FORMALIZED MATHEMATICS Volume 13, Number 1, 2005 University of Białystok

Partial Sum of Some Series

Ming Liang QingDao QiuShi College of Vocation and Technology

Yuzhong Ding QingDao University of Science and Technology

Summary. Solving the partial sum of some often used series.

MML Identifier: SERIES 2.

The articles [2], [1], [4], [3], [5], [7], and [6] provide the notation and terminology for this paper.

In this paper n is a natural number and s is a sequence of real numbers. Next we state a number of propositions:

(1) |(−1)^{n}| = 1.

(2) (n+1)^{3} = n^{3} +3·n^{2} +3·n+1 and (n+1)^{4} = n^{4} +4·n^{3} +6·n^{2} +4·n+1 and (n + 1)^{5} = n^{5} + 5 · n^{4} + 10 · n^{3} + 10 · n^{2} + 5 · n + 1.

(3) If for every n holds s(n) = n, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)}{2}.

(4) If for every n holds s(n) = 2 · n, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = n · (n + 1).

(5) If for every n holds s(n) = 2 · n + 1, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (n + 1)^{2}.

(6) If for every n holds s(n) = n · (n + 1), then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(n+2)}{3}.

(7) If for every n holds s(n) = n · (n + 1) · (n + 2), then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(n+2)·(n+3)}{4}.

(8) If for every n holds s(n) = n · (n + 1) · (n + 2) · (n + 3), then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(n+2)·(n+3)·(n+4)}{5}.

(9) If for every n holds s(n) = \frac{1}{n·(n+1)}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 1 − \frac{1}{n+1}.

2 ming liang and yuzhong ding

(10) If for every n holds s(n) = \frac{1}{n·(n+1)·(n+2)}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1}{4} − \frac{1}{2·(n+1)·(n+2)}.

(11) If for every n holds s(n) = \frac{1}{n·(n+1)·(n+2)·(n+3)}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1}{18} − \frac{1}{3·(n+1)·(n+2)·(n+3)}.

(12) If for every n holds s(n) = n^{2}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(2·n+1)}{6}.

(13) If for every n holds s(n) = (−1)^{n+1} · n^{2}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{(−1)^{n+1}·n·(n+1)}{2}.

(14) If for every n such that n ≥ 1 holds s(n) = (2 · n − 1)^{2} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(4·n^{2}−1)}{3}.

(15) If for every n holds s(n) = n^{3}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n^{2}·(n+1)^{2}}{4} .

(16) If for every n such that n ≥ 1 holds s(n) = (2·n−1)^{3} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = n^{2} · (2 · n^{2} − 1).

(17) If for every n holds s(n) = n^{4}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(2·n+1)·((3·n^{2}+3·n)−1)}{30}.

(18) If for every n holds s(n) = (−1)^{n+1} · n^{4}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{(−1)^{n+1}·n·(n+1)·((n^{2}+n)−1)}{2}.

(19) If for every n holds s(n) = n^{5}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n^{2}·(n+1)^{2}·((2·n^{2}+2·n)−1)}{12}.

(20) If for every n holds s(n) = n^{6}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(2·n+1)·(((3·n^{4}+6·n^{3})−3·n)+1)}{42}.

(21) If for every n holds s(n) = n^{7}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n^{2}·(n+1)^{2}·(((3·n^{4}+6·n^{3})−n^{2}−4·n)+2)}{24}.

(22) If for every n holds s(n) = n · (n + 1)^{2}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(n+2)·(3·n+5)}{12}.

(23) If for every n holds s(n) = n · (n + 1)^{2} · (n + 2), then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{n·(n+1)·(n+2)·(n+3)·(2·n+3)}{10}.

(24) If for every n holds s(n) = n · (n + 1) · 2^{n}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 2^{n+1} · ((n^{2} − n) + 2) − 4.

(25) Suppose that for every n such that n ≥ 2 holds s(n) = \frac{1}{(n−1)·(n+1)} and s(0) = 0 and s(1) = 0. Let given n. If n ≥ 2, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{3}{4} − \frac{1}{2·n} − \frac{1}{2·(n+1)}.

(26) If for every n such that n ≥ 1 holds s(n) = \frac{1}{(2·n−1)·(2·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}{2·n+1}.

(27) If for every n such that n ≥ 1 holds s(n) = \frac{1}{(3·n−2)·(3·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}{3·n+1}.

partial sum of some series 3

(28) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{1}{(2·n−1)·(2·n+1)·(2·n+3)} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1}{12} − \frac{1}{4·(2·n+1)·(2·n+3)}.

(29) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{1}{(3·n−2)·(3·n+1)·(3·n+4)} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1}{24} − \frac{1}{6·(3·n+1)·(3·n+4)}.

(30) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{2·n−1}{n·(n+1)·(n+2)} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (\frac{3}{4} − \frac{2}{n+2}) + \frac{1}{2·(n+1)·(n+2)}.

(31) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{n+2}{n·(n+1)·(n+3)} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{29}{36} − \frac{1}{n+3} − \frac{3}{2·(n+2)·(n+3)} − \frac{4}{3·(n+1)·(n+2)·(n+3)}.

(32) If for every n holds s(n) = \frac{(n+1)·2^{n}}{(n+2)·(n+3)}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{2^{n+1}}{n+3} − \frac{1}{2}.

(33) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{n^{2}·4^{n}}{(n+1)·(n+2)} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{2}{3} + \frac{(n−1)·4^{n+1}}{3·(n+2)} .

(34) If for every n such that n ≥ 1 holds s(n) = \frac{n+2}{n·(n+1)·2^{n}} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 1 − \frac{1}{(n+1)·2^{n}} .

(35) Suppose that for every n such that n ≥ 1 holds s(n) = \frac{2·n+3}{n·(n+1)·3^{n}} and s(0) = 0. Let given n. If n ≥ 1, then (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 1 − \frac{1}{(n+1)·3^{n}} .

(36) If for every n holds s(n) = \frac{(−1)^{n}·2^{n+1}}{(2^{n+1}+(−1)^{n+1})·(2^{n+2}+(−1)^{n+2})}, then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1}{3} + \frac{(−1)^{n+2}}{3·(2^{n+2}+(−1)^{n+2})}.

(37) If for every n holds s(n) = n! · n, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = (n + 1)! − 1.

(38) If for every n holds s(n) = \frac{n}{(n+1)!}, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 1 − \frac{1}{(n+1)!}.

(39) If for every n such that n ≥ 1 holds s(n) = \frac{(n^{2}+n)−1}{(n+2)!} and s(0) = 0, then for every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = \frac{1}{2} − \frac{n+1}{(n+2)!}.

(40) If for every n such that n ≥ 1 holds s(n) = \frac{n·2^{n}}{(n+2)!} and s(0) = 0, then for

every n such that n ≥ 1 holds (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n) = 1 − \frac{2^{n+1}}{(n+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.

4 ming liang and yuzhong ding

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

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

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

Substitution in First-Order Formulas: Elementary Properties^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349-360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article introduces the basic concepts of substitution of a variable for a variable in a first-order formula. The contents of this article correspond to Chapter III par. 8, Definition 8.1, 8.2 of Ebbinghaus, Flum, Thomas.

MML Identifier: SUBSTUT1.

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

1. Preliminaries

For simplicity, we follow the rules: a, b are sets, i, k are natural numbers, x, y are bound variables, P is a k-ary predicate symbol, l_{1} is a variables list of k, l_{2} is a finite sequence of elements of Var, and p is a formula.

The functor vSUB is defined by:

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

(Def. 1) vSUB = BoundVar ˙→ BoundVar .

One can check that vSUB is non empty. A CQC-substitution is an element of vSUB. Let us note that vSUB is functional. In the sequel S_{1} is a CQC-substitution. Let us consider S_{1}. The functor^{@}S_{1} yielding a partial function from BoundVar

to BoundVar is defined as follows: (Def. 2) ^{@}S_{1} = S_{1}.

Next we state the proposition

(1) If a ∈ dom S_{1}, then S_{1}(a) ∈ BoundVar . Let l be a finite sequence of elements of Var and let us consider S_{1}. The func-

tor CQC-subst(l, S_{1}) yields a finite sequence of elements of Var and is defined as follows:

(Def. 3) len CQC-subst(l, S_{1}) = len l and for every k such that 1 ≤ k and k ≤ len l holds if l(k) ∈ dom S_{1}, then (CQC-subst(l, S_{1}))(k) = S_{1}(l(k)) and if l(k) ∉ dom S_{1}, then (CQC-subst(l, S_{1}))(k) = l(k).

Let l be a finite sequence of elements of BoundVar. The functor ^{@}l yielding a finite sequence of elements of Var is defined by:

(Def. 4) ^{@}l = l.

Let l be a finite sequence of elements of BoundVar and let us consider S_{1}. The functor CQC-subst(l, S_{1}) yields a finite sequence of elements of BoundVar and is defined as follows:

(Def. 5) CQC-subst(l, S_{1}) = CQC-subst(^{@}l, S_{1}).

Let us consider S_{1} and let X be a set. Then S_{1}↾X is a CQC-substitution. One can verify that there exists a CQC-substitution which is finite. Let us consider x, p, S_{1}. The functor RestrictSub(x, p, S_{1}) yielding a finite

CQC-substitution is defined by: (Def. 6) RestrictSub(x, p, S_{1}) = S_{1}↾{y : y ∈ snb(p) ∧ y is an element of dom S_{1} ∧

y ≠ x ∧ y ≠ S_{1}(y)}. Let us consider l_{2}. The functor BoundVars(l_{2}) yielding an element of 2^{BoundVar}

is defined as follows: (Def. 7) BoundVars(l_{2}) = {l_{2}(k) : 1 ≤ k ∧ k ≤ len l_{2} ∧ l_{2}(k) ∈ BoundVar}.

Let us consider p. The functor BoundVars(p) yielding an element of 2^{BoundVar} is defined by the condition (Def. 8).

(Def. 8) There exists a function F from WFF into 2^{BoundVar} such that (i) BoundVars(p) = F (p), and (ii) for every element p of WFF and for all elements d_{1}, d_{2} of 2^{BoundVar}

holds if p = VERUM, then F (p) = ∅_{BoundVar} and if p is atomic, then F (p) = BoundVars(Args(p)) and if p is negative and d_{1} = F (Arg(p)), then F (p) = d_{1} and if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = d_{1} ∪ d_{2} and if p is universal and d_{1} = F (Scope(p)), then F (p) = d_{1} ∪ {Bound(p)}.

One can prove the following propositions:

(2) BoundVars(VERUM) = ∅.

(3) For every formula p such that p is atomic holds BoundVars(p) = BoundVars(Args(p)).

(4) For every formula p such that p is negative holds BoundVars(p) = BoundVars(Arg(p)).

(5) For every formula p such that p is conjunctive holds BoundVars(p) = BoundVars(LeftArg(p)) ∪ BoundVars(RightArg(p)).

(6) For every formula p such that p is universal holds BoundVars(p) = BoundVars(Scope(p)) ∪ {Bound(p)}.

Let us consider p. One can check that BoundVars(p) is finite. Let us consider p. The functor DomBoundVars(p) yielding a finite subset of

ℕ is defined as follows: (Def. 9) DomBoundVars(p) = {i : x_{i} ∈ BoundVars(p)}.

In the sequel f_{1} denotes a finite CQC-substitution. Let us consider f_{1}. The functor Sub-Var(f_{1}) yields a finite subset of ℕ and

is defined as follows: (Def. 10) Sub-Var(f_{1}) = {i : x_{i} ∈ rng f_{1}}.

Let us consider p, f_{1}. The functor NSub(p, f_{1}) yields a non empty subset of ℕ and is defined as follows:

(Def. 11) NSub(p, f_{1}) = ℕ \ (DomBoundVars(p) ∪ Sub-Var(f_{1})).

Let us consider f_{1}, p. The functor upVar(f_{1}, p) yielding a natural number is defined as follows:

(Def. 12) upVar(f_{1}, p) = min NSub(p, f_{1}).

Let us consider x, p, f_{1}. Let us assume that there exists S_{1} such that f_{1} = RestrictSub(x, ∀_{x}p, S_{1}). The functor ExpandSub(x, p, f_{1}) yielding a CQCsubstitution is defined by:

(Def. 13) ExpandSub(x, p, f_{1}) = \{ \begin{matrix}f_{1} ∪ \{⟨⟨x, x_{upVar(f_{1},p)}⟩⟩\}, if x ∈ rng f_{1}, \\ f_{1} ∪ \{⟨⟨x, x⟩⟩\}, otherwise. \end{matrix}

Let us consider p, S_{1}, b. The predicate b = PQSub(p, S_{1}) is defined as follows:

(Def. 14) If p is universal, then b = ExpandSub(Bound(p), Scope(p),

RestrictSub(Bound(p), p, S_{1})) and if p is not universal, then b = ∅. The function QSub is defined as follows:

(Def. 15) a ∈ QSub iff there exist p, S_{1}, b such that a = ⟨⟨⟨⟨p, S_{1}⟩⟩, b⟩⟩ and b = PQSub(p, S_{1}).

2. Definition and Properties of the Formula – Substitution – Construction

In the sequel e denotes an element of vSUB. We now state the proposition (7)(i) [: WFF, vSUB :] is a subset of [: [: ℕ, ℕ:]^{∗}, vSUB :], (ii) for every natural number k and for every k-ary predicate symbol p and

for every list of variables l_{1} of the length k and for every element e of vSUB holds ⟨⟨⟨p⟩^{⌢} l_{1}, e⟩⟩ ∈ [: WFF, vSUB :],

(iii) for every element e of vSUB holds ⟨⟨⟨⟨⟨0, 0⟩⟩⟩, e⟩⟩ ∈ [: WFF, vSUB :], (iv) for every finite sequence p of elements of [: ℕ, ℕ:] and for every element e

of vSUB such that ⟨⟨p, e⟩⟩ ∈ [: WFF, vSUB :] holds ⟨⟨⟨⟨⟨1, 0⟩⟩⟩^{⌢}p, e⟩⟩ ∈ [: WFF, vSUB :],

(v) for all finite sequences p, q of elements of [: ℕ, ℕ:] and for every element e of vSUB such that ⟨⟨p, e⟩⟩ ∈ [: WFF, vSUB :] and ⟨⟨q, e⟩⟩ ∈ [: WFF, vSUB :] holds ⟨⟨⟨⟨⟨2, 0⟩⟩⟩^{⌢} p^{⌢} q, e⟩⟩ ∈ [: WFF, vSUB :], and

(vi) for every bound variable x and for every finite sequence p of elements of [: ℕ, ℕ:] and for every element e of vSUB such that ⟨⟨p, QSub(⟨⟨⟨⟨⟨3, 0⟩⟩⟩^{⌢} ⟨x⟩^{⌢}p, e⟩⟩)⟩⟩ ∈ [: WFF, vSUB :] holds ⟨⟨⟨⟨⟨3, 0⟩⟩⟩^{⌢}⟨x⟩^{⌢}p, e⟩⟩ ∈ [: WFF, vSUB :].

Let I_{1} be a set. We say that I_{1} is QC-Sub-closed if and only if the conditions (Def. 16) are satisfied.

(Def. 16)(i) I_{1} is a subset of [: [: ℕ, ℕ:]^{∗}, vSUB :],

(ii) for every natural number k and for every k-ary predicate symbol p and for every list of variables l_{1} of the length k and for every element e of vSUB holds ⟨⟨⟨p⟩^{⌢} l_{1}, e⟩⟩ ∈ I_{1},

(iii) for every element e of vSUB holds ⟨⟨⟨⟨⟨0, 0⟩⟩⟩, e⟩⟩ ∈ I_{1}, (iv) for every finite sequence p of elements of [: ℕ, ℕ:] and for every element

e of vSUB such that ⟨⟨p, e⟩⟩ ∈ I_{1} holds ⟨⟨⟨⟨⟨1, 0⟩⟩⟩^{⌢} p, e⟩⟩ ∈ I_{1},

(v) for all finite sequences p, q of elements of [: ℕ, ℕ:] and for every element e of vSUB such that ⟨⟨p, e⟩⟩ ∈ I_{1} and ⟨⟨q, e⟩⟩ ∈ I_{1} holds ⟨⟨⟨⟨⟨2, 0⟩⟩⟩ ^{⌢} p ^{⌢} q, e⟩⟩ ∈ I_{1}, and

(vi) for every bound variable x and for every finite sequence p of elements of [: ℕ, ℕ:] and for every element e of vSUB such that ⟨⟨p, QSub(⟨⟨⟨⟨⟨3, 0⟩⟩⟩^{⌢} ⟨x⟩^{⌢} p, e⟩⟩)⟩⟩ ∈ I_{1} holds ⟨⟨⟨⟨⟨3, 0⟩⟩⟩^{⌢} ⟨x⟩^{⌢} p, e⟩⟩ ∈ I_{1}.

Let us mention that there exists a set which is QC-Sub-closed and non empty. The non empty set QC-Sub-WFF is defined as follows:

(Def. 17) QC-Sub-WFF is QC-Sub-closed and for every non empty set D such

that D is QC-Sub-closed holds QC-Sub-WFF ⊆ D. In the sequel S, S′, S_{2}, S_{3}, S′_{1}, S′_{2} are elements of QC-Sub-WFF. Next we state the proposition

(8) There exist p, e such that S = ⟨⟨p, e⟩⟩.

Let us note that QC-Sub-WFF is QC-Sub-closed. Let P be a predicate symbol, let l be a finite sequence of elements of Var, and

let us consider e. Let us assume that Arity(P) = len l. The functor SubP(P, l, e) yields an element of QC-Sub-WFF and is defined as follows:

(Def. 18) SubP(P, l, e) = ⟨⟨P [l], e⟩⟩.

We now state the proposition

(9) Let k be a natural number, P be a k-ary predicate symbol, and l_{1} be a list of variables of the length k. Then SubP(P, l_{1}, e) = ⟨⟨P [l_{1}], e⟩⟩.

Let us consider S. We say that S is sub-verum if and only if: (Def. 19) There exists e such that S = ⟨⟨ VERUM, e⟩⟩.

Let us consider S. Then S_{1} is an element of WFF. Then S_{2} is an element of vSUB.

The following proposition is true

(10) S = ⟨⟨S_{1}, S_{2}⟩⟩. Let us consider S. The functor SubNot(S) yields an element of QC-Sub-WFF

and is defined as follows: (Def. 20) SubNot(S) = ⟨⟨¬(S_{1}), S_{2}⟩⟩.

Let us consider S, S′. Let us assume that S_{2} = S′_{2}. The functor SubAnd(S, S′) yields an element of QC-Sub-WFF and is defined by:

(Def. 21) SubAnd(S, S′) = ⟨⟨S_{1} ∧ S′_{1}, S_{2}⟩⟩.

In the sequel B denotes an element of [: QC-Sub-WFF, BoundVar :]. Let us consider B. Then B_{1} is an element of QC-Sub-WFF. Then B_{2} is an

element of BoundVar.

Let us consider B. We say that B is quantifiable if and only if: (Def. 22) There exists e such that (B_{1})_{2} = QSub(⟨⟨∀_{B_{2}}((B_{1})_{1}), e⟩⟩).

Let us consider B. Let us assume that B is quantifiable. An element of vSUB is called a second q.-component of B if:

(Def. 23) (B_{1})_{2} = QSub(⟨⟨∀_{B_{2}}((B_{1})_{1}), it⟩⟩).

In the sequel S_{4} is a second q.-component of B. Let us consider B, S_{4}. Let us assume that B is quantifiable. The functor

SubAll(B, S_{4}) yields an element of QC-Sub-WFF and is defined by: (Def. 24) SubAll(B, S_{4}) = ⟨⟨∀_{B_{2}}((B_{1})_{1}), S_{4}⟩⟩.

Let us consider S, x. Then ⟨⟨S, x⟩⟩ is an element of [: QC-Sub-WFF, BoundVar :]. The scheme SubQCInd concerns a unary predicate P, and states that:

For every element S of QC-Sub-WFF holds P[S] provided the following conditions are satisfied:

• Let k be a natural number, P be a k-ary predicate symbol, l_{1} be a list of variables of the length k, and e be an element of vSUB. Then P[SubP(P, l_{1}, e)], • For every element S of QC-Sub-WFF such that S is sub-verum

holds P[S],

• For every element S of QC-Sub-WFF such that P[S] holds P[SubNot(S)],

• For all elements S, S′ of QC-Sub-WFF such that S_{2} = S′_{2} and P[S] and P[S′] holds P[SubAnd(S, S′)], and

• Let x be a bound variable, S be an element of QC-Sub-WFF, and S_{4} be a second q.-component of ⟨⟨S, x⟩⟩. If ⟨⟨S, x⟩⟩ is quantifiable and P[S], then P[SubAll(⟨⟨S, x⟩⟩, S_{4})].

Let us consider S. We say that S is sub-atomic if and only if the condition (Def. 25) is satisfied.

(Def. 25) There exists a natural number k and there exists a k-ary predicate symbol P and there exists a list of variables l_{1} of the length k and there exists an element e of vSUB such that S = SubP(P, l_{1}, e).

One can prove the following proposition

(11) If S is sub-atomic, then S_{1} is atomic.

Let k be a natural number, let P be a k-ary predicate symbol, let l_{1} be a list of variables of the length k, and let e be an element of vSUB. One can verify that SubP(P, l_{1}, e) is sub-atomic.

Let us consider S. We say that S is sub-negative if and only if: (Def. 26) There exists S′ such that S = SubNot(S′).

We say that S is sub-conjunctive if and only if: (Def. 27) There exist S_{2}, S_{3} such that S = SubAnd(S_{2}, S_{3}) and (S_{2})_{2} = (S_{3})_{2}.

Let A be a set. We say that A is sub-universal if and only if: (Def. 28) There exist B, S_{4} such that A = SubAll(B, S_{4}) and B is quantifiable.

Next we state the proposition

(12) Every S is either sub-verum, sub-atomic, sub-negative, sub-conjunctive,

or sub-universal. Let us consider S. Let us assume that S is sub-atomic. The functor

SubArguments(S) yields a finite sequence of elements of Var and is defined by the condition (Def. 29).

(Def. 29) There exists a natural number k and there exists a k-ary predicate symbol P and there exists a list of variables l_{1} of the length k and there exists an element e of vSUB such that SubArguments(S) = l_{1} and S = SubP(P, l_{1}, e).

Let us consider S. Let us assume that S is sub-negative. The functor SubArgument(S) yields an element of QC-Sub-WFF and is defined as follows:

(Def. 30) S = SubNot(SubArgument(S)).

Let us consider S. Let us assume that S is sub-conjunctive. The functor SubLeftArgument(S) yields an element of QC-Sub-WFF and is defined by:

(Def. 31) There exists S′ such that S = SubAnd(SubLeftArgument(S), S′) and

(SubLeftArgument(S))_{2} = S′_{2}. Let us consider S. Let us assume that S is sub-conjunctive. The func-

tor SubRightArgument(S) yielding an element of QC-Sub-WFF is defined as follows:

(Def. 32) There exists S′ such that S = SubAnd(S′, SubRightArgument(S)) and

S′_{2} = (SubRightArgument(S))_{2}. Let A be a set. Let us assume that A is sub-universal. The functor

SubBound(A) yields a bound variable and is defined as follows: (Def. 33) There exist B, S_{4} such that A = SubAll(B, S_{4}) and B_{2} = SubBound(A)

and B is quantifiable. Let A be a set. Let us assume that A is sub-universal. The functor

SubScope(A) yielding an element of QC-Sub-WFF is defined as follows: (Def. 34) There exist B, S_{4} such that A = SubAll(B, S_{4}) and B_{1} = SubScope(A)

and B is quantifiable. Let us consider S. One can verify that SubNot(S) is sub-negative. The following propositions are true:

(13) If (S_{2})_{2} = (S_{3})_{2}, then SubAnd(S_{2}, S_{3}) is sub-conjunctive.

