, y, r. Suppose y > 0 and U = Ball([x, y], r) ∩ (y ≥ 0)-plane. Then there exists a continuous function f from Niemytzki plane into 𝕀 such that f([x, y]) = 0 and for all real numbers a, b holds if [a, b] ∈ U^{c}, then f([a, b]) = 1 and if [a, b] ∈ U, then f([a, b]) = \frac{|[x,y]−[a,b]|}{r}.

(86) Niemytzki plane is a T_{1} space.

(87) Niemytzki plane is Tychonoff.

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On the Partial Product and Partial Sum of Series and Related Basic Inequalities

Fuguo Ge

Qingdao University of Science

and Technology China

Xiquan Liang Qingdao University of Science and Technology China

Summary. This article introduced some important inequalities on partial sum and partial product, as well as some basic inequalities.

MML identifier: SERIES 5, version: 7.6.01 4.50.934

The notation and terminology used in this paper are introduced in the following papers: [2], [1], [9], [6], [3], [5], [7], [8], and [4].

For simplicity, we adopt the following rules: a, b, c, d are positive real numbers, m, u, w, x, y, z are real numbers, n, k are natural numbers, and s, s_{1} are sequences of real numbers.

Next we state a number of propositions:

(1) (a + b) · (\frac{1}{a} + \frac{1}{b}) ≥ 4.

(2) a^{4} + b^{4} ≥ a^{3} · b + a · b^{3}.

(3) If a < b, then 1 < \frac{b+c}{a+c}.

(4) If a < b, then \frac{a}{b} < \sqrt{\frac{a}{b}}.

(5) If a < b, then \sqrt{\frac{a}{b}} < \frac{b+\sqrt{\frac{a^{2}+b^{2}}{2}}}{a+\sqrt{\frac{a^{2}+b^{2}}{2}}} .

(6) If a < b, then \frac{a}{b} < \frac{b+\sqrt{\frac{a^{2}+b^{2}}{2}}}{a+\sqrt{\frac{a^{2}+b^{2}}{2}}} .

(7) _{\frac{2}{\frac{1}{a} + \frac{1}{b}}} ≤ \sqrt{a · b}.

(8) \frac{a+b}{2} ≤ \sqrt{\frac{a^{2}+b^{2}}{2}} .

(9) x + y ≤ \sqrt{2 · (x^{2} + y^{2})}.

(10) _{\frac{2}{\frac{1}{a} + \frac{1}{b}}} ≤ \frac{a+b}{2}.

(11) \sqrt{a · b} ≤ \sqrt{\frac{a^{2}+b^{2}}{2}} .

(12) _{\frac{2}{\frac{1}{a} + \frac{1}{b}}} ≤ \sqrt{\frac{a^{2}+b^{2}}{2}} .

(13) If |x| < 1 and |y| < 1, then |\frac{x+y}{1+x·y} | ≤ 1.

(14) \frac{|x+y|}{1+|x+y|} ≤ \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}.

(15) \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} > 1.

(16) \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} < 2.

(17) If a+b > c and b+c > a and a+c > b, then \frac{1}{(a+b)−c} + \frac{1}{(b+c)−a} + \frac{1}{(c+a)−b} ≥ \frac{9}{a+b+c}.

(18) \sqrt{(a + b) · (c + d)} ≥ \sqrt{a · c} + \sqrt{b · d}.

(19) (a · b + c · d) · (a · c + b · d) ≥ 4 · a · b · c · d.

(20) \frac{a}{b} + \frac{b}{c} + \frac{c}{a} ≥ 3.

(21) If a · b + b · c + c · a = 1, then a + b + c ≥ \sqrt{3}.

(22) \frac{(b+c)−a}{a} + \frac{(c+a)−b}{b} + \frac{(a+b)−c}{c} ≥ 3.

(23) (a + \frac{1}{a}) · (b + \frac{1}{b}) ≥ (\sqrt{a · b} + \frac{1}{\sqrt{a·b}})^{2}.

(24) \frac{b·c}{a} + \frac{a·c}{b} + \frac{a·b}{c} ≥ a + b + c.

(25) If x > y and y > z, then x^{2} · y + y^{2} · z + z^{2} · x > x · y^{2} + y · z^{2} + z · x^{2}.

(26) If a > b and b > c, then \frac{b}{a−b} > \frac{c}{a−c}.

(27) If b > a and c > d, then \frac{c}{c+a} > \frac{d}{d+b}.

(28) m · x + z · y ≤ \sqrt{m^{2} + z^{2}} · \sqrt{x^{2} + y^{2}}.

(29) (m · x + u · y + w · z)^{2} ≤ (m^{2} + u^{2} + w^{2}) · (x^{2} + y^{2} + z^{2}).

(30) \frac{9·a·b·c}{a^{2}+b^{2}+c^{2}} ≤ a + b + c.

(31) a + b + c ≤ \sqrt{\frac{a^{2}+a·b+b^{2}}{3}} + \sqrt{\frac{b^{2}+b·c+c^{2}}{3}} + \sqrt{\frac{c^{2}+c·a+a^{2}}{3}} .

(32) \sqrt{\frac{a^{2}+a·b+b^{2}}{3}} + \sqrt{\frac{b^{2}+b·c+c^{2}}{3}} + \sqrt{\frac{c^{2}+c·a+a^{2}}{3}} ≤ \sqrt{\frac{a^{2}+b^{2}}{2}} + \sqrt{\frac{b^{2}+c^{2}}{2}} + \sqrt{\frac{c^{2}+a^{2}}{2}} .

(33) \sqrt{\frac{a^{2}+b^{2}}{2}} + \sqrt{\frac{b^{2}+c^{2}}{2}} + \sqrt{\frac{c^{2}+a^{2}}{2}} ≤ \sqrt{3 · (a^{2} + b^{2} + c^{2})}.

(34) \sqrt{3 · (a^{2} + b^{2} + c^{2})} ≤ \frac{b·c}{a} + \frac{c·a}{b} + \frac{a·b}{c}.

(35) If a + b = 1, then (\frac{1}{a^{2}} − 1) · (\frac{1}{b^{2}} − 1) ≥ 9.

(36) If a + b = 1, then a · b + \frac{1}{a·b} ≥ \frac{17}{4}.

(37) If a + b + c = 1, then \frac{1}{a} + \frac{1}{b} + \frac{1}{c} ≥ 9.

(38) If a + b + c = 1, then (\frac{1}{a} − 1) · (\frac{1}{b} − 1) · (\frac{1}{c} − 1) ≥ 8.

(39) If a + b + c = 1, then (1 + \frac{1}{a}) · (1 + \frac{1}{b}) · (1 + \frac{1}{c}) ≥ 64.

(40) If x + y + z = 1, then x^{2} + y^{2} + z^{2} ≥ \frac{1}{3}.

(41) If x + y + z = 1, then x · y + y · z + z · x ≤ \frac{1}{3}.

(42) If a · b · c = 1, then \sqrt{a} + \sqrt{b} + \sqrt{c} ≤ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.

(43) If a > b and b > c, then a^{2·a} · b^{2·b} · c^{2·c} > a^{b+c} · b^{a+c} · c^{a+b}.

(44) If n ≥ 1, then a^{n+1} + b^{n+1} ≥ a^{n} · b + a · b^{n}.

(45) If a^{2} + b^{2} = c^{2} and n ≥ 3, then a^{n+2} + b^{n+2} < c^{n+2}.

(46) If n ≥ 1, then (1 + \frac{1}{n+1})^{n} < (1 + \frac{1}{n})^{n+1}.

(47) If n ≥ 1 and k ≥ 1, then (a^{k} + b^{k}) · (a^{n} + b^{n}) ≤ 2 · (a^{k+n} + b^{k+n}).

(48) If for every n holds s(n) = \frac{1}{\sqrt{n+1}}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) < 2 · \sqrt{n + 1}.

(49) If for every n holds s(n) = \frac{1}{(n+1)^{2}} , then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) ≤ 2 − \frac{1}{n+1}.

(50) If for every n holds s(n) = \frac{1}{(n+1)^{2}} , then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) < 2.

(51) If for every n holds s(n) < 1, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) < n + 1.

(52) If for every n holds s(n) > 0 and s(n) < 1, then for every n holds (the partial product of s)(n) ≥ (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) − n.

(53) If for every n holds s(n) > 0 and s_{1}(n) = \frac{1}{s(n)}, then for every n holds (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n) > 0.

(54) If for every n holds s(n) > 0 and s_{1}(n) = \frac{1}{s(n)}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n) ≥ (n + 1)^{2}.

(55) If for every n such that n ≥ 1 holds s(n) = \sqrt{n} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) < \frac{1}{6} · (4 · n + 3) · \sqrt{n}.

(56) If for every n such that n ≥ 1 holds s(n) = \sqrt{n} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) > \frac{2}{3} · n · \sqrt{n}.

(57) Suppose that for every n such that n ≥ 1 holds s(n) = 1 + \frac{1}{2·n+1} and s(0) = 1. Let given n. If n ≥ 1, then (the partial product of s)(n) > \frac{1}{2} · \sqrt{2 · n + 3}.

(58) If for every n such that n ≥ 1 holds s(n) = \sqrt{n · (n + 1)} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) > \frac{n·(n+1)}{2}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Fuguo Ge and Xiquan Liang. On the partial product of series and related basic inequalities. Formalized Mathematics, 13(3):413–416, 2005.

[5] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[6] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890,

528 fuguo ge and xiquan liang

[7] Konrad Raczkowski and Andrzej Nȩdzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[8] Konrad Raczkowski and Andrzej Nȩdzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[9] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

Several Differentiable Formulas of Special Functions. Part II

Yan Zhang Qingdao University of Science and Technology China

Bo Li Qingdao University of Science and Technology China

Xiquan Liang Qingdao University of Science and Technology China

Summary. In this article, we give several other differentiable formulas of special functions.

MML identifier: FDIFF 6, version: 7.6.01 4.50.934

The papers [11], [13], [14], [1], [8], [10], [2], [4], [7], [5], [6], [9], [15], [3], and [12] provide the notation and terminology for this paper.

For simplicity, we use the following convention: x, a denote real numbers, n denotes a natural number, Z denotes an open subset of ℝ, and f, f_{1}, f_{2} denote partial functions from ℝ to ℝ.

One can prove the following propositions:

(1) If a > 0, then exp(x · log_{e} a) = a_{ℝ}^{x}.

(2) If a > 0, then exp(−x · log_{e} a) = a_{ℝ}^{−x}.

(3) Suppose Z ⊆ dom(f_{1} − f_{2}) and for every x such that x ∈ Z holds f_{1}(x) = a^{2} and f_{2} = _{ℤ}^{2}. Then f_{1} − f_{2} is differentiable on Z and for every x such that x ∈ Z holds (f_{1} − f_{2})′_{↾Z}(x) = −2 · x.

(4) Suppose Z ⊆ dom(\frac{f_{1}+f_{2}}{f_{1}−f_{2}} ) and f_{2} = _{ℤ}^{2} and for every x such that x ∈ Z holds f_{1}(x) = a^{2} and (f_{1} − f_{2})(x) ≠ 0. Then \frac{f_{1}+f_{2}}{f_{1}−f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{f_{1}+f_{2}}{f_{1}−f_{2}} )′_{↾Z}(x) = \frac{4·a^{2}·x}{(a^{2}−x^{2})^{2}} .

(5) Suppose Z ⊆ dom f and f = log (e) · \frac{f_{1}+f_{2}}{f_{1}−f_{2}} and f_{2} = _{ℤ}^{2} and for every x such that x ∈ Z holds f_{1}(x) = a^{2} and (f_{1}−f_{2})(x) > 0 and a ≠ 0. Then f is differentiable on Z and for every x such that x ∈ Z holds f′_{↾Z}(x) = \frac{4·a^{2}·x}{a^{4}−x^{4}} .

(6) Suppose Z ⊆ dom(\frac{1}{4·a^{2}} f) and f = log (e) · \frac{f_{1}+f_{2}}{f_{1}−f_{2}} and f_{2} = _{ℤ}^{2} and for every x such that x ∈ Z holds f_{1}(x) = a^{2} and (f_{1} − f_{2})(x) > 0 and a ≠ 0. Then \frac{1}{4·a^{2}} f is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{4·a^{2}} f)′_{↾Z}(x) = \frac{x}{a^{4}−x^{4}} .

(7) Suppose Z ⊆ dom(\frac{f_{1}}{f_{2}+f_{1}} ) and f_{1} = _{ℤ}^{2} and for every x such that x ∈ Z holds f_{2}(x) = 1 and x ≠ 0. Then \frac{f_{1}}{f_{2}+f_{1}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{f_{1}}{f_{2}+f_{1}} )′_{↾Z}(x) = \frac{2·x}{(1+x^{2})^{2}} .

(8) Suppose Z ⊆ dom(\frac{1}{2} f) and f = log (e) · \frac{f_{1}}{f_{2}+f_{1}} and f_{1} = _{ℤ}^{2} and for every x such that x ∈ Z holds f_{2}(x) = 1 and x ≠ 0. Then \frac{1}{2} f is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{2} f)′_{↾Z}(x) = \frac{1}{x·(1+x^{2})}.

(9) Suppose Z ⊆ dom(log (e) · _{ℤ}^{n}) and for every x such that x ∈ Z holds x > 0. Then log (e) · _{ℤ}^{n} is differentiable on Z and for every x such that x ∈ Z holds (log (e) · _{ℤ}^{n})′_{↾Z}(x) = \frac{n}{x}.

(10) Suppose Z ⊆ dom(\frac{1}{f_{2}} + log (e) · \frac{f_{1}}{f_{2}} ) and for every x such that x ∈ Z holds f_{2}(x) = x and f_{2}(x) > 0 and f_{1}(x) = x − 1 and f_{1}(x) > 0. Then \frac{1}{f_{2}} + log (e) · \frac{f_{1}}{f_{2}} is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{f_{2}} + log (e) · \frac{f_{1}}{f_{2}} )′_{↾Z}(x) = \frac{1}{x^{2}·(x−1)}.

(11) Suppose Z ⊆ dom(exp ·f) and for every x such that x ∈ Z holds f(x) = x·log_{e} a and a > 0. Then exp ·f is differentiable on Z and for every x such that x ∈ Z holds (exp ·f)′_{↾Z}(x) = (a_{ℝ}^{x}) · log_{e} a.

(12) Suppose Z ⊆ dom(\frac{1}{log_{e} a} ((exp ·f_{1}) f_{2})) and for every x such that x ∈ Z holds f_{1}(x) = x · log_{e} a and f_{2}(x) = x − \frac{1}{log_{e} a} and a > 0 and a ≠ 1. Then \frac{1}{log_{e} a} ((exp ·f_{1}) f_{2}) is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{log_{e} a} ((exp ·f_{1}) f_{2}))′_{↾Z}(x) = x · a_{ℝ}^{x}.

(13) Suppose Z ⊆ dom(\frac{1}{1+log_{e} a} ((exp ·f) exp)) and for every x such that x ∈ Z holds f(x) = x · log_{e} a and a > 0 and a ≠ \frac{1}{e}. Then \frac{1}{1+log_{e} a} ((exp ·f) exp) is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{1+log_{e} a} ((exp ·f) exp))′_{↾Z}(x) = (a_{ℝ}^{x}) · exp(x).

(14) Suppose Z ⊆ dom(exp ·f) and for every x such that x ∈ Z holds f(x) = −x. Then exp ·f is differentiable on Z and for every x such that x ∈ Z holds (exp ·f)′_{↾Z}(x) = −exp(−x).

(15) Suppose Z ⊆ dom(f_{1} (exp ·f_{2})) and for every x such that x ∈ Z holds f_{1}(x) = −x − 1 and f_{2}(x) = −x. Then f_{1} (exp ·f_{2}) is differentiable on Z and for every x such that x ∈ Z holds (f_{1} (exp ·f_{2}))′_{↾Z}(x) = \frac{x}{exp x}.