(14) If B is quantifiable, then SubAll(B, S_{4}) is sub-universal.

(15) If SubNot(S) = SubNot(S′), then S = S′.

(16) SubArgument(SubNot(S)) = S.

(17) If (S_{2})_{2} = (S_{3})_{2} and (S′_{1})_{2} = (S′_{2})_{2} and SubAnd(S_{2}, S_{3}) = SubAnd(S′_{1}, S′_{2}), then S_{2} = S′_{1} and S_{3} = S′_{2}.

(18) If (S_{2})_{2} = (S_{3})_{2}, then SubLeftArgument(SubAnd(S_{2}, S_{3})) = S_{2}.

(19) If (S_{2})_{2} = (S_{3})_{2}, then SubRightArgument(SubAnd(S_{2}, S_{3})) = S_{3}.

(20) Let B_{1}, B_{2} be elements of [: QC-Sub-WFF, BoundVar :], S_{5} be a second q.-component of B_{1}, and S_{6} be a second q.-component of B_{2}. If B_{1} is quantifiable and B_{2} is quantifiable and SubAll(B_{1}, S_{5}) = SubAll(B_{2}, S_{6}), then B_{1} = B_{2}.

(21) If B is quantifiable, then SubScope(SubAll(B, S_{4})) = B_{1}.

The scheme SubQCInd2 concerns a unary predicate P, and states that:

For every element S of QC-Sub-WFF holds P[S] provided the following requirement is met:

• Let S be an element of QC-Sub-WFF. Then (i) if S is sub-atomic, then P[S], (ii) if S is sub-verum, then P[S], (iii) if S is sub-negative and P[SubArgument(S)], then P[S], (iv) if S is sub-conjunctive and P[SubLeftArgument(S)] and P[SubRightArgument(S)], then P[S], and

(v) if S is sub-universal and P[SubScope(S)], then P[S]. One can prove the following propositions:

(22) If S is sub-negative, then len(^{@}((SubArgument(S))_{1})) < len(^{@}(S_{1})).

(23) If S is sub-conjunctive, then len(^{@}((SubLeftArgument(S))_{1})) < len(^{@}(S_{1})) and len(^{@}((SubRightArgument(S))_{1})) < len(^{@}(S_{1})).

(24) If S is sub-universal, then len(^{@}((SubScope(S))_{1})) < len(^{@}(S_{1})). (25)(i) If S is sub-verum, then (^{@}(S_{1}))(1)_{1} = 0, (ii) if S is sub-atomic, then there exists a natural number k such that

(^{@}(S_{1}))(1) is a k-ary predicate symbol, (iii) if S is sub-negative, then (^{@}(S_{1}))(1)_{1} = 1, (iv) if S is sub-conjunctive, then (^{@}(S_{1}))(1)_{1} = 2, and

(v) if S is sub-universal, then (^{@}(S_{1}))(1)_{1} = 3.

(26) If S is sub-atomic, then (^{@}(S_{1}))(1)_{1} ≠ 0 and (^{@}(S_{1}))(1)_{1} ≠ 1 and (^{@}(S_{1}))(1)_{1} ≠ 2 and (^{@}(S_{1}))(1)_{1} ≠ 3.

(27) There exists no S which satisfies any of the following conditions: (i) it is sub-atomic and sub-negative, (ii) it is sub-atomic and sub-conjunctive, (iii) it is sub-atomic and sub-universal, (iv) it is sub-negative and sub-conjunctive,

(v) it is sub-negative and sub-universal, (vi) it is sub-conjunctive and sub-universal, (vii) it is sub-verum and sub-atomic, (viii) it is sub-verum and sub-negative, (ix) it is sub-verum and sub-conjunctive,

(x) it is sub-verum and sub-universal.

Now we present two schemes. The scheme SubFuncEx deals with a non empty set A, an element B of A, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A, and states that:

There exists a function F from QC-Sub-WFF into A such that for every element S of QC-Sub-WFF and for all elements d_{1}, d_{2} of A holds

(i) if S is sub-verum, then F (S) = B, (ii) if S is sub-atomic, then F (S) = F(S), (iii) if S is sub-negative and d_{1} = F (SubArgument(S)), then F (S) = G(d_{1}), (iv) if S is sub-conjunctive and d_{1} = F (SubLeftArgument(S)) and d_{2} = F (SubRightArgument(S)), then F (S) = H(d_{1}, d_{2}), and

(v) if S is sub-universal and d_{1} = F (SubScope(S)), then F (S) = I(S, d_{1})

for all values of the parameters.

The scheme SubQCFuncUniq deals with a non empty set A, a function B from QC-Sub-WFF into A, a function C from QC-Sub-WFF into A, an element D of A, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A, and states that:

B = C provided the parameters satisfy the following conditions:

• Let S be an element of QC-Sub-WFF and d_{1}, d_{2} be elements of A. Then

(i) if S is sub-verum, then B(S) = D, (ii) if S is sub-atomic, then B(S) = F(S), (iii) if S is sub-negative and d_{1} = B(SubArgument(S)), then B(S) = G(d_{1}), (iv) if S is sub-conjunctive and d_{1} = B(SubLeftArgument(S)) and d_{2} = B(SubRightArgument(S)), then B(S) = H(d_{1}, d_{2}), and

(v) if S is sub-universal and d_{1} = B(SubScope(S)), then B(S) = I(S, d_{1}), and

• Let S be an element of QC-Sub-WFF and d_{1}, d_{2} be elements of A. Then

(i) if S is sub-verum, then C(S) = D, (ii) if S is sub-atomic, then C(S) = F(S), (iii) if S is sub-negative and d_{1} = C(SubArgument(S)), then C(S) = G(d_{1}), (iv) if S is sub-conjunctive and d_{1} = C(SubLeftArgument(S)) and d_{2} = C(SubRightArgument(S)), then C(S) = H(d_{1}, d_{2}), and

(v) if S is sub-universal and d_{1} = C(SubScope(S)), then C(S) = I(S, d_{1}).

Let us consider S. The functor ^{@}S yielding an element of [: WFF, vSUB :] is defined as follows:

(Def. 35) ^{@}S = S.

In the sequel Z denotes an element of [: WFF, vSUB :].

Let us consider Z. Then Z_{1} is an element of WFF. Then Z_{2} is a CQCsubstitution.

Let us consider Z. The functor S-Bound(Z) yields a bound variable and is defined by:

(Def. 36) S-Bound(Z) = \{ \begin{matrix}x_{upVar(RestrictSub(Bound(Z_{1}),Z_{1},Z_{2}),Scope(Z_{1}))}, \\ if Bound(Z_{1}) ∈ rng RestrictSub(Bound(Z_{1}), Z_{1}, Z_{2}), \\ Bound(Z_{1}), otherwise. \end{matrix}

Let us consider S, p. The functor Quant(S, p) yielding an element of WFF is defined by:

(Def. 37) Quant(S, p) = ∀_{S-Bound(^{@}S)}p.

3. Definition and Properties of Substitution

Let S be an element of QC-Sub-WFF. The functor CQCSub(S) yielding an element of WFF is defined by the condition (Def. 38).

(Def. 38) There exists a function F from QC-Sub-WFF into WFF such that (i) CQCSub(S) = F (S), and (ii) for every element S′ of QC-Sub-WFF holds if S′ is sub-

verum, then F (S′) = VERUM and if S′ is sub-atomic, then F (S′) = PredSym(S′_{1})[CQC-subst(SubArguments(S′), S′_{2})] and if S′ is sub-negative, then F (S′) = ¬F (SubArgument(S′)) and if S′ is sub-conjunctive, then F (S′) = F (SubLeftArgument(S′)) ∧ F (SubRightArgument(S′)) and if S′ is sub-universal, then F (S′) = Quant(S′, F (SubScope(S′))).

We now state several propositions:

(28) If S is sub-negative, then CQCSub(S) = ¬ CQCSub(SubArgument(S)).

(29) CQCSub(SubNot(S)) = ¬ CQCSub(S).

(30) If S is sub-conjunctive, then CQCSub(S) = CQCSub(SubLeftArgument(S)) ∧ CQCSub(SubRightArgument(S)).

(31) If (S_{2})_{2} = (S_{3})_{2}, then CQCSub(SubAnd(S_{2}, S_{3})) = CQCSub(S_{2}) ∧ CQCSub(S_{3}).

(32) If S is sub-universal, then CQCSub(S) =

Quant(S, CQCSub(SubScope(S))). The subset CQC-Sub-WFF of QC-Sub-WFF is defined by:

(Def. 39) CQC-Sub-WFF = {S : S_{1} is an element of CQC-WFF}. Let us observe that CQC-Sub-WFF is non empty. Next we state several propositions:

(33) If S is sub-verum, then CQCSub(S) is an element of CQC-WFF.

(34) Let h be a finite sequence. Then h is a variables list of k if and only if h is a finite sequence of elements of BoundVar and len h = k.

(35) CQCSub(SubP(P, l_{1}, e)) is an element of CQC-WFF.

(36) If CQCSub(S) is an element of CQC-WFF, then CQCSub(SubNot(S)) is an element of CQC-WFF.

(37) If (S_{2})_{2} = (S_{3})_{2} and CQCSub(S_{2}) is an element of CQC-WFF and CQCSub(S_{3}) is an element of CQC-WFF, then CQCSub(SubAnd(S_{2}, S_{3})) is an element of CQC-WFF.

In the sequel x_{1} denotes a second q.-component of ⟨⟨S, x⟩⟩. We now state the proposition

(38) If CQCSub(S) is an element of CQC-WFF and ⟨⟨S, x⟩⟩ is quantifiable, then CQCSub(SubAll(⟨⟨S, x⟩⟩, x_{1})) is an element of CQC-WFF.

substitution in first-order formulas: … 15

In the sequel S is an element of CQC-Sub-WFF. The scheme SubCQCInd concerns a unary predicate P, and states that:

For every S holds P[S] provided the following requirement is met:

• Let S, S′ be elements of CQC-Sub-WFF, x be a bound variable, S_{4} be a second q.-component of ⟨⟨S, x⟩⟩, k be a natural number, l_{1} be a variables list of k, P be a k-ary predicate symbol, and e be an element of vSUB. Then

(i) P[SubP(P, l_{1}, e)], (ii) if S is sub-verum, then P[S], (iii) if P[S], then P[SubNot(S)], (iv) if S_{2} = S′_{2} and P[S] and P[S′], then P[SubAnd(S, S′)], and (v) if ⟨⟨S, x⟩⟩ is quantifiable and P[S], then P[SubAll(⟨⟨S, x⟩⟩, S_{4})].

Let us consider S. Then CQCSub(S) is an element of CQC-WFF. 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. A classical first order language. Formalized Mathematics, 1(4):669–676, 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. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[8] Czesław Byliński and Grzegorz Bancerek. Variables in formulae of the first order language. Formalized Mathematics, 1(3):459–469, 1990.

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

[10] Agata Darmochwał and Andrzej Trybulec. Similarity of formulae. Formalized Mathematics, 2(5):635–642, 1991.

[11] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

[13] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[14] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

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

16 patrick braselmann and peter koepke

Coincidence Lemma and Substitution Lemma^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349–360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article establishes further concepts of substitution of a variable for a variable in a first-order formula. The main result is the substitution lemma. The contents of this article correspond to Chapter III par. 5, 5.1 Coincidence Lemma and Chapter III par. 8, 8.3 Substitution Lemma of Ebbinghaus, Flum, Thomas.

MML Identifier: SUBLEMMA.

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

1. Preliminaries

For simplicity, we adopt the following rules: a, b are sets, i, k are natural numbers, p, q are elements of CQC-WFF, x, y are bound variables, A is a non empty set, J is an interpretation of A, v, w are elements of V(A), P, P′ are

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

k-ary predicate symbols, l_{1}, l′_{1} are variables lists of k, l_{2} is a finite sequence of elements of Var, S_{1}, S′_{1} are CQC-substitutions, and S, S_{2}, S_{3} are elements of CQC-Sub-WFF.

Next we state two propositions:

(1) For all functions f, g, h, h_{1}, h_{2} such that dom h_{1} ⊆ dom h and dom h_{2} ⊆ dom h holds f+·g+·h = f+·h_{1}+·(g+·h_{2})+·h.

(2) For every function v_{1} such that x ∈ dom v_{1} holds v_{1}↾(dom v_{1} \ {x})+·(x⟼ . v_{1}(x)) = v_{1}.

Let us consider A. A value substitution of A is a partial function from BoundVar to A.

In the sequel v_{2}, v_{1}, v_{3} are value substitutions of A. Let us consider A, v, v_{2}. The functor v(v_{2}) yields an element of V(A) and

is defined by: (Def. 1) v(v_{2}) = v+·v_{2}.

Let us consider S. Then S_{1} is an element of CQC-WFF. Let us consider S, A, v. The functor ValS(v, S) yielding a value substitution

of A is defined by: (Def. 2) ValS(v, S) = (^{@}(S_{2})) · v.

The following proposition is true

(3) If S is sub-verum, then CQCSub(S) = VERUM . Let us consider S, A, v, J. The predicate J, v ⊧ S is defined as follows:

(Def. 3) J, v ⊧ S_{1}.

The following propositions are true:

(4) If S is sub-verum, then for every v holds J, v ⊧ CQCSub(S) iff J, v(ValS(v, S)) ⊧ S.

(5) If i ∈ dom l_{1}, then l_{1}(i) is a bound variable.

(6) If S is sub-atomic, then CQCSub(S) = PredSym(S_{1})[CQC-Subst(SubArguments(S), S_{2})].

(7) If SubArguments(SubP(P, l_{1}, S_{1})) = SubArguments(SubP(P′, l′_{1}, S′_{1})), then l_{1} = l′_{1}.

(8) SubArguments(SubP(P, l_{1}, S_{1})) = l_{1}. Let us consider k, P, l_{1}, S_{1}. Then SubP(P, l_{1}, S_{1}) is an element of CQC-Sub-WFF. We now state three propositions:

(9) CQCSub(SubP(P, l_{1}, S_{1})) = P [CQC-Subst(l_{1}, S_{1})].

(10) P [CQC-Subst(l_{1}, S_{1})] is an element of CQC-WFF.

(11) CQC-Subst(l_{1}, S_{1}) is a variables list of k.

Let us consider k, l_{1}, S_{1}. Then CQC-Subst(l_{1}, S_{1}) is a variables list of k. One can prove the following propositions:

(12) If x ∉ dom(S_{2}), then v(ValS(v, S))(x) = v(x).

(13) If x ∈ dom(S_{2}), then v(ValS(v, S))(x) = (ValS(v, S))(x).

(14) v(ValS(v, SubP(P, l_{1}, S_{1}))) ∗ l_{1} = v ∗ CQC-Subst(l_{1}, S_{1}).

(15) (SubP(P, l_{1}, S_{1}))_{1} = P [l_{1}].

(16) For every v holds J, v ⊧ CQCSub(SubP(P, l_{1}, S_{1})) iff J, v(ValS(v, SubP(P, l_{1}, S_{1}))) ⊧ SubP(P, l_{1}, S_{1}).

(17) (SubNot(S))_{1} = ¬(S_{1}) and (SubNot(S))_{2} = S_{2}.

Let us consider S. Then SubNot(S) is an element of CQC-Sub-WFF.

We now state three propositions:

(18) J, v(ValS(v, S)) |̸= S iff J, v(ValS(v, S)) ⊧ SubNot(S).

(19) ValS(v, S) = ValS(v, SubNot(S)).

(20) If for every v holds J, v ⊧ CQCSub(S) iff J, v(ValS(v, S)) ⊧ S, then for every v holds J, v ⊧ CQCSub(SubNot(S)) iff J, v(ValS(v, SubNot(S))) ⊧ SubNot(S).

Let us consider S_{2}, S_{3}. Let us assume that (S_{2})_{2} = (S_{3})_{2}. The functor CQCSubAnd(S_{2}, S_{3}) yielding an element of CQC-Sub-WFF is defined as follows:

(Def. 4) CQCSubAnd(S_{2}, S_{3}) = SubAnd(S_{2}, S_{3}).

Next we state several propositions:

(21) If (S_{2})_{2} = (S_{3})_{2}, then (CQCSubAnd(S_{2}, S_{3}))_{1} = (S_{2})_{1} ∧ (S_{3})_{1} and (CQCSubAnd(S_{2}, S_{3}))_{2} = (S_{2})_{2}.

(22) If (S_{2})_{2} = (S_{3})_{2}, then (CQCSubAnd(S_{2}, S_{3}))_{2} = (S_{2})_{2}.

(23) If (S_{2})_{2} = (S_{3})_{2}, then ValS(v, S_{2}) = ValS(v, CQCSubAnd(S_{2}, S_{3})) and ValS(v, S_{3}) = ValS(v, CQCSubAnd(S_{2}, S_{3})).

(24) If (S_{2})_{2} = (S_{3})_{2}, then CQCSub(CQCSubAnd(S_{2}, S_{3})) = CQCSub(S_{2})∧ CQCSub(S_{3}).

(25) If (S_{2})_{2} = (S_{3})_{2}, then J, v(ValS(v, S_{2})) ⊧ S_{2} and J, v(ValS(v, S_{3})) ⊧ S_{3} iff J, v(ValS(v, CQCSubAnd(S_{2}, S_{3}))) ⊧ CQCSubAnd(S_{2}, S_{3}).

(26) Suppose (S_{2})_{2} = (S_{3})_{2} and for every v holds J, v ⊧ CQCSub(S_{2}) iff J, v(ValS(v, S_{2})) ⊧ S_{2} and for every v holds J, v ⊧ CQCSub(S_{3}) iff J, v(ValS(v, S_{3})) ⊧ S_{3}. Let given v. Then J, v ⊧ CQCSub(CQCSubAnd(S_{2}, S_{3})) if and only if J, v(ValS(v, CQCSubAnd(S_{2}, S_{3}))) ⊧ CQCSubAnd(S_{2}, S_{3}).

In the sequel B is an element of [: QC-Sub-WFF, BoundVar :] and S_{4} is a second q.-component of B.

The following proposition is true

(27) If B is quantifiable, then (SubAll(B, S_{4}))_{1} = ∀_{B_{2}}((B_{1})_{1}) and

(SubAll(B, S_{4}))_{2} = S_{4}. Let B be an element of [: QC-Sub-WFF, BoundVar :]. We say that B is

CQC-WFF-like if and only if:

(Def. 5) B_{1} ∈ CQC-Sub-WFF .

Let us observe that there exists an element of [: QC-Sub-WFF, BoundVar :] which is CQC-WFF-like.

Let us consider S, x. Then ⟨⟨S, x⟩⟩ is a CQC-WFF-like element of [: QC-Sub-WFF, BoundVar :]. In the sequel B denotes a CQC-WFF-like element of [: QC-Sub-WFF, BoundVar :], x_{1} denotes a second q.-component of ⟨⟨S, x⟩⟩,

and S_{4} denotes a second q.-component of B.

Let us consider B. Then B_{1} is an element of CQC-Sub-WFF. Let us consider B, S_{4}. Let us assume that B is quantifiable. The func-

tor CQCSubAll(B, S_{4}) yields an element of CQC-Sub-WFF and is defined as follows:

(Def. 6) CQCSubAll(B, S_{4}) = SubAll(B, S_{4}).

We now state the proposition

(28) If B is quantifiable, then CQCSubAll(B, S_{4}) is sub-universal. Let us consider S. Let us assume that S is sub-universal. The functor

CQCSubScope(S) yielding an element of CQC-Sub-WFF is defined as follows: (Def. 7) CQCSubScope(S) = SubScope(S).

Let us consider S_{2}, p. Let us assume that S_{2} is sub-universal and p = CQCSub(CQCSubScope(S_{2})). The functor CQCQuant(S_{2}, p) yielding an element of CQC-WFF is defined as follows:

(Def. 8) CQCQuant(S_{2}, p) = Quant(S_{2}, p).

The following two propositions are true:

(29) If S is sub-universal, then CQCSub(S) = CQCQuant(S, CQCSub(CQCSubScope(S))).

(30) If B is quantifiable, then CQCSubScope(CQCSubAll(B, S_{4})) = B_{1}.

2. The Substitution Lemma

The following propositions are true:

(31) If ⟨⟨S, x⟩⟩ is quantifiable, then CQCSubScope(CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) = S and CQCQuant(CQCSubAll(⟨⟨S, x⟩⟩, x_{1}), CQCSub(CQCSubScope (CQCSubAll(⟨⟨S, x⟩⟩, x_{1})))) = CQCQuant(CQCSubAll(⟨⟨S, x⟩⟩, x_{1}), CQCSub(S)).

(32) If ⟨⟨S, x⟩⟩ is quantifiable, then CQCQuant(CQCSubAll(⟨⟨S, x⟩⟩, x_{1}), CQCSub(S)) = ∀_{S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩,x_{1}))} CQCSub(S).

(33) If x ∈ dom(S_{2}), then v((^{@}(S_{2}))(x)) = v(ValS(v, S))(x).

(34) If x ∈ dom(^{@}(S_{2})), then (^{@}(S_{2}))(x) is a bound variable.

(35) [: WFF, vSUB :] ⊆ dom QSub .

In the sequel B_{1} denotes an element of [: QC-Sub-WFF, BoundVar :] and S_{5} denotes a second q.-component of B_{1}.

We now state a number of propositions:

(36) If B is quantifiable and B_{1} is quantifiable and SubAll(B, S_{4}) = SubAll(B_{1}, S_{5}), then B_{2} = (B_{1})_{2} and S_{4} = S_{5}.

(37) If B is quantifiable and B_{1} is quantifiable and CQCSubAll(B, S_{4}) = SubAll(B_{1}, S_{5}), then B_{2} = (B_{1})_{2} and S_{4} = S_{5}.

(38) If ⟨⟨S, x⟩⟩ is quantifiable, then SubBound(CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) = x.

(39) If ⟨⟨S, x⟩⟩ is quantifiable and x ∈ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}), then S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) ∉ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}) and S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) ∉ BoundVars(S_{1}).

(40) If ⟨⟨S, x⟩⟩ is quantifiable and x ∉ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}), then S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) ∉ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}).

(41) If ⟨⟨S, x⟩⟩ is quantifiable, then S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) ∉ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}).

(42) If ⟨⟨S, x⟩⟩ is quantifiable, then S_{2} = ExpandSub(x, S_{1}, RestrictSub(x, ∀_{x}(S_{1}), x_{1})).

(43) snb(VERUM) ⊆ BoundVars(VERUM).

(44) snb(P [l_{1}]) ⊆ BoundVars(P [l_{1}]).

(45) If snb(p) ⊆ BoundVars(p), then snb(¬p) ⊆ BoundVars(¬p).

(46) If snb(p) ⊆ BoundVars(p) and snb(q) ⊆ BoundVars(q), then snb(p∧q) ⊆ BoundVars(p ∧ q).

(47) If snb(p) ⊆ BoundVars(p), then snb(∀_{x}p) ⊆ BoundVars(∀_{x}p).

(48) For every p holds snb(p) ⊆ BoundVars(p).

Let us consider A, let a be an element of A, and let us consider x. The functor x↾a yields a value substitution of A and is defined as follows:

(Def. 9) x↾a = x⟼ . a. In the sequel a denotes an element of A. The following propositions are true:

(49) If x ≠ b, then v(x↾a)(b) = v(b).

(50) If x = y, then v(x↾a)(y) = a.

(51) J, v ⊧ ∀_{x}p iff for every a holds J, v(x↾a) ⊧ p.

Let us consider S, x, x_{1}, A, v. The functor NExVal(v, S, x, x_{1}) yielding a value substitution of A is defined as follows:

(Def. 10) NExVal(v, S, x, x_{1}) = (^{@}RestrictSub(x, ∀_{x}(S_{1}), x_{1})) · v.

Let us consider A and let v, w be value substitutions of A. Then v+·w is a value substitution of A.

One can prove the following propositions:

(52) If ⟨⟨S, x⟩⟩ is quantifiable and x ∈ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}), then S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) = x_{upVar(RestrictSub(x,∀_{x}(S_{1}),x_{1}),S_{1})}.

(53) If ⟨⟨S, x⟩⟩ is quantifiable and x ∉ rng RestrictSub(x, ∀_{x}(S_{1}), x_{1}), then S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) = x.

(54) If ⟨⟨S, x⟩⟩ is quantifiable, then for every a holds ValS(v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a), S) = NExVal(v(S-Bound (^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a), S, x, x_{1})+·x↾a and dom RestrictSub(x, ∀_{x}(S_{1}), x_{1}) misses {x}.

(55) Suppose ⟨⟨S, x⟩⟩ is quantifiable. Then for every a holds J, v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a)(ValS(v(S-Bound(^{@}CQCSubAll (⟨⟨S, x⟩⟩, x_{1}))↾a), S)) ⊧ S if and only if for every a holds J, v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a)(NExVal(v(S-Bound (^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a), S, x, x_{1})+·x↾a) ⊧ S.

(56) If ⟨⟨S, x⟩⟩ is quantifiable, then for every a holds NExVal(v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a), S, x, x_{1}) = NExVal(v, S, x, x_{1}).

(57) Suppose ⟨⟨S, x⟩⟩ is quantifiable. Then for every a holds J, v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a)(NExVal(v(S-Bound (^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a), S, x, x_{1})+·x↾a) ⊧ S if and only if for every a holds J, v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a)(NExVal(v, S, x, x_{1}) +·x↾a) ⊧ S.

3. The Coincidence Lemma

The following propositions are true:

(58) If rng l_{2} ⊆ BoundVar, then snb(l_{2}) = rng l_{2}.

(59) dom v = BoundVar and dom(x↾a) = {x}.

(60) v ∗ l_{1} = l_{1} · (v↾ snb(l_{1})).

(61) For all v, w such that v↾ snb(P [l_{1}]) = w↾ snb(P [l_{1}]) holds J, v ⊧ P [l_{1}] iff J, w ⊧ P [l_{1}].

(62) Suppose that for all v, w such that v↾ snb(p) = w↾ snb(p) holds J, v ⊧ p iff J, w ⊧ p. Let given v, w. If v↾ snb(¬p) = w↾ snb(¬p), then J, v ⊧ ¬p iff J, w ⊧ ¬p.

(63) Suppose that

(i) for all v, w such that v↾ snb(p) = w↾ snb(p) holds J, v ⊧ p iff J, w ⊧ p, and

(ii) for all v, w such that v↾ snb(q) = w↾ snb(q) holds J, v ⊧ q iff J, w ⊧ q. Let given v, w. If v↾ snb(p ∧ q) = w↾ snb(p ∧ q), then J, v ⊧ p ∧ q iff J, w ⊧ p ∧ q.

(64) For every set X such that X ⊆ BoundVar holds dom(v↾X) = dom(v(x↾a)↾X) and dom(v↾X) = X.

(65) If v↾ snb(p) = w↾ snb(p), then v(x↾a)↾ snb(p) = w(x↾a)↾ snb(p).

(66) snb(p) ⊆ snb(∀_{x}p) ∪ {x}.

(67) If v↾(snb(p) \ {x}) = w↾(snb(p) \ {x}), then v(x↾a)↾ snb(p) = w(x↾a)↾ snb(p).

(68) Suppose that for all v, w such that v↾ snb(p) = w↾ snb(p) holds J, v ⊧ p iff J, w ⊧ p. Let given v, w. If v↾ snb(∀_{x}p) = w↾ snb(∀_{x}p), then J, v ⊧ ∀_{x}p iff J, w ⊧ ∀_{x}p.

(69) For all v, w such that v↾ snb(VERUM) = w↾ snb(VERUM) holds J, v ⊧ VERUM iff J, w ⊧ VERUM .

(70) For every p and for all v, w such that v↾ snb(p) = w↾ snb(p) holds J, v ⊧ p iff J, w ⊧ p.

(71) If ⟨⟨S, x⟩⟩ is quantifiable, then v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a) (NExVal(v, S, x, x_{1})+·x↾a)↾ snb(S_{1}) = v(NExVal(v, S, x, x_{1})+·x↾a)↾ snb(S_{1}).

(72) If ⟨⟨S, x⟩⟩ is quantifiable, then for every a holds J, v(S-Bound(^{@}CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))↾a)(NExVal(v, S, x, x_{1})+·x↾a) ⊧ S iff for every a holds J, v(NExVal(v, S, x, x_{1})+·x↾a) ⊧ S.

(73) dom NExVal(v, S, x, x_{1}) = dom RestrictSub(x, ∀_{x}(S_{1}), x_{1}).

(74) If ⟨⟨S, x⟩⟩ is quantifiable, then v(NExVal(v, S, x, x_{1})+·x↾a) = v(NExVal(v, S, x, x_{1}))(x↾a).

(75) If ⟨⟨S, x⟩⟩ is quantifiable, then for every a holds J, v(NExVal(v, S, x, x_{1})+·x↾a) ⊧ S iff for every a holds J, v(NExVal(v, S, x, x_{1}))(x↾a) ⊧ S.

(76) For every a holds J, v(NExVal(v, S, x, x_{1}))(x↾a) ⊧ S iff for every a holds J, v(NExVal(v, S, x, x_{1}))(x↾a) ⊧ S_{1}.

(77) Let given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(VERUM) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then J, v(v_{2}) ⊧ VERUM if and only if J, v(v_{2}+·v_{1}+·v_{3}) ⊧ VERUM .

(78) Let given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(l_{1}) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then v(v_{2}) ∗ l_{1} = v(v_{2}+·v_{1}+·v_{3}) ∗ l_{1}.

(79) Let given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(P [l_{1}]) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then J, v(v_{2}) ⊧ P [l_{1}] if and only if J, v(v_{2}+·v_{1}+·v_{3}) ⊧ P [l_{1}].

(80) Suppose that for all v, v_{2}, v_{1}, v_{3} such that for every y such that y ∈ dom v_{1} holds y ∉ snb(p) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3} holds J, v(v_{2}) ⊧ p iff J, v(v_{2}+·v_{1}+·v_{3}) ⊧ p. Let given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(¬p) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then J, v(v_{2}) ⊧ ¬p if and only if J, v(v_{2}+·v_{1}+·v_{3}) ⊧ ¬p.

(81) Suppose that

(i) for all v, v_{2}, v_{1}, v_{3} such that for every y such that y ∈ dom v_{1} holds y ∉ snb(p) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3} holds J, v(v_{2}) ⊧ p iff J, v(v_{2}+·v_{1}+·v_{3}) ⊧ p, and

(ii) for all v, v_{2}, v_{1}, v_{3} such that for every y such that y ∈ dom v_{1} holds y ∉ snb(q) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3} holds J, v(v_{2}) ⊧ q iff J, v(v_{2}+·v_{1}+·v_{3}) ⊧ q. Let given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(p ∧ q) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then J, v(v_{2}) ⊧ p ∧ q if and only if J, v(v_{2}+·v_{1}+·v_{3}) ⊧ p ∧ q.

(82) If for every y such that y ∈ dom v_{1} holds y ∉ snb(∀_{x}p), then for every y such that y ∈ dom v_{1} \ {x} holds y ∉ snb(p).

(83) Let v_{1} be a function. Suppose for every y such that y ∈ dom v_{1} holds v_{1}(y) = v(y) and dom v_{2} misses dom v_{1}. Let given y. If y ∈ dom v_{1} \ {x}, then (v_{1}↾(dom v_{1} \ {x}))(y) = v(v_{2})(y).

(84) Suppose that for all v, v_{2}, v_{1}, v_{3} such that for every y such that y ∈ dom v_{1} holds y ∉ snb(p) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3} holds J, v(v_{2}) ⊧ p iff J, v(v_{2}+·v_{1}+·v_{3}) ⊧ p. Let given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(∀_{x}p) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then J, v(v_{2}) ⊧ ∀_{x}p if and only if J, v(v_{2}+·v_{1}+·v_{3}) ⊧ ∀_{x}p.

(85) Let given p and given v, v_{2}, v_{1}, v_{3}. Suppose for every y such that y ∈ dom v_{1} holds y ∉ snb(p) and for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y) and dom v_{2} misses dom v_{3}. Then J, v(v_{2}) ⊧ p if and only if J, v(v_{2}+·v_{1}+·v_{3}) ⊧ p.

Let us consider p. The functor RSub1 p yields a set and is defined by: (Def. 11) b ∈ RSub1 p iff there exists x such that x = b and x ∉ snb(p).

Let us consider p, S_{1}. The functor RSub2(p, S_{1}) yielding a set is defined as follows:

(Def. 12) b ∈ RSub2(p, S_{1}) iff there exists x such that x = b and x ∈ snb(p) and

x = (^{@}S_{1})(x). Next we state several propositions:

(86) dom((^{@}S_{1})↾ RSub1 p) misses dom((^{@}S_{1})↾ RSub2(p, S_{1})).

(87) ^{@}RestrictSub(x, ∀_{x}p, S_{1}) = (^{@}S_{1}) \ ((^{@}S_{1})↾ RSub1 ∀_{x}p+·(^{@}S_{1})↾ RSub2(∀_{x}p, S_{1})).

(88) dom(^{@}RestrictSub(x, p, S_{1})) misses dom((^{@}S_{1})↾ RSub1 p) ∪ dom((^{@}S_{1})↾ RSub2(p, S_{1})).

(89) If ⟨⟨S, x⟩⟩ is quantifiable, then ^{@}((CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))_{2}) = (^{@}RestrictSub(x, ∀_{x}(S_{1}), x_{1}))+·(^{@}x_{1})↾ RSub1 ∀_{x}(S_{1})+·(^{@}x_{1})↾ RSub2 (∀_{x}(S_{1}), x_{1}).

(90) Suppose ⟨⟨S, x⟩⟩ is quantifiable. Then there exist v_{1}, v_{3} such that (i) for every y such that y ∈ dom v_{1} holds y ∉ snb(∀_{x}(S_{1})), (ii) for every y such that y ∈ dom v_{3} holds v_{3}(y) = v(y), (iii) dom NExVal(v, S, x, x_{1}) misses dom v_{3}, and (iv) v(ValS(v, CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))) = v(NExVal(v, S, x, x_{1})+·v_{1}+·v_{3}).

(91) If ⟨⟨S, x⟩⟩ is quantifiable, then for every v holds J, v(NExVal(v, S, x, x_{1})) ⊧ ∀_{x}(S_{1}) iff J, v(ValS(v, CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))) ⊧ CQCSubAll(⟨⟨S, x⟩⟩, x_{1}).

(92) Suppose ⟨⟨S, x⟩⟩ is quantifiable and for every v holds J, v ⊧ CQCSub(S) iff J, v(ValS(v, S)) ⊧ S. Let given v. Then J, v ⊧ CQCSub(CQCSubAll(⟨⟨S, x⟩⟩, x_{1})) if and only if J, v(ValS(v, CQCSubAll(⟨⟨S, x⟩⟩, x_{1}))) ⊧ CQCSubAll(⟨⟨S, x⟩⟩, x_{1}).

The scheme SubCQCInd1 concerns a unary predicate P, and states that:

For every S holds P[S] provided the following condition is met:

• Let S, S′ be elements of CQC-Sub-WFF, x be a bound variable, S_{4} be a second q.-component of ⟨⟨S, x⟩⟩, k be a natural number, l_{1} be a variables list of k, P be a k-ary predicate symbol, and e be an element of vSUB. Then

(i) P[SubP(P, l_{1}, e)], (ii) if S is sub-verum, then P[S], (iii) if P[S], then P[SubNot(S)], (iv) if S_{2} = S′_{2} and P[S] and P[S′], then P[CQCSubAnd(S, S′)], and

(v) if ⟨⟨S, x⟩⟩ is quantifiable and P[S], then P[CQCSubAll(⟨⟨S, x⟩⟩, S_{4})].

Next we state the proposition

(93) For all S, v holds J, v ⊧ CQCSub(S) iff J, v(ValS(v, S)) ⊧ S.

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[16] Edmund Woronowicz. Interpretation and satisfiability in the first order logic. Formalized Mathematics, 1(4):739–743, 1990.

[17] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 1990.

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

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

Substitution in First-Order Formulas. Part II.

The Construction of First-Order Formulas^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349-360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article establishes that every substitution can be applied to every formula as in Chapter III par. 8, Definition 8.1, 8.2 of Ebbinghaus, Flum, Thomas. After that, it is observed that substitution doesn’t change the number of quantifiers of a formula. Then further details about substitution and some results about the construction of formulas are proven.

MML Identifier: SUBSTUT2.

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

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

1. Further Properties of Substitution

For simplicity, we adopt the following convention: i, k, n denote natural numbers, p, q, r, s denote elements of CQC-WFF, x, y denote bound variables, P denotes a k-ary predicate symbol, l, l_{1} denote variables lists of k, S_{1} denotes a CQC-substitution, and S, S_{2} denote elements of CQC-Sub-WFF.

Next we state several propositions:

(1) For every S_{1} there exists S such that S_{1} = VERUM and S_{2} = S_{1}.

(2) For every S_{1} there exists S such that S_{1} = P [l_{1}] and S_{2} = S_{1}.

(3) Let k, l be natural numbers. Suppose P is a k-ary predicate symbol and a l-ary predicate symbol. Then k = l.

(4) If for every S_{1} there exists S such that S_{1} = p and S_{2} = S_{1}, then for every S_{1} there exists S such that S_{1} = ¬p and S_{2} = S_{1}.

(5) Suppose for every S_{1} there exists S such that S_{1} = p and S_{2} = S_{1} and for every S_{1} there exists S such that S_{1} = q and S_{2} = S_{1}. Let given S_{1}. Then there exists S such that S_{1} = p ∧ q and S_{2} = S_{1}.

Let us consider p, S_{1}. Then ⟨⟨p, S_{1}⟩⟩ is an element of [: WFF, vSUB :]. We now state several propositions:

(6) dom RestrictSub(x, ∀_{x}p, S_{1}) misses {x}.

(7) If x ∈ rng RestrictSub(x, ∀_{x}p, S_{1}), then S-Bound(⟨⟨∀_{x}p, S_{1}⟩⟩) = x_{upVar(RestrictSub(x,∀_{x}p,S_{1}),p)}.

(8) If x ∉ rng RestrictSub(x, ∀_{x}p, S_{1}), then S-Bound(⟨⟨∀_{x}p, S_{1}⟩⟩) = x.

(9) ExpandSub(x, p, RestrictSub(x, ∀_{x}p, S_{1})) = (^{@}RestrictSub(x, ∀_{x}p, S_{1}))+·x↾ S-Bound(⟨⟨∀_{x}p, S_{1}⟩⟩).

(10) If S_{2} = (^{@}RestrictSub(x, ∀_{x}p, S_{1}))+·x↾ S-Bound(⟨⟨∀_{x}p, S_{1}⟩⟩) and S_{1} = p, then ⟨⟨S, x⟩⟩ is quantifiable and there exists S_{2} such that S_{2} = ⟨⟨∀_{x}p, S_{1}⟩⟩.

(11) If for every S_{1} there exists S such that S_{1} = p and S_{2} = S_{1}, then for every S_{1} there exists S such that S_{1} = ∀_{x}p and S_{2} = S_{1}.

(12) For all p, S_{1} there exists S such that S_{1} = p and S_{2} = S_{1}.

Let us consider p, S_{1}. Then ⟨⟨p, S_{1}⟩⟩ is an element of CQC-Sub-WFF. Let us consider x, y. The functor Sbst(x, y) yielding a CQC-substitution is

defined by: (Def. 1) Sbst(x, y) = x⟼ . y.

2. Facts about Substitution and Quantifiers of a Formula

Let us consider p, x, y. The functor p(x, y) yields an element of CQC-WFF and is defined as follows:

(Def. 2) p(x, y) = CQCSub(⟨⟨p, Sbst(x, y)⟩⟩).

In this article we present several logical schemes. The scheme CQCInd1 concerns a unary predicate P, and states that:

For every p holds P[p] provided the parameters meet the following conditions:

• For every p such that the number of quantifiers in p = 0 holds P[p], and

• Let given k. Suppose that for every p such that the number of quantifiers in p = k holds P[p]. Let given p. If the number of quantifiers in p = k + 1, then P[p].

The scheme CQCInd2 concerns a unary predicate P, and states that:

For every p holds P[p] provided the following conditions are met:

• For every p such that the number of quantifiers in p ≤ 0 holds P[p], and

• Let given k. Suppose that for every p such that the number of quantifiers in p ≤ k holds P[p]. Let given p. If the number of quantifiers in p ≤ k + 1, then P[p].

We now state three propositions:

(13) VERUM(x, y) = VERUM .

(14) P [l](x, y) = P [CQC-Subst(l, Sbst(x, y))] and the number of quantifiers in P [l] = the number of quantifiers in P [l](x, y).

(15) The number of quantifiers in P [l] = the number of quantifiers in

CQCSub(⟨⟨P [l], S_{1}⟩⟩). Let S be an element of QC-Sub-WFF. Then S_{2} is a CQC-substitution. Next we state several propositions:

(16) ⟨⟨¬p, S_{1}⟩⟩ = SubNot(⟨⟨p, S_{1}⟩⟩). (17)(i) (¬p)(x, y) = ¬p(x, y), and (ii) if the number of quantifiers in p = the number of quantifiers in p(x,

y), then the number of quantifiers in ¬p = the number of quantifiers in (¬p)(x, y).

(18) Suppose that for every S_{1} holds the number of quantifiers in p = the number of quantifiers in CQCSub(⟨⟨p, S_{1}⟩⟩). Let given S_{1}. Then the number of quantifiers in ¬p = the number of quantifiers in CQCSub(⟨⟨¬p, S_{1}⟩⟩).

(19) ⟨⟨p ∧ q, S_{1}⟩⟩ = CQCSubAnd(⟨⟨p, S_{1}⟩⟩, ⟨⟨q, S_{1}⟩⟩). (20)(i) (p ∧ q)(x, y) = p(x, y) ∧ q(x, y), and (ii) if the number of quantifiers in p = the number of quantifiers in p(x,

y) and the number of quantifiers in q = the number of quantifiers in q(x, y), then the number of quantifiers in p ∧ q = the number of quantifiers in (p ∧ q)(x, y).

(21) Suppose that (i) for every S_{1} holds the number of quantifiers in p = the number of

quantifiers in CQCSub(⟨⟨p, S_{1}⟩⟩), and (ii) for every S_{1} holds the number of quantifiers in q = the number of

quantifiers in CQCSub(⟨⟨q, S_{1}⟩⟩). Let given S_{1}. Then the number of quantifiers in p ∧ q = the number of quantifiers in CQCSub(⟨⟨p ∧ q, S_{1}⟩⟩).

The function CFQ from CQC-Sub-WFF into vSUB is defined as follows: (Def. 3) CFQ = QSub ↾ CQC-Sub-WFF .

Let us consider p, x, S_{1}. The functor QScope(p, x, S_{1}) yielding a CQC-WFF-like element of [: QC-Sub-WFF, BoundVar :] is defined by:

(Def. 4) QScope(p, x, S_{1}) = ⟨⟨⟨⟨p, CFQ(⟨⟨∀_{x}p, S_{1}⟩⟩)⟩⟩, x⟩⟩.

Let us consider p, x, S_{1}. The functor Qsc(p, x, S_{1}) yielding a second q.- component of QScope(p, x, S_{1}) is defined by:

(Def. 5) Qsc(p, x, S_{1}) = S_{1}.

The following propositions are true:

(22) ⟨⟨∀_{x}p, S_{1}⟩⟩ = CQCSubAll(QScope(p, x, S_{1}), Qsc(p, x, S_{1})) and QScope(p, x, S_{1}) is quantifiable.

(23) Suppose that for every S_{1} holds the number of quantifiers in p = the number of quantifiers in CQCSub(⟨⟨p, S_{1}⟩⟩). Let given S_{1}. Then the number of quantifiers in ∀_{x}p = the number of quantifiers in CQCSub(⟨⟨∀_{x}p, S_{1}⟩⟩).

(24) The number of quantifiers in VERUM = the number of quantifiers in CQCSub(⟨⟨ VERUM, S_{1}⟩⟩).

(25) For all p, S_{1} holds the number of quantifiers in p = the number of quantifiers in CQCSub(⟨⟨p, S_{1}⟩⟩).

(26) If p is atomic, then there exist k, P, l_{1} such that p = P [l_{1}].

The scheme CQCInd3 concerns a unary predicate P, and states that: For every p such that the number of quantifiers in p = 0 holds P[p]

provided the following condition is satisfied:

• Let given r, s, x, k, l be a variables list of k, and P be a k-ary predicate symbol. Then P[VERUM] and P[P [l]] and if P[r], then P[¬r] and if P[r] and P[s], then P[r ∧ s].

3. Results about the Construction of Formulas

In the sequel F_{1}, F_{2}, F_{3} denote formulae and L denotes a finite sequence. Let G, H be formulae. Let us assume that G is a subformula of H. A finite

sequence is called a path from G to H if it satisfies the conditions (Def. 6).

(Def. 6)(i) 1 ≤ len it, (ii) it(1) = G, (iii) it(len it) = H, and (iv) for every k such that 1 ≤ k and k < len it there exist elements G_{1}, H_{1}

of WFF such that it(k) = G_{1} and it(k + 1) = H_{1} and G_{1} is an immediate constituent of H_{1}.

The following propositions are true:

(27) Let L be a path from F_{1} to F_{2}. Suppose F_{1} is a subformula of F_{2} and 1 ≤ i and i ≤ len L. Then there exists F_{3} such that F_{3} = L(i) and F_{3} is a subformula of F_{2}.

(28) For every path L from F_{1} to p such that F_{1} is a subformula of p and 1 ≤ i and i ≤ len L holds L(i) is an element of CQC-WFF.

(29) Let L be a path from q to p. Suppose the number of quantifiers in p ≤ n and q is a subformula of p and 1 ≤ i and i ≤ len L. Then there exists r such that r = L(i) and the number of quantifiers in r ≤ n.

(30) If the number of quantifiers in p = n and q is a subformula of p, then the number of quantifiers in q ≤ n.

(31) For all n, p such that for every q such that q is a subformula of p holds the number of quantifiers in q = n holds n = 0.

(32) Let given p. Suppose that for every q such that q is a subformula of p and for all x, r holds q ≠ ∀_{x}r. Then the number of quantifiers in p = 0.

(33) Let given p. Suppose that for every q such that q is a subformula of p holds the number of quantifiers in q ≠ 1. Then the number of quantifiers in p = 0.

(34) Suppose 1 ≤ the number of quantifiers in p. Then there exists q such that q is a subformula of p and the number of quantifiers in q = 1.

References

[1] Grzegorz Bancerek. Connectives and subformulae of the first order language. Formalized Mathematics, 1(3):451–458, 1990.

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

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

[4] Patrick Braselmann and Peter Koepke. Coincidence lemma and substitution lemma. Formalized Mathematics, 13(1):17–26, 2005.

[5] Patrick Braselmann and Peter Koepke. Substitution in first-order formulas: Elementary properties. Formalized Mathematics, 13(1):5–15, 2005.

[6] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[10] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[11] Czesław Byliński and Grzegorz Bancerek. Variables in formulae of the first order language. Formalized Mathematics, 1(3):459–469, 1990.

[12] Agata Darmochwał and Andrzej Trybulec. Similarity of formulae. Formalized Mathematics, 2(5):635–642, 1991.

[13] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

[14] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

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

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

A Sequent Calculus for First-Order Logic^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349–360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article introduces a sequent calculus for first-order logic. The correctness of this calculus is shown and some important inferences are derived. The contents of this article correspond to Chapter IV of Ebbinghaus, Flum, Thomas.

MML Identifier: CALCUL 1.