(16) Suppose Z ⊆ dom(−exp ·f) and for every x such that x ∈ Z holds f(x) = −x · log_{e} a and a > 0. Then −exp ·f is differentiable on Z and for every x such that x ∈ Z holds (−exp ·f)′_{↾Z}(x) = (a_{ℝ}^{−x}) · log_{e} a.

(17) Suppose Z ⊆ dom(\frac{1}{log_{e} a} ((−exp ·f_{1}) f_{2})) and for every x such that x ∈ Z holds f_{1}(x) = −x · log_{e} a and f_{2}(x) = x + \frac{1}{log_{e} a} and a > 0 and a ≠ 1. Then \frac{1}{log_{e} a} ((−exp ·f_{1}) f_{2}) is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{log_{e} a} ((−exp ·f_{1}) f_{2}))′_{↾Z}(x) = \frac{x}{a_{ℝ}^{x}}.

(18) Suppose Z ⊆ dom(\frac{1}{log_{e} a−1} \frac{exp ·f}{exp} ) and for every x such that x ∈ Z holds f(x) = x · log_{e} a and a > 0 and a ≠ e. Then \frac{1}{log_{e} a−1} \frac{exp ·f}{exp} is differentiable

on Z and for every x such that x ∈ Z holds (\frac{1}{log_{e} a−1} \frac{exp ·f}{exp} )′_{↾Z}(x) = \frac{a_{ℝ}^{x}}{exp(x)}.

(19) Suppose Z ⊆ dom(\frac{1}{1−log_{e} a} \frac{exp}{exp ·f}) and for every x such that x ∈ Z holds f(x) = x · log_{e} a and a > 0 and a ≠ e. Then \frac{1}{1−log_{e} a} \frac{exp}{exp ·f} is differentiable

on Z and for every x such that x ∈ Z holds (\frac{1}{1−log_{e} a} \frac{exp}{exp ·f})′_{↾Z}(x) = \frac{exp(x)}{a_{ℝ}^{x}}.

(20) Suppose Z ⊆ dom(log (e) · (exp +f)) and for every x such that x ∈ Z holds f(x) = 1. Then log (e)·(exp +f) is differentiable on Z and for every x such that x ∈ Z holds (log (e) · (exp +f))′_{↾Z}(x) = \frac{exp(x)}{exp(x)+1}.

(21) Suppose Z ⊆ dom(log (e) · (exp −f)) and for every x such that x ∈ Z holds f(x) = 1 and (exp −f)(x) > 0. Then log (e) · (exp −f) is differentiable on Z and for every x such that x ∈ Z holds (log (e) · (exp −f))′_{↾Z}(x) = \frac{exp(x)}{exp(x)−1}.

(22) Suppose Z ⊆ dom(−log (e) · (f − exp)) and for every x such that x ∈ Z holds f(x) = 1 and (f − exp)(x) > 0. Then −log (e) · (f − exp) is differentiable on Z and for every x such that x ∈ Z holds (−log (e) · (f − exp))′_{↾Z}(x) = \frac{exp(x)}{1−exp(x)}.

(23) Suppose Z ⊆ dom((_{ℤ}^{2}) · exp +f) and for every x such that x ∈ Z holds f(x) = 1. Then (_{ℤ}^{2}) · exp +f is differentiable on Z and for every x such that x ∈ Z holds ((_{ℤ}^{2}) · exp +f)′_{↾Z}(x) = 2 · exp(2 · x).

(24) Suppose Z ⊆ dom(\frac{1}{2} (log (e) · f)) and f = (_{ℤ}^{2}) · exp +f_{1} and for every x such that x ∈ Z holds f_{1}(x) = 1. Then \frac{1}{2} (log (e) · f) is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{2} (log (e)·f))′_{↾Z}(x) = \frac{exp x}{exp x+exp(−x)}.

(25) Suppose Z ⊆ dom((_{ℤ}^{2}) · exp −f) and for every x such that x ∈ Z holds f(x) = 1. Then (_{ℤ}^{2}) · exp −f is differentiable on Z and for every x such that x ∈ Z holds ((_{ℤ}^{2}) · exp −f)′_{↾Z}(x) = 2 · exp(2 · x).

(26) Suppose Z ⊆ dom(\frac{1}{2} (log (e) · f)) and f = (_{ℤ}^{2}) · exp −f_{1} and for every x such that x ∈ Z holds f_{1}(x) = 1 and f(x) > 0. Then \frac{1}{2} (log (e) · f) is differentiable on Z and for every x such that x ∈ Z holds (\frac{1}{2} (log (e) · f))′_{↾Z}(x) = \frac{exp x}{exp x−exp(−x)}.

(27) Suppose Z ⊆ dom((_{ℤ}^{2}) · (exp −f)) and for every x such that x ∈ Z holds f(x) = 1. Then (_{ℤ}^{2}) · (exp −f) is differentiable on Z and for every x such that x ∈ Z holds ((_{ℤ}^{2}) · (exp −f))′_{↾Z}(x) = 2 · exp(x) · (exp(x) − 1).

(28) Suppose Z ⊆ dom f and f = log (e) · \frac{(_{ℤ}^{2})·(exp −f_{1})}{exp} and for every x such that x ∈ Z holds f_{1}(x) = 1 and (exp −f_{1})(x) > 0. Then f is differentiable on Z and for every x such that x ∈ Z holds f′_{↾Z}(x) = \frac{exp(x)+1}{exp(x)−1}.

(29) Suppose Z ⊆ dom((_{ℤ}^{2}) · (exp +f)) and for every x such that x ∈ Z holds f(x) = 1. Then (_{ℤ}^{2}) · (exp +f) is differentiable on Z and for every x such that x ∈ Z holds ((_{ℤ}^{2}) · (exp +f))′_{↾Z}(x) = 2 · exp(x) · (exp(x) + 1).

(30) Suppose Z ⊆ dom f and f = log (e) · \frac{(_{ℤ}^{2})·(exp +f_{1})}{exp} and for every x such that x ∈ Z holds f_{1}(x) = 1. Then f is differentiable on Z and for every x such that x ∈ Z holds f′_{↾Z}(x) = \frac{exp(x)−1}{exp(x)+1}.

(31) Suppose Z ⊆ dom((_{ℤ}^{2}) · (f − exp)) and for every x such that x ∈ Z holds f(x) = 1. Then (_{ℤ}^{2}) · (f − exp) is differentiable on Z and for every x such that x ∈ Z holds ((_{ℤ}^{2}) · (f − exp))′_{↾Z}(x) = −2 · exp(x) · (1 − exp(x)).

(32) Suppose Z ⊆ dom f and f = log (e) · \frac{exp}{(_{ℤ}^{2})·(f_{1}−exp)} and for every x such that x ∈ Z holds f_{1}(x) = 1 and (f_{1} − exp)(x) > 0. Then f is differentiable on Z and for every x such that x ∈ Z holds f′_{↾Z}(x) = \frac{1+exp(x)}{1−exp(x)}.

(33) Suppose Z ⊆ dom f and f = log (e) · \frac{exp}{(_{ℤ}^{2})·(f_{1}+exp)} and for every x such that x ∈ Z holds f_{1}(x) = 1. Then f is differentiable on Z and for every x such that x ∈ Z holds f′_{↾Z}(x) = \frac{1−exp(x)}{1+exp(x)}.

(34) Suppose Z ⊆ dom(log (e)·f) and f = exp + exp ·f_{1} and for every x such that x ∈ Z holds f_{1}(x) = −x. Then log (e) · f is differentiable on Z and for every x such that x ∈ Z holds (log (e) · f)′_{↾Z}(x) = \frac{exp x−exp(−x)}{exp x+exp(−x)}.

(35) Suppose Z ⊆ dom(log (e)·f) and f = exp − exp ·f_{1} and for every x such that x ∈ Z holds f_{1}(x) = −x and f(x) > 0. Then log (e) · f is differentiable on Z and for every x such that x ∈ Z holds (log (e) · f)′_{↾Z}(x) = \frac{exp x+exp(−x)}{exp x−exp(−x)}.

(36) Suppose Z ⊆ dom(\frac{2}{3} ((ℝ^{\frac{3}{2}}) · (f + exp))) and for every x such that x ∈ Z

holds f(x) = 1. Then \frac{2}{3} ((ℝ^{\frac{3}{2}})·(f +exp)) is differentiable on Z and for every

x such that x ∈ Z holds (\frac{2}{3} ((ℝ^{\frac{3}{2}}) · (f + exp)))′_{↾Z}(x) = exp(x) · (1 + exp(x))ℝ^{\frac{1}{2}}.

(37) Suppose Z ⊆ dom(\frac{2}{3·log_{e} a} ((ℝ^{\frac{3}{2}}) · (f + exp ·f_{1}))) and for every x such that x ∈ Z holds f(x) = 1 and f_{1}(x) = x · log_{e} a and a > 0 and a ≠ 1. Then \frac{2}{3·log_{e} a} ((ℝ^{\frac{3}{2}}) · (f + exp ·f_{1})) is differentiable on Z and for every x such that

x ∈ Z holds (\frac{2}{3·log_{e} a} ((ℝ^{\frac{3}{2}}) · (f + exp ·f_{1})))′_{↾Z}(x) = (a_{ℝ}^{x}) · (1 + a_{ℝ}^{x})ℝ^{\frac{1}{2}}.

(38) Suppose Z ⊆ dom((−\frac{1}{2}) ((the function cos) ·f)) and for every x such

that x ∈ Z holds f(x) = 2 · x. Then

(i) (−\frac{1}{2}) ((the function cos) ·f) is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((−\frac{1}{2}) ((the function cos) ·f))′_{↾Z}(x) =

sin(2 · x).

(39) Suppose that (i) Z ⊆ dom(2 ((ℝ^{\frac{1}{2}}) · (f − the function cos))), and (ii) for every x such that x ∈ Z holds f(x) = 1 and (the function sin)(x) > 0

and (the function cos)(x) < 1 and (the function cos)(x) > −1. Then

(iii) 2 ((ℝ^{\frac{1}{2}}) · (f − the function cos)) is differentiable on Z, and

(iv) for every x such that x ∈ Z holds (2 ((ℝ^{\frac{1}{2}})·(f −the function cos)))′_{↾Z}(x) =

(1 + (the function cos)(x))ℝ^{\frac{1}{2}}.

(40) Suppose that (i) Z ⊆ dom((−2) ((ℝ^{\frac{1}{2}}) · (f + the function cos))), and (ii) for every x such that x ∈ Z holds f(x) = 1 and (the function sin)(x) > 0

and (the function cos)(x) < 1 and (the function cos)(x) > −1. Then

(iii) (−2) ((ℝ^{\frac{1}{2}}) · (f + the function cos)) is differentiable on Z, and

(iv) for every x such that x ∈ Z holds ((−2) ((ℝ^{\frac{1}{2}}) · (f + the function

cos)))′_{↾Z}(x) = (1 − (the function cos)(x))ℝ^{\frac{1}{2}}.

(41) Suppose Z ⊆ dom(\frac{1}{2} (log (e) · f)) and f = f_{1} + 2 (the function sin) and

for every x such that x ∈ Z holds f_{1}(x) = 1 and f(x) > 0. Then

(i) \frac{1}{2} (log (e) · f) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (\frac{1}{2} (log (e) · f))′_{↾Z}(x) =

\frac{(the function cos)(x)}{1+2·(the function sin)(x)}.

(42) Suppose Z ⊆ dom((−\frac{1}{2}) (log (e) · f)) and f = f_{1} + 2 (the function cos)

and for every x such that x ∈ Z holds f_{1}(x) = 1 and f(x) > 0. Then

(i) (−\frac{1}{2}) (log (e) · f) is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((−\frac{1}{2}) (log (e) · f))′_{↾Z}(x) =

\frac{(the function sin)(x)}{1+2·(the function cos)(x)}.

(43) Suppose Z ⊆ dom(\frac{1}{4·a} ((the function sin) ·f)) and for every x such that

x ∈ Z holds f(x) = 2 · a · x and a ≠ 0. Then

(i) \frac{1}{4·a} ((the function sin) ·f) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (\frac{1}{4·a} ((the function sin) ·f))′_{↾Z}(x) =

\frac{1}{2} · cos(2 · a · x).

(44) Suppose Z ⊆ dom(f_{1} − \frac{1}{4·a} ((the function sin) ·f)) and for every x such that x ∈ Z holds f_{1}(x) = \frac{x}{2} and f(x) = 2 · a · x and a ≠ 0. Then (i) f_{1} − \frac{1}{4·a} ((the function sin) ·f) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (f_{1}−\frac{1}{4·a} ((the function sin) ·f))′_{↾Z}(x) =

(sin(a · x))^{2}.

(45) Suppose Z ⊆ dom(f_{1} + \frac{1}{4·a} ((the function sin) ·f)) and for every x such

that x ∈ Z holds f_{1}(x) = \frac{x}{2} and f(x) = 2 · a · x and a ≠ 0. Then

(i) f_{1} + \frac{1}{4·a} ((the function sin) ·f) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (f_{1}+\frac{1}{4·a} ((the function sin) ·f))′_{↾Z}(x) =

(cos(a · x))^{2}.

(46) Suppose Z ⊆ dom(\frac{1}{n} ((_{ℤ}^{n}) · (the function cos))) and n > 0. Then (i) \frac{1}{n} ((_{ℤ}^{n}) · (the function cos)) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (\frac{1}{n} ((_{ℤ}^{n}) · (the function cos)))′_{↾Z}(x) =

−((the function cos)(x)_{ℤ}^{n−1}) · (the function sin)(x).

(47) Suppose Z ⊆ dom(\frac{1}{3} ((_{ℤ}^{3}) · (the function cos))−the function cos) and

n > 0. Then

(i) \frac{1}{3} ((_{ℤ}^{3}) · (the function cos))−the function cos is differentiable on Z, and (ii) for every x such that x ∈ Z holds (\frac{1}{3} ((_{ℤ}^{3}) · (the function cos))−the

function cos)′_{↾Z}(x) = (the function sin)(x)^{3}.

(48) Suppose Z ⊆ dom((the function sin)−\frac{1}{3} ((_{ℤ}^{3}) · (the function sin))) and

n > 0. Then

(i) (the function sin)−\frac{1}{3} ((_{ℤ}^{3})·(the function sin)) is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((the function sin)−\frac{1}{3} ((_{ℤ}^{3}) · (the

function sin)))′_{↾Z}(x) = (the function cos)(x)^{3}.

(49) Suppose Z ⊆ dom((the function sin) · log (e)). Then (i) (the function sin) · log (e) is differentiable on Z, and (ii) for every x such that x ∈ Z holds ((the function sin) · log (e))′_{↾Z}(x) =

\frac{(the function cos)(log_{e} x)}{x}.

(50) Suppose Z ⊆ dom(−(the function cos) · log (e)). Then (i) −(the function cos) · log (e) is differentiable on Z, and (ii) for every x such that x ∈ Z holds (−(the function cos) · log (e))′_{↾Z}(x) =

\frac{(the function sin)(log_{e} x)}{x}.

References

[1] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[2] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics,

1(1):35–40, 1990.

[3] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized

Mathematics, 1(4):703–709, 1990.

[4] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathemat-

ics, 1(2):269–272, 1990.

[5] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125–

130, 1991.

[6] Konrad Raczkowski and Andrzej Nȩdzusiak. Real exponents and logarithms. Formalized

Mathematics, 2(2):213–216, 1991.

several differentiable formulas … 535

[7] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[8] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[9] Yasunari Shidama. The Taylor expansions. Formalized Mathematics, 12(2):195–200, 2004.

[10] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[12] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[13] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[14] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[15] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

536 yan zhang and bo li and xiquan liang

On the Calculus of Binary Arithmetics. Part II

Shunichi Kobayashi Matsumoto University Nagano, Japan

Summary. In this paper, we introduce binary arithmetic and its related operations. We include some theorems concerning logical operators.

MML identifier: BINARI 6, version: 7.6.01 4.50.934

The terminology and notation used in this paper are introduced in the following articles: [4], [3], [2], and [1].

In this paper x, y, z denote boolean sets. Next we state a number of propositions:

(1) true ⇒ x = x.

(2) false ⇒ x = true.

(3) x ⇒ x = true and ¬(x ⇒ x) = false.

(4) ¬(x ⇒ y) = x ∧ ¬y.

(5) x ⇒ ¬x = ¬x and ¬(x ⇒ ¬x) = x.

(6) ¬x ⇒ x = x.

(7) true ⇔ x = x.

(8) false ⇔ x = ¬x.

(9) x ⇔ x = true and ¬(x ⇔ x) = false.

(10) ¬x ⇔ x = false.

(11) x ∧ (y ⇔ z) = x ∧ (¬y ∨ z) ∧ (¬z ∨ y).

(12) x ∧ (y ′nand′ z) = x ∧ ¬y ∨ x ∧ ¬z.

(13) x ∧ (y ′nor′ z) = x ∧ ¬y ∧ ¬z.

(14) x ∧ (x ∧ y) = x ∧ y.

(15) x ∧ (x ∨ y) = x ∨ x ∧ y.

(16) x ∧ (x ⊕ y) = x ∧ ¬y.

(17) x ∧ (x ⇒ y) = x ∧ y.

(18) x ∧ (x ⇔ y) = x ∧ y.

(19) x ∧ (x ′nand′ y) = x ∧ ¬y.

(20) x ∧ (x ′nor′ y) = false.

(21) x ∨ (y ⊕ z) = x ∨ ¬y ∧ z ∨ y ∧ ¬z.

(22) x ∨ (y ⇔ z) = (x ∨ ¬y ∨ z) ∧ (x ∨ ¬z ∨ y).

(23) x ∨ (y ′nand′ z) = x ∨ ¬y ∨ ¬z.

(24) x∨(y ′nor′ z) = (x∨¬y)∧(x∨¬z) and x∨(y ′nor′ z) = (y ⇒ x)∧(z ⇒ x).

(25) x ∨ (x ∨ y) = x ∨ y.

(26) x ∨ (x ⇒ y) = true.

(27) x ∨ (x ⇔ y) = y ⇒ x.

(28) x ∨ (x ′nand′ y) = true.

(29) x ∨ (x ′nor′ y) = y ⇒ x.

(30) x ⇒ y ⊕ z = ¬x ∨ ¬y ∧ z ∨ y ∧ ¬z.

(31) x ⇒ y ⇔ z = (¬x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ ¬z).

(32) x ⇒ y ′nand′ z = ¬x ∨ ¬y ∨ ¬z.

(33) x ⇒ y ′nor′ z = (¬x ∨ ¬y) ∧ (¬x ∨ ¬z) and x ⇒ y ′nor′ z = (x ⇒ ¬y) ∧ (x ⇒ ¬z).

(34) x ⇒ x ∧ y = x ⇒ y.

(35) x ⇒ x ∨ y = true.

(36) x ⇒ x ⊕ y = ¬x ∨ ¬y.

(37) x ⇒ x ⇒ y = x ⇒ y.

(38) x ⇒ x ⇔ y = x ⇒ y and x ⇒ x ⇔ y = x ⇒ x ⇒ y.

(39) x ⇒ x ′nand′ y = ¬(x ∧ y).

(40) x ⇒ x ′nor′ y = ¬x.

(41) x ′nand′ (y ⇒ z) = (¬x ∨ y) ∧ (¬x ∨ ¬z) and x ′nand′ (y ⇒ z) = (x ⇒ y) ∧ (x ⇒ ¬z).

(42) x ′nand′ (y ⇔ z) = ¬(x ∧ (¬y ∨ z) ∧ (¬z ∨ y)).

(43) x ′nand′ (y ′nand′ z) = (¬x ∨ y) ∧ (¬x ∨ z) and x ′nand′ (y ′nand′ z) = (x ⇒ y) ∧ (x ⇒ z).

(44) x ′nand′ (y ′nor′ z) = ¬x ∨ y ∨ z.

(45) x ′nand′ x ∧ y = ¬(x ∧ y).

(46) x ′nand′ (x ⊕ y) = x ⇒ y.

(47) x ′nand′ (x ⇒ y) = ¬(x ∧ y).

(48) x ′nand′ (x ⇔ y) = ¬(x ∧ y).

(49) x ′nand′ (x ′nand′ y) = x ⇒ y.

(50) x ′nand′ (x ′nor′ y) = true.

(51) x ′nor′ (y ⊕ z) = ¬(x ∨ ¬y ∧ z ∨ y ∧ ¬z).

(52) x ′nor′ (y ⇔ z) = ¬((x ∨ ¬y ∨ z) ∧ (x ∨ ¬z ∨ y)).

(53) x ′nor′ (y ′nand′ z) = ¬x ∧ y ∧ z.

(54) x ′nor′ (y ′nor′ z) = ¬x ∧ y ∨ ¬x ∧ z.

(55) x ′nor′ x ∧ y = ¬x.

(56) x ′nor′ (x ∨ y) = ¬x ∧ ¬y.

(57) x ′nor′ (x ⊕ y) = ¬x ∧ ¬y.

(58) x ′nor′ (x ⇒ y) = false.

(59) x ′nor′ (x ⇔ y) = ¬x ∧ y.

(60) x ′nor′ (x ′nand′ y) = false.

(61) x ′nor′ (x ′nor′ y) = ¬x ∧ y.

(62) x ⊕ y ∧ z = (x ∨ y ∧ z) ∧ (¬x ∨ ¬(y ∧ z)).

(63) x ⊕ x ∧ y = x ∧ ¬y.

(64) x ⊕ (x ∨ y) = ¬x ∧ y.

(65) ¬x ∧ (x ⊕ y) = ¬x ∧ y.

(66) x ∧ ¬(x ⊕ y) = x ∧ y.

(67) x ⊕ (x ⊕ y) = y.

(68) x ∧ ¬(x ⇒ y) = x ∧ ¬y.

(69) x ⊕ (x ⇒ y) = ¬x ∨ ¬y.

(70) ¬x ∧ (x ⇔ y) = ¬x ∧ ¬y.

(71) x ∧ ¬(x ⇔ y) = x ∧ ¬y.

(72) x ⊕ (x ⇔ y) = ¬y.

(73) x ⊕ (x ′nand′ y) = x ⇒ y.

(74) x ⊕ (x ′nor′ y) = y ⇒ x.

(75) ¬x ∧ (x ⇒ y) = ¬x ∨ ¬x ∧ y.

(76) ¬x ∧ (y ⇔ z) = ¬x ∧ (¬y ∨ z) ∧ (¬z ∨ y).

(77) ¬x ∧ (x ⇔ y) = ¬x ∧ ¬y ∧ (¬x ∨ y).

(78) ¬x ∧ (x ′nand′ y) = ¬x ∨ ¬x ∧ ¬y.

(79) ¬x ∧ (x ′nor′ y) = ¬x ∧ ¬y.

(80) ¬x ∨ (x ⇒ y) = ¬x ∨ y.

(81) ¬x ∨ (x ⇔ y) = ¬x ∨ y.

(82) ¬x ∨ (x ′nand′ y) = ¬x ∨ ¬y.

(83) ¬x ⊕ (x ⇒ y) = x ∧ y.

(84) ¬x ⊕ (y ⇒ x) = x ∧ (x ∨ ¬y) ∨ ¬x ∧ y.

(85) ¬(x ⇒ y) = x ∧ ¬y.

540 shunichi kobayashi

(86) ¬(x ⇔ y) = x ∧ ¬y ∨ y ∧ ¬x.

(87) ¬x ⊕ (x ⇔ y) = y.

References

[1] Shunichi Kobayashi. On the calculus of binary arithmetics. Formalized Mathematics, 11(4):417–419, 2003.

[2] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

[3] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[4] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 1990.

Some Properties of Some Special Matrices

Xiaopeng Yue Qingdao University of Science and Technology China

Xiquan Liang Qingdao University of Science and Technology China

Zhongpin Sun Qingdao University of Science and Technology China

Summary. This article describes definitions of reversible matrix, symmetrical matrix, antisymmetric matrix, orthogonal matrix and their main properties.

MML identifier: MATRIX 6, version: 7.6.01 4.50.934

The terminology and notation used in this paper have been introduced in the following articles: [8], [3], [11], [12], [1], [10], [9], [6], [2], [4], [5], [13], and [7].

For simplicity, we adopt the following convention: n denotes a natural number, K denotes a field, a denotes an element of K, and M, M_{1}, M_{2}, M_{3}, M_{4} denote matrices over K of dimension n.

Let n be a natural number, let K be a field, and let M_{1}, M_{2} be matrices over K of dimension n. We say that M_{1} is permutable with M_{2} if and only if:

(Def. 1) M_{1} · M_{2} = M_{2} · M_{1}.

Let us note that the predicate M_{1} is permutable with M_{2} is symmetric.

Let n be a natural number, let K be a field, and let M_{1}, M_{2} be matrices over K of dimension n. We say that M_{1} is reverse of M_{2} if and only if:

(Def. 2) M_{1} · M_{2} = M_{2} · M_{1} and M_{1} · M_{2} = ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}.

Let us note that the predicate M_{1} is reverse of M_{2} is symmetric.

Let n be a natural number, let K be a field, and let M_{1} be a matrix over K of dimension n. We say that M_{1} is reversible if and only if:

(Def. 3) There exists a matrix M_{2} over K of dimension n such that M_{1} is reverse of M_{2}.

Let us consider n, K and let M_{1} be a matrix over K of dimension n. Then −M_{1} is a matrix over K of dimension n.

Let us consider n, K and let M_{1}, M_{2} be matrices over K of dimension n. Then M_{1} + M_{2} is a matrix over K of dimension n.

Let us consider n, K and let M_{1}, M_{2} be matrices over K of dimension n. Then M_{1} − M_{2} is a matrix over K of dimension n.

Let us consider n, K and let M_{1}, M_{2} be matrices over K of dimension n. Then M_{1} · M_{2} is a matrix over K of dimension n.

The following propositions are true:

(1) For every field K and for every matrix A over K such that

len A > 0 and width A > 0 holds ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(len A)×(len A)} · A =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(len A)×(width A)}.

(2) For every field K and for every matrix A over K such that

len A > 0 and width A > 0 holds A · ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(width A)×(width A)} =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(len A)×(width A)}.

(3) If n > 0, then M_{1} is permutable with ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n}.

(4) If M_{1} is permutable with M_{2} and M_{2} is permutable with M_{3} and M_{1} is permutable with M_{3}, then M_{1} is permutable with M_{2} · M_{3}.

(5) If M_{1} is permutable with M_{2} and permutable with M_{3} and n > 0, then M_{1} is permutable with M_{2} + M_{3}.

(6) M_{1} is permutable with ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}.

(7) If M_{2} is reverse of M_{3} and M_{1} is reverse of M_{3}, then M_{1} = M_{2}. Let n be a natural number, let K be a field, and let M_{1} be a matrix over K

of dimension n. Let us assume that M_{1} is reversible. The functor M_{1}^{⌣} yields a matrix over K of dimension n and is defined by:

(Def. 4) M_{1}^{⌣} is reverse of M_{1}. We now state a number of propositions:

(8) (( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n})^{⌣} = ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} and ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} is

reversible.

(9) ((( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n})^{⌣})^{⌣} = ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}.

(10) If n > 0, then (( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n})^{T} = ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}.

(11) Let K be a field, n be a natural number, and M be a matrix over K

of dimension n. If M = (( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n})^{T} and n > 0, then M^{⌣} =

( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}.

(12) If M_{1}^{T} = M_{2} and M_{3} is reverse of M_{1} and M = M_{3}^{T} and n > 0, then M_{2} is reverse of M.

(13) If M is reversible and n > 0 and M_{1} = M^{T} and M_{2} = (M^{⌣})^{T}, then M_{1}^{⌣} = M_{2}.

(14) Let K be a field, n be a natural number, and M_{1}, M_{2}, M_{3}, M_{4} be matrices over K of dimension n. If M_{3} is reverse of M_{1} and M_{4} is reverse of M_{2}, then M_{3} · M_{4} is reverse of M_{2} · M_{1}.

(15) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. If M_{2} is reverse of M_{1}, then M_{1} is permutable with M_{2}.

(16) If M is reversible, then M^{⌣} is reversible and (M^{⌣})^{⌣} = M.

(17) If n > 0 and M_{1} · M_{2} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n} and M_{1} is reversible, then

M_{1} is permutable with M_{2}.

(18) If n > 0 and M_{1} = M_{1} · M_{2} and M_{1} is reversible, then M_{1} is permutable with M_{2}.

(19) If n > 0 and M_{1} = M_{2} · M_{1} and M_{1} is reversible, then M_{1} is permutable

with M_{2}. Let n be a natural number, let K be a field, and let M_{1} be a matrix over K

of dimension n. We say that M_{1} is symmetrical if and only if: (Def. 5) M_{1}^{T} = M_{1}.

The following propositions are true:

(20) If n > 0, then ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} is symmetrical.

(21) If n > 0, then (( \begin{matrix}a & … & a \\ ⋮ & ... & ⋮ \\ a & … & a \end{matrix} )^{n×n})^{T} = ( \begin{matrix}a & … & a \\ ⋮ & ... & ⋮ \\ a & … & a \end{matrix} )^{n×n} .

(22) If n > 0, then ( \begin{matrix}a & … & a \\ ⋮ & ... & ⋮ \\ a & … & a \end{matrix} )^{n×n} is symmetrical.

(23) If n > 0 and M_{1} is symmetrical and M_{2} is symmetrical, then M_{1} is permutable with M_{2} iff M_{1} · M_{2} is symmetrical.

(24) If n > 0, then (M_{1} + M_{2})^{T} = M_{1}^{T} + M_{2}^{T}.

(25) If n > 0 and M_{1} is symmetrical and M_{2} is symmetrical, then M_{1} + M_{2} is symmetrical.

(26) Suppose that

(i) M_{1} is an upper triangular matrix over K of dimension n and a lower triangular matrix over K of dimension n, and

(ii) n > 0.

Then M_{1} is symmetrical.

(27) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. If n > 0, then (−M_{1})^{T} = −M_{1}^{T}.

(28) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. If M_{1} is symmetrical and n > 0, then −M_{1} is symmetrical.

(29) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. Suppose n > 0 and M_{1} is symmetrical and M_{2} is symmetrical. Then M_{1} − M_{2} is symmetrical.

Let n be a natural number, let K be a field, and let M_{1} be a matrix over K of dimension n. We say that M_{1} is antisymmetric if and only if:

(Def. 6) M_{1}^{T} = −M_{1}.

We now state a number of propositions:

(30) Let K be a Fanoian field, n be a natural number, and M_{1} be a matrix over K of dimension n. If M_{1} is symmetrical and antisymmetric and n > 0,

then M_{1} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n}.

(31) Let K be a Fanoian field, n, i be natural numbers, and M_{1} be a matrix over K of dimension n. If M_{1} is antisymmetric and n > 0 and i ∈ Seg n, then M_{1} ∘ (i, i) = 0_{K}.

(32) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. Suppose n > 0 and M_{1} is antisymmetric and M_{2} is antisymmetric. Then M_{1} + M_{2} is antisymmetric.

(33) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. If M_{1} is antisymmetric and n > 0, then −M_{1} is antisymmetric.

(34) Let K be a field, n be a natural number, and M_{1}, M_{2} be matrices over K of dimension n. Suppose n > 0 and M_{1} is antisymmetric and M_{2} is antisymmetric. Then M_{1} − M_{2} is antisymmetric.

(35) If M_{2} = M_{1} − M_{1}^{T} and n > 0, then M_{2} is antisymmetric.

(36) If n > 0, then M_{1} is permutable with M_{2} iff (M_{1} + M_{2}) · (M_{1} + M_{2}) = M_{1} · M_{1} + M_{1} · M_{2} + M_{1} · M_{2} + M_{2} · M_{2}.

(37) If n > 0 and M_{1} is reversible and M_{2} is reversible, then M_{1} · M_{2} is reversible and (M_{1} · M_{2})^{⌣} = M_{2}^{⌣} · M_{1}^{⌣}.

(38) If n > 0 and M_{1} is reversible and M_{2} is reversible and M_{1} is permutable with M_{2}, then M_{1} · M_{2} is reversible and (M_{1} · M_{2})^{⌣} = M_{1}^{⌣} · M_{2}^{⌣}.