The notation and terminology used here are introduced in the following papers: [18], [11], [20], [4], [9], [14], [15], [3], [1], [2], [8], [23], [12], [21], [13], [24], [10], [17], [22], [16], [19], [6], [7], and [5].

1. Preliminaries

For simplicity, we adopt the following rules: a, b, c, d denote sets, i, j, m, n denote natural numbers, p, q, r denote elements of CQC-WFF, x, y denote bound variables, X denotes a subset of CQC-WFF, A denotes a non empty set,

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

J denotes an interpretation of A, v, w denote elements of V(A), S_{1} denotes a CQC-substitution, and f, g denote finite sequences of elements of CQC-WFF.

Let g be a finite sequence and let N be a set. Observe that g↾N is finite subsequence-like.

Let D be a non empty set and let f be a finite sequence of elements of D. The functor Ant(f) yields a finite sequence of elements of D and is defined as follows:

(Def. 1)(i) For every i such that len f = i+1 holds Ant(f) = f↾ Seg i if len f > 0, (ii) Ant(f) = ∅, otherwise. Let D be a non empty set and let f be a finite sequence of elements of D.

Let us assume that len f > 0. The functor Suc(f) yielding an element of D is defined as follows:

(Def. 2) Suc(f) = f(len f).

Let D be a non empty set, let p be an element of D, and let f be a finite sequence of elements of D. We say that p is a tail of f if and only if:

(Def. 3) There exists i such that i ∈ dom f and f(i) = p.

Let us consider f, g. We say that f is a subsequence of g if and only if: (Def. 4) There exists a subset N of ℕ such that f ⊆ Seq(g↾N).

We now state several propositions:

(1) If f is a subsequence of g, then rng f ⊆ rng g and there exists a subset N of ℕ such that rng f ⊆ rng(g↾N).

(2) If len f > 0, then len Ant(f) + 1 = len f and len Ant(f) < len f.

(3) If len f > 0, then f = (Ant(f)) ^{⌢} ⟨Suc(f)⟩ and rng f = rng Ant(f) ∪ {Suc(f)}.

(4) If len f > 1, then len Ant(f) > 0.

(5) Suc(f ^{⌢} ⟨p⟩) = p and Ant(f ^{⌢} ⟨p⟩) = f.

In the sequel f_{1}, f_{2} are finite sequences.

We now state several propositions:

(6) len f_{1} ≤ len(f_{1}^{⌢}f_{2}) and len f_{2} ≤ len(f_{1}^{⌢}f_{2}) and if f_{1} ≠ ∅, then 1 ≤ len f_{1} and len f_{2} < len(f_{2} ^{⌢} f_{1}).

(7) Seq((f ^{⌢} g)↾ dom f) = (f ^{⌢} g)↾ dom f.

(8) f is a subsequence of f ^{⌢} g.

(9) 1 < len(f_{1} ^{⌢} ⟨b⟩^{⌢} ⟨c⟩).

(10) 1 ≤ len(f_{1} ^{⌢} ⟨b⟩) and len(f_{1} ^{⌢} ⟨b⟩) ∈ dom(f_{1} ^{⌢} ⟨b⟩).

(11) If 0 < m, then len Sgm(Seg n ∪ {n + m}) = n + 1.

(12) If 0 < m, then dom Sgm(Seg n ∪ {n + m}) = Seg(n + 1).

(13) If 0 < len f, then f is a subsequence of (Ant(f))^{⌢} g^{⌢} ⟨Suc(f)⟩.

(14) 1 ∈ dom⟨c, d⟩ and 2 ∈ dom⟨c, d⟩ and (f^{⌢}⟨c, d⟩)(len f +1) = c and (f^{⌢}⟨c, d⟩)(len f + 2) = d.

2. A Sequent Calculus

Let us consider f. The functor snb(f) yielding an element of 2^{BoundVar} is defined by:

(Def. 5) a ∈ snb(f) iff there exist i, p such that i ∈ dom f and p = f(i) and

a ∈ snb(p). The set of CQC-WFF-sequences is defined as follows:

(Def. 6) a ∈ the set of CQC-WFF-sequences iff a is a finite sequence of elements

of CQC-WFF. In the sequel P_{1}, P_{2} denote finite sequences of elements of [: the set of CQC-

WFF-sequences, 𝕂:].

Let us consider P_{1} and let n be a natural number. We say that step n in P_{1} is correct if and only if:

(Def. 7)(i) There exists f such that Suc(f) is a tail of Ant(f) and P_{1}(n)_{1} = f if

P_{1}(n)_{2} = 0, (ii) there exists f such that P_{1}(n)_{1} = f ^{⌢} ⟨VERUM⟩ if P_{1}(n)_{2} = 1, (iii) there exist i, f, g such that 1 ≤ i and i < n and Ant(f) is a subsequence

of Ant(g) and Suc(f) = Suc(g) and P_{1}(i)_{1} = f and P_{1}(n)_{1} = g if P_{1}(n)_{2} = 2,

(iv) there exist i, j, f, g such that 1 ≤ i and i < n and 1 ≤ j and j < i and len f > 1 and len g > 1 and Ant(Ant(f)) = Ant(Ant(g)) and ¬ Suc(Ant(f)) = Suc(Ant(g)) and Suc(f) = Suc(g) and f = P_{1}(j)_{1} and g = P_{1}(i)_{1} and (Ant(Ant(f)))^{⌢} ⟨Suc(f)⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 3,

(v) there exist i, j, f, g, p such that 1 ≤ i and i < n and 1 ≤ j and j < i and len f > 1 and Ant(f) = Ant(g) and Suc(Ant(f)) = ¬p and ¬ Suc(f) = Suc(g) and f = P_{1}(j)_{1} and g = P_{1}(i)_{1} and (Ant(Ant(f)))^{⌢} ⟨p⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 4,

(vi) there exist i, j, f, g such that 1 ≤ i and i < n and 1 ≤ j and j < i and Ant(f) = Ant(g) and f = P_{1}(j)_{1} and g = P_{1}(i)_{1} and (Ant(f))^{⌢} ⟨Suc(f) ∧ Suc(g)⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 5,

(vii) there exist i, f, p, q such that 1 ≤ i and i < n and p ∧ q = Suc(f) and

f = P_{1}(i)_{1} and (Ant(f))^{⌢} ⟨p⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 6, (viii) there exist i, f, p, q such that 1 ≤ i and i < n and p ∧ q = Suc(f) and

f = P_{1}(i)_{1} and (Ant(f))^{⌢} ⟨q⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 7, (ix) there exist i, f, p, x, y such that 1 ≤ i and i < n and Suc(f) = ∀_{x}p

and f = P_{1}(i)_{1} and (Ant(f))^{⌢} ⟨p(x, y)⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 8, (x) there exist i, f, p, x, y such that 1 ≤ i and i < n and Suc(f) = p(x,

y) and y ∉ snb(Ant(f)) and y ∉ snb(∀_{x}p) and f = P_{1}(i)_{1} and (Ant(f))^{⌢} ⟨∀_{x}p⟩ = P_{1}(n)_{1} if P_{1}(n)_{2} = 9.

Let us consider P_{1}. We say that P_{1} is a formal proof if and only if: (Def. 8) P_{1} ≠ ∅ and for every n such that 1 ≤ n and n ≤ len P_{1} holds step n in

P_{1} is correct. Let us consider f. The predicate ⊢ f is defined by:

(Def. 9) There exists P_{1} such that P_{1} is a formal proof and f = P_{1}(len P_{1})_{1}.

Let us consider p, X. We say that p is formally provable from X if and only if:

(Def. 10) There exists f such that rng Ant(f) ⊆ X and Suc(f) = p and ⊢ f.

Let us consider X, let us consider A, let us consider J, and let us consider v. The predicate J, v ⊧ X is defined as follows:

(Def. 11) If p ∈ X, then J, v ⊧ p.

Let us consider X, p. The predicate X ⊧ p is defined as follows: (Def. 12) If J, v ⊧ X, then J, v ⊧ p.

Let us consider p. The predicate ⊨ p is defined as follows: (Def. 13) ∅_{CQC-WFF} ⊧ p.

Let us consider f, A, J, v. The predicate J, v ⊧ f is defined as follows: (Def. 14) J, v ⊧ rng f.

Let us consider f, p. The predicate f ⊧ p is defined by: (Def. 15) If J, v ⊧ f, then J, v ⊧ p.

One can prove the following propositions:

(15) If Suc(f) is a tail of Ant(f), then Ant(f) ⊧ Suc(f).

(16) If Ant(f) is a subsequence of Ant(g) and Suc(f) = Suc(g) and Ant(f) ⊧ Suc(f), then Ant(g) ⊧ Suc(g).

(17) If len f > 0, then J, v ⊧ Ant(f) and J, v ⊧ Suc(f) iff J, v ⊧ f.

(18) If len f > 1 and len g > 1 and Ant(Ant(f)) = Ant(Ant(g)) and ¬ Suc(Ant(f)) = Suc(Ant(g)) and Suc(f) = Suc(g) and Ant(f) ⊧ Suc(f) and Ant(g) ⊧ Suc(g), then Ant(Ant(f)) ⊧ Suc(f).

(19) If len f > 1 and Ant(f) = Ant(g) and ¬p = Suc(Ant(f)) and ¬ Suc(f) = Suc(g) and Ant(f) ⊧ Suc(f) and Ant(g) ⊧ Suc(g), then Ant(Ant(f)) ⊧ p.

(20) If Ant(f) = Ant(g) and Ant(f) ⊧ Suc(f) and Ant(g) ⊧ Suc(g), then Ant(f) ⊧ Suc(f) ∧ Suc(g).

(21) If Suc(f) = p ∧ q and Ant(f) ⊧ p ∧ q, then Ant(f) ⊧ p.

(22) If Suc(f) = p ∧ q and Ant(f) ⊧ p ∧ q, then Ant(f) ⊧ q.

(23) J, v ⊧ ⟨⟨p, S_{1}⟩⟩ iff J, v ⊧ p.

In the sequel a is an element of A.

We now state several propositions:

(24) J, v ⊧ p(x, y) iff there exists a such that v(y) = a and J, v(x↾a) ⊧ p.

(25) If Suc(f) = ∀_{x}p and Ant(f) ⊧ Suc(f), then for every y holds Ant(f) ⊧ p(x, y).

(26) For every set X such that X ⊆ BoundVar holds if x ∉ X, then v(x↾a)↾X = v↾X.

(27) For all v, w such that v↾ snb(f) = w↾ snb(f) holds J, v ⊧ f iff J, w ⊧ f.

(28) If y ∉ snb(∀_{x}p), then v(y↾a)(x↾a)↾ snb(p) = v(x↾a)↾ snb(p).

(29) If Suc(f) = p(x, y) and Ant(f) ⊧ Suc(f) and y ∉ snb(Ant(f)) and y ∉ snb(∀_{x}p), then Ant(f) ⊧ ∀_{x}p.

(30) Ant(f ^{⌢} ⟨VERUM⟩) ⊧ Suc(f ^{⌢} ⟨VERUM⟩).

(31) Suppose 1 ≤ n and n ≤ len P_{1}. Then P_{1}(n)_{2} = 0 or P_{1}(n)_{2} = 1 or P_{1}(n)_{2} = 2 or P_{1}(n)_{2} = 3 or P_{1}(n)_{2} = 4 or P_{1}(n)_{2} = 5 or P_{1}(n)_{2} = 6 or P_{1}(n)_{2} = 7 or P_{1}(n)_{2} = 8 or P_{1}(n)_{2} = 9.

(32) If p is formally provable from X, then X ⊧ p.

3. Derived Rules

Next we state a number of propositions:

(33) If Suc(f) is a tail of Ant(f), then ⊢ f.

(34) If 1 ≤ n and n ≤ len P_{1}, then step n in P_{1} is correct iff step n in P_{1} ^{⌢} P_{2} is correct.

(35) If 1 ≤ n and n ≤ len P_{2} and step n in P_{2} is correct, then step n + len P_{1} in P_{1} ^{⌢} P_{2} is correct.

(36) If Ant(f) is a subsequence of Ant(g) and Suc(f) = Suc(g) and ⊢ f, then ⊢ g.

(37) If 1 < len f and 1 < len g and Ant(Ant(f)) = Ant(Ant(g)) and ¬ Suc(Ant(f)) = Suc(Ant(g)) and Suc(f) = Suc(g) and ⊢ f and ⊢ g, then ⊢ (Ant(Ant(f)))^{⌢} ⟨Suc(f)⟩.

(38) If len f > 1 and Ant(f) = Ant(g) and Suc(Ant(f)) = ¬p and ¬ Suc(f) = Suc(g) and ⊢ f and ⊢ g, then ⊢ (Ant(Ant(f)))^{⌢} ⟨p⟩.

(39) If Ant(f) = Ant(g) and ⊢ f and ⊢ g, then ⊢ (Ant(f))^{⌢}⟨Suc(f)∧Suc(g)⟩.

(40) If p ∧ q = Suc(f) and ⊢ f, then ⊢ (Ant(f))^{⌢} ⟨p⟩.

(41) If p ∧ q = Suc(f) and ⊢ f, then ⊢ (Ant(f))^{⌢} ⟨q⟩.

(42) If Suc(f) = ∀_{x}p and ⊢ f, then ⊢ (Ant(f))^{⌢} ⟨p(x, y)⟩.

(43) If Suc(f) = p(x, y) and y ∉ snb(Ant(f)) and y ∉ snb(∀_{x}p) and ⊢ f, then ⊢ (Ant(f))^{⌢} ⟨∀_{x}p⟩.

(44) If ⊢ f and ⊢ (Ant(f))^{⌢} ⟨¬ Suc(f)⟩, then ⊢ (Ant(f))^{⌢} ⟨p⟩.

(45) If 1 ≤ len f and ⊢ f and ⊢ f ^{⌢} ⟨p⟩, then ⊢ (Ant(f))^{⌢} ⟨p⟩.

(46) If ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨¬q⟩^{⌢} ⟨¬p⟩.

(47) If ⊢ f ^{⌢} ⟨¬p⟩^{⌢} ⟨¬q⟩, then ⊢ f ^{⌢} ⟨q⟩^{⌢} ⟨p⟩.

(48) If ⊢ f ^{⌢} ⟨¬p⟩^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨¬q⟩^{⌢} ⟨p⟩.

(49) If ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨¬q⟩, then ⊢ f ^{⌢} ⟨q⟩^{⌢} ⟨¬p⟩.

(50) If ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨r⟩ and ⊢ f ^{⌢} ⟨q⟩^{⌢} ⟨r⟩, then ⊢ f ^{⌢} ⟨p ∨ q⟩^{⌢} ⟨r⟩.

(51) If ⊢ f ^{⌢} ⟨p⟩, then ⊢ f ^{⌢} ⟨p ∨ q⟩.

(52) If ⊢ f ^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨p ∨ q⟩.

(53) If ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨r⟩ and ⊢ f ^{⌢} ⟨q⟩^{⌢} ⟨r⟩, then ⊢ f ^{⌢} ⟨p ∨ q⟩^{⌢} ⟨r⟩.

(54) If ⊢ f ^{⌢} ⟨p⟩, then ⊢ f ^{⌢} ⟨¬¬p⟩.

(55) If ⊢ f ^{⌢} ⟨¬¬p⟩, then ⊢ f ^{⌢} ⟨p⟩.

(56) If ⊢ f ^{⌢} ⟨p ⇒ q⟩ and ⊢ f ^{⌢} ⟨p⟩, then ⊢ f ^{⌢} ⟨q⟩.

(57) (¬p)(x, y) = ¬p(x, y).

(58) If there exists y such that ⊢ f ^{⌢} ⟨p(x, y)⟩, then ⊢ f ^{⌢} ⟨∃_{x}p⟩.

(59) snb(f ^{⌢} g) = snb(f) ∪ snb(g).

(60) snb(⟨p⟩) = snb(p).

(61) If ⊢ f^{⌢} ⟨p(x, y)⟩^{⌢} ⟨q⟩ and y ∉ snb(f^{⌢} ⟨∃_{x}p⟩^{⌢} ⟨q⟩), then ⊢ f^{⌢} ⟨∃_{x}p⟩^{⌢} ⟨q⟩.

(62) snb(f) = ⋃{snb(p) : ⋁_{i} (i ∈ dom f ∧ p = f(i))}.

(63) snb(f) is finite.

(64) \overline{\overline{BoundVar}} = ℵ_{0} and BoundVar is not finite.

(65) There exists x such that x ∉ snb(f).

(66) If ⊢ f ^{⌢} ⟨∀_{x}p⟩, then ⊢ f ^{⌢} ⟨∀_{x}¬¬p⟩.

(67) If ⊢ f ^{⌢} ⟨∀_{x}¬¬p⟩, then ⊢ f ^{⌢} ⟨∀_{x}p⟩.

(68) ⊢ f ^{⌢} ⟨∀_{x}p⟩ iff ⊢ f ^{⌢} ⟨¬∃_{x}¬p⟩.

References

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

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

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

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

[5] Patrick Braselmann and Peter Koepke. Coincidence lemma and substitution lemma. Formalized Mathematics, 13(1):17–26, 2005.

[6] Patrick Braselmann and Peter Koepke. Substitution in first-order formulas: Elementary properties. Formalized Mathematics, 13(1):5–15, 2005.

[7] Patrick Braselmann and Peter Koepke. Substitution in first-order formulas. Part II. The construction of first-order formulas. Formalized Mathematics, 13(1):27–32, 2005.

[8] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676,

a sequent calculus for first-order logic 39

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

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

[11] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

[13] Agata Darmochwał. A first–order predicate calculus. Formalized Mathematics, 1(4):689– 695, 1990.

[14] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

[16] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[17] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

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

[21] Edmund Woronowicz. Interpretation and satisfiability in the first order logic. Formalized Mathematics, 1(4):739–743, 1990.

[22] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 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.

40 patrick braselmann and peter koepke

Consequences of the Sequent Calculus^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349-360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The first main result of the present article is that the derivablility of a sequent doesn’t depend on the ordering of the antecedent. The second main result says: if a sequent is derivable, then the formulas in the antecendent only need to occur once.

MML Identifier: CALCUL 2.

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

1. f is a Subsequence of g^{f}

For simplicity, we adopt the following convention: p, q denote elements of CQC-WFF, k, m, n, i denote natural numbers, f, g denote finite sequences of elements of CQC-WFF, and a, b, b_{1}, b_{2}, c denote natural numbers.

Let m, n be natural numbers. The functor seq(m, n) yielding a set is defined as follows:

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

(Def. 1) seq(m, n) = {k : 1 + m ≤ k ∧ k ≤ n + m}. Let m, n be natural numbers. Then seq(m, n) is a subset of ℕ. One can prove the following propositions:

(1) c ∈ seq(a, b) iff 1 + a ≤ c and c ≤ b + a.

(2) seq(a, 0) = ∅.

(3) b = 0 or b + a ∈ seq(a, b).

(4) b_{1} ≤ b_{2} iff seq(a, b_{1}) ⊆ seq(a, b_{2}).

(5) seq(a, b) ∪ {a + b + 1} = seq(a, b + 1).

(6) seq(m, n) ≈ n.

Let us consider m, n. Observe that seq(m, n) is finite.

Let us consider f. Observe that len f is finite.

Next we state a number of propositions:

(7) seq(m, n) ⊆ Seg(m + n).

(8) Seg n misses seq(n, m).

(9) For all finite sequences f, g holds Seg len(f ^{⌢} g) = Seg len f ∪ seq(len f, len g).

(10) len Sgm seq(len g, len f) = len f.

(11) dom Sgm seq(len g, len f) = dom f.

(12) rng Sgm seq(len g, len f) = seq(len g, len f).

(13) If i ∈ dom Sgm seq(len g, len f), then (Sgm seq(len g, len f))(i) = len g+i.

(14) seq(len g, len f) ⊆ dom(g^{⌢} f).

(15) dom((g^{⌢} f)↾ seq(len g, len f)) = seq(len g, len f).

(16) Seq((g^{⌢} f)↾ seq(len g, len f)) = Sgm seq(len g, len f) · (g^{⌢} f).

(17) dom Seq((g^{⌢} f)↾ seq(len g, len f)) = dom f.

(18) f is a subsequence of g^{⌢} f.

Let D be a non empty set, let f be a finite sequence of elements of D, and let P be a permutation of dom f. The functor Per(f, P) yielding a finite sequence of elements of D is defined as follows:

(Def. 2) Per(f, P) = P · f.

In the sequel P denotes a permutation of dom f.

The following propositions are true:

(19) dom Per(f, P) = dom f.

(20) If ⊢ f ^{⌢} ⟨p⟩, then ⊢ g^{⌢} f ^{⌢} ⟨p⟩.

2. The Ordering of the Antecedent is Irrelevant

Let us consider f. The functor Begin(f) yielding an element of CQC-WFF is defined by:

(Def. 3) Begin(f) = \{ \begin{matrix}f(1), if 1 ≤ len f, \\ VERUM, otherwise. \end{matrix}

Let us consider f. Let us assume that 1 ≤ len f. The functor Impl(f) yields an element of CQC-WFF and is defined by the condition (Def. 4).

(Def. 4) There exists a finite sequence F of elements of CQC-WFF such that (i) Impl(f) = F (len f), (ii) len F = len f, (iii) F (1) = Begin(f) or len f = 0, and (iv) for every n such that 1 ≤ n and n < len f there exist p, q such that

p = f(n + 1) and q = F (n) and F (n + 1) = p ⇒ q. We now state a number of propositions:

(21) ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨p⟩.

(22) If ⊢ f ^{⌢} ⟨p ∧ q⟩, then ⊢ f ^{⌢} ⟨p⟩.

(23) If ⊢ f ^{⌢} ⟨p ∧ q⟩, then ⊢ f ^{⌢} ⟨q⟩.

(24) If ⊢ f ^{⌢} ⟨p⟩ and ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨q⟩.

(25) If ⊢ f ^{⌢} ⟨p⟩ and ⊢ f ^{⌢} ⟨¬p⟩, then ⊢ f ^{⌢} ⟨q⟩.

(26) If ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨q⟩ and ⊢ f ^{⌢} ⟨¬p⟩^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨q⟩.

(27) If ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨p ⇒ q⟩.

(28) If 1 ≤ len g and ⊢ f ^{⌢} g, then ⊢ f ^{⌢} ⟨Impl(Rev(g))⟩.

(29) If ⊢ (Per(f, P))^{⌢} ⟨Impl(Rev(f ^{⌢} ⟨p⟩))⟩, then ⊢ (Per(f, P))^{⌢} ⟨p⟩.

(30) If ⊢ f ^{⌢} ⟨p⟩, then ⊢ (Per(f, P))^{⌢} ⟨p⟩.

3. Multiple Occurrence in the Antecedent is Irrelevant

Let us consider n and let c be a set. We introduce IdFinS(c, n) as a synonym of n ↦ c.

We now state the proposition

(31) For every set c such that 1 ≤ n holds rng IdFinS(c, n) = rng⟨c⟩. Let D be a non empty set, let n be a natural number, and let p be an element

of D. Then IdFinS(p, n) is a finite sequence of elements of D.

The following proposition is true

(32) If 1 ≤ n and ⊢ f ^{⌢} IdFinS(p, n)^{⌢} ⟨q⟩, then ⊢ f ^{⌢} ⟨p⟩^{⌢} ⟨q⟩.

References

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

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

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

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

[5] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

[7] Patrick Braselmann and Peter Koepke. A sequent calculus for first-order logic. Formalized Mathematics, 13(1):33–39, 2005.

[8] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

[9] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

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

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

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

[13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 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] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

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

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

Equivalences of Inconsistency and Henkin Models^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349–360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag, New York Inc. The present article establishes some equivalences of inconsistency. It is proved that a countable union of consistent sets is consistent. Then the concept of a Henkin model is introduced. The contents of this article correspond to Chapter IV, par. 7 and Chapter V, par. 1 of Ebbinghaus, Flum, Thomas.