(39) If n > 0 and M_{1} is reversible and M_{2} is reversible and M_{1} · M_{2} =

( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}, then M_{1} is reverse of M_{2}.

(40) If n > 0 and M_{1} is reversible and M_{2} is reversible and M_{2} · M_{1} =

( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n}, then M_{1} is reverse of M_{2}.

(41) If n > 0 and M_{1} is reversible and permutable with M_{2}, then M_{1}^{⌣} is permutable with M_{2}.

546 xiaopeng yue et al.

Let n be a natural number, let K be a field, and let M_{1} be a matrix over K of dimension n. We say that M_{1} is orthogonal if and only if:

(Def. 7) M_{1} is reversible and M_{1}^{T} = M_{1}^{⌣}.

The following propositions are true:

(42) If n > 0, then M_{1} · M_{1}^{T} = ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} and M_{1} is reversible iff M_{1} is orthogonal.

(43) If n > 0, then M_{1} is reversible and M_{1}^{T} · M_{1} = ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} iff M_{1} is orthogonal.

(44) If n > 0 and M_{1} is orthogonal, then M_{1}^{T} · M_{1} = M_{1} · M_{1}^{T}.

(45) If n > 0 and M_{1} is orthogonal and permutable with M_{2} and M_{3} = M_{1}^{T}, then M_{3} is permutable with M_{2}.

(46) If n > 0 and M_{1} is reversible and M_{2} is reversible, then M_{1} · M_{2} is reversible and (M_{1} · M_{2})^{⌣} = M_{2}^{⌣} · M_{1}^{⌣}.

(47) If n > 0 and M_{1} is orthogonal and M_{2} is orthogonal, then M_{1} · M_{2} is orthogonal.

(48) If n > 0 and M_{1} is orthogonal and permutable with M_{2} and M_{3} = M_{1}^{T}, then M_{3} is permutable with M_{2}.

(49) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} + M_{1} is permutable with M_{2}.

(50) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} + M_{2} is permutable with M_{2}.

(51) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} + M_{1} is permutable with M_{2} + M_{2}.

(52) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} + M_{2} is permutable with M_{2} + M_{2}.

(53) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} + M_{2} is permutable with M_{1} + M_{2}.

(54) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} · M_{2} is permutable with M_{2}.

(55) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} · M_{1} is permutable with M_{2}.

(56) If n > 0 and M_{1} is permutable with M_{2}, then M_{1} · M_{1} is permutable with M_{2} · M_{2}.

(57) If n > 0 and M_{1} is permutable with M_{2} and M_{3} = M_{1}^{T} and M_{4} = M_{2}^{T},

some properties … 547

then M_{3} is permutable with M_{4}.

(58) Suppose n > 0 and M_{1} is reversible and M_{2} is reversible and M_{3} is reversible. Then M_{1} · M_{2} · M_{3} is reversible and (M_{1} · M_{2} · M_{3})^{⌣} = M_{3}^{⌣} · M_{2}^{⌣} · M_{1}^{⌣}.

(59) If n > 0 and M_{1} is orthogonal and M_{2} is orthogonal and M_{3} is orthogonal, then M_{1} · M_{2} · M_{3} is orthogonal.

(60) If n > 0, then ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} is orthogonal.

(61) If n > 0 and M_{1} is orthogonal and M_{2} is orthogonal, then M_{1}^{⌣} · M_{2} is orthogonal.

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[2] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[3] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[4] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

[5] Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711–717, 1991.

[6] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[7] Yatsuka Nakamura and Hiroshi Yamazaki. Calculation of matrices of field elements. Part I. Formalized Mathematics, 11(4):385–391, 2003.

[8] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[9] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[10] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291– 296, 1990.

[11] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[12] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[13] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1–8, 1993.

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Generalized Full Adder Circuits (GFAs). Part I

Shin’nosuke Yamaguchi Kyushu Institute of Technology Fukuoka, Japan Katsumi Wasaki Shinshu University Nagano, Japan

Nobuhiro Shimoi Tokyo National College of Technology Tokyo, Japan

Summary. In the article we formalized the concept of the Generalized Full Addition and Subtraction circuits (GFAs), defined the structures of calculation units for the redundant signed digit (RSD) operations, and proved the stability of the circuits. Generally, 1-bit binary full adder assumes positive weights to all of its three binary inputs and two outputs. We obtained four type of 1-bit GFA to constract the RSD arithmetic logical units that we generalized full adder to have both positive and negative weights to inputs and outputs.

      MML identifier: GFACIRC1, version: 7.6.01 4.50.934

The articles [15], [14], [18], [13], [1], [21], [5], [6], [7], [2], [4], [16], [20], [8], [12], [17], [11], [10], [9], [3], and [19] provide the terminology and notation for this paper.

1. Preliminaries

In this article we present several logical schemes. The scheme 1AryBooleEx deals with a unary functor F yielding an element of Boolean, and states that:

There exists a function f from Boolean^{1} into Boolean such that for every element x of Boolean holds f(⟨x⟩) = F(x)

for all values of the parameter.

The scheme 1AryBooleUniq deals with a unary functor F yielding an element of Boolean, and states that:

Let f_{1}, f_{2} be functions from Boolean^{1} into Boolean. Suppose for every element x of Boolean holds f_{1}(⟨x⟩) = F(x) and for every element x of Boolean holds f_{2}(⟨x⟩) = F(x). Then f_{1} = f_{2}

for all values of the parameter.

The scheme 1AryBooleDef deals with a unary functor F yielding an element of Boolean, and states that:

(i) There exists a function f from Boolean^{1} into Boolean such that for every element x of Boolean holds f(⟨x⟩) = F(x), and (ii) for all functions f_{1}, f_{2} from Boolean^{1} into Boolean such that for every element x of Boolean holds f_{1}(⟨x⟩) = F(x) and for every element x of Boolean holds f_{2}(⟨x⟩) = F(x) holds f_{1} = f_{2}

for all values of the parameter.

The function inv1 from Boolean^{1} into Boolean is defined by: (Def. 1) For every element x of Boolean holds (inv1)(⟨x⟩) = ¬x.

Next we state the proposition

(1) For every element x of Boolean holds (inv1)(⟨x⟩) = ¬x and (inv1)(⟨x⟩) = nand_{2}(⟨x, x⟩) and (inv1)(⟨0⟩) = 1 and (inv1)(⟨1⟩) = 0.

The function buf1 from Boolean^{1} into Boolean is defined by: (Def. 2) For every element x of Boolean holds (buf1)(⟨x⟩) = x.

One can prove the following proposition

(2) For every element x of Boolean holds (buf1)(⟨x⟩) = x and (buf1)(⟨x⟩) = and_{2}(⟨x, x⟩) and (buf1)(⟨0⟩) = 0 and (buf1)(⟨1⟩) = 1.

The function and2c from Boolean^{2} into Boolean is defined by: (Def. 3) For all elements x, y of Boolean holds (and2c)(⟨x, y⟩) = x ∧ ¬y.

Next we state the proposition

(3) Let x, y be elements of Boolean. Then (and2c)(⟨x, y⟩) = x ∧ ¬y and (and2c)(⟨x, y⟩) = (and_{2a})(⟨y, x⟩) and (and2c)(⟨x, y⟩) = (nor_{2a})(⟨x, y⟩) and (and2c)(⟨0, 0⟩) = 0 and (and2c)(⟨0, 1⟩) = 0 and (and2c)(⟨1, 0⟩) = 1 and (and2c)(⟨1, 1⟩) = 0.

The function xor2c from Boolean^{2} into Boolean is defined by: (Def. 4) For all elements x, y of Boolean holds (xor2c)(⟨x, y⟩) = x ⊕ ¬y.

We now state several propositions:

(4) Let x, y be elements of Boolean. Then (xor2c)(⟨x, y⟩) = x ⊕ ¬y and (xor2c)(⟨x, y⟩) = (xor_{2a})(⟨x, y⟩) and (xor2c)(⟨x, y⟩) = or_{2}(⟨(and_{2b})(⟨x, y⟩), and_{2}(⟨x, y⟩)⟩) and (xor2c)(⟨0, 0⟩) = 1 and (xor2c)(⟨0, 1⟩) = 0 and (xor2c)(⟨1, 0⟩) = 0 and (xor2c)(⟨1, 1⟩) = 1.

(5) For all elements x, y of Boolean holds ¬(x ⊕ y) = ¬x ⊕ y and ¬(x ⊕ y) = x ⊕ ¬y and ¬x ⊕ ¬y = x ⊕ y.

(6) For all elements x, y of Boolean holds (inv1)(⟨xor_{2}(⟨x, y⟩)⟩) = (xor_{2a})(⟨x, y⟩) and (inv1)(⟨xor_{2}(⟨x, y⟩)⟩) = (xor2c)(⟨x, y⟩) and xor_{2}(⟨(inv1)(⟨x⟩), (inv1)(⟨y⟩)⟩) = xor_{2}(⟨x, y⟩).

(7) For all elements x, y, z of Boolean holds ¬(x ⊕ ¬y ⊕ z) = x ⊕ ¬y ⊕ ¬z.

(8) For all elements x, y, z of Boolean holds (inv1)(⟨xor_{2}(⟨(xor2c)(⟨x, y⟩), z⟩)⟩) = (xor2c)(⟨(xor2c)(⟨x, y⟩), z⟩).

(9) For all elements x, y, z of Boolean holds ¬x ⊕ y ⊕ ¬z = x ⊕ ¬y ⊕ ¬z.

(10) For all elements x, y, z of Boolean holds (xor2c)(⟨(xor_{2a})(⟨x, y⟩), z⟩) = (xor2c)(⟨(xor2c)(⟨x, y⟩), z⟩).

(11) For all elements x, y, z of Boolean holds ¬(¬x ⊕ ¬y ⊕ ¬z) = x ⊕ y ⊕ z.

(12) For all elements x, y, z of Boolean holds (inv1)(⟨(xor2c)(⟨(xor_{2b})(⟨x, y⟩), z⟩)⟩) = xor_{2}(⟨xor_{2}(⟨x, y⟩), z⟩).

2. Generalized Full Adder (GFA) Circuit (TYPE-0)

Let x, y, z be sets. The functor GFA0CarryIStr(x, y, z) yields an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined by:

(Def. 5) GFA0CarryIStr(x, y, z) = 1GateCircStr(⟨x, y⟩, and_{2})+· 1GateCircStr(⟨y,

z⟩, and_{2})+· 1GateCircStr(⟨z, x⟩, and_{2}). Let x, y, z be sets. The functor GFA0CarryICirc(x, y, z) yields a strict

Boolean circuit of GFA0CarryIStr(x, y, z) with denotation held in gates and is defined as follows:

(Def. 6) GFA0CarryICirc(x, y, z) = 1GateCircuit(x, y, and_{2})+· 1GateCircuit(y, z,

and_{2})+· 1GateCircuit(z, x, and_{2}). Let x, y, z be sets. The functor GFA0CarryStr(x, y, z) yields an unsplit

non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined as follows:

(Def. 7) GFA0CarryStr(x, y, z) = GFA0CarryIStr(x, y, z)+· 1GateCircStr(⟨⟨⟨⟨x,

y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩⟩, or_{3}). Let x, y, z be sets. The functor GFA0CarryCirc(x, y, z) yields a strict

Boolean circuit of GFA0CarryStr(x, y, z) with denotation held in gates and is defined as follows:

(Def. 8) GFA0CarryCirc(x, y, z) = GFA0CarryICirc(x, y, z)+· 1GateCircuit(⟨⟨⟨x,

y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩, or_{3}). Let x, y, z be sets. The functor GFA0CarryOutput(x, y, z) yielding an

element of InnerVertices(GFA0CarryStr(x, y, z)) is defined as follows: (Def. 9) GFA0CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩,

and_{2} ⟩⟩⟩, or_{3} ⟩⟩.

One can prove the following propositions:

(13) For all sets x, y, z holds InnerVertices(GFA0CarryIStr(x, y, z)) = {⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩}.

(14) For all sets x, y, z holds InnerVertices(GFA0CarryStr(x, y, z)) = {⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩} ∪ {GFA0CarryOutput(x, y, z)}.

(15) For all sets x, y, z holds InnerVertices(GFA0CarryStr(x, y, z)) is a binary relation.

(16) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ holds InputVertices(GFA0CarryIStr(x, y, z)) = {x, y, z}.

(17) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ holds InputVertices(GFA0CarryStr(x, y, z)) = {x, y, z}.

(18) For all non pair sets x, y, z holds InputVertices(GFA0CarryStr(x, y, z)) has no pairs.

(19) Let x, y, z be sets. Then x ∈ the carrier of GFA0CarryStr(x, y, z) and y ∈ the carrier of GFA0CarryStr(x, y, z) and z ∈ the carrier of GFA0CarryStr(x, y, z) and ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ ∈ the carrier of GFA0CarryStr(x, y, z) and ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ ∈ the carrier of GFA0CarryStr(x, y, z) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ the carrier of GFA0CarryStr(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩⟩, or_{3} ⟩⟩ ∈ the carrier of GFA0CarryStr(x, y, z).

(20) For all sets x, y, z holds ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ ∈ InnerVertices(GFA0CarryStr(x, y, z)) and ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ ∈ InnerVertices(GFA0CarryStr(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ InnerVertices(GFA0CarryStr(x, y, z)) and GFA0CarryOutput (x, y, z) ∈ InnerVertices(GFA0CarryStr(x, y, z)).

(21) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ holds x ∈ InputVertices(GFA0CarryStr(x, y, z)) and y ∈ InputVertices(GFA0CarryStr(x, y, z)) and z ∈ InputVertices(GFA0CarryStr(x, y, z)).

(22) For all non pair sets x, y, z holds InputVertices(GFA0CarryStr(x, y, z)) = {x, y, z}.

(23) Let x, y, z be sets, s be a state of GFA0CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, and_{2} ⟩⟩) = a_{1} ∧ a_{2} and (Following(s))(⟨⟨⟨y, z⟩, and_{2} ⟩⟩) = a_{2} ∧ a_{3} and (Following(s))(⟨⟨⟨z, x⟩, and_{2} ⟩⟩) = a_{3} ∧ a_{1}.

(24) Let x, y, z be sets, s be a state of GFA0CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(⟨⟨⟨x, y⟩, and_{2} ⟩⟩) and a_{2} = s(⟨⟨⟨y, z⟩, and_{2} ⟩⟩) and a_{3} = s(⟨⟨⟨z, x⟩, and_{2} ⟩⟩), then (Following(s))(GFA0CarryOutput(x, y, z)) = a_{1} ∨ a_{2} ∨ a_{3}.

(25) Let x, y, z be sets. Suppose x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩. Let s be a state of GFA0CarryCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA0CarryOutput(x, y, z)) = a_{1} ∧ a_{2} ∨ a_{2} ∧ a_{3} ∨ a_{3} ∧ a_{1} and (Following(s, 2))(⟨⟨⟨x, y⟩, and_{2} ⟩⟩) = a_{1} ∧ a_{2} and (Following(s, 2))(⟨⟨⟨y, z⟩, and_{2} ⟩⟩) = a_{2} ∧ a_{3} and (Following(s, 2))(⟨⟨⟨z, x⟩, and_{2} ⟩⟩) = a_{3} ∧ a_{1}.

(26) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ and for every state s of GFA0CarryCirc(x, y, z) holds Following(s, 2) is stable.

Let x, y, z be sets. The functor GFA0AdderStr(x, y, z) yielding an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined as follows:

(Def. 10) GFA0AdderStr(x, y, z) = 2GatesCircStr(x, y, z, xor_{2}).

Let x, y, z be sets. The functor GFA0AdderCirc(x, y, z) yielding a strict Boolean circuit of GFA0AdderStr(x, y, z) with denotation held in gates is defined by:

(Def. 11) GFA0AdderCirc(x, y, z) = 2GatesCircuit(x, y, z, xor_{2}).