MML Identifier: HENMODEL.

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

1. Preliminaries and Equivalences of Inconsistency

For simplicity, we use the following convention: a denotes a set, X, Y denote subsets of CQC-WFF, k, m, n denote natural numbers, p, q denote elements of

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

CQC-WFF, P denotes a k-ary predicate symbol, l_{1} denotes a variables list of k, and f, g denote finite sequences of elements of CQC-WFF.

Let D be a non empty set and let X be a subset of 2^{D}. Then ⋃ X is a subset of D.

In the sequel A is a non empty finite subset of ℕ. The following two propositions are true:

(1) Let f be a function from n into A. Suppose there exists m such that succ m = n and f is one-to-one and rng f = A and for all n, m such that m ∈ dom f and n ∈ dom f and n < m holds f(n) ∈ f(m). Then f(⋃ n) = ⋃ rng f.

(2) ⋃ A ∈ A and for every a such that a ∈ A holds a ∈ ⋃ A or a = ⋃ A. Let A be a set. The functor min^{∗}A yielding a natural number is defined by:

(Def. 1)(i) min^{∗}A ∈ A and for every k such that k ∈ A holds min^{∗}A ≤ k if A is

a non empty subset of ℕ, (ii) min^{∗}A = 0, otherwise. In the sequel C denotes a non empty set. Next we state the proposition

(3) Let f be a function from ℕ into C and X be a finite set. Suppose for all n, m such that m ∈ dom f and n ∈ dom f and n < m holds f(n) ⊆ f(m) and X ⊆ ⋃ rng f. Then there exists k such that X ⊆ f(k).

Let us consider X, p. The predicate X ⊢ p is defined as follows: (Def. 2) There exists f such that rng f ⊆ X and ⊢ f ^{⌢} ⟨p⟩.

Let us consider X. We say that X is consistent if and only if: (Def. 3) For every p holds X ⊬ p or X ⊬ ¬p.

Let us consider X. We introduce X is inconsistent as an antonym of X is consistent.

Let f be a finite sequence of elements of CQC-WFF. We say that f is consistent if and only if:

(Def. 4) For every p holds ⊬ f ^{⌢} ⟨p⟩ or ⊬ f ^{⌢} ⟨¬p⟩.

Let f be a finite sequence of elements of CQC-WFF. We introduce f is inconsistent as an antonym of f is consistent.

Next we state several propositions:

(4) If X is consistent and rng g ⊆ X, then g is consistent.

(5) If ⊢ f ^{⌢} ⟨p⟩, then ⊢ f ^{⌢} g^{⌢} ⟨p⟩.

(6) X is inconsistent iff for every p holds X ⊢ p.

(7) If X is inconsistent, then there exists Y such that Y ⊆ X and Y is finite and inconsistent.

(8) If X ∪{p} ⊢ q, then there exists g such that rng g ⊆ X and ⊢ g^{⌢}⟨p⟩^{⌢}⟨q⟩.

(9) X ⊢ p iff X ∪ {¬p} is inconsistent.

(10) X ⊢ ¬p iff X ∪ {p} is inconsistent.

2. Unions of Consistent Sets

We now state the proposition

(11) Let f be a function from ℕ into 2^{CQC-WFF}. Suppose that for all n, m such that m ∈ dom f and n ∈ dom f and n < m holds f(n) is consistent and f(n) ⊆ f(m). Then ⋃ rng f is consistent.

3. Construction of a Henkin Model

In the sequel A is a non empty set, v is an element of V(A), and J is an interpretation of A.

We now state two propositions:

(12) If X is inconsistent, then for all J, v holds J, v |̸= X.

(13) {VERUM} is consistent.

Let us observe that there exists a subset of CQC-WFF which is consistent. In the sequel C_{1} denotes a consistent subset of CQC-WFF. The non empty set HCar is defined by:

(Def. 5) HCar = BoundVar .

Let P be an element of PredSym and let l_{1} be a variables list of Arity(P). Then P [l_{1}] is an element of CQC-WFF.

Let us consider C_{1}. An interpretation of HCar is said to be a Henkin interpretation of C_{1} if it satisfies the condition (Def. 6).

(Def. 6) Let P be an element of PredSym and r be an element of Rel(HCar). Suppose it(P) = r. Let given a. Then a ∈ r if and only if there exists a variables list l_{1} of Arity(P) such that a = l_{1} and C_{1} ⊢ P [l_{1}].

The element valH of V(HCar) is defined as follows: (Def. 7) valH = id_{BoundVar}.

4. Some Properties of the Henkin Model

In the sequel J_{1} is a Henkin interpretation of C_{1}. We now state four propositions:

(14) valH ∗l_{1} = l_{1}.

(15) ⊢ f ^{⌢} ⟨VERUM⟩.

(16) J_{1}, valH ⊧ VERUM iff C_{1} ⊢ VERUM .

(17) J_{1}, valH ⊧ P [l_{1}] iff C_{1} ⊢ P [l_{1}].

References

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

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

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

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

[5] Patrick Braselmann and Peter Koepke. A sequent calculus for first-order logic. Formalized Mathematics, 13(1):33–39, 2005.

[6] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 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. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

[11] Agata Darmochwał. A first–order predicate calculus. Formalized Mathematics, 1(4):689– 695, 1990.

[12] Agata Darmochwał and Andrzej Trybulec. Similarity of formulae. Formalized Mathematics, 2(5):635–642, 1991.

[13] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

[15] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[16] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

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

[20] Edmund Woronowicz. Interpretation and satisfiability in the first order logic. Formalized Mathematics, 1(4):739–743, 1990.

[21] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 1990.

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

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

Gödel’s Completeness Theorem^{1}

Patrick Braselmann University of Bonn

Peter Koepke University of Bonn

Summary. This article is part of a series of Mizar articles which constitute a formal proof (of a basic version) of Kurt Gödel’s famous completeness theorem (K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930), 349–360). The completeness theorem provides the theoretical basis for a uniform formalization of mathematics as in the Mizar project. We formalize first-order logic up to the completeness theorem as in H. D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, 1984, Springer Verlag New York Inc. The present article contains the proof of a simplified completeness theorem for a countable relational language without equality.

MML Identifier: GOEDELCP.

The notation and terminology used in this paper are introduced in the following articles: [19], [13], [21], [2], [4], [11], [16], [1], [17], [10], [23], [14], [22], [24], [12], [15], [18], [20], [3], [8], [5], [9], [7], and [6].

1. Henkin’s Theorem

For simplicity, we adopt the following convention: X, Y denote subsets of CQC-WFF, n denotes a natural number, p, q denote elements of CQC-WFF, x, y denote bound variables, A denotes a non empty set, J denotes an interpretation of A, v denotes an element of V(A), f_{1} denotes a finite sequence of

1This research was carried out within the project “Wissensformate” and was financially supported by the Mathematical Institute of the University of Bonn (http://www.- wissensformate.uni-bonn.de). Preparation of the Mizar code was part of the first author’s graduate work under the supervision of the second author. The authors thank Jip Veldman for his work on the final version of this article.

elements of CQC-WFF, C_{1}, C_{2}, C_{3} denote consistent subsets of CQC-WFF, J_{1} denotes a Henkin interpretation of C_{1}, and a denotes an element of A.

Let us consider X. We say that X is negation faithful if and only if: (Def. 1) X ⊢ p or X ⊢ ¬p.

Let us consider X. We say that X has examples if and only if: (Def. 2) For all x, p there exists y such that X ⊢ ¬∃_{x}p ∨ p(x, y).

One can prove the following propositions:

(1) If C_{1} is negation faithful, then C_{1} ⊢ p iff C_{1} ⊬ ¬p.

(2) For every finite sequence f of elements of CQC-WFF such that ⊢ f ^{⌢} ⟨¬p ∨ q⟩ and ⊢ f ^{⌢} ⟨p⟩ holds ⊢ f ^{⌢} ⟨q⟩.

(3) If X has examples, then X ⊢ ∃_{x}p iff there exists y such that X ⊢ p(x, y).

(4) Suppose if C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ p iff C_{1} ⊢ p. Suppose C_{1} is negation faithful and has examples. Then J_{1}, valH ⊧ ¬p if and only if C_{1} ⊢ ¬p.

(5) If ⊢ f_{1} ^{⌢} ⟨p⟩ and ⊢ f_{1} ^{⌢} ⟨q⟩, then ⊢ f_{1} ^{⌢} ⟨p ∧ q⟩.

(6) X ⊢ p and X ⊢ q iff X ⊢ p ∧ q.

(7) Suppose that

(i) if C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ p iff C_{1} ⊢ p, and

(ii) if C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ q iff C_{1} ⊢ q. Suppose C_{1} is negation faithful and has examples. Then J_{1}, valH ⊧ p ∧ q if and only if C_{1} ⊢ p ∧ q.

(8) Let given p. Suppose the number of quantifiers in p ≤ 0. If C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ p iff C_{1} ⊢ p.

(9) J, v ⊧ ∃_{x}p iff there exists a such that J, v(x↾a) ⊧ p.

(10) J_{1}, valH ⊧ ∃_{x}p iff there exists y such that J_{1}, valH ⊧ p(x, y).

(11) J, v ⊧ ¬∃_{x}¬p iff J, v ⊧ ∀_{x}p.

(12) X ⊢ ¬∃_{x}¬p iff X ⊢ ∀_{x}p.

(13) The number of quantifiers in ∃_{x}p = (the number of quantifiers in p) + 1.

(14) The number of quantifiers in p = the number of quantifiers in p(x, y). In the sequel a denotes a set. The following three propositions are true:

(15) Let given p. Suppose the number of quantifiers in p = 1. If C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ p iff C_{1} ⊢ p.

(16) Let given n. Suppose that for every p such that the number of quantifiers in p ≤ n holds if C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ p iff C_{1} ⊢ p. Let given p. Suppose the number of quantifiers in p ≤ n + 1. If C_{1} is negation faithful and has examples, then J_{1}, valH ⊧ p iff C_{1} ⊢ p.

(17) For every p such that C_{1} is negation faithful and has examples holds J_{1}, valH ⊧ p iff C_{1} ⊢ p.

2. Satisfiability of Consistent Sets of Formulas with Finitely Many Free Variables

The following proposition is true

(18) WFF is countable.

The subset ExCl of CQC-WFF is defined by: (Def. 3) a ∈ ExCl iff there exist x, p such that a = ∃_{x}p.

The following propositions are true:

(19) CQC-WFF is countable.

(20) ExCl is non empty and ExCl is countable.

Let p be an element of WFF. Let us assume that p is existential. The functor ExBound(p) yielding a bound variable is defined as follows:

(Def. 4) There exists an element q of WFF such that p = ∃_{ExBound(p)}q.

Let p be an element of CQC-WFF. Let us assume that p is existential. The functor ExScope(p) yielding an element of CQC-WFF is defined by:

(Def. 5) There exists x such that p = ∃_{x} ExScope(p).

Let F be a function from ℕ into CQC-WFF and let a be a natural number. The bound in F (a) yields a bound variable and is defined as follows:

(Def. 6) If p = F (a), then the bound in F (a) = ExBound(p).

Let F be a function from ℕ into CQC-WFF and let a be a natural number. The scope of F (a) yields an element of CQC-WFF and is defined by:

(Def. 7) If p = F (a), then the scope of F (a) = ExScope(p).

Let us consider X. The functor snb(X) yields an element of 2^{BoundVar} and is defined by:

(Def. 8) snb(X) = ⋃{snb(p) : p ∈ X}.

Next we state a number of propositions:

(21) If p ∈ X, then X ⊢ p.

(22) ExBound(∃_{x}p) = x and ExScope(∃_{x}p) = p.

(23) X ⊢ VERUM .

(24) X ⊢ ¬ VERUM iff X is inconsistent.

(25) For all finite sequences f, g of elements of CQC-WFF such that 0 < len f and ⊢ f ^{⌢} ⟨p⟩ holds ⊢ (Ant(f))^{⌢} g^{⌢} ⟨Suc(f)⟩^{⌢} ⟨p⟩.

(26) snb({p}) = snb(p).

(27) snb(X ∪ Y) = snb(X) ∪ snb(Y).

(28) For every element A of 2^{BoundVar} such that A is finite there exists x such that x ∉ A.

(29) If X ⊆ Y, then snb(X) ⊆ snb(Y).

(30) For every finite sequence f of elements of CQC-WFF holds snb(rng f) = snb(f).

(31) If snb(C_{1}) is finite, then there exists C_{2} such that C_{1} ⊆ C_{2} and C_{2} has examples.

(32) If X ⊢ p and X ⊆ Y, then Y ⊢ p.

(33) If C_{1} has examples, then there exists C_{2} such that C_{1} ⊆ C_{2} and C_{2} is negation faithful and has examples.

In the sequel J_{2} denotes a Henkin interpretation of C_{3}, J denotes an interpretation of A, and v denotes an element of V(A).

We now state the proposition

(34) If snb(C_{1}) is finite, then there exist C_{3}, J_{2} such that J_{2}, valH ⊧ C_{1}.

3. Gödel’s Completeness Theorem

We now state four propositions:

(35) If J, v ⊧ X and Y ⊆ X, then J, v ⊧ Y.

(36) If snb(X) is finite, then snb(X ∪ {p}) is finite.

(37) If X ⊧ p, then J, v |̸= X ∪ {¬p}.

(38) If snb(X) is finite and X ⊧ p, then X ⊢ p.

References

[1] Grzegorz Bancerek. Connectives and subformulae of the first order language. Formalized Mathematics, 1(3):451–458, 1990.

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

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

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

[5] Patrick Braselmann and Peter Koepke. Coincidence lemma and substitution lemma. Formalized Mathematics, 13(1):17–26, 2005.

[6] Patrick Braselmann and Peter Koepke. Equivalences of inconsistency and Henkin models. Formalized Mathematics, 13(1):45–48, 2005.

[7] Patrick Braselmann and Peter Koepke. A sequent calculus for first-order logic. Formalized Mathematics, 13(1):33–39, 2005.

[8] Patrick Braselmann and Peter Koepke. Substitution in first-order formulas: Elementary properties. Formalized Mathematics, 13(1):5–15, 2005.

[9] Patrick Braselmann and Peter Koepke. Substitution in first-order formulas. Part II. The construction of first-order formulas. Formalized Mathematics, 13(1):27–32, 2005.

[10] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

gödel’s completeness theorem 53

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

[13] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

[15] Agata Darmochwał and Andrzej Trybulec. Similarity of formulae. Formalized Mathematics, 2(5):635–642, 1991.

[16] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

[18] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

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

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

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

[22] Edmund Woronowicz. Interpretation and satisfiability in the first order logic. Formalized Mathematics, 1(4):739–743, 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.

54 patrick braselmann and peter koepke

Propositional Calculus for Boolean Valued Functions. Part VIII

Shunichi Kobayashi Matsumoto University Nagano

Summary. In this paper, we proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC26.

The articles [5], [6], [8], [7], [9], [1], [4], [3], and [2] provide the notation and terminology for this paper.

In this paper Y denotes a non empty set and a, b, c denote elements of Boolean^{Y}.

Let p, q be boolean-valued functions. The functor p ′nand′ q yielding a function is defined as follows:

(Def. 1) dom(p ′nand′ q) = dom p ∩ dom q and for every set x such that x ∈

dom(p ′nand′ q) holds (p ′nand′ q)(x) = p(x) ′nand′ q(x). Let us observe that the functor p ′nand′ q is commutative. The functor p ′nor′ q yielding a function is defined as follows:

(Def. 2) dom(p ′nor′ q) = dom p ∩ dom q and for every set x such that x ∈

dom(p ′nor′ q) holds (p ′nor′ q)(x) = p(x) ′nor′ q(x). Let us note that the functor p ′nor′ q is commutative.

Let p, q be boolean-valued functions. Note that p ′nand′ q is boolean-valued and p ′nor′ q is boolean-valued.

Let A be a non empty set and let p, q be elements of Boolean^{A}. Then p ′nand′ q is an element of Boolean^{A} and it can be characterized by the condition:

(Def. 3) For every element x of A holds (p ′nand′ q)(x) = p(x) ′nand′ q(x).

Then p ′nor′ q is an element of Boolean^{A} and it can be characterized by the condition:

(Def. 4) For every element x of A holds (p ′nor′ q)(x) = p(x) ′nor′ q(x).

Let us consider Y and let a, b be elements of BVF(Y). Then a ′nand′ b is an element of BVF(Y). Then a ′nor′ b is an element of BVF(Y).

We now state a number of propositions:

(1) a ′nand′ b = ¬(a ∧ b).

(2) a ′nor′ b = ¬(a ∨ b).

(3) true(Y) ′nand′ a = ¬a.

(4) false(Y) ′nand′ a = true(Y).

(5) false(Y) ′nand′ false(Y) = true(Y) and false(Y) ′nand′ true(Y) = true(Y) and true(Y) ′nand′ true(Y) = false(Y).

(6) a ′nand′ a = ¬a and ¬(a ′nand′ a) = a.

(7) ¬(a ′nand′ b) = a ∧ b.

(8) a ′nand′ ¬a = true(Y) and ¬(a ′nand′ ¬a) = false(Y).

(9) a ′nand′ b ∧ c = ¬(a ∧ b ∧ c).

(10) a ′nand′ b ∧ c = a ∧ b ′nand′ c.

(11) a ′nand′ (b ∨ c) = ¬(a ∧ b) ∧ ¬(a ∧ c).

(12) a ′nand′ (b ⊕ c) = a ∧ b ⇔ a ∧ c.

(13) a ′nand′ (b ′nand′ c) = ¬a ∨ b ∧ c and a ′nand′ (b ′nand′ c) = a ⇒ b ∧ c.

(14) a ′nand′ (b ′nor′ c) = ¬a ∨ b ∨ c and a ′nand′ (b ′nor′ c) = a ⇒ b ∨ c.

(15) a ′nand′ (b ⇔ c) = a ⇒ b ⊕ c.

(16) a ′nand′ a ∧ b = a ′nand′ b.

(17) a ′nand′ (a ∨ b) = ¬a ∧ ¬(a ∧ b).

(18) a ′nand′ (a ⇔ b) = a ⇒ a ⊕ b.

(19) a ′nand′ (a ′nand′ b) = ¬a ∨ b and a ′nand′ (a ′nand′ b) = a ⇒ b.

(20) a ′nand′ (a ′nor′ b) = true(Y).

(21) a ′nand′ (a ⇔ b) = ¬a ∨ ¬b.

(22) a ∧ b = a ′nand′ b ′nand′ (a ′nand′ b).

(23) a ′nand′ b ′nand′ (a ′nand′ c) = a ∧ (b ∨ c).

(24) a ′nand′ (b ⇒ c) = (¬a ∨ b) ∧ ¬(a ∧ c).

(25) a ′nand′ (a ⇒ b) = ¬(a ∧ b).

(26) true(Y) ′nor′ a = false(Y).

(27) false(Y) ′nor′ a = ¬a.

(28) false(Y) ′nor′ false(Y) = true(Y) and false(Y) ′nor′ true(Y) = false(Y) and true(Y) ′nor′ true(Y) = false(Y).

(29) a ′nor′ a = ¬a and ¬(a ′nor′ a) = a.

(30) ¬(a ′nor′ b) = a ∨ b.

(31) a ′nor′ ¬a = false(Y) and ¬(a ′nor′ ¬a) = true(Y).

(32) ¬a ∧ (a ⊕ b) = ¬a ∧ b.

(33) a ′nor′ b ∧ c = ¬(a ∨ b) ∨ ¬(a ∨ c).

(34) a ′nor′ (b ∨ c) = ¬(a ∨ b ∨ c).

(35) a ′nor′ (b ⇔ c) = ¬a ∧ (b ⊕ c).

(36) a ′nor′ (b ⇒ c) = ¬a ∧ b ∧ ¬c.

(37) a ′nor′ (b ′nand′ c) = ¬a ∧ b ∧ c.

(38) a ′nor′ (b ′nor′ c) = ¬a ∧ (b ∨ c).

(39) a ′nor′ a ∧ b = ¬(a ∧ (a ∨ b)).

(40) a ′nor′ (a ∨ b) = ¬(a ∨ b).

(41) a ′nor′ (a ⇔ b) = ¬a ∧ b.

(42) a ′nor′ (a ⇒ b) = false(Y).

(43) a ′nor′ (a ′nand′ b) = false(Y).

(44) a ′nor′ (a ′nor′ b) = ¬a ∧ b.

(45) false(Y) ⇔ false(Y) = true(Y).

(46) false(Y) ⇔ true(Y) = false(Y).

(47) true(Y) ⇔ true(Y) = true(Y).

(48) a ⇔ a = true(Y) and ¬(a ⇔ a) = false(Y).

(49) a ⇔ a ∨ b = a ∨ ¬b.

(50) a ∧ (b ′nand′ c) = a ∧ ¬b ∨ a ∧ ¬c.

(51) a ∨ (b ′nand′ c) = a ∨ ¬b ∨ ¬c.

(52) a ⊕ (b ′nand′ c) = ¬a ∧ ¬(b ∧ c) ∨ a ∧ b ∧ c.

(53) a ⇔ b ′nand′ c = a ∧ ¬(b ∧ c) ∨ ¬a ∧ b ∧ c.

(54) a ⇒ b ′nand′ c = ¬(a ∧ b ∧ c).

(55) a ′nor′ (b ′nand′ c) = ¬(a ∨ ¬b ∨ ¬c).

(56) a ∧ (a ′nand′ b) = a ∧ ¬b.

(57) a ∨ (a ′nand′ b) = true(Y).

(58) a ⊕ (a ′nand′ b) = ¬a ∨ b.

(59) a ⇔ a ′nand′ b = a ∧ ¬b.

(60) a ⇒ a ′nand′ b = ¬(a ∧ b).

(61) a ′nor′ (a ′nand′ b) = false(Y).

(62) a ∧ (b ′nor′ c) = a ∧ ¬b ∧ ¬c.

(63) a ∨ (b ′nor′ c) = (a ∨ ¬b) ∧ (a ∨ ¬c).

(64) a ⊕ (b ′nor′ c) = (a ∨ ¬(b ∨ c)) ∧ (¬a ∨ b ∨ c).

(65) a ⇔ b ′nor′ c = (a ∨ b ∨ c) ∧ (¬a ∨ ¬(b ∨ c)).

(66) a ⇒ b ′nor′ c = ¬(a ∧ (b ∨ c)).

58 shunichi kobayashi

(67) a ′nand′ (b ′nor′ c) = ¬a ∨ b ∨ c.

(68) a ∧ (a ′nor′ b) = false(Y).

(69) a ∨ (a ′nor′ b) = a ∨ ¬b.

(70) a ⊕ (a ′nor′ b) = a ∨ ¬b.

(71) a ⇔ a ′nor′ b = ¬a ∧ b.

(72) a ⇒ a ′nor′ b = ¬(a ∨ a ∧ b).

(73) a ′nand′ (a ′nor′ b) = true(Y).

Acknowledgments

This research was partially supported by the research funds of the Matsumoto University.

References

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

[2] Shunichi Kobayashi. On the calculus of binary arithmetics. Formalized Mathematics, 11(4):417–419, 2003.

[3] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

[4] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

[7] Edmund Woronowicz. Interpretation and satisfiability in the first order logic. Formalized Mathematics, 1(4):739–743, 1990.

[8] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 1990.

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

Hölder’s Inequality and Minkowski’s Inequality

Yasumasa Suzuki Take, Yokosuka-shi Japan

Summary. In this article, Hölder’s inequality and Minkowski’s inequality are proved. These equalities are basic ones of functional analysis.

MML Identifier: HOLDER 1.

The papers [12], [13], [14], [3], [1], [11], [4], [2], [7], [5], [6], [10], [8], and [9] provide the notation and terminology for this paper.