Let x, y, z be sets. The functor GFA0AdderOutput(x, y, z) yielding an element of InnerVertices(GFA0AdderStr(x, y, z)) is defined by:

(Def. 12) GFA0AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor_{2}).

Next we state a number of propositions:

(27) For all sets x, y, z holds InnerVertices(GFA0AdderStr(x, y, z)) = {⟨⟨⟨x, y⟩, xor_{2} ⟩⟩} ∪ {GFA0AdderOutput(x, y, z)}.

(28) For all sets x, y, z holds InnerVertices(GFA0AdderStr(x, y, z)) is a binary relation.

(29) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ holds InputVertices(GFA0AdderStr(x, y, z)) = {x, y, z}.

(30) For all non pair sets x, y, z holds InputVertices(GFA0AdderStr(x, y, z)) has no pairs.

(31) Let x, y, z be sets. Then (i) x ∈ the carrier of GFA0AdderStr(x, y, z), (ii) y ∈ the carrier of GFA0AdderStr(x, y, z), (iii) z ∈ the carrier of GFA0AdderStr(x, y, z), (iv) ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ the carrier of GFA0AdderStr(x, y, z), and

(v) ⟨⟨⟨⟨⟨⟨x, y⟩, xor_{2} ⟩⟩, z⟩, xor_{2} ⟩⟩ ∈ the carrier of GFA0AdderStr(x, y, z).

(32) For all sets x, y, z holds ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ InnerVertices(GFA0AdderStr(x, y, z)) and GFA0AdderOutput(x, y, z) ∈ InnerVertices(GFA0AdderStr(x, y, z)).

(33) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ holds x ∈ InputVertices(GFA0AdderStr(x, y, z)) and y ∈ InputVertices(GFA0AdderStr(x, y, z)) and z ∈ InputVertices(GFA0AdderStr(x, y, z)).

(34) For all non pair sets x, y, z holds InputVertices(GFA0AdderStr(x, y, z)) = {x, y, z}.

(35) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA0AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, xor_{2} ⟩⟩) = a_{1} ⊕ a_{2} and (Following(s))(x) = a_{1} and (Following(s))(y) = a_{2} and (Following(s))(z) = a_{3}.

(36) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA0AdderCirc(x, y, z) and a_{4}, a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{4} = s(⟨⟨⟨x, y⟩, xor_{2} ⟩⟩) and a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s))(GFA0AdderOutput(x, y, z)) = a_{4} ⊕ a_{3}.

(37) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA0AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA0AdderOutput(x, y, z)) = a_{1} ⊕ a_{2} ⊕ a_{3} and (Following(s, 2))(⟨⟨⟨x, y⟩, xor_{2} ⟩⟩) = a_{1} ⊕ a_{2} and (Following(s, 2))(x) = a_{1} and (Following(s, 2))(y) = a_{2} and (Following(s, 2))(z) = a_{3}.

(38) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and for every state s of

GFA0AdderCirc(x, y, z) holds Following(s, 2) is stable. Let x, y, z be sets. The functor BitGFA0Str(x, y, z) yielding an unsplit

non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined as follows:

(Def. 13) BitGFA0Str(x, y, z) = GFA0AdderStr(x, y, z)+· GFA0CarryStr(x, y, z).

Let x, y, z be sets. The functor BitGFA0Circ(x, y, z) yielding a strict Boolean circuit of BitGFA0Str(x, y, z) with denotation held in gates is defined by:

(Def. 14) BitGFA0Circ(x, y, z) = GFA0AdderCirc(x, y, z)+· GFA0CarryCirc(x, y, z).

We now state several propositions:

(39) For all sets x, y, z holds InnerVertices(BitGFA0Str(x, y, z)) = {⟨⟨⟨x, y⟩, xor_{2} ⟩⟩}∪{GFA0AdderOutput(x, y, z)}∪{⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩} ∪ {GFA0CarryOutput(x, y, z)}.

(40) For all sets x, y, z holds InnerVertices(BitGFA0Str(x, y, z)) is a binary relation.

(41) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ holds InputVertices(BitGFA0Str(x, y, z)) = {x, y, z}.

(42) For all non pair sets x, y, z holds InputVertices(BitGFA0Str(x, y, z)) = {x, y, z}.

(43) For all non pair sets x, y, z holds InputVertices(BitGFA0Str(x, y, z)) has no pairs.

(44) Let x, y, z be sets. Then x ∈ the carrier of BitGFA0Str(x, y, z) and y ∈ the carrier of BitGFA0Str(x, y, z) and z ∈ the carrier of BitGFA0Str(x, y, z) and ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ the carrier of BitGFA0Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, xor_{2} ⟩⟩, z⟩, xor_{2} ⟩⟩ ∈ the carrier of BitGFA0Str(x, y, z) and ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ ∈ the carrier of BitGFA0Str(x, y, z) and ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ ∈ the carrier of BitGFA0Str(x, y, z) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ the carrier of BitGFA0Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩⟩, or_{3} ⟩⟩ ∈ the carrier of BitGFA0Str(x, y, z).

(45) Let x, y, z be sets. Then ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ InnerVertices(BitGFA0Str(x, y, z)) and GFA0AdderOutput(x, y, z) ∈ InnerVertices(BitGFA0Str(x, y, z)) and ⟨⟨⟨x, y⟩, and_{2} ⟩⟩ ∈ InnerVertices(BitGFA0Str(x, y, z)) and ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ ∈ InnerVertices(BitGFA0Str(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ InnerVertices(BitGFA0Str(x, y, z)) and GFA0CarryOutput(x, y, z) ∈ InnerVertices(BitGFA0Str(x, y, z)).

(46) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩. Then x ∈ InputVertices(BitGFA0Str(x, y, z)) and y ∈ InputVertices(BitGFA0Str(x, y, z)) and z ∈ InputVertices(BitGFA0Str(x, y, z)).

Let x, y, z be sets. The functor BitGFA0CarryOutput(x, y, z) yielding an element of InnerVertices(BitGFA0Str(x, y, z)) is defined as follows:

(Def. 15) BitGFA0CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2} ⟩⟩, ⟨⟨⟨z, x⟩,

and_{2} ⟩⟩⟩, or_{3} ⟩⟩. Let x, y, z be sets. The functor BitGFA0AdderOutput(x, y, z) yielding an

element of InnerVertices(BitGFA0Str(x, y, z)) is defined as follows:

(Def. 16) BitGFA0AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor_{2}).

One can prove the following two propositions:

(47) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩. Let s be a state of BitGFA0Circ(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA0AdderOutput(x, y, z)) = a_{1} ⊕ a_{2} ⊕ a_{3} and (Following(s, 2))(GFA0CarryOutput(x, y, z)) = a_{1} ∧ a_{2} ∨ a_{2} ∧ a_{3} ∨ a_{3} ∧ a_{1}.

(48) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2} ⟩⟩. Let s be a state of BitGFA0Circ(x, y, z). Then Following(s, 2) is stable.

3. Generalized Full Adder (GFA) Circuit (TYPE-1)

Let x, y, z be sets. The functor GFA1CarryIStr(x, y, z) yielding an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined by:

(Def. 17) GFA1CarryIStr(x, y, z) = 1GateCircStr(⟨x, y⟩, and2c)+· 1GateCircStr(⟨y,

z⟩, and_{2a})+· 1GateCircStr(⟨z, x⟩, and_{2}). Let x, y, z be sets. The functor GFA1CarryICirc(x, y, z) yields a strict

Boolean circuit of GFA1CarryIStr(x, y, z) with denotation held in gates and is defined as follows:

(Def. 18) GFA1CarryICirc(x, y, z) = 1GateCircuit(x, y, and2c)+· 1GateCircuit(y,

z, and_{2a})+· 1GateCircuit(z, x, and_{2}). Let x, y, z be sets. The functor GFA1CarryStr(x, y, z) yielding an unsplit

non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined by:

(Def. 19) GFA1CarryStr(x, y, z) = GFA1CarryIStr(x, y, z)+· 1GateCircStr(⟨⟨⟨⟨x,

y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩⟩, or_{3}). Let x, y, z be sets. The functor GFA1CarryCirc(x, y, z) yielding a strict

Boolean circuit of GFA1CarryStr(x, y, z) with denotation held in gates is defined by:

(Def. 20) GFA1CarryCirc(x, y, z) = GFA1CarryICirc(x, y, z)+· 1GateCircuit(⟨⟨⟨x,

y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩, or_{3}). Let x, y, z be sets. The functor GFA1CarryOutput(x, y, z) yielding an

element of InnerVertices(GFA1CarryStr(x, y, z)) is defined as follows: (Def. 21) GFA1CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩,

and_{2} ⟩⟩⟩, or_{3} ⟩⟩. We now state a number of propositions:

(49) For all sets x, y, z holds InnerVertices(GFA1CarryIStr(x, y, z)) = {⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩}.

(50) For all sets x, y, z holds InnerVertices(GFA1CarryStr(x, y, z)) = {⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩}∪{GFA1CarryOutput(x, y, z)}.

(51) For all sets x, y, z holds InnerVertices(GFA1CarryStr(x, y, z)) is a binary relation.

(52) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩ holds InputVertices(GFA1CarryIStr(x, y, z)) = {x, y, z}.

(53) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩ holds InputVertices(GFA1CarryStr(x, y, z)) = {x, y, z}.

(54) For all non pair sets x, y, z holds InputVertices(GFA1CarryStr(x, y, z)) has no pairs.

(55) Let x, y, z be sets. Then x ∈ the carrier of GFA1CarryStr(x, y, z) and y ∈ the carrier of GFA1CarryStr(x, y, z) and z ∈ the carrier of GFA1CarryStr(x, y, z) and ⟨⟨⟨x, y⟩, and2c ⟩⟩ ∈ the carrier of GFA1CarryStr(x, y, z) and ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ ∈ the carrier of GFA1CarryStr(x, y, z) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ the carrier of GFA1CarryStr(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩⟩, or_{3} ⟩⟩ ∈ the carrier of GFA1CarryStr(x, y, z).

(56) For all sets x, y, z holds ⟨⟨⟨x, y⟩, and2c ⟩⟩ ∈ InnerVertices(GFA1CarryStr(x, y, z)) and ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ ∈ InnerVertices(GFA1CarryStr(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ InnerVertices(GFA1CarryStr(x, y, z)) and GFA1CarryOutput (x, y, z) ∈ InnerVertices(GFA1CarryStr(x, y, z)).

(57) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩ holds x ∈ InputVertices(GFA1CarryStr(x, y, z)) and y ∈ InputVertices(GFA1CarryStr(x, y, z)) and z ∈ InputVertices(GFA1CarryStr(x, y, z)).

(58) For all non pair sets x, y, z holds InputVertices(GFA1CarryStr(x, y, z)) = {x, y, z}.

(59) Let x, y, z be sets, s be a state of GFA1CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, and2c ⟩⟩) = a_{1} ∧ ¬a_{2} and (Following(s))(⟨⟨⟨y, z⟩, and_{2a} ⟩⟩) = ¬a_{2} ∧ a_{3} and (Following(s))(⟨⟨⟨z, x⟩, and_{2} ⟩⟩) = a_{3} ∧ a_{1}.

(60) Let x, y, z be sets, s be a state of GFA1CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(⟨⟨⟨x, y⟩, and2c ⟩⟩) and a_{2} = s(⟨⟨⟨y, z⟩, and_{2a} ⟩⟩) and a_{3} = s(⟨⟨⟨z, x⟩, and_{2} ⟩⟩), then (Following(s))(GFA1CarryOutput(x, y, z)) = a_{1} ∨ a_{2} ∨ a_{3}.

(61) Let x, y, z be sets. Suppose x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩. Let s be a state of GFA1CarryCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA1CarryOutput(x, y, z)) = a_{1}∧¬a_{2}∨ ¬a_{2} ∧ a_{3} ∨ a_{3} ∧ a_{1} and (Following(s, 2))(⟨⟨⟨x, y⟩, and2c ⟩⟩) = a_{1} ∧ ¬a_{2} and (Following(s, 2))(⟨⟨⟨y, z⟩, and_{2a} ⟩⟩) = ¬a_{2} ∧ a_{3} and (Following(s, 2))(⟨⟨⟨z, x⟩, and_{2} ⟩⟩) = a_{3} ∧ a_{1}.

(62) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩ and for every state s of GFA1CarryCirc(x, y, z) holds Following(s, 2) is stable.

Let x, y, z be sets. The functor GFA1AdderStr(x, y, z) yields an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined as follows:

(Def. 22) GFA1AdderStr(x, y, z) = 2GatesCircStr(x, y, z, xor2c).

Let x, y, z be sets. The functor GFA1AdderCirc(x, y, z) yielding a strict Boolean circuit of GFA1AdderStr(x, y, z) with denotation held in gates is defined by:

(Def. 23) GFA1AdderCirc(x, y, z) = 2GatesCircuit(x, y, z, xor2c).

Let x, y, z be sets. The functor GFA1AdderOutput(x, y, z) yields an element of InnerVertices(GFA1AdderStr(x, y, z)) and is defined as follows:

(Def. 24) GFA1AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor2c).

We now state a number of propositions:

(63) For all sets x, y, z holds InnerVertices(GFA1AdderStr(x, y, z)) = {⟨⟨⟨x, y⟩, xor2c ⟩⟩} ∪ {GFA1AdderOutput(x, y, z)}.

(64) For all sets x, y, z holds InnerVertices(GFA1AdderStr(x, y, z)) is a binary relation.

(65) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ holds InputVertices(GFA1AdderStr(x, y, z)) = {x, y, z}.

(66) For all non pair sets x, y, z holds InputVertices(GFA1AdderStr(x, y, z)) has no pairs.

(67) Let x, y, z be sets. Then (i) x ∈ the carrier of GFA1AdderStr(x, y, z), (ii) y ∈ the carrier of GFA1AdderStr(x, y, z), (iii) z ∈ the carrier of GFA1AdderStr(x, y, z), (iv) ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ the carrier of GFA1AdderStr(x, y, z), and

(v) ⟨⟨⟨⟨⟨⟨x, y⟩, xor2c ⟩⟩, z⟩, xor2c ⟩⟩ ∈ the carrier of GFA1AdderStr(x, y, z).

(68) For all sets x, y, z holds ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ InnerVertices(GFA1AdderStr(x, y, z)) and GFA1AdderOutput(x, y, z) ∈ InnerVertices(GFA1AdderStr(x, y, z)).

(69) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ holds x ∈ InputVertices(GFA1AdderStr(x, y, z)) and y ∈ InputVertices(GFA1AdderStr(x, y, z)) and z ∈ InputVertices(GFA1AdderStr(x, y, z)).

(70) For all non pair sets x, y, z holds InputVertices(GFA1AdderStr(x, y, z)) = {x, y, z}.

(71) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA1AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, xor2c ⟩⟩) = a_{1}⊕¬a_{2} and (Following(s))(x) = a_{1} and (Following(s))(y) = a_{2} and (Following(s))(z) = a_{3}.

(72) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA1AdderCirc(x, y, z) and a_{4}, a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{4} = s(⟨⟨⟨x, y⟩, xor2c ⟩⟩) and a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s))(GFA1AdderOutput(x, y, z)) = a_{4} ⊕ ¬a_{3}.

(73) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA1AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA1AdderOutput(x, y, z)) = a_{1} ⊕ ¬a_{2} ⊕ ¬a_{3} and (Following(s, 2))(⟨⟨⟨x, y⟩, xor2c ⟩⟩) = a_{1}⊕¬a_{2} and (Following(s, 2))(x) = a_{1} and (Following(s, 2))(y) = a_{2} and (Following(s, 2))(z) = a_{3}.

(74) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA1AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s, 2))(GFA1AdderOutput(x, y, z)) = ¬(a_{1} ⊕ ¬a_{2} ⊕ a_{3}).

(75) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and for every state s of

GFA1AdderCirc(x, y, z) holds Following(s, 2) is stable. Let x, y, z be sets. The functor BitGFA1Str(x, y, z) yields an unsplit non

void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined as follows:

(Def. 25) BitGFA1Str(x, y, z) = GFA1AdderStr(x, y, z)+· GFA1CarryStr(x, y, z).