1. Hölder’s Inequality

In this paper a, b, p, q are real numbers. Let x be a real number. One can verify that [x, +∞[ is non empty. Next we state several propositions:

(1) For all real numbers p, q such that 0 < p and 0 < q and for every real number a such that 0 ≤ a holds a^{p} · a^{q} = a^{p+q}.

(2) For all real numbers p, q such that 0 < p and 0 < q and for every real number a such that 0 ≤ a holds (a^{p})^{q} = a^{p·q}.

(3) For every real number p such that 0 < p and for all real numbers a, b such that 0 ≤ a and a ≤ b holds a^{p} ≤ b^{p}.

(4) If 1 < p and \frac{1}{p} + \frac{1}{q} = 1 and 0 < a and 0 < b, then a · b ≤ \frac{a_{ℝ}^{p}}{p} + \frac{b_{ℝ}^{q}}{q} and

a · b = \frac{a_{ℝ}^{p}}{p} + \frac{b_{ℝ}^{q}}{q} iff a_{ℝ}^{p} = b_{ℝ}^{q}.

(5) If 1 < p and \frac{1}{p} + \frac{1}{q} = 1 and 0 ≤ a and 0 ≤ b, then a · b ≤ \frac{a^{p}}{p} + \frac{b^{q}}{q} and a · b = \frac{a^{p}}{p} + \frac{b^{q}}{q} iff a^{p} = b^{q}.

2. Minkowski’s Inequality

Next we state several propositions:

(6) Let p, q be real numbers. Suppose 1 < p and \frac{1}{p} + \frac{1}{q} = 1. Let a, b, a_{1}, b_{1}, a_{2} be sequences of real numbers. Suppose that for every natural number n holds a_{1}(n) = |a(n)|^{p} and b_{1}(n) = |b(n)|^{q} and a_{2}(n) = |a(n)·b(n)|. Let n be a natural number. Then (∑_{α=0}^{κ}(a_{2})(α))_{κ∈ℕ}(n) ≤ (∑_{α=0}^{κ}(a_{1})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} · (∑_{α=0}^{κ}(b_{1})(α))_{κ∈ℕ}(n)^{\frac{1}{q}} .

(7) Let p be a real number. Suppose 1 < p. Let a, b, a_{1}, b_{2}, a_{2} be sequences of real numbers. Suppose that for every natural number n holds a_{1}(n) = |a(n)|^{p} and b_{2}(n) = |b(n)|^{p} and a_{2}(n) = |a(n) + b(n)|^{p}. Let n be a natural number. Then (∑_{α=0}^{κ}(a_{2})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} ≤ (∑_{α=0}^{κ}(a_{1})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} + (∑_{α=0}^{κ}(b_{2})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} .

(8) Let a, b be sequences of real numbers. Suppose for every natural number n holds a(n) ≤ b(n) and b is convergent and a is non-decreasing. Then a is convergent and lim a ≤ lim b.

(9) Let a, b, c be sequences of real numbers. Suppose for every natural number n holds a(n) ≤ b(n) + c(n) and b is convergent and c is convergent and a is non-decreasing. Then a is convergent and lim a ≤ lim b + lim c.

(10) Let p be a real number. Suppose 0 < p. Let a, a_{1} be sequences of real numbers. Suppose a is convergent and for every natural number n holds 0 ≤ a(n) and for every natural number n holds a_{1}(n) = a(n)^{p}. Then a_{1} is convergent and lim a_{1} = (lim a)^{p}.

(11) Let p be a real number. Suppose 0 < p. Let a, a_{1} be sequences of real numbers. Suppose a is summable and for every natural number n holds 0 ≤ a(n) and for every natural number n holds a_{1}(n) = (∑_{α=0}^{κ} a(α))_{κ∈ℕ}(n)^{p}. Then a_{1} is convergent and lim a_{1} = (∑ a)^{p} and a_{1} is non-decreasing and for every natural number n holds a_{1}(n) ≤ (∑ a)^{p}.

(12) Let p, q be real numbers. Suppose 1 < p and \frac{1}{p} + \frac{1}{q} = 1. Let a, b, a_{1}, b_{1}, a_{2} be sequences of real numbers. Suppose for every natural number n holds a_{1}(n) = |a(n)|^{p} and b_{1}(n) = |b(n)|^{q} and a_{2}(n) = |a(n) · b(n)| and a_{1} is summable and b_{1} is summable. Then a_{2} is summable and ∑ a_{2} ≤ (∑ a_{1})^{\frac{1}{p}} · (∑ b_{1})^{\frac{1}{q}} .

(13) Let p be a real number. Suppose 1 < p. Let a, b, a_{1}, b_{2}, a_{2} be sequences

of real numbers. Suppose that

(i) for every natural number n holds a_{1}(n) = |a(n)|^{p} and b_{2}(n) = |b(n)|^{p} and a_{2}(n) = |a(n) + b(n)|^{p},

(ii) a_{1} is summable, and (iii) b_{2} is summable.

Then a_{2} is summable and (∑ a_{2})^{\frac{1}{p}} ≤ (∑ a_{1})^{\frac{1}{p}} + (∑ b_{2})^{\frac{1}{p}} . hölder’s inequality and minkowski’s … 61

References

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

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

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

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

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

[6] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

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

[8] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

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

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

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

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

62 yasumasa suzuki

The Banach Space l^{p}

Yasumasa Suzuki Take, Yokosuka-shi Japan

Summary. We introduce the arithmetic addition and multiplication in the set of l^{p} real sequences and also introduce the norm. This set has the structure of the Banach space.

MML Identifier: LP SPACE.

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

1. The Real Norm Space of l^{p} Real Sequences

Let x be a sequence of real numbers and let p be a real number. The functor x^{p} yielding a sequence of real numbers is defined as follows:

(Def. 1) For every natural number n holds x^{p}(n) = |x(n)|^{p}.

Let p be a real number. Let us assume that p ≥ 1. The functor l^{p} yielding a non empty subset of the carrier of the linear space of real sequences is defined as follows:

(Def. 2) For every set x holds x ∈ l^{p} iff x ∈ the set of real sequences and

(id_{seq}(x))^{p} is summable. In the sequel a, b, c are real numbers. We now state several propositions:

(1) If a ≥ 0 and a < b and c > 0, then a^{c} < b^{c}.

(2) Let p be a real number. Suppose 1 ≤ p. Let a, b be sequences of real numbers and n be a natural number. Then (∑_{α=0}^{κ}((a+b)^{p})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} ≤ (∑_{α=0}^{κ}(a^{p})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} + (∑_{α=0}^{κ}(b^{p})(α))_{κ∈ℕ}(n)^{\frac{1}{p}} .

(3) Let a, b be sequences of real numbers and p be a real number. Suppose 1 ≤ p and a^{p} is summable and b^{p} is summable. Then (a + b)^{p} is summable and (∑((a + b)^{p}))^{\frac{1}{p}} ≤ (∑(a^{p}))^{\frac{1}{p}} + (∑(b^{p}))^{\frac{1}{p}} .

(4) For every real number p such that 1 ≤ p holds l^{p} is linearly closed.

(5) Let p be a real number. Suppose 1 ≤ p. Then ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences)⟩ is a subspace of the linear space of real sequences.

(6) Let p be a real number. Suppose 1 ≤ p. Then ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences)⟩ is Abelian, addassociative, right zeroed, right complementable, and real linear space-like.

(7) Let p be a real number. Suppose 1 ≤ p. Then ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences)⟩ is a real linear space.

Let p be a real number. The functor l^{p}-norm yielding a function from l^{p} into ℝ is defined by:

(Def. 3) For every set x such that x ∈ l^{p} holds l^{p}-norm(x) = (∑((id_{seq}(x))^{p}))^{\frac{1}{p}} . The following two propositions are true:

(8) Let p be a real number. Suppose 1 ≤ p. Then ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩ is a real linear space.

(9) Let p be a real number. Suppose p ≥ 1. Then ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩ is a subspace of the linear space of real sequences.

2. The Banach Space of l^{p} Real Sequences

Next we state several propositions:

(10) Let p be a real number. Suppose 1 ≤ p. Let l_{1} be a non empty normed structure. Suppose l_{1} = ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩. Then the carrier of l_{1} = l^{p} and for every set x holds x is a vector of l_{1} iff x is a sequence of real numbers and (id_{seq}(x))^{p} is summable and 0_{(l_{1})} = Zeroseq and for every vector x of l_{1} holds x = id_{seq}(x) and for all vectors x, y of l_{1} holds x + y = id_{seq}(x) + id_{seq}(y) and for every real number r and for every

the banach space l^{p} 65

vector x of l_{1} holds r · x = r id_{seq}(x) and for every vector x of l_{1} holds −x = −id_{seq}(x) and id_{seq}(−x) = −id_{seq}(x) and for all vectors x, y of l_{1} holds x−y = id_{seq}(x)−id_{seq}(y) and for every vector x of l_{1} holds (id_{seq}(x))^{p} is summable and for every vector x of l_{1} holds ∥x∥ = (∑((id_{seq}(x))^{p}))^{\frac{1}{p}} .

(11) Let p be a real number. Suppose p ≥ 1. Let r_{1} be a sequence of real numbers. Suppose that for every natural number n holds r_{1}(n) = 0. Then r_{1}^{p} is summable and (∑(r_{1}^{p}))^{\frac{1}{p}} = 0.

(12) Let p be a real number. Suppose 1 ≤ p. Let r_{1} be a sequence of real numbers. Suppose r_{1}^{p} is summable and (∑(r_{1}^{p}))^{\frac{1}{p}} = 0. Let n be a natural number. Then r_{1}(n) = 0.

(13) Let p be a real number. Suppose 1 ≤ p. Let l_{1} be a non empty normed structure. Suppose l_{1} = ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩. Let x, y be points of l_{1} and a be a real number. Then ∥x∥ = 0 iff x = 0_{(l_{1})} and 0 ≤ ∥x∥ and ∥x + y∥ ≤ ∥x∥ + ∥y∥ and ∥a · x∥ = |a| · ∥x∥.

(14) Let p be a real number. Suppose p ≥ 1. Let l_{1} be a non empty normed structure. Suppose l_{1} = ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩. Then l_{1} is real normed space-like.

(15) Let p be a real number. Suppose p ≥ 1. Let l_{1} be a non empty normed structure. Suppose l_{1} = ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩. Then l_{1} is a real normed space.

(16) Let p be a real number. Suppose 1 ≤ p. Let l_{1} be a real normed space. Suppose l_{1} = ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩. Let v_{1} be a sequence of l_{1}. If v_{1} is Cauchy sequence by norm, then v_{1} is convergent.

Let p be a real number. Let us assume that 1 ≤ p. The functor l^{p}-space yielding a real Banach space is defined by the condition (Def. 4).

(Def. 4) l^{p}-space = ⟨l^{p}, Zero (l^{p}, the linear space of real sequences), Add (l^{p}, the linear space of real sequences), Mult (l^{p}, the linear space of real sequences), l^{p}-norm⟩.

References

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

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

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

66 yasumasa suzuki

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

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249–253, 2003.

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

[8] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

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

[10] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

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

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

[13] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.

[14] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.

[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] 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.

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

Lebesgue Integral of Simple Valued Function^{1}

Yasunari Shidama Shinshu University

Nagano

Noboru Endou Gifu National College of Technology

Summary. In this article, the authors introduce Lebesgue integral of simple valued function.

MML Identifier: MESFUNC3.

The terminology and notation used in this paper are introduced in the following papers: [23], [12], [25], [21], [26], [10], [11], [3], [22], [24], [7], [14], [1], [2], [20], [4], [5], [6], [8], [9], [19], [13], [15], [16], [17], and [18].

1. Integral of Simple Valued Function

The following propositions are true:

(1) Let n, m be natural numbers, a be a function from [: Seg n, Seg m:] into ℝ, and p, q be finite sequences of elements of ℝ. Suppose that

(i) dom p = Seg n, (ii) for every natural number i such that i ∈ dom p there exists a finite

sequence r of elements of ℝ such that dom r = Seg m and p(i) = ∑ r and for every natural number j such that j ∈ dom r holds r(j) = a(⟨⟨i, j⟩⟩),

(iii) dom q = Seg m, and (iv) for every natural number j such that j ∈ dom q there exists a finite

sequence s of elements of ℝ such that dom s = Seg n and q(j) = ∑ s and for every natural number i such that i ∈ dom s holds s(i) = a(⟨⟨i, j⟩⟩). Then ∑ p = ∑ q.

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

(2) Let F be a finite sequence of elements of \overline{ℝ} and f be a finite sequence of elements of ℝ. If F = f, then ∑ F = ∑ f.

(3) Let X be a non empty set, S be a σ-field of subsets of X, and f be a partial function from X to \overline{ℝ}. Suppose f is simple function in S. Then there exists a finite sequence F of separated subsets of S and there exists a finite sequence a of elements of \overline{ℝ} such that

(i) dom f = ⋃ rng F, (ii) dom F = dom a, (iii) for every natural number n such that n ∈ dom F and for every set x

such that x ∈ F (n) holds f(x) = a(n), and (iv) for every set x such that x ∈ dom f there exists a finite sequence a_{1} of

elements of \overline{ℝ} such that dom a_{1} = dom a and for every natural number n such that n ∈ dom a_{1} holds a_{1}(n) = a(n) · χ_{F (n),X}(x).

(4) Let X be a set and F be a finite sequence of elements of X. Then F is disjoint valued if and only if for all natural numbers i, j such that i ∈ dom F and j ∈ dom F and i ≠ j holds F (i) misses F (j).

(5) Let X be a non empty set, A be a set, S be a σ-field of subsets of X, F be a finite sequence of separated subsets of S, and G be a finite sequence of elements of S. Suppose dom G = dom F and for every natural number i such that i ∈ dom G holds G(i) = A ∩ F (i). Then G is a finite sequence of separated subsets of S.

(6) Let X be a non empty set, A be a set, and F, G be finite sequences of elements of X. Suppose dom G = dom F and for every natural number i such that i ∈ dom G holds G(i) = A ∩ F (i). Then ⋃ rng G = A ∩ ⋃ rng F.

(7) Let X be a set, F be a finite sequence of elements of X, and i be a natural number. If i ∈ dom F, then F (i) ⊆ ⋃ rng F and F (i) ∩ ⋃ rng F = F (i).

(8) Let X be a non empty set, S be a σ-field of subsets of X, M be a σmeasure on S, and F be a finite sequence of separated subsets of S. Then dom F = dom(M · F).

(9) Let X be a non empty set, S be a σ-field of subsets of X, M be a σmeasure on S, and F be a finite sequence of separated subsets of S. Then M(⋃ rng F) = ∑(M · F).

(10) Let F, G be finite sequences of elements of \overline{ℝ} and a be an extended real

number. Suppose that

(i) a ≠ +∞ and a ≠ −∞ or for every natural number i such that i ∈ dom F holds F (i) < 0\overline{ℝ} or for every natural number i such that i ∈ dom F holds 0\overline{ℝ} < F (i),

(ii) dom F = dom G, and (iii) for every natural number i such that i ∈ dom G holds G(i) = a · F (i).

Then ∑ G = a · ∑ F.

(11) Every finite sequence of elements of ℝ is a finite sequence of elements of

\overline{ℝ}. Let X be a non empty set, let S be a σ-field of subsets of X, let f be a

partial function from X to \overline{ℝ}, let F be a finite sequence of separated subsets of S, and let a be a finite sequence of elements of \overline{ℝ}. We say that F and a are re-presentation of f if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) dom f = ⋃ rng F, (ii) dom F = dom a, and (iii) for every natural number n such that n ∈ dom F and for every set x

such that x ∈ F (n) holds f(x) = a(n). One can prove the following propositions:

(12) Let X be a non empty set, S be a σ-field of subsets of X, and f be a partial function from X to \overline{ℝ}. Suppose f is simple function in S. Then there exists a finite sequence F of separated subsets of S and there exists a finite sequence a of elements of \overline{ℝ} such that F and a are re-presentation of f.

(13) Let X be a non empty set, S be a σ-field of subsets of X, and F be a finite sequence of separated subsets of S. Then there exists a finite sequence G of separated subsets of S such that

(i) ⋃ rng F = ⋃ rng G, and (ii) for every natural number n such that n ∈ dom G holds G(n) ≠ ∅ and

there exists a natural number m such that m ∈ dom F and F (m) = G(n).

(14) Let X be a non empty set, S be a σ-field of subsets of X, and f be a partial function from X to \overline{ℝ}. Suppose f is simple function in S 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 exists a finite sequence a of elements of \overline{ℝ} such that

(i) F and a are re-presentation of f, (ii) a(1) = 0\overline{ℝ}, and (iii) for every natural number n such that 2 ≤ n and n ∈ dom a holds

0\overline{ℝ} < a(n) and a(n) < +∞.

(15) 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}, F be a finite sequence of separated subsets of S, a be a finite sequence of elements of \overline{ℝ}, and x be an element of X. Suppose F and a are re-presentation of f and x ∈ dom f. Then there exists a finite sequence a_{1} of elements of \overline{ℝ} such that dom a_{1} = dom a and for every natural number n such that n ∈ dom a_{1} holds a_{1}(n) = a(n) · χ_{F (n),X}(x) and f(x) = ∑ a_{1}.

(16) Let p be a finite sequence of elements of \overline{ℝ} and q be a finite sequence of elements of ℝ. If p = q, then ∑ p = ∑ q.

70 yasunari shidama and noboru endou

(17) Let p be a finite sequence of elements of \overline{ℝ}. Suppose for every natural number n such that n ∈ dom p holds 0\overline{ℝ} ≤ p(n) and there exists a natural number n such that n ∈ dom p and p(n) = +∞. Then ∑ p = +∞.

Let X be a non empty set, let S be a σ-field of subsets of X, let M be a σ-measure on S, and let f be a partial function from X to \overline{ℝ}. Let us assume 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). The functor integral(X, S, M, f) yielding an element of \overline{ℝ} is defined by the condition (Def. 2).

(Def. 2) 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 re-presentation of f, (ii) a(1) = 0\overline{ℝ}, (iii) for every natural number n such that 2 ≤ n and n ∈ dom a holds

0\overline{ℝ} < a(n) and a(n) < +∞, (iv) dom x = dom F,

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

(vi) integral(X, S, M, f) = ∑ x.

2. Additional Lemma

We now state the proposition

(18) Let a be a finite sequence of elements of \overline{ℝ} and p, N be elements of \overline{ℝ}. Suppose N = len a and for every natural number n such that n ∈ dom a holds a(n) = p. Then ∑ a = N · p.

References

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

[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. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.

[8] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

[9] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

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

lebesgue integral of simple valued function 71

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

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

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

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

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

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

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

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

[20] Andrzej Nȩdzusiak. Probability. Formalized Mathematics, 1(4):745–749, 1990.

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

[22] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

[24] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1(5):979–981, 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.

72 yasunari shidama and noboru endou

Inverse Trigonometric Functions Arcsin and Arccos^{1}

Artur Korniłowicz University of Białystok Yasunari Shidama Shinshu University Nagano

Summary. Notions of inverse sine and inverse cosine have been introduced. Their basic properties have been proved.

MML Identifier: SIN COS6.

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

1. Preliminaries

In this paper r, s are real numbers and i is an integer number. We now state two propositions:

(1) If 0 ≤ r and r < s, then ⌊\frac{r}{s}⌋ = 0.

(2) For every function f and for all sets X, Y such that f↾X is one-to-one and Y ⊆ X holds f↾Y is one-to-one.

2. Functions sine and cosine

We now state four propositions:

(3) −1 ≤ sin r.

(4) sin r ≤ 1.

1The paper was written during the first author’s post-doctoral fellowship granted by the Shinshu University, Japan.

(5) −1 ≤ cos r.

(6) cos r ≤ 1.

One can check that π is positive.

The following propositions are true:

(7) sin(−\frac{π}{2} ) = −1 and (the function sin)(−\frac{π}{2} ) = −1.

(8) (The function sin)(r) = (the function sin)(r + 2 · π · i).

(9) cos(−\frac{π}{2} ) = 0 and (the function cos)(−\frac{π}{2} ) = 0.

(10) (The function cos)(r) = (the function cos)(r + 2 · π · i).

(11) If 2 · π · i < r and r < π + 2 · π · i, then sin r > 0.

(12) If π + 2 · π · i < r and r < 2 · π + 2 · π · i, then sin r < 0.

(13) If −\frac{π}{2} + 2 · π · i < r and r < \frac{π}{2} + 2 · π · i, then cos r > 0.

(14) If \frac{π}{2} + 2 · π · i < r and r < \frac{3}{2} · π + 2 · π · i, then cos r < 0.

(15) If \frac{3}{2} · π + 2 · π · i < r and r < 2 · π + 2 · π · i, then cos r > 0.

(16) If 2 · π · i ≤ r and r ≤ π + 2 · π · i, then sin r ≥ 0.

(17) If π + 2 · π · i ≤ r and r ≤ 2 · π + 2 · π · i, then sin r ≤ 0.

(18) If −\frac{π}{2} + 2 · π · i ≤ r and r ≤ \frac{π}{2} + 2 · π · i, then cos r ≥ 0.

(19) If \frac{π}{2} + 2 · π · i ≤ r and r ≤ \frac{3}{2} · π + 2 · π · i, then cos r ≤ 0.

(20) If \frac{3}{2} · π + 2 · π · i ≤ r and r ≤ 2 · π + 2 · π · i, then cos r ≥ 0.

(21) If 2 · π · i ≤ r and r < 2 · π + 2 · π · i and sin r = 0, then r = 2 · π · i or r = π + 2 · π · i.

(22) If 2 · π · i ≤ r and r < 2 · π + 2 · π · i and cos r = 0, then r = \frac{π}{2} + 2 · π · i or r = \frac{3}{2} · π + 2 · π · i.

(23) If sin r = −1, then r = \frac{3}{2} · π + 2 · π · ⌊\frac{r}{2·π} ⌋.

(24) If sin r = 1, then r = \frac{π}{2} + 2 · π · ⌊\frac{r}{2·π} ⌋.

(25) If cos r = −1, then r = π + 2 · π · ⌊\frac{r}{2·π} ⌋.

(26) If cos r = 1, then r = 2 · π · ⌊\frac{r}{2·π} ⌋.

(27) If 0 ≤ r and r ≤ 2 · π and sin r = −1, then r = \frac{3}{2} · π.

(28) If 0 ≤ r and r ≤ 2 · π and sin r = 1, then r = \frac{π}{2} .

(29) If 0 ≤ r and r ≤ 2 · π and cos r = −1, then r = π.

(30) If 0 ≤ r and r < \frac{π}{2} , then sin r < 1.

(31) If 0 ≤ r and r < \frac{3}{2} · π, then sin r > −1.

(32) If \frac{3}{2} · π < r and r ≤ 2 · π, then sin r > −1.

(33) If \frac{π}{2} < r and r ≤ 2 · π, then sin r < 1.

(34) If 0 < r and r < 2 · π, then cos r < 1.

(35) If 0 ≤ r and r < π, then cos r > −1.

(36) If π < r and r ≤ 2 · π, then cos r > −1.

(37) If 2 · π · i ≤ r and r < \frac{π}{2} + 2 · π · i, then sin r < 1.

(38) If 2 · π · i ≤ r and r < \frac{3}{2} · π + 2 · π · i, then sin r > −1.

(39) If \frac{3}{2} · π + 2 · π · i < r and r ≤ 2 · π + 2 · π · i, then sin r > −1.

(40) If \frac{π}{2} + 2 · π · i < r and r ≤ 2 · π + 2 · π · i, then sin r < 1.

(41) If 2 · π · i < r and r < 2 · π + 2 · π · i, then cos r < 1.

(42) If 2 · π · i ≤ r and r < π + 2 · π · i, then cos r > −1.

(43) If π + 2 · π · i < r and r ≤ 2 · π + 2 · π · i, then cos r > −1.

(44) If cos(2 · π · r) = 1, then r ∈ ℤ.

(45) (The function sin) ^{∘}[−\frac{π}{2} , \frac{π}{2} ] = [−1, 1].

(46) (The function sin) ^{∘}]−\frac{π}{2} , \frac{π}{2} [ = ]−1, 1[.

(47) (The function sin) ^{∘}[\frac{π}{2} , \frac{3}{2} · π] = [−1, 1].

(48) (The function sin) ^{∘}]\frac{π}{2} , \frac{3}{2} · π[ = ]−1, 1[.

(49) (The function cos) ^{∘}[0, π] = [−1, 1].

(50) (The function cos) ^{∘}]0, π[ = ]−1, 1[.

(51) (The function cos) ^{∘}[π, 2 · π] = [−1, 1].

(52) (The function cos) ^{∘}]π, 2 · π[ = ]−1, 1[.

(53) The function sin is increasing on [−\frac{π}{2} + 2 · π · i, \frac{π}{2} + 2 · π · i].

(54) The function sin is decreasing on [\frac{π}{2} + 2 · π · i, \frac{3}{2} · π + 2 · π · i].

(55) The function cos is decreasing on [2 · π · i, π + 2 · π · i].

(56) The function cos is increasing on [π + 2 · π · i, 2 · π + 2 · π · i].

(57) (The function sin)↾[−\frac{π}{2} + 2 · π · i, \frac{π}{2} + 2 · π · i] is one-to-one.

(58) (The function sin)↾[\frac{π}{2} + 2 · π · i, \frac{3}{2} · π + 2 · π · i] is one-to-one. One can check that (the function sin)↾[−\frac{π}{2} , \frac{π}{2} ] is one-to-one and (the function

sin)↾[\frac{π}{2} , \frac{3}{2} · π] is one-to-one. One can check the following observations:

∗ (the function sin)↾[−\frac{π}{2} , 0] is one-to-one,

∗ (the function sin)↾[0, \frac{π}{2} ] is one-to-one,

∗ (the function sin)↾[\frac{π}{2} , π] is one-to-one,

∗ (the function sin)↾[π, \frac{3}{2} · π] is one-to-one, and

∗ (the function sin)↾[\frac{3}{2} · π, 2 · π] is one-to-one.

One can verify the following observations:

∗ (the function sin)↾]−\frac{π}{2} , \frac{π}{2} [ is one-to-one,

∗ (the function sin)↾]\frac{π}{2} , \frac{3}{2} · π[ is one-to-one,

∗ (the function sin)↾]−\frac{π}{2} , 0[ is one-to-one,

∗ (the function sin)↾]0, \frac{π}{2} [ is one-to-one,

∗ (the function sin)↾]\frac{π}{2} , π[ is one-to-one,

∗ (the function sin)↾]π, \frac{3}{2} · π[ is one-to-one, and

∗ (the function sin)↾]\frac{3}{2} · π, 2 · π[ is one-to-one.

Next we state two propositions:

(59) (The function cos)↾[2 · π · i, π + 2 · π · i] is one-to-one.

(60) (The function cos)↾[π + 2 · π · i, 2 · π + 2 · π · i] is one-to-one. Let us note that (the function cos)↾[0, π] is one-to-one and (the function

cos)↾[π, 2 · π] is one-to-one.

One can check the following observations:

∗ (the function cos)↾[0, \frac{π}{2} ] is one-to-one,

∗ (the function cos)↾[\frac{π}{2} , π] is one-to-one,

∗ (the function cos)↾[π, \frac{3}{2} · π] is one-to-one, and

∗ (the function cos)↾[\frac{3}{2} · π, 2 · π] is one-to-one. One can check the following observations:

∗ (the function cos)↾]0, π[ is one-to-one,

∗ (the function cos)↾]π, 2 · π[ is one-to-one,

∗ (the function cos)↾]0, \frac{π}{2} [ is one-to-one,

∗ (the function cos)↾]\frac{π}{2} , π[ is one-to-one,

∗ (the function cos)↾]π, \frac{3}{2} · π[ is one-to-one, and

∗ (the function cos)↾]\frac{3}{2} · π, 2 · π[ is one-to-one. The following proposition is true

(61) If 2 · π · i ≤ r and r < 2 · π + 2 · π · i and 2 · π · i ≤ s and s < 2 · π + 2 · π · i and sin r = sin s and cos r = cos s, then r = s.

3. Function arcsin

The function arcsin is a partial function from ℝ to ℝ and is defined by: (Def. 1) The function arcsin = ((the function sin)↾[−\frac{π}{2} , \frac{π}{2} ])^{−1}.

Let r be a set. The functor arcsin r is defined by: (Def. 2) arcsin r = (the function arcsin)(r).

Let r be a set. Then arcsin r is a real number. Next we state two propositions:

(62) (The function arcsin) ^{−1} = (the function sin)↾[−\frac{π}{2} , \frac{π}{2} ].

(63) rng (the function arcsin) = [−\frac{π}{2} , \frac{π}{2} ].

Let us note that the function arcsin is one-to-one. The following propositions are true:

(64) dom (the function arcsin) = [−1, 1].

(65) ((The function sin)↾[−\frac{π}{2} , \frac{π}{2} ] qua function) ·(the function arcsin) = id_{[−1,1]}.

(66) (The function arcsin) ·((the function sin)↾[−\frac{π}{2} , \frac{π}{2} ]) = id_{[−1,1]}.

(67) ((The function sin)↾[−\frac{π}{2} , \frac{π}{2} ]) · (the function arcsin) = id_{[− \frac{π}{2} , \frac{π}{2} ]}.

(68) (The function arcsin qua function) ·((the function sin)↾[−\frac{π}{2} , \frac{π}{2} ]) = id_{[− \frac{π}{2} , \frac{π}{2} ]}.

(69) If −1 ≤ r and r ≤ 1, then sin arcsin r = r.

(70) If −\frac{π}{2} ≤ r and r ≤ \frac{π}{2} , then arcsin sin r = r.

(71) arcsin(−1) = −\frac{π}{2} .

(72) arcsin 0 = 0.

(73) arcsin 1 = \frac{π}{2} .

(74) If −1 ≤ r and r ≤ 1 and arcsin r = −\frac{π}{2} , then r = −1.

(75) If −1 ≤ r and r ≤ 1 and arcsin r = 0, then r = 0.

(76) If −1 ≤ r and r ≤ 1 and arcsin r = \frac{π}{2} , then r = 1.

(77) If −1 ≤ r and r ≤ 1, then −\frac{π}{2} ≤ arcsin r and arcsin r ≤ \frac{π}{2} .

(78) If −1 < r and r < 1, then −\frac{π}{2} < arcsin r and arcsin r < \frac{π}{2} .

(79) If −1 ≤ r and r ≤ 1, then arcsin r = −arcsin(−r).

(80) If 0 ≤ s and r^{2} + s^{2} = 1, then cos arcsin r = s.

(81) If s ≤ 0 and r^{2} + s^{2} = 1, then cos arcsin r = −s.

(82) If −1 ≤ r and r ≤ 1, then cos arcsin r = \sqrt{1 − r^{2}}.

(83) The function arcsin is increasing on [−1, 1].

(84) The function arcsin is differentiable on ]−1, 1[ and if −1 < r and r < 1, then (the function arcsin)′(r) = \frac{1}{\sqrt{1−r^{2}}} .

(85) The function arcsin is continuous on [−1, 1].

4. Function arccos

The function arccos is a partial function from ℝ to ℝ and is defined by: (Def. 3) The function arccos = ((the function cos)↾[0, π])^{−1}.

Let r be a set. The functor arccos r is defined by: (Def. 4) arccos r = (the function arccos)(r).

Let r be a set. Then arccos r is a real number.

One can prove the following two propositions:

(86) (The function arccos) ^{−1} = (the function cos)↾[0, π].

(87) rng (the function arccos) = [0, π].

Let us note that the function arccos is one-to-one.

The following propositions are true:

(88) dom (the function arccos) = [−1, 1].

(89) ((The function cos)↾[0, π] qua function) ·(the function arccos) = id_{[−1,1]}.

(90) (The function arccos) ·((the function cos)↾[0, π]) = id_{[−1,1]}.

(91) ((The function cos)↾[0, π]) · (the function arccos) = id_{[0,π]}.

(92) (The function arccos qua function) ·((the function cos)↾[0, π]) = id_{[0,π]}.

(93) If −1 ≤ r and r ≤ 1, then cos arccos r = r.

(94) If 0 ≤ r and r ≤ π, then arccos cos r = r.

(95) arccos(−1) = π.

(96) arccos 0 = \frac{π}{2} .

(97) arccos 1 = 0.

(98) If −1 ≤ r and r ≤ 1 and arccos r = 0, then r = 1.

(99) If −1 ≤ r and r ≤ 1 and arccos r = \frac{π}{2} , then r = 0.

(100) If −1 ≤ r and r ≤ 1 and arccos r = π, then r = −1.

(101) If −1 ≤ r and r ≤ 1, then 0 ≤ arccos r and arccos r ≤ π.

(102) If −1 < r and r < 1, then 0 < arccos r and arccos r < π.

(103) If −1 ≤ r and r ≤ 1, then arccos r = π − arccos(−r).

(104) If 0 ≤ s and r^{2} + s^{2} = 1, then sin arccos r = s.

(105) If s ≤ 0 and r^{2} + s^{2} = 1, then sin arccos r = −s.

(106) If −1 ≤ r and r ≤ 1, then sin arccos r = \sqrt{1 − r^{2}}.

(107) The function arccos is decreasing on [−1, 1].

(108) The function arccos is differentiable on ]−1, 1[ and if −1 < r and r < 1, then (the function arccos)′(r) = −\frac{1}{\sqrt{1−r^{2}}} .

(109) The function arccos is continuous on [−1, 1].

(110) If −1 ≤ r and r ≤ 1, then arcsin r + arccos r = \frac{π}{2} .

(111) If −1 ≤ r and r ≤ 1, then arccos(−r) − arcsin r = \frac{π}{2} .

(112) If −1 ≤ r and r ≤ 1, then arccos r − arcsin(−r) = \frac{π}{2} .

References

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

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 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. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

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

[7] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

[8] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

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

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

inverse trigonometric functions arcsin and … 79

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

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

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

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

[17] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

80 artur korniłowicz and yasunari shidama

On Some Points of a Simple Closed Curve^{1}

Artur Korniłowicz University of Białystok

MML Identifier: JORDAN21.

The notation and terminology used here are introduced in the following papers: [26], [28], [2], [13], [1], [29], [5], [18], [17], [3], [14], [24], [9], [23], [4], [25], [7], [10], [11], [12], [19], [20], [22], [21], [6], [8], [15], [16], and [27].

1. On the Subsets of E_{T}^{2}

For simplicity, we follow the rules: C denotes a simple closed curve, P denotes a subset of E_{T}^{2}, R denotes a non empty subset of E_{T}^{2}, p denotes a point of E_{T}^{2}, and i, j, k, m, n denote natural numbers.

One can prove the following propositions:

(1) For every point p of E_{T}^{n} holds {p} is Bounded.

(2) For all real numbers s_{1}, t and for every subset P of E_{T}^{2} such that P = {[s, t]; s ranges over real numbers: s_{1} < s} holds P is convex.

(3) For all real numbers s_{2}, t and for every subset P of E_{T}^{2} such that P = {[s, t]; s ranges over real numbers: s < s_{2}} holds P is convex.

(4) For all real numbers s, t_{1} and for every subset P of E_{T}^{2} such that P = {[s, t]; t ranges over real numbers: t_{1} < t} holds P is convex.

(5) For all real numbers s, t_{2} and for every subset P of E_{T}^{2} such that P = {[s, t]; t ranges over real numbers: t < t_{2}} holds P is convex.

(6) NorthHalfline p \ {p} is convex.

(7) SouthHalfline p \ {p} is convex.

(8) WestHalfline p \ {p} is convex.

(9) EastHalfline p \ {p} is convex.

1The paper has been completed during the author’s post-doctoral fellowship granted by the Shinshu University, Japan.

(10) For every subset A of the carrier of E_{T}^{2} holds UBD A misses A.

(11) Let P be a subset of the carrier of E_{T}^{2} and 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 p_{1} ≠ q_{1} and p_{2} ≠ q_{2}. Then p_{1} ∉ Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) and p_{2} ∉ Segment(P, p_{1}, p_{2}, q_{1}, q_{2}).

(12) proj2^{∘}(C ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2})) is not empty.

(13) For every compact subset C of E_{T}^{2} holds proj2^{∘}(C ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2})) is closed, lower bounded, and upper bounded.

2. Gauges

The following propositions are true:

(14) ⟨⟨1, 1⟩⟩ ∈ the indices of Gauge(R, n).

(15) ⟨⟨1, 2⟩⟩ ∈ the indices of Gauge(R, n).

(16) ⟨⟨2, 1⟩⟩ ∈ the indices of Gauge(R, n).

(17) Let C be a non vertical non horizontal compact subset of E_{T}^{2}. Suppose m > k and ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, k) and ⟨⟨i, j + 1⟩⟩ ∈ the indices of Gauge(C, k). Then ρ(Gauge(C, m) ∘ (i, j), Gauge(C, m) ∘ (i, j + 1)) < ρ(Gauge(C, k) ∘ (i, j), Gauge(C, k) ∘ (i, j + 1)).

(18) For every non vertical non horizontal compact subset C of E_{T}^{2} such that m > k holds ρ(Gauge(C, m)∘(1, 1), Gauge(C, m)∘(1, 2)) < ρ(Gauge(C, k)∘ (1, 1), Gauge(C, k) ∘ (1, 2)).

(19) Let C be a non vertical non horizontal compact subset of E_{T}^{2}. Suppose m > k and ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, k) and ⟨⟨i + 1, j⟩⟩ ∈ the indices of Gauge(C, k). Then ρ(Gauge(C, m) ∘ (i, j), Gauge(C, m) ∘ (i + 1, j)) < ρ(Gauge(C, k) ∘ (i, j), Gauge(C, k) ∘ (i + 1, j)).

(20) For every non vertical non horizontal compact subset C of E_{T}^{2} such that m > k holds ρ(Gauge(C, m)∘(1, 1), Gauge(C, m)∘(2, 1)) < ρ(Gauge(C, k)∘ (1, 1), Gauge(C, k) ∘ (2, 1)).

(21) Let r, t be real numbers. Suppose r > 0 and t > 0. Then there exists a natural number n such that i < n and ρ(Gauge(C, n)∘(1, 1), Gauge(C, n)∘ (1, 2)) < r and ρ(Gauge(C, n) ∘ (1, 1), Gauge(C, n) ∘ (2, 1)) < t.

3. Middle Points

We now state four propositions:

(22) UpperMiddlePoint C ∈ C.

(23) LowerMiddlePoint C ∈ C.

(24) (LowerMiddlePoint C)_{2} ≠ (UpperMiddlePoint C)_{2}.

(25) LowerMiddlePoint C ≠ UpperMiddlePoint C.

4. UpperArc and LowerArc

Next we state several propositions:

(26) W-bound(C) = W-bound(UpperArc(C)).

(27) E-bound(C) = E-bound(UpperArc(C)).

(28) W-bound(C) = W-bound(LowerArc(C)).

(29) E-bound(C) = E-bound(LowerArc(C)).

(30) UpperArc(C) ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2}) is not empty and proj2^{∘}(UpperArc(C) ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2})) is not empty.

(31) LowerArc(C) ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2}) is not empty and proj2^{∘}(LowerArc(C) ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2})) is not empty.

(32) For every compact connected subset P of E_{T}^{2} such that P ⊆ C and W_{min}(C) ∈ P and E_{max}(C) ∈ P holds UpperArc(C) ⊆ P or LowerArc(C) ⊆ P.

5. UMP and LMP

Let P be a subset of the carrier of E_{T}^{2}. The functor UMP P yielding a point of E_{T}^{2} is defined by:

(Def. 1) UMP P = [\frac{E-bound(P)+W-bound(P)}{2},

sup(proj2^{∘}(P ∩ VerticalLine(\frac{E-bound(P)+W-bound(P)}{2})))]. The functor LMP P yielding a point of E_{T}^{2} is defined as follows:

(Def. 2) LMP P = [\frac{E-bound(P)+W-bound(P)}{2},

inf(proj2^{∘}(P ∩ VerticalLine(\frac{E-bound(P)+W-bound(P)}{2})))]. We now state a number of propositions:

(33) (UMP P)_{1} = \frac{W-bound(P)+E-bound(P)}{2}.

(34) (UMP P)_{2} = sup(proj2^{∘}(P ∩ VerticalLine(\frac{E-bound(P)+W-bound(P)}{2}))).

(35) (LMP P)_{1} = \frac{W-bound(P)+E-bound(P)}{2}.

(36) (LMP P)_{2} = inf(proj2^{∘}(P ∩ VerticalLine(\frac{E-bound(P)+W-bound(P)}{2}))).

(37) For every non vertical compact subset C of E_{T}^{2} holds UMP C ≠ W_{min}(C).

(38) For every non vertical compact subset C of E_{T}^{2} holds UMP C ≠ E_{max}(C).

(39) For every non vertical compact subset C of E_{T}^{2} holds LMP C ≠ W_{min}(C).

(40) For every non vertical compact subset C of E_{T}^{2} holds LMP C ≠ E_{max}(C).

(41) For every compact subset C of E_{T}^{2} such that p ∈ C ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2}) holds p_{2} ≤ (UMP C)_{2}.

(42) For every compact subset C of E_{T}^{2} such that p ∈ C ∩ VerticalLine(\frac{W-bound(C)+E-bound(C)}{2}) holds (LMP C)_{2} ≤ p_{2}.

(43) UMP C ∈ C.

(44) LMP C ∈ C.

(45) L(UMP P, [\frac{W-bound(P)+E-bound(P)}{2}, N-bound(P)]) is vertical.

(46) L(LMP P, [\frac{W-bound(P)+E-bound(P)}{2}, S-bound(P)]) is vertical.

(47) L(UMP C, [\frac{W-bound(C)+E-bound(C)}{2}, N-bound(C)]) ∩ C = \{UMP C\}.

(48) L(LMP C, [\frac{W-bound(C)+E-bound(C)}{2}, S-bound(C)]) ∩ C = \{LMP C\}.

(49) (LMP C)_{2} < (UMP C)_{2}.

(50) UMP C ≠ LMP C.

(51) S-bound(C) < (UMP C)_{2}.

(52) (UMP C)_{2} ≤ N-bound(C).

(53) S-bound(C) ≤ (LMP C)_{2}.

(54) (LMP C)_{2} < N-bound(C).

(55) L(UMP C, [\frac{W-bound(C)+E-bound(C)}{2}, N-bound(C)]) misses L(LMP C, [\frac{W-bound(C)+E-bound(C)}{2}, S-bound(C)]).

(56) Let A, B be subsets of E_{T}^{2}. Suppose A ⊆ B and W-bound(A) + E-bound(A) = W-bound(B) + E-bound(B) and A ∩ VerticalLine(\frac{W-bound(A)+E-bound(A)}{2}) is non empty and proj2^{∘}(B ∩ VerticalLine(\frac{W-bound(A)+E-bound(A)}{2})) is upper bounded. Then (UMP A)_{2} ≤ (UMP B)_{2}.

(57) Let A, B be subsets of E_{T}^{2}. Suppose A ⊆ B and W-bound(A) + E-bound(A) = W-bound(B) + E-bound(B) and A ∩ VerticalLine(\frac{W-bound(A)+E-bound(A)}{2}) is non empty and proj2^{∘}(B ∩ VerticalLine(\frac{W-bound(A)+E-bound(A)}{2})) is lower bounded. Then (LMP B)_{2} ≤ (LMP A)_{2}.

(58) Let A, B be subsets of E_{T}^{2}. Suppose A ⊆ B and UMP B ∈ A and A ∩ VerticalLine(\frac{W-bound(A)+E-bound(A)}{2}) is non empty and proj2^{∘}(B ∩ VerticalLine(\frac{W-bound(B)+E-bound(B)}{2})) is upper bounded and W-bound(A) + E-bound(A) = W-bound(B) + E-bound(B). Then UMP A = UMP B.

(59) Let A, B be subsets of E_{T}^{2}. Suppose A ⊆ B and LMP B ∈ A and A ∩ VerticalLine(\frac{W-bound(A)+E-bound(A)}{2}) is non empty and proj2^{∘}(B ∩ VerticalLine(\frac{W-bound(B)+E-bound(B)}{2})) is lower bounded and W-bound(A)+ E-bound(A) = W-bound(B) + E-bound(B). Then LMP A = LMP B.

(60) (UMP UpperArc(C))_{2} ≤ N-bound(C).

(61) S-bound(C) ≤ (LMP LowerArc(C))_{2}.

(62) LMP C ∉ LowerArc(C) or UMP C ∉ LowerArc(C).

(63) LMP C ∉ UpperArc(C) or UMP C ∉ UpperArc(C).

(64) If 0 < n, then sup(proj2^{∘}(L̃(Cage(C, n)) ∩ L(Gauge(C, n) ∘ (Center Gauge(C, n), 1), Gauge(C, n) ∘ (Center Gauge(C, n), len Gauge(C, n))))) = sup(proj2^{∘}(L̃(Cage(C, n)) ∩ VerticalLine (\frac{E-bound(L̃(Cage(C,n)))+W-bound(L̃(Cage(C,n)))}{2}))).

(65) If 0 < n, then inf(proj2^{∘}(L̃(Cage(C, n)) ∩ L(Gauge(C, n) ∘ (Center Gauge(C, n), 1), Gauge(C, n) ∘ (Center Gauge(C, n), len Gauge(C, n))))) = inf(proj2^{∘}(L̃(Cage(C, n)) ∩ VerticalLine (\frac{E-bound(L̃(Cage(C,n)))+W-bound(L̃(Cage(C,n)))}{2}))).

(66) If 0 < n, then UMP L̃(Cage(C, n)) = [\frac{E-bound(L̃(Cage(C,n)))+W-bound(L̃(Cage(C,n)))}{2}, sup(proj2^{∘}(L̃(Cage(C, n)) ∩L(Gauge(C, n) ∘ (Center Gauge(C, n), 1), Gauge(C, n)∘ (Center Gauge(C, n), len Gauge(C, n)))))].

(67) If 0 < n, then LMP L̃(Cage(C, n)) = [\frac{E-bound(L̃(Cage(C,n)))+W-bound(L̃(Cage(C,n)))}{2}, inf(proj2^{∘}(L̃(Cage(C, n)) ∩L(Gauge(C, n) ∘ (Center Gauge(C, n), 1), Gauge(C, n)∘ (Center Gauge(C, n), len Gauge(C, n)))))].

(68) (UMP C)_{2} < (UMP L̃(Cage(C, n)))_{2}.

(69) (LMP C)_{2} > (LMP L̃(Cage(C, n)))_{2}.

(70) UMP UpperArc(L̃(Cage(C, n))) ∈ UpperArc(L̃(Cage(C, n))).

(71) LMP LowerArc(L̃(Cage(C, n))) ∈ LowerArc(L̃(Cage(C, n))).

(72) If 0 < n, then there exists a natural number i such that 1 ≤ i and i ≤ len Gauge(C, n) and UMP L̃(Cage(C, n)) = Gauge(C, n) ∘ (Center Gauge(C, n), i).

(73) If 0 < n, then there exists a natural number i such that 1 ≤ i and i ≤ len Gauge(C, n) and LMP L̃(Cage(C, n)) = Gauge(C, n) ∘ (Center Gauge(C, n), i).

(74) If 0 < n, then UMP L̃(Cage(C, n)) = UMP UpperArc(L̃(Cage(C, n))).

(75) If 0 < n, then LMP L̃(Cage(C, n)) = LMP LowerArc(L̃(Cage(C, n))).

(76) If 0 < n, then (UMP C)_{2} < (UMP UpperArc(L̃(Cage(C, n))))_{2}.