Let x, y, z be sets. The functor BitGFA1Circ(x, y, z) yielding a strict Boolean circuit of BitGFA1Str(x, y, z) with denotation held in gates is defined by:

(Def. 26) BitGFA1Circ(x, y, z) = GFA1AdderCirc(x, y, z)+· GFA1CarryCirc(x, y, z).

We now state several propositions:

(76) For all sets x, y, z holds InnerVertices(BitGFA1Str(x, y, z)) = {⟨⟨⟨x, y⟩, xor2c ⟩⟩}∪{GFA1AdderOutput(x, y, z)}∪{⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩} ∪ {GFA1CarryOutput(x, y, z)}.

(77) For all sets x, y, z holds InnerVertices(BitGFA1Str(x, y, z)) is a binary relation.

(78) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩ holds InputVertices(BitGFA1Str(x, y, z)) = {x, y, z}.

(79) For all non pair sets x, y, z holds InputVertices(BitGFA1Str(x, y, z)) = {x, y, z}.

(80) For all non pair sets x, y, z holds InputVertices(BitGFA1Str(x, y, z)) has no pairs.

(81) Let x, y, z be sets. Then x ∈ the carrier of BitGFA1Str(x, y, z) and y ∈ the carrier of BitGFA1Str(x, y, z) and z ∈ the carrier of BitGFA1Str(x, y, z) and ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ the carrier of BitGFA1Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, xor2c ⟩⟩, z⟩, xor2c ⟩⟩ ∈ the carrier of BitGFA1Str(x, y, z) and ⟨⟨⟨x, y⟩, and2c ⟩⟩ ∈ the carrier of BitGFA1Str(x, y, z) and ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ ∈ the carrier of BitGFA1Str(x, y, z) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ the carrier of BitGFA1Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2} ⟩⟩⟩, or_{3} ⟩⟩ ∈ the carrier of BitGFA1Str(x, y, z).

(82) Let x, y, z be sets. Then ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ InnerVertices(BitGFA1Str(x, y, z)) and GFA1AdderOutput(x, y, z) ∈ InnerVertices(BitGFA1Str(x, y, z)) and ⟨⟨⟨x, y⟩, and2c ⟩⟩ ∈ InnerVertices(BitGFA1Str(x, y, z)) and ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ ∈ InnerVertices(BitGFA1Str(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ ∈ InnerVertices(BitGFA1Str(x, y, z)) and GFA1CarryOutput(x, y, z) ∈ InnerVertices(BitGFA1Str(x, y, z)).

(83) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩. Then x ∈ InputVertices(BitGFA1Str(x, y, z)) and y ∈ InputVertices(BitGFA1Str(x, y, z)) and z ∈ InputVertices(BitGFA1Str(x, y, z)).

Let x, y, z be sets. The functor BitGFA1CarryOutput(x, y, z) yielding an element of InnerVertices(BitGFA1Str(x, y, z)) is defined as follows:

(Def. 27) BitGFA1CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and2c ⟩⟩, ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩, ⟨⟨⟨z,

x⟩, and_{2} ⟩⟩⟩, or_{3} ⟩⟩. Let x, y, z be sets. The functor BitGFA1AdderOutput(x, y, z) yielding an

element of InnerVertices(BitGFA1Str(x, y, z)) is defined as follows: (Def. 28) BitGFA1AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor2c).

The following two propositions are true:

(84) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩. Let s be a state of BitGFA1Circ(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA1AdderOutput(x, y, z)) = ¬(a_{1} ⊕ ¬a_{2} ⊕ a_{3}) and (Following(s, 2))(GFA1CarryOutput(x, y, z)) = a_{1}∧¬a_{2}∨¬a_{2}∧a_{3}∨a_{3}∧a_{1}.

(85) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2a} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and2c ⟩⟩. Let s be a state of BitGFA1Circ(x, y, z). Then Following(s, 2) is stable.

4. Generalized Full Adder (GFA) Circuit (TYPE-2)

Let x, y, z be sets. The functor GFA2CarryIStr(x, y, z) yields an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined by:

(Def. 29) GFA2CarryIStr(x, y, z) = 1GateCircStr(⟨x, y⟩, and_{2a})+· 1GateCircStr(⟨y, z⟩, and2c)+· 1GateCircStr(⟨z, x⟩, and_{2b}).

Let x, y, z be sets. The functor GFA2CarryICirc(x, y, z) yielding a strict Boolean circuit of GFA2CarryIStr(x, y, z) with denotation held in gates is defined as follows:

(Def. 30) GFA2CarryICirc(x, y, z) = 1GateCircuit(x, y, and_{2a})+· 1GateCircuit(y,

z, and2c)+· 1GateCircuit(z, x, and_{2b}). Let x, y, z be sets. The functor GFA2CarryStr(x, y, z) yielding an unsplit

non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined as follows:

(Def. 31) GFA2CarryStr(x, y, z) = GFA2CarryIStr(x, y, z)+· 1GateCircStr(⟨⟨⟨⟨x,

y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3}). Let x, y, z be sets. The functor GFA2CarryCirc(x, y, z) yields a strict

Boolean circuit of GFA2CarryStr(x, y, z) with denotation held in gates and is defined as follows:

(Def. 32) GFA2CarryCirc(x, y, z) = GFA2CarryICirc(x, y, z)+· 1GateCircuit(⟨⟨⟨x,

y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩, nor_{3}). Let x, y, z be sets. The functor GFA2CarryOutput(x, y, z) yields an element

of InnerVertices(GFA2CarryStr(x, y, z)) and is defined by: (Def. 33) GFA2CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩,

and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩. We now state a number of propositions:

(86) For all sets x, y, z holds InnerVertices(GFA2CarryIStr(x, y, z)) = {⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩}.

(87) For all sets x, y, z holds InnerVertices(GFA2CarryStr(x, y, z)) = {⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩}∪{GFA2CarryOutput(x, y, z)}.

(88) For all sets x, y, z holds InnerVertices(GFA2CarryStr(x, y, z)) is a binary relation.

(89) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ holds InputVertices(GFA2CarryIStr(x, y, z)) = {x, y, z}.

(90) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ holds InputVertices(GFA2CarryStr(x, y, z)) = {x, y, z}.

(91) For all non pair sets x, y, z holds InputVertices(GFA2CarryStr(x, y, z)) has no pairs.

(92) Let x, y, z be sets. Then x ∈ the carrier of GFA2CarryStr(x, y, z) and y ∈ the carrier of GFA2CarryStr(x, y, z) and z ∈ the carrier of GFA2CarryStr(x, y, z) and ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ ∈ the carrier of GFA2CarryStr(x, y, z) and ⟨⟨⟨y, z⟩, and2c ⟩⟩ ∈ the carrier of GFA2CarryStr(x, y, z) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ the carrier of GFA2CarryStr(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩ ∈ the carrier of GFA2CarryStr(x, y, z).

(93) For all sets x, y, z holds ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ ∈ InnerVertices(GFA2CarryStr(x, y, z)) and ⟨⟨⟨y, z⟩, and2c ⟩⟩ ∈ InnerVertices(GFA2CarryStr(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ InnerVertices(GFA2CarryStr(x, y, z)) and GFA2CarryOutput (x, y, z) ∈ InnerVertices(GFA2CarryStr(x, y, z)).

(94) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ holds x ∈ InputVertices(GFA2CarryStr(x, y, z)) and y ∈ InputVertices(GFA2CarryStr(x, y, z)) and z ∈ InputVertices(GFA2CarryStr(x, y, z)).

(95) For all non pair sets x, y, z holds InputVertices(GFA2CarryStr(x, y, z)) = {x, y, z}.

(96) Let x, y, z be sets, s be a state of GFA2CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) = ¬a_{1} ∧ a_{2} and (Following(s))(⟨⟨⟨y, z⟩, and2c ⟩⟩) = a_{2} ∧ ¬a_{3} and (Following(s))(⟨⟨⟨z, x⟩, and_{2b} ⟩⟩) = ¬a_{3} ∧ ¬a_{1}.

(97) Let x, y, z be sets, s be a state of GFA2CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) and a_{2} = s(⟨⟨⟨y, z⟩, and2c ⟩⟩) and a_{3} = s(⟨⟨⟨z, x⟩, and_{2b} ⟩⟩), then (Following(s))(GFA2CarryOutput(x, y, z)) = ¬(a_{1} ∨ a_{2} ∨ a_{3}).

(98) Let x, y, z be sets. Suppose x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩. Let s be a state of GFA2CarryCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA2CarryOutput(x, y, z)) = ¬(¬a_{1} ∧ a_{2} ∨a_{2} ∧¬a_{3} ∨¬a_{3} ∧¬a_{1}) and (Following(s, 2))(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) = ¬a_{1} ∧a_{2} and (Following(s, 2))(⟨⟨⟨y, z⟩, and2c ⟩⟩) = a_{2}∧¬a_{3} and (Following(s, 2))(⟨⟨⟨z, x⟩, and_{2b} ⟩⟩) = ¬a_{3} ∧ ¬a_{1}.

(99) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ and for every state s of GFA2CarryCirc(x, y, z) holds Following(s, 2) is stable.

Let x, y, z be sets. The functor GFA2AdderStr(x, y, z) yields an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined as follows:

(Def. 34) GFA2AdderStr(x, y, z) = 2GatesCircStr(x, y, z, xor2c).

Let x, y, z be sets. The functor GFA2AdderCirc(x, y, z) yielding a strict Boolean circuit of GFA2AdderStr(x, y, z) with denotation held in gates is defined as follows:

(Def. 35) GFA2AdderCirc(x, y, z) = 2GatesCircuit(x, y, z, xor2c).

Let x, y, z be sets. The functor GFA2AdderOutput(x, y, z) yields an element of InnerVertices(GFA2AdderStr(x, y, z)) and is defined by:

(Def. 36) GFA2AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor2c).

One can prove the following propositions:

(100) For all sets x, y, z holds InnerVertices(GFA2AdderStr(x, y, z)) = {⟨⟨⟨x, y⟩, xor2c ⟩⟩} ∪ {GFA2AdderOutput(x, y, z)}.

(101) For all sets x, y, z holds InnerVertices(GFA2AdderStr(x, y, z)) is a binary relation.

(102) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ holds InputVertices(GFA2AdderStr(x, y, z)) = {x, y, z}.

(103) For all non pair sets x, y, z holds InputVertices(GFA2AdderStr(x, y, z)) has no pairs.

(104) Let x, y, z be sets. Then (i) x ∈ the carrier of GFA2AdderStr(x, y, z), (ii) y ∈ the carrier of GFA2AdderStr(x, y, z), (iii) z ∈ the carrier of GFA2AdderStr(x, y, z), (iv) ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ the carrier of GFA2AdderStr(x, y, z), and

(v) ⟨⟨⟨⟨⟨⟨x, y⟩, xor2c ⟩⟩, z⟩, xor2c ⟩⟩ ∈ the carrier of GFA2AdderStr(x, y, z).

(105) For all sets x, y, z holds ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ InnerVertices(GFA2AdderStr(x, y, z)) and GFA2AdderOutput(x, y, z) ∈ InnerVertices(GFA2AdderStr(x, y, z)).

(106) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ holds x ∈ InputVertices(GFA2AdderStr(x, y, z)) and y ∈ InputVertices(GFA2AdderStr(x, y, z)) and z ∈ InputVertices(GFA2AdderStr(x, y, z)).

(107) For all non pair sets x, y, z holds InputVertices(GFA2AdderStr(x, y, z)) = {x, y, z}.

(108) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA2AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, xor2c ⟩⟩) = a_{1}⊕¬a_{2} and (Following(s))(x) = a_{1} and (Following(s))(y) = a_{2} and (Following(s))(z) = a_{3}.

(109) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA2AdderCirc(x, y, z) and a_{4}, a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{4} = s(⟨⟨⟨x, y⟩, xor2c ⟩⟩) and a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s))(GFA2AdderOutput(x, y, z)) = a_{4} ⊕ ¬a_{3}.

(110) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA2AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA2AdderOutput(x, y, z)) = a_{1} ⊕ ¬a_{2} ⊕ ¬a_{3} and (Following(s, 2))(⟨⟨⟨x, y⟩, xor2c ⟩⟩) = a_{1}⊕¬a_{2} and (Following(s, 2))(x) = a_{1} and (Following(s, 2))(y) = a_{2} and (Following(s, 2))(z) = a_{3}.

(111) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩. Let s be a state of GFA2AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s, 2))(GFA2AdderOutput(x, y, z)) = ¬a_{1} ⊕ a_{2} ⊕ ¬a_{3}.

(112) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and for every state s of

GFA2AdderCirc(x, y, z) holds Following(s, 2) is stable. Let x, y, z be sets. The functor BitGFA2Str(x, y, z) yielding an unsplit

non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined as follows:

(Def. 37) BitGFA2Str(x, y, z) = GFA2AdderStr(x, y, z)+· GFA2CarryStr(x, y, z).

Let x, y, z be sets. The functor BitGFA2Circ(x, y, z) yields a strict Boolean circuit of BitGFA2Str(x, y, z) with denotation held in gates and is defined by:

(Def. 38) BitGFA2Circ(x, y, z) = GFA2AdderCirc(x, y, z)+· GFA2CarryCirc(x, y, z).

Next we state several propositions:

(113) For all sets x, y, z holds InnerVertices(BitGFA2Str(x, y, z)) = {⟨⟨⟨x, y⟩, xor2c ⟩⟩}∪{GFA2AdderOutput(x, y, z)}∪{⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩} ∪ {GFA2CarryOutput(x, y, z)}.

(114) For all sets x, y, z holds InnerVertices(BitGFA2Str(x, y, z)) is a binary relation.

(115) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ holds InputVertices(BitGFA2Str(x, y, z)) = {x, y, z}.

(116) For all non pair sets x, y, z holds InputVertices(BitGFA2Str(x, y, z)) = {x, y, z}.

(117) For all non pair sets x, y, z holds InputVertices(BitGFA2Str(x, y, z)) has no pairs.

(118) Let x, y, z be sets. Then x ∈ the carrier of BitGFA2Str(x, y, z) and y ∈ the carrier of BitGFA2Str(x, y, z) and z ∈ the carrier of BitGFA2Str(x, y, z) and ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ the carrier of BitGFA2Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, xor2c ⟩⟩, z⟩, xor2c ⟩⟩ ∈ the carrier of BitGFA2Str(x, y, z) and ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ ∈ the carrier of BitGFA2Str(x, y, z) and ⟨⟨⟨y, z⟩, and2c ⟩⟩ ∈ the carrier of BitGFA2Str(x, y, z) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ the carrier of BitGFA2Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩ ∈ the carrier of BitGFA2Str(x, y, z).

(119) Let x, y, z be sets. Then ⟨⟨⟨x, y⟩, xor2c ⟩⟩ ∈ InnerVertices(BitGFA2Str(x, y, z)) and GFA2AdderOutput(x, y, z) ∈ InnerVertices(BitGFA2Str(x, y, z)) and ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ ∈ InnerVertices(BitGFA2Str(x, y, z)) and ⟨⟨⟨y, z⟩, and2c ⟩⟩ ∈ InnerVertices(BitGFA2Str(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ InnerVertices(BitGFA2Str(x, y, z)) and GFA2CarryOutput(x, y, z) ∈

InnerVertices(BitGFA2Str(x, y, z)).

(120) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩. Then x ∈ InputVertices(BitGFA2Str(x, y, z)) and y ∈ InputVertices(BitGFA2Str(x, y, z)) and z ∈ InputVertices(BitGFA2Str(x, y, z)).

Let x, y, z be sets. The functor BitGFA2CarryOutput(x, y, z) yields an element of InnerVertices(BitGFA2Str(x, y, z)) and is defined by:

(Def. 39) BitGFA2CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, z⟩, and2c ⟩⟩, ⟨⟨⟨z,

x⟩, and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩. Let x, y, z be sets. The functor BitGFA2AdderOutput(x, y, z) yielding an

element of InnerVertices(BitGFA2Str(x, y, z)) is defined by: (Def. 40) BitGFA2AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor2c).

Next we state two propositions:

(121) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩. Let s be a state of BitGFA2Circ(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA2AdderOutput(x, y, z)) = ¬a_{1} ⊕ a_{2} ⊕ ¬a_{3} and (Following(s, 2))(GFA2CarryOutput(x, y, z)) = ¬(¬a_{1} ∧ a_{2} ∨ a_{2} ∧ ¬a_{3} ∨ ¬a_{3} ∧ ¬a_{1}).

(122) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor2c ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and2c ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩. Let s be a state of BitGFA2Circ(x, y, z). Then Following(s, 2) is stable.

5. Generalized Full Adder (GFA) Circuit (TYPE-3)

Let x, y, z be sets. The functor GFA3CarryIStr(x, y, z) yielding an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined by:

(Def. 41) GFA3CarryIStr(x, y, z) = 1GateCircStr(⟨x, y⟩, and_{2b})+· 1GateCircStr(⟨y,

z⟩, and_{2b})+· 1GateCircStr(⟨z, x⟩, and_{2b}). Let x, y, z be sets. The functor GFA3CarryICirc(x, y, z) yielding a strict

Boolean circuit of GFA3CarryIStr(x, y, z) with denotation held in gates is defined by:

(Def. 42) GFA3CarryICirc(x, y, z) = 1GateCircuit(x, y, and_{2b})+· 1GateCircuit(y,

z, and_{2b})+· 1GateCircuit(z, x, and_{2b}). Let x, y, z be sets. The functor GFA3CarryStr(x, y, z) yielding an unsplit

non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined by:

(Def. 43) GFA3CarryStr(x, y, z) = GFA3CarryIStr(x, y, z)+· 1GateCircStr(⟨⟨⟨⟨x,

y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3}). Let x, y, z be sets. The functor GFA3CarryCirc(x, y, z) yielding a strict

Boolean circuit of GFA3CarryStr(x, y, z) with denotation held in gates is defined by:

(Def. 44) GFA3CarryCirc(x, y, z) = GFA3CarryICirc(x, y, z)+· 1GateCircuit(⟨⟨⟨x,

y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩, nor_{3}). Let x, y, z be sets. The functor GFA3CarryOutput(x, y, z) yields an element

of InnerVertices(GFA3CarryStr(x, y, z)) and is defined as follows: (Def. 45) GFA3CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩,

and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩. The following propositions are true:

(123) For all sets x, y, z holds InnerVertices(GFA3CarryIStr(x, y, z)) = {⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩}.

(124) For all sets x, y, z holds InnerVertices(GFA3CarryStr(x, y, z)) = {⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩}∪{GFA3CarryOutput(x, y, z)}.

(125) For all sets x, y, z holds InnerVertices(GFA3CarryStr(x, y, z)) is a binary relation.

(126) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ holds InputVertices(GFA3CarryIStr(x, y, z)) = {x, y, z}.

(127) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ holds InputVertices(GFA3CarryStr(x, y, z)) = {x, y, z}.

(128) For all non pair sets x, y, z holds InputVertices(GFA3CarryStr(x, y, z)) has no pairs.

(129) Let x, y, z be sets. Then x ∈ the carrier of GFA3CarryStr(x, y, z) and y ∈ the carrier of GFA3CarryStr(x, y, z) and z ∈ the carrier of GFA3CarryStr(x, y, z) and ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ ∈ the carrier of GFA3CarryStr(x, y, z) and ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ ∈ the carrier of GFA3CarryStr(x, y, z) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ the carrier of GFA3CarryStr(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩ ∈ the carrier of GFA3CarryStr(x, y, z).

(130) For all sets x, y, z holds ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ ∈ InnerVertices(GFA3CarryStr(x, y, z)) and ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ ∈ InnerVertices(GFA3CarryStr(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ InnerVertices(GFA3CarryStr(x, y, z)) and GFA3CarryOutput (x, y, z) ∈ InnerVertices(GFA3CarryStr(x, y, z)).

(131) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ holds x ∈ InputVertices(GFA3CarryStr(x, y, z)) and y ∈ InputVertices(GFA3CarryStr(x, y, z)) and

z ∈ InputVertices(GFA3CarryStr(x, y, z)).

(132) For all non pair sets x, y, z holds InputVertices(GFA3CarryStr(x, y, z)) = {x, y, z}.

(133) Let x, y, z be sets, s be a state of GFA3CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, and_{2b} ⟩⟩) = ¬a_{1} ∧ ¬a_{2} and (Following(s))(⟨⟨⟨y, z⟩, and_{2b} ⟩⟩) = ¬a_{2} ∧ ¬a_{3} and (Following(s))(⟨⟨⟨z, x⟩, and_{2b} ⟩⟩) = ¬a_{3} ∧ ¬a_{1}.

(134) Let x, y, z be sets, s be a state of GFA3CarryCirc(x, y, z), and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(⟨⟨⟨x, y⟩, and_{2b} ⟩⟩) and a_{2} = s(⟨⟨⟨y, z⟩, and_{2b} ⟩⟩) and a_{3} = s(⟨⟨⟨z, x⟩, and_{2b} ⟩⟩), then (Following(s))(GFA3CarryOutput(x, y, z)) = ¬(a_{1} ∨ a_{2} ∨ a_{3}).

(135) Let x, y, z be sets. Suppose x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩. Let s be a state of GFA3CarryCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA3CarryOutput(x, y, z)) = ¬(¬a_{1} ∧ ¬a_{2} ∨ ¬a_{2} ∧ ¬a_{3} ∨ ¬a_{3} ∧ ¬a_{1}) and (Following(s, 2))(⟨⟨⟨x, y⟩, and_{2b} ⟩⟩) = ¬a_{1} ∧ ¬a_{2} and (Following(s, 2))(⟨⟨⟨y, z⟩, and_{2b} ⟩⟩) = ¬a_{2} ∧ ¬a_{3} and (Following(s, 2))(⟨⟨⟨z, x⟩, and_{2b} ⟩⟩) = ¬a_{3} ∧ ¬a_{1}.

(136) For all sets x, y, z such that x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ and for every state s of GFA3CarryCirc(x, y, z) holds Following(s, 2) is stable.

Let x, y, z be sets. The functor GFA3AdderStr(x, y, z) yields an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined by:

(Def. 46) GFA3AdderStr(x, y, z) = 2GatesCircStr(x, y, z, xor_{2}).

Let x, y, z be sets. The functor GFA3AdderCirc(x, y, z) yielding a strict Boolean circuit of GFA3AdderStr(x, y, z) with denotation held in gates is defined by:

(Def. 47) GFA3AdderCirc(x, y, z) = 2GatesCircuit(x, y, z, xor_{2}).

Let x, y, z be sets. The functor GFA3AdderOutput(x, y, z) yielding an element of InnerVertices(GFA3AdderStr(x, y, z)) is defined by:

(Def. 48) GFA3AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor_{2}).

One can prove the following propositions:

(137) For all sets x, y, z holds InnerVertices(GFA3AdderStr(x, y, z)) = {⟨⟨⟨x, y⟩, xor_{2} ⟩⟩} ∪ {GFA3AdderOutput(x, y, z)}.

(138) For all sets x, y, z holds InnerVertices(GFA3AdderStr(x, y, z)) is a binary relation.

(139) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ holds InputVertices(GFA3AdderStr(x, y, z)) = {x, y, z}.

(140) For all non pair sets x, y, z holds InputVertices(GFA3AdderStr(x, y, z)) has no pairs.

(141) Let x, y, z be sets. Then (i) x ∈ the carrier of GFA3AdderStr(x, y, z), (ii) y ∈ the carrier of GFA3AdderStr(x, y, z), (iii) z ∈ the carrier of GFA3AdderStr(x, y, z), (iv) ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ the carrier of GFA3AdderStr(x, y, z), and

(v) ⟨⟨⟨⟨⟨⟨x, y⟩, xor_{2} ⟩⟩, z⟩, xor_{2} ⟩⟩ ∈ the carrier of GFA3AdderStr(x, y, z).

(142) For all sets x, y, z holds ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ InnerVertices(GFA3AdderStr(x, y, z)) and GFA3AdderOutput(x, y, z) ∈ InnerVertices(GFA3AdderStr(x, y, z)).

(143) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ holds x ∈ InputVertices(GFA3AdderStr(x, y, z)) and y ∈ InputVertices(GFA3AdderStr(x, y, z)) and z ∈ InputVertices(GFA3AdderStr(x, y, z)).

(144) For all non pair sets x, y, z holds InputVertices(GFA3AdderStr(x, y, z)) = {x, y, z}.

(145) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA3AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s))(⟨⟨⟨x, y⟩, xor_{2} ⟩⟩) = a_{1} ⊕ a_{2} and (Following(s))(x) = a_{1} and (Following(s))(y) = a_{2} and (Following(s))(z) = a_{3}.

(146) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA3AdderCirc(x, y, z) and a_{4}, a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{4} = s(⟨⟨⟨x, y⟩, xor_{2} ⟩⟩) and a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s))(GFA3AdderOutput(x, y, z)) = a_{4} ⊕ a_{3}.

(147) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA3AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA3AdderOutput(x, y, z)) = a_{1} ⊕ a_{2} ⊕ a_{3} and (Following(s, 2))(⟨⟨⟨x, y⟩, xor_{2} ⟩⟩) = a_{1} ⊕ a_{2} and (Following(s, 2))(x) = a_{1} and (Following(s, 2))(y) = a_{2} and (Following(s, 2))(z) = a_{3}.

(148) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩. Let s be a state of GFA3AdderCirc(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z), then (Following(s, 2))(GFA3AdderOutput(x, y, z)) = ¬(¬a_{1} ⊕ ¬a_{2} ⊕ ¬a_{3}).

(149) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and for every state s of

GFA3AdderCirc(x, y, z) holds Following(s, 2) is stable. Let x, y, z be sets. The functor BitGFA3Str(x, y, z) yielding an unsplit

non void strict non empty many sorted signature with arity held in gates and

Boolean denotation held in gates is defined by: (Def. 49) BitGFA3Str(x, y, z) = GFA3AdderStr(x, y, z)+· GFA3CarryStr(x, y, z).

Let x, y, z be sets. The functor BitGFA3Circ(x, y, z) yields a strict Boolean circuit of BitGFA3Str(x, y, z) with denotation held in gates and is defined as follows:

(Def. 50) BitGFA3Circ(x, y, z) = GFA3AdderCirc(x, y, z)+· GFA3CarryCirc(x, y, z).

One can prove the following propositions:

(150) For all sets x, y, z holds InnerVertices(BitGFA3Str(x, y, z)) = {⟨⟨⟨x, y⟩, xor_{2} ⟩⟩} ∪ {GFA3AdderOutput(x, y, z)} ∪ {⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩} ∪ {GFA3CarryOutput(x, y, z)}.

(151) For all sets x, y, z holds InnerVertices(BitGFA3Str(x, y, z)) is a binary relation.

(152) For all sets x, y, z such that z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ holds InputVertices(BitGFA3Str(x, y, z)) = {x, y, z}.

(153) For all non pair sets x, y, z holds InputVertices(BitGFA3Str(x, y, z)) = {x, y, z}.

(154) For all non pair sets x, y, z holds InputVertices(BitGFA3Str(x, y, z)) has no pairs.

(155) Let x, y, z be sets. Then x ∈ the carrier of BitGFA3Str(x, y, z) and y ∈ the carrier of BitGFA3Str(x, y, z) and z ∈ the carrier of BitGFA3Str(x, y, z) and ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ the carrier of BitGFA3Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, xor_{2} ⟩⟩, z⟩, xor_{2} ⟩⟩ ∈ the carrier of BitGFA3Str(x, y, z) and ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ ∈ the carrier of BitGFA3Str(x, y, z) and ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ ∈ the carrier of BitGFA3Str(x, y, z) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ the carrier of BitGFA3Str(x, y, z) and ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩ ∈ the carrier of BitGFA3Str(x, y, z).

(156) Let x, y, z be sets. Then ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ ∈ InnerVertices(BitGFA3Str(x, y, z)) and GFA3AdderOutput(x, y, z) ∈ InnerVertices(BitGFA3Str(x, y, z)) and ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩ ∈ InnerVertices(BitGFA3Str(x, y, z)) and ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ ∈ InnerVertices(BitGFA3Str(x, y, z)) and ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ ∈ InnerVertices(BitGFA3Str(x, y, z)) and GFA3CarryOutput(x, y, z) ∈ InnerVertices(BitGFA3Str(x, y, z)).

(157) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩. Then x ∈ InputVertices(BitGFA3Str(x, y, z)) and y ∈ InputVertices(BitGFA3Str(x, y, z)) and z ∈ InputVertices(BitGFA3Str(x, y, z)).

Let x, y, z be sets. The functor BitGFA3CarryOutput(x, y, z) yields an element of InnerVertices(BitGFA3Str(x, y, z)) and is defined by:

570 shin’nosuke yamaguchi et al.

(Def. 51) BitGFA3CarryOutput(x, y, z) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2b} ⟩⟩, ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩, ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩⟩, nor_{3} ⟩⟩.

Let x, y, z be sets. The functor BitGFA3AdderOutput(x, y, z) yielding an element of InnerVertices(BitGFA3Str(x, y, z)) is defined by:

(Def. 52) BitGFA3AdderOutput(x, y, z) = 2GatesCircOutput(x, y, z, xor_{2}). Next we state two propositions:

(158) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩. Let s be a state of BitGFA3Circ(x, y, z) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(z). Then (Following(s, 2))(GFA3AdderOutput(x, y, z)) = ¬(¬a_{1} ⊕ ¬a_{2} ⊕ ¬a_{3}) and (Following(s, 2))(GFA3CarryOutput(x, y, z)) = ¬(¬a_{1} ∧ ¬a_{2} ∨ ¬a_{2} ∧ ¬a_{3} ∨ ¬a_{3} ∧ ¬a_{1}).

(159) Let x, y, z be sets. Suppose z ≠ ⟨⟨⟨x, y⟩, xor_{2} ⟩⟩ and x ≠ ⟨⟨⟨y, z⟩, and_{2b} ⟩⟩ and y ≠ ⟨⟨⟨z, x⟩, and_{2b} ⟩⟩ and z ≠ ⟨⟨⟨x, y⟩, and_{2b} ⟩⟩. Let s be a state of BitGFA3Circ(x, y, z). Then Following(s, 2) is stable.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Grzegorz Bancerek and Yatsuka Nakamura. Full adder circuit. Part I. Formalized Mathematics, 5(3):367–380, 1996.

[4] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[9] Yatsuka Nakamura and Grzegorz Bancerek. Combining of circuits. Formalized Mathematics, 5(2):283–295, 1996.

[10] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Introduction to circuits, II. Formalized Mathematics, 5(2):273–278, 1996.

[11] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, II. Formalized Mathematics, 5(2):215–220, 1996.

[12] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[13] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[14] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[15] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[16] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15–22, 1993.

[17] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37–42, 1996.

[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[19] Katsumi Wasaki and Pauline N. Kawamoto. 2’s complement circuit. Formalized Mathematics, 6(2):189–197, 1997.

generalized full adder circuits (gfas)…. 571

[20] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 1990.

[21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received December 7, 2005 572 shin’nosuke yamaguchi et al.

Quotient Rings

Artur Korniłowicz Institute of Computer Science University of Białystok Sosnowa 64, 15-887 Białystok, Poland

Summary. The notions of prime ideals and maximal ideals of a ring are introduced. Quotient rings are defined. Characterisation of prime and maximal ideals using quotient rings are proved.

MML identifier: RING 1, version: 7.6.01 4.50.934

The articles [18], [10], [22], [17], [2], [19], [6], [23], [24], [7], [9], [8], [25], [15], [3], [4], [5], [14], [20], [16], [13], [21], [11], [12], and [1] provide the terminology and notation for this paper.

1. Preliminaries

Let S be a non empty 1-sorted structure. Note that Ω_{S} is non proper. The following propositions are true:

(1) Let L be an add-associative right zeroed right complementable non empty loop structure and a, b be elements of L. Then (a − b) + b = a.

(2) Let L be an add-associative right zeroed right complementable Abelian non empty loop structure and b, c be elements of L. Then c = b − (b − c).

(3) Let L be an add-associative right zeroed right complementable Abelian non empty loop structure and a, b, c be elements of L. Then a−b−(c−b) = a − c.

2. Ideals

Let K be a non empty groupoid and let S be a subset of K. We say that S is quasi-prime if and only if:

(Def. 1) For all elements a, b of K such that a · b ∈ S holds a ∈ S or b ∈ S.

Let K be a non empty multiplicative loop structure and let S be a subset of K. We say that S is prime if and only if:

(Def. 2) S is proper and quasi-prime.

Let R be a non empty double loop structure and let I be a subset of R. We say that I is quasi-maximal if and only if:

(Def. 3) For every ideal J of R such that I ⊆ J holds J = I or J is non proper.

Let R be a non empty double loop structure and let I be a subset of R. We say that I is maximal if and only if:

(Def. 4) I is proper and quasi-maximal.

Let K be a non empty multiplicative loop structure. Note that every subset of K which is prime is also proper and quasi-prime and every subset of K which is proper and quasi-prime is also prime.

Let R be a non empty double loop structure. One can verify that every subset of R which is maximal is also proper and quasi-maximal and every subset of R which is proper and quasi-maximal is also maximal.

Let R be a non empty loop structure. One can verify that Ω_{R} is add closed. Let R be a non empty groupoid. Observe that Ω_{R} is left ideal and right

ideal.

We now state the proposition

(4) For every integral domain R holds {0_{R}} is prime.

3. Equivalence Relation

In the sequel R denotes a ring, I denotes an ideal of R, and a, b denote elements of R.

Let R be a ring and let I be an ideal of R. The functor ≈_{I} yielding a binary relation on R is defined by:

(Def. 5) For all elements a, b of R holds ⟨⟨a, b⟩⟩ ∈ ≈_{I} iff a − b ∈ I.

Let R be a ring and let I be an ideal of R. One can verify that ≈_{I} is non empty, total, symmetric, and transitive.

We now state several propositions:

(5) a ∈ [b]_{≈_{I}} iff a − b ∈ I.

(6) [a]_{≈_{I}} = [b]_{≈_{I}} iff a − b ∈ I.

(7) [a]_{≈_{Ω_{R}}} = the carrier of R.

(8) ≈_{Ω_{R}} = {the carrier of R}.

(9) [a]_{≈_{{0_{R}}}} = {a}.

(10) ≈_{{0_{R}}} = rng(singleton_{the carrier of R}).

4. Quotient Ring

Let R be a ring and let I be an ideal of R. The functor R/I yields a strict double loop structure and is defined by the conditions (Def. 6).

(Def. 6)(i) The carrier of R/I = Classes(≈_{I}), (ii) the unity of R/I = [1_{R}]_{≈_{I}} , (iii) the zero of R/I = [0_{R}]_{≈_{I}} , (iv) for all elements x, y of R/I there exist elements a, b of R such that

x = [a]_{≈_{I}} and y = [b]_{≈_{I}} and (the addition of R/I)(x, y) = [a + b]_{≈_{I}} , and (v) for all elements x, y of R/I there exist elements a, b of R such that

x = [a]_{≈_{I}} and y = [b]_{≈_{I}} and (the multiplication of R/I)(x, y) = [a · b]_{≈_{I}} . Let R be a ring and let I be an ideal of R. Note that R/I is non empty. In the sequel x, y denote elements of R/I. We now state several propositions:

(11) There exists an element a of R such that x = [a]_{≈_{I}} .

(12) [a]_{≈_{I}} is an element of R/I.

(13) If x = [a]_{≈_{I}} and y = [b]_{≈_{I}} , then x + y = [a + b]_{≈_{I}} .

(14) If x = [a]_{≈_{I}} and y = [b]_{≈_{I}} , then x · y = [a · b]_{≈_{I}} .

(15) [1_{R}]_{≈_{I}} = 1_{R/I} .

Let R be a ring and let I be an ideal of R. Observe that R/I is Abelian, add-associative, and right zeroed.

Let R be a commutative ring and let I be an ideal of R. Note that R/I is commutative.

The following propositions are true:

(16) I is proper iff R/I is non degenerated.

(17) I is quasi-prime iff R/I is integral domain-like.

(18) For every commutative ring R and for every ideal I of R holds I is prime iff R/I is an integral domain.

(19) If R is commutative and I is quasi-maximal, then R/I is field-like.

(20) If R/I is field-like, then I is quasi-maximal.

(21) For every commutative ring R and for every ideal I of R holds I is maximal iff R/I is a skew field.

576 artur korniłowicz

Let R be a non degenerated commutative ring. One can check that every ideal of R which is maximal is also prime.

Let R be a non degenerated ring. Note that there exists an ideal of R which is maximal.

Let R be a non degenerated commutative ring and let I be a quasi-prime ideal of R. Observe that R/I is integral domain-like.

Let R be a non degenerated commutative ring and let I be a quasi-maximal ideal of R. Observe that R/I is field-like.

References

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[5] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[6] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[9] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[10] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[11] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[12] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[13] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

[14] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[15] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[16] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.

[17] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[20] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[21] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291– 296, 1990.

[22] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[23] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[24] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[25] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Completeness of the Real Euclidean Space

Noboru Endou Gifu National College of Technology Gifu, Japan Yasunari Shidama Shinshu University Nagano, Japan

MML identifier: REAL NS1, version: 7.6.01 4.50.934

The terminology and notation used here are introduced in the following articles: [21], [8], [24], [25], [6], [26], [7], [3], [14], [2], [5], [1], [20], [22], [4], [23], [15], [16], [13], [12], [11], [9], [18], [10], [19], and [17].

1. The Real Euclidean Space as a Real Linear Space

In this paper n is a natural number.

Let n be a natural number. The functor ⟨E^{n}, ∥ · ∥⟩ yields a strict non empty normed structure and is defined by the conditions (Def. 1).

(Def. 1)(i) The carrier of ⟨E^{n}, ∥ · ∥⟩ = R^{n},

(ii) the zero of ⟨E^{n}, ∥ · ∥⟩ = ⟨\underbrace{0, …, 0}_{n}⟩,

(iii) for all elements a, b of R^{n} holds (the addition of ⟨E^{n}, ∥·∥⟩)(a, b) = a+b, (iv) for every element r of ℝ and for every element x of R^{n} holds (the

external multiplication of ⟨E^{n}, ∥ · ∥⟩)(r, x) = r · x, and

(v) for every element x of R^{n} holds (the norm of ⟨E^{n}, ∥ · ∥⟩)(x) = |x|.

Let n be a natural number. Note that the addition of ⟨E^{n}, ∥ · ∥⟩ is commutative and associative.

Let n be a non empty natural number. Note that ⟨E^{n}, ∥ · ∥⟩ is non trivial. One can prove the following propositions:

(1) For every vector x of ⟨E^{n}, ∥ · ∥⟩ and for every element y of R^{n} such that x = y holds ∥x∥ = |y|.

(2) Let n be a natural number, x, y be vectors of ⟨E^{n}, ∥ · ∥⟩, and a, b be elements of R^{n}. If x = a and y = b, then x + y = a + b.

(3) For every vector x of ⟨E^{n}, ∥ · ∥⟩ and for every element y of R^{n} and for every real number a such that x = y holds a · x = a · y.

Let n be a natural number. Note that ⟨E^{n}, ∥ · ∥⟩ is real normed space-like, real linear space-like, Abelian, add-associative, right zeroed, and right complementable.

One can prove the following propositions:

(4) For every vector x of ⟨E^{n}, ∥ · ∥⟩ and for every element a of R^{n} such that x = a holds −x = −a.

(5) For all vectors x, y of ⟨E^{n}, ∥ · ∥⟩ and for all elements a, b of R^{n} such that x = a and y = b holds x − y = a − b.

(6) For every finite sequence f of elements of ℝ such that dom f = Seg n holds f is an element of R^{n}.

(7) Let n be a natural number and x be an element of R^{n}. Suppose that for every natural number i such that i ∈ Seg n holds 0 ≤ x(i). Then 0 ≤ ∑ x and for every natural number i such that i ∈ Seg n holds x(i) ≤ ∑ x.

(8) For every element x of R^{n} and for every natural number i such that i ∈ Seg n holds |x(i)| ≤ |x|.

(9) Let x be a point of ⟨E^{n}, ∥ · ∥⟩ and y be an element of R^{n}. If x = y, then for every natural number i such that i ∈ Seg n holds |y(i)| ≤ ∥x∥.

(10) For every element x of R^{n+1} holds |x|^{2} = |x↾n|^{2} + x(n + 1)^{2}. Let n be a natural number, let f be a function from ℕ into R^{n}, and let k

be a natural number. Then f(k) is an element of R^{n}.

We now state two propositions:

(11) Let n be a natural number, x be a point of ⟨E^{n}, ∥ · ∥⟩, x_{2} be an element of R^{n}, s_{1} be a sequence of ⟨E^{n}, ∥ · ∥⟩, and x_{1} be a function from ℕ into R^{n}. Suppose x_{2} = x and x_{1} = s_{1}. Then s_{1} is convergent and lim s_{1} = x if and only if for every natural number i such that i ∈ Seg n there exists a sequence r_{1} of real numbers such that for every natural number k holds r_{1}(k) = x_{1}(k)(i) and r_{1} is convergent and x_{2}(i) = lim r_{1}.

(12) For every sequence f of ⟨E^{n}, ∥ · ∥⟩ such that f is Cauchy sequence by

norm holds f is convergent. Let us consider n. Note that ⟨E^{n}, ∥ · ∥⟩ is complete.

2. The Real Euclidean Space as a Real Normed Space

Let n be a natural number. The functor ⟨E^{n}, (·|·)⟩ yields a strict non empty unitary space structure and is defined by the conditions (Def. 2).

(Def. 2)(i) The RLS structure of ⟨E^{n}, (·|·)⟩ = the RLS structure of ⟨E^{n}, ∥·∥⟩, and

(ii) for all elements x, y of R^{n} holds (the scalar product of ⟨E^{n}, (·|·)⟩)(x, y) = ∑(x • y).

Let n be a non empty natural number. One can verify that ⟨E^{n}, (·|·)⟩ is non trivial.

Let n be a natural number. Observe that ⟨E^{n}, (·|·)⟩ is real unitary space-like, real linear space-like, Abelian, add-associative, right zeroed, and right complementable.

The following propositions are true:

(13) Let n be a natural number, a be a real number, x_{3}, y_{1} be points of ⟨E^{n}, ∥ · ∥⟩, and x_{4}, y_{2} be points of ⟨E^{n}, (·|·)⟩. If x_{3} = x_{4} and y_{1} = y_{2}, then x_{3} + y_{1} = x_{4} + y_{2} and −x_{3} = −x_{4} and a · x_{3} = a · x_{4}.

(14) For every natural number n and for every point x_{3} of ⟨E^{n}, ∥ · ∥⟩ and for every point x_{4} of ⟨E^{n}, (·|·)⟩ such that x_{3} = x_{4} holds ∥x_{3}∥^{2} = (x_{4}|x_{4}).

(15) Let n be a natural number and f be a set. Then f is a sequence of ⟨E^{n}, ∥ · ∥⟩ if and only if f is a sequence of ⟨E^{n}, (·|·)⟩.

(16) Let n be a natural number, s_{2} be a sequence of ⟨E^{n}, ∥ · ∥⟩, and s_{3} be a

sequence of ⟨E^{n}, (·|·)⟩ such that s_{2} = s_{3}. Then

(i) if s_{2} is convergent, then s_{3} is convergent and lim s_{2} = lim s_{3}, and (ii) if s_{3} is convergent, then s_{2} is convergent and lim s_{2} = lim s_{3}.

(17) Let n be a natural number, s_{2} be a sequence of ⟨E^{n}, ∥ · ∥⟩, and s_{3} be a sequence of ⟨E^{n}, (·|·)⟩. If s_{2} = s_{3} and s_{2} is Cauchy sequence by norm, then s_{3} is Cauchy.

(18) Let n be a natural number, s_{2} be a sequence of ⟨E^{n}, ∥ · ∥⟩, and s_{3} be a sequence of ⟨E^{n}, (·|·)⟩. If s_{2} = s_{3} and s_{3} is Cauchy, then s_{2} is Cauchy sequence by norm.

Let us consider n. Note that ⟨E^{n}, (·|·)⟩ is Hilbert.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[8] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[9] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[10] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[11] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

580 noboru endou and yasunari shidama

[13] Library Committee of the Association of Mizar Users. Binary operations on numbers. To appear in Formalized Mathematics.

[14] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[15] Jan Popiołek. Introduction to Banach and Hilbert spaces – part I. Formalized Mathematics, 2(4):511–516, 1991.

[16] Jan Popiołek. Introduction to Banach and Hilbert spaces – part II. Formalized Mathematics, 2(4):517–521, 1991.

[17] Jan Popiołek. Introduction to Banach and Hilbert spaces – part III. Formalized Mathematics, 2(4):523–526, 1991.

[18] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[19] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.

[20] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291– 296, 1990.

[24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[26] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

completeness of the real euclidean space 581

.

Index of MML Identifiers

BINARI 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 CARD FIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 CIRCLED1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 COMPLSP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 EUCLIDLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 FDIFF 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 FDIFF 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 FDIFF 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 FILEREC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 FINTOPO5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 GFACIRC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 JGRAPH 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 JORDAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 JORDAN24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 MATRIX 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 MATRIXC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 MESFUNC4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 PROB 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 REAL NS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 RELSET 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 RINFSUP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 RING 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 SERIES 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 SERIES 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 SERIES 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 SETLIM 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 SIN COS7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 SIN COS8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 TAYLOR 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 TIETZE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 TOPGEN 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 TOPGEN 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Contents Formaliz. Math. 13 (4)

Set Sequences and Monotone Class

By Bo Zhang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

A Theory of Sequential Files

By Hirofumi Fukura and Yatsuka Nakamura . . . . . . . . . . . 443

Circled Sets, Circled Hull, and Circled Family

By Fahui Zhai and Jianbing Cao and Xiquan Liang . . . . . . 447

On the Borel Families of Subsets of Topological Spaces

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Linearity of Lebesgue Integral of Simple Valued Function

By Noboru Endou and Yasunari Shidama . . . . . . . . . . . . . . . 463

The Fashoda Meet Theorem for Continuous Mappings

By Yatsuka Nakamura et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Tietze Extension Theorem

By Artur Korniłowicz et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Homeomorphisms of Jordan Curves

By Adam Naumowicz and Grzegorz Bancerek . . . . . . . . . . 477

Jordan Curve Theorem

By Artur Korniłowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

The Inner Product and Conjugate of Matrix of Complex Numbers

By Wenpai Chang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Partial Sum and Partial Product of Some Series

By Jianbing Cao and Fahui Zhai and Xiquan Liang . . . . . . 501

Some Differentiable Formulas of Special Functions

By Jianbing Cao and Fahui Zhai and Xiquan Liang . . . . . . 505

Formulas and Identities of Hyperbolic Functions

By Pacharapokin Chanapat and Hiroshi Yamazaki . . . . . . 511

Continued on inside back cover

Niemytzki Plane - an Example of Tychonoff Space Which Is Not

T_{4}

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

On the Partial Product and Partial Sum of Series and Related

Basic Inequalities

By Fuguo Ge and Xiquan Liang . . . . . . . . . . . . . . . . . . . . . . . . 525

Several Differentiable Formulas of Special Functions. Part II

By Yan Zhang and Bo Li and Xiquan Liang . . . . . . . . . . . . . 529

On the Calculus of Binary Arithmetics. Part II

By Shunichi Kobayashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Some Properties of Some Special Matrices

By Xiaopeng Yue et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

Generalized Full Adder Circuits (GFAs). Part I

By Shin’nosuke Yamaguchi et al. . . . . . . . . . . . . . . . . . . . . . . . . 549

Quotient Rings

By Artur Korniłowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Completeness of the Real Euclidean Space

By Noboru Endou and Yasunari Shidama . . . . . . . . . . . . . . . 577

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582