(77) If 0 < n, then (LMP LowerArc(L̃(Cage(C, n))))_{2} < (LMP C)_{2}.

(78) If i ≤ j, then (UMP L̃(Cage(C, j)))_{2} ≤ (UMP L̃(Cage(C, i)))_{2}.

(79) If i ≤ j, then (LMP L̃(Cage(C, i)))_{2} ≤ (LMP L̃(Cage(C, j)))_{2}.

(80) If 0 < i and i ≤ j, then (UMP UpperArc(L̃(Cage(C, j))))_{2} ≤ (UMP UpperArc(L̃(Cage(C, i))))_{2}.

86 artur korniłowicz

(81) If 0 < i and i ≤ j, then (LMP LowerArc(L̃(Cage(C, i))))_{2} ≤ (LMP LowerArc(L̃(Cage(C, j))))_{2}.

Acknowledgments

The author would like to acknowledge Professor Andrzej Trybulec for his continuous encouragement to complete the proof of the Jordan Curve Theorem in Mizar as well as his precious suggestions regarding the proof itself.

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 from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Gauges. Formalized Mathematics, 8(1):25–27, 1999.

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

[8] Czesław Byliński and Mariusz Żynel. Cages – the external approximation of Jordan’s curve. Formalized Mathematics, 9(1):19–24, 2001.

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

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

[11] 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.

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

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

[14] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

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

[16] Artur Korniłowicz, Robert Milewski, Adam Naumowicz, and Andrzej Trybulec. Gauges and cages. Part I. Formalized Mathematics, 9(3):501–509, 2001.

[17] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

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

[19] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

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

[21] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.

[22] Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1–13, 1999.

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

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

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

on some points of a simple closed curve 87

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

[27] Andrzej Trybulec. On the minimal distance between sets in Euclidean space. Formalized Mathematics, 10(3):153–158, 2002.

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

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

88 artur korniłowicz

On Some Points of a Simple Closed Curve. Part II

Artur Korniłowicz^{1} University of Białystok

Adam Grabowski^{2} University of Białystok

Summary. In the paper we formalize some lemmas needed by the proof of the Jordan Curve Theorem according to [23]. We show basic properties of the upper and the lower approximations of a simple closed curve (as its compactness and connectedness) and some facts about special points of such approximations.

MML Identifier: JORDAN22.

The notation and terminology used in this paper are introduced in the following papers: [25], [28], [1], [24], [29], [4], [16], [15], [2], [12], [22], [7], [27], [21], [13], [3], [5], [8], [9], [10], [18], [19], [20], [26], [6], [11], [17], and [14].

1. Properties of the Approximations

In this paper C denotes a simple closed curve and i denotes a natural number. We now state two propositions:

(1) (UpperAppr(C))(i) ⊆ \overline{RightComp(Cage(C, 0))}.

(2) (LowerAppr(C))(i) ⊆ \overline{RightComp(Cage(C, 0))}.

Let C be a simple closed curve. One can verify that UpperArc(C) is connected and LowerArc(C) is connected.

We now state two propositions:

(3) (UpperAppr(C))(i) is compact and connected.

(4) (LowerAppr(C))(i) is compact and connected.

1The paper has been completed during the first author’s post-doctoral fellowship granted by the Shinshu University, Japan.

2This work has been partially supported by the KBN grant 4 T11C 039 24 and the FP6 IST grant TYPES No. 510996.

90 artur korniłowicz and adam grabowski

Let C be a simple closed curve. Observe that NorthArc(C) is compact and SouthArc(C) is compact.

2. On Special Points of Approximations

One can prove the following propositions:

(5) W_{min}(C) ∈ NorthArc(C).

(6) E_{max}(C) ∈ NorthArc(C).

(7) W_{min}(C) ∈ SouthArc(C).

(8) E_{max}(C) ∈ SouthArc(C).

(9) UMP C ∈ NorthArc(C).

(10) LMP C ∈ SouthArc(C).

(11) NorthArc(C) ⊆ C.

(12) SouthArc(C) ⊆ C.

(13) LMP C ∈ LowerArc(C) and UMP C ∈ UpperArc(C) or UMP C ∈ LowerArc(C) and LMP C ∈ UpperArc(C).

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

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

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

[6] Czesław Byliński and Mariusz Żynel. Cages – the external approximation of Jordan’s curve. Formalized Mathematics, 9(1):19–24, 2001.

[7] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 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}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

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

[11] Adam Grabowski. On the Kuratowski limit operators. Formalized Mathematics, 11(4):399–409, 2003.

[12] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

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

[14] Artur Korniłowicz. On some points of a simple closed curve. Formalized Mathematics, 13(1):81–87, 2005.

[15] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

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

on some points of a simple closed curve. … 91

[17] Robert Milewski. On the upper and lower approximations of the curve. Formalized Mathematics, 11(4):425–430, 2003.

[18] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[19] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

[20] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.

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

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

[23] Yukio Takeuchi and Yatsuka Nakamura. On the Jordan curve theorem. Technical Report 19804, Dept. of Information Eng., Shinshu University, 500 Wakasato, Nagano city, Japan, April 1980.

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

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

[26] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

[27] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5(3):317–322, 1996.

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

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

92 artur korniłowicz and adam grabowski

Uniform Continuity of Functions on Normed Complex Linear Spaces

Noboru Endou Gifu National College of Technology

MML Identifier: NCFCONT2.

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

1. Uniform Continuity of Functions on Real and Complex Normed Linear Spaces

For simplicity, we follow the rules: X, X_{1} denote sets, r, s denote real numbers, z denotes a complex number, R_{1} denotes a real normed space, and C_{1}, C_{2}, C_{3} denote complex normed spaces.

Let X be a set, let C_{2}, C_{3} be complex normed spaces, and let f be a partial function from C_{2} to C_{3}. We say that f is uniformly continuous on X if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) X ⊆ dom f, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all points x_{1}, x_{2} of C_{2} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds ∥f_{x_{1}} − f_{x_{2}}∥ < r.

Let X be a set, let R_{1} be a real normed space, let C_{1} be a complex normed space, and let f be a partial function from C_{1} to R_{1}. We say that f is uniformly continuous on X if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) X ⊆ dom f, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all points x_{1}, x_{2} of C_{1} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds ∥f_{x_{1}} − f_{x_{2}}∥ < r.

Let X be a set, let R_{1} be a real normed space, let C_{1} be a complex normed space, and let f be a partial function from R_{1} to C_{1}. We say that f is uniformly continuous on X if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) X ⊆ dom f, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all points x_{1}, x_{2} of R_{1} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds ∥f_{x_{1}} − f_{x_{2}}∥ < r.

Let X be a set, let C_{1} be a complex normed space, and let f be a partial function from the carrier of C_{1} to ℂ. We say that f is uniformly continuous on X if and only if the conditions (Def. 4) are satisfied.

(Def. 4)(i) X ⊆ dom f, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all points x_{1}, x_{2} of C_{1} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds |f_{x_{1}} − f_{x_{2}}| < r.

Let X be a set, let C_{1} be a complex normed space, and let f be a partial function from the carrier of C_{1} to ℝ. We say that f is uniformly continuous on X if and only if the conditions (Def. 5) are satisfied.

(Def. 5)(i) X ⊆ dom f, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all points x_{1}, x_{2} of C_{1} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds |f_{x_{1}} − f_{x_{2}}| < r.

Let X be a set, let R_{1} be a real normed space, and let f be a partial function from the carrier of R_{1} to ℂ. We say that f is uniformly continuous on X if and only if the conditions (Def. 6) are satisfied.

(Def. 6)(i) X ⊆ dom f, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all points x_{1}, x_{2} of R_{1} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds |f_{x_{1}} − f_{x_{2}}| < r.

Next we state a number of propositions:

(1) Let f be a partial function from C_{2} to C_{3}. Suppose f is uniformly continuous on X and X_{1} ⊆ X. Then f is uniformly continuous on X_{1}.

(2) Let f be a partial function from C_{1} to R_{1}. Suppose f is uniformly continuous on X and X_{1} ⊆ X. Then f is uniformly continuous on X_{1}.

(3) Let f be a partial function from R_{1} to C_{1}. Suppose f is uniformly continuous on X and X_{1} ⊆ X. Then f is uniformly continuous on X_{1}.

(4) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3}. Suppose f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}. Then f_{1} + f_{2} is uniformly continuous on X ∩ X_{1}.

(5) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1}. Suppose f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}. Then f_{1} + f_{2} is

uniformly continuous on X ∩ X_{1}.

(6) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1}. Suppose f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}. Then f_{1} + f_{2} is uniformly continuous on X ∩ X_{1}.

(7) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3}. Suppose f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}. Then f_{1} − f_{2} is uniformly continuous on X ∩ X_{1}.

(8) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1}. Suppose f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}. Then f_{1} − f_{2} is uniformly continuous on X ∩ X_{1}.

(9) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1}. Suppose f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}. Then f_{1} − f_{2} is uniformly continuous on X ∩ X_{1}.

(10) Let f be a partial function from C_{2} to C_{3}. If f is uniformly continuous on X, then z f is uniformly continuous on X.

(11) Let f be a partial function from C_{1} to R_{1}. If f is uniformly continuous on X, then r f is uniformly continuous on X.

(12) Let f be a partial function from R_{1} to C_{1}. If f is uniformly continuous on X, then z f is uniformly continuous on X.

(13) Let f be a partial function from C_{2} to C_{3}. If f is uniformly continuous on X, then −f is uniformly continuous on X.

(14) Let f be a partial function from C_{1} to R_{1}. If f is uniformly continuous on X, then −f is uniformly continuous on X.

(15) Let f be a partial function from R_{1} to C_{1}. If f is uniformly continuous on X, then −f is uniformly continuous on X.

(16) Let f be a partial function from C_{2} to C_{3}. If f is uniformly continuous on X, then ∥f∥ is uniformly continuous on X.

(17) Let f be a partial function from C_{1} to R_{1}. If f is uniformly continuous on X, then ∥f∥ is uniformly continuous on X.

(18) Let f be a partial function from R_{1} to C_{1}. If f is uniformly continuous on X, then ∥f∥ is uniformly continuous on X.

(19) For every partial function f from C_{2} to C_{3} such that f is uniformly continuous on X holds f is continuous on X.

(20) For every partial function f from C_{1} to R_{1} such that f is uniformly continuous on X holds f is continuous on X.

(21) For every partial function f from R_{1} to C_{1} such that f is uniformly continuous on X holds f is continuous on X.

(22) Let f be a partial function from the carrier of C_{1} to ℂ. If f is uniformly continuous on X, then f is continuous on X.

(23) Let f be a partial function from the carrier of C_{1} to ℝ. If f is uniformly continuous on X, then f is continuous on X.

(24) Let f be a partial function from the carrier of R_{1} to ℂ. If f is uniformly continuous on X, then f is continuous on X.

(25) For every partial function f from C_{2} to C_{3} such that f is Lipschitzian on X holds f is uniformly continuous on X.

(26) For every partial function f from C_{1} to R_{1} such that f is Lipschitzian on X holds f is uniformly continuous on X.

(27) For every partial function f from R_{1} to C_{1} such that f is Lipschitzian on X holds f is uniformly continuous on X.

(28) Let f be a partial function from C_{2} to C_{3} and Y be a subset of C_{2}. Suppose Y is compact and f is continuous on Y. Then f is uniformly continuous on Y.

(29) Let f be a partial function from C_{1} to R_{1} and Y be a subset of C_{1}. Suppose Y is compact and f is continuous on Y. Then f is uniformly continuous on Y.

(30) Let f be a partial function from R_{1} to C_{1} and Y be a subset of R_{1}. Suppose Y is compact and f is continuous on Y. Then f is uniformly continuous on Y.

(31) Let f be a partial function from C_{2} to C_{3} and Y be a subset of C_{2}. Suppose Y ⊆ dom f and Y is compact and f is uniformly continuous on Y. Then f^{∘}Y is compact.

(32) Let f be a partial function from C_{1} to R_{1} and Y be a subset of C_{1}. Suppose Y ⊆ dom f and Y is compact and f is uniformly continuous on Y. Then f^{∘}Y is compact.

(33) Let f be a partial function from R_{1} to C_{1} and Y be a subset of R_{1}. Suppose Y ⊆ dom f and Y is compact and f is uniformly continuous on Y. Then f^{∘}Y is compact.

(34) Let f be a partial function from the carrier of C_{1} to ℝ and Y be a subset of C_{1}. Suppose Y ≠ ∅ and Y ⊆ dom f and Y is compact and f is uniformly continuous on Y. Then there exist points x_{1}, x_{2} of C_{1} such that x_{1} ∈ Y and x_{2} ∈ Y and f_{x_{1}} = sup(f^{∘}Y) and f_{x_{2}} = inf(f^{∘}Y).

(35) Let f be a partial function from C_{2} to C_{3}. If X ⊆ dom f and f is a constant on X, then f is uniformly continuous on X.

(36) Let f be a partial function from C_{1} to R_{1}. If X ⊆ dom f and f is a constant on X, then f is uniformly continuous on X.

(37) Let f be a partial function from R_{1} to C_{1}. If X ⊆ dom f and f is a constant on X, then f is uniformly continuous on X.

uniform continuity of functions on normed … 97

2. Contraction Mapping Principle on Normed Complex Linear Spaces

Let M be a complex Banach space. A function from the carrier of M into the carrier of M is said to be a contraction of M if:

(Def. 7) There exists a real number L such that 0 < L and L < 1 and for all points x, y of M holds ∥it(x) − it(y)∥ ≤ L · ∥x − y∥.

One can prove the following four propositions:

(38) For every complex normed space X and for all points x, y of X holds ∥x − y∥ > 0 iff x ≠ y.

(39) For every complex normed space X and for all points x, y of X holds ∥x − y∥ = ∥y − x∥.

(40) Let X be a complex Banach space and f be a function from X into X. Suppose f is a contraction of X. Then there exists a point x_{3} of X such that f(x_{3}) = x_{3} and for every point x of X such that f(x) = x holds x_{3} = x.

(41) Let X be a complex Banach space and f be a function from X into X. Given a natural number n_{0} such that f^{n_{0}} is a contraction of X. Then there exists a point x_{3} of X such that f(x_{3}) = x_{3} and for every point x of X such that f(x) = x holds x_{3} = x.

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

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98 noboru endou

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

[18] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

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Introduction to Real Linear Topological Spaces^{1}

Czesław Byliński University of Białystok

MML Identifier: RLTOPSP1.

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

1. Preliminaries

In this paper X is a non empty RLS structure and r, s, t are real numbers. Let us note that there exists a real number which is non zero. We now state a number of propositions: (2)^{2} Let T be a non empty topological space, X be a non empty subset of

T, and F_{1} be a family of subsets of T. Suppose F_{1} is a cover of X. Let x be a point of T. If x ∈ X, then there exists a subset W of T such that x ∈ W and W ∈ F_{1}.

(4)^{3} Let X be a non empty loop structure, M, N be subsets of X, and F be a family of subsets of X. If F = {x + N; x ranges over points of X: x ∈ M}, then M + N = ⋃ F.

(5) Let X be an add-associative right zeroed right complementable non empty loop structure and M be a subset of X. Then 0_{X} + M = M.

(6) Let X be an add-associative non empty loop structure, x, y be points of X, and M be a subset of X. Then (x + y) + M = x + (y + M).

1This work has been partially supported by the KBN grant 4 T11C 039 24.

2The proposition (1) has been removed.

3The proposition (3) has been removed.

(7) Let X be an add-associative non empty loop structure, x be a point of X, and M, N be subsets of X. Then (x + M) + N = x + (M + N).

(8) Let X be a non empty loop structure, M, N be subsets of X, and x be a point of X. If M ⊆ N, then x + M ⊆ x + N.

(9) Let X be a non empty real linear space, M be a subset of X, and x be a point of X. If x ∈ M, then 0_{X} ∈ −x + M.

(10) For every non empty loop structure X and for all subsets M, N, V of X such that M ⊆ N holds M + V ⊆ N + V.

(11) For every non empty loop structure X and for all subsets V_{1}, V_{2}, W_{1}, W_{2} of X such that V_{1} ⊆ W_{1} and V_{2} ⊆ W_{2} holds V_{1} + V_{2} ⊆ W_{1} + W_{2}.

(12) For every non empty real linear space X and for all subsets V_{1}, V_{2} of X such that 0_{X} ∈ V_{2} holds V_{1} ⊆ V_{1} + V_{2}.

(13) For every non empty real linear space X and for every real number r holds r · {0_{X}} = {0_{X}}.

(14) Let X be a non empty real linear space, M be a subset of X, and r be a non zero real number. If 0_{X} ∈ r · M, then 0_{X} ∈ M.

(15) Let X be a non empty real linear space, M, N be subsets of X, and r be a non zero real number. Then (r · M) ∩ (r · N) = r · (M ∩ N).

(16) Let X be a non empty topological space, x be a point of X, A be a neighbourhood of x, and B be a subset of X. If A ⊆ B, then B is a neighbourhood of x.

Let V be a non empty real linear space and let M be a subset of V. Let us observe that M is convex if and only if:

(Def. 1) For all points u, v of V and for every real number r such that 0 ≤ r and

r ≤ 1 and u ∈ M and v ∈ M holds r · u + (1 − r) · v ∈ M. One can prove the following proposition

(17) Let X be a non empty real linear space, M be a convex subset of X, and r_{1}, r_{2} be real numbers. If 0 ≤ r_{1} and 0 ≤ r_{2}, then r_{1} · M + r_{2} · M = (r_{1} + r_{2}) · M.

Let X be a non empty real linear space and let M be an empty subset of X. One can check that conv M is empty.

Next we state several propositions:

(18) For every non empty real linear space X and for every convex subset M of X holds conv M = M.

(19) For every non empty real linear space X and for every subset M of X and for every real number r holds r · conv M = conv r · M.

(20) For every non empty real linear space X and for all subsets M_{1}, M_{2} of X such that M_{1} ⊆ M_{2} holds Convex-Family M_{2} ⊆ Convex-Family M_{1}.

(21) For every non empty real linear space X and for all subsets M_{1}, M_{2} of X such that M_{1} ⊆ M_{2} holds conv M_{1} ⊆ conv M_{2}.

(22) Let X be a non empty real linear space, M be a convex subset of X,

and r be a real number. If 0 ≤ r and r ≤ 1 and 0_{X} ∈ M, then r · M ⊆ M. Let X be a non empty real linear space and let v, w be points of X. The

functor L(v, w) yields a subset of X and is defined as follows: (Def. 2) L(v, w) = {(1 − r) · v + r · w : 0 ≤ r ∧ r ≤ 1}.

Let X be a non empty real linear space and let v, w be points of X. Note that L(v, w) is non empty and convex.

Next we state the proposition

(23) Let X be a non empty real linear space and M be a subset of X. Then M is convex if and only if for all points u, w of X such that u ∈ M and w ∈ M holds L(u, w) ⊆ M.

Let V be a non empty RLS structure and let P be a family of subsets of V. We say that P is convex-membered if and only if:

(Def. 3) For every subset M of V such that M ∈ P holds M is convex.

Let V be a non empty RLS structure. One can verify that there exists a family of subsets of V which is non empty and convex-membered.

We now state the proposition

(24) For every non empty RLS structure V and for every convex-membered

family F of subsets of V holds ⋂ F is convex. Let X be a non empty RLS structure and let A be a subset of X. The

functor −A yielding a subset of X is defined by: (Def. 4) −A = (−1) · A.

One can prove the following proposition

(25) Let X be a non empty real linear space, M, N be subsets of X, and v

be a point of X. Then v + M meets N if and only if v ∈ N + −M. Let X be a non empty RLS structure and let A be a subset of X. We say

that A is symmetric if and only if: (Def. 5) A = −A.

Let X be a non empty real linear space. Observe that there exists a subset of X which is non empty and symmetric.

One can prove the following proposition

(26) Let X be a non empty real linear space, A be a symmetric subset of X,

and x be a point of X. If x ∈ A, then −x ∈ A. Let X be a non empty RLS structure and let A be a subset of X. We say

that A is circled if and only if: (Def. 6) For every real number r such that |r| ≤ 1 holds r · A ⊆ A.

Let X be a non empty real linear space. Note that ∅_{X} is circled.

We now state the proposition

(27) For every non empty real linear space X holds {0_{X}} is circled. Let X be a non empty real linear space. Observe that there exists a subset

of X which is non empty and circled.

The following proposition is true

(28) For every non empty real linear space X and for every non empty circled

subset B of X holds 0_{X} ∈ B. Let X be a non empty real linear space and let A, B be circled subsets of

X. One can verify that A + B is circled.

We now state the proposition

(29) Let X be a non empty real linear space, A be a circled subset of X, and

r be a real number. If |r| = 1, then r · A = A. Let X be a non empty real linear space. One can check that every subset of

X which is circled is also symmetric.

Let X be a non empty real linear space and let M be a circled subset of X. One can check that conv M is circled.

Let X be a non empty RLS structure and let F be a family of subsets of X. We say that F is circled-membered if and only if:

(Def. 7) For every subset V of X such that V ∈ F holds V is circled.

Let V be a non empty real linear space. Note that there exists a family of subsets of V which is non empty and circled-membered.

The following two propositions are true:

(30) For every non empty real linear space X and for every circled-membered family F of subsets of X holds ⋃ F is circled.

(31) For every non empty real linear space X and for every circled-membered family F of subsets of X holds ⋂ F is circled.

2. Real Linear Topological Space

We introduce real linear topological structures which are extensions of RLS structure and topological structure and are systems

⟨ a carrier, a zero, an addition, an external multiplication, a topology ⟩, where the carrier is a set, the zero is an element of the carrier, the addition is a binary operation on the carrier, the external multiplication is a function from [: ℝ, the carrier :] into the carrier, and the topology is a family of subsets of the carrier.

Let X be a non empty set, let O be an element of X, let F be a binary operation on X, let G be a function from [: ℝ, X:] into X, and let T be a family of subsets of X. Observe that ⟨X, O, F, G, T ⟩ is non empty.

Let us note that there exists a real linear topological structure which is strict and non empty.

Let X be a non empty real linear topological structure. We say that X is add-continuous if and only if the condition (Def. 8) is satisfied.

(Def. 8) Let x_{1}, x_{2} be points of X and V be a subset of X. Suppose V is open and x_{1} +x_{2} ∈ V. Then there exist subsets V_{1}, V_{2} of X such that V_{1} is open and V_{2} is open and x_{1} ∈ V_{1} and x_{2} ∈ V_{2} and V_{1} + V_{2} ⊆ V.

We say that X is mult-continuous if and only if the condition (Def. 9) is satisfied. (Def. 9) Let a be a real number, x be a point of X, and V be a subset of X.

Suppose V is open and a · x ∈ V. Then there exists a positive real number r and there exists a subset W of X such that W is open and x ∈ W and for every real number s such that |s − a| < r holds s · W ⊆ V.

Let us note that there exists a non empty real linear topological structure which is non empty, strict, add-continuous, mult-continuous, topological spacelike, Abelian, add-associative, right zeroed, right complementable, and real linear space-like.

A linear topological space is an add-continuous mult-continuous topological space-like Abelian add-associative right zeroed right complementable real linear space-like non empty real linear topological structure.

One can prove the following two propositions:

(32) Let X be a non empty linear topological space, x_{1}, x_{2} be points of X, and V be a neighbourhood of x_{1} + x_{2}. Then there exists a neighbourhood V_{1} of x_{1} and there exists a neighbourhood V_{2} of x_{2} such that V_{1} + V_{2} ⊆ V.

(33) Let X be a non empty linear topological space, a be a real number, x be a point of X, and V be a neighbourhood of a · x. Then there exists a positive real number r and there exists a neighbourhood W of x such that for every real number s if |s − a| < r, then s · W ⊆ V.

Let X be a non empty real linear topological structure and let a be a point of X. The functor transl(a, X) yields a map from X into X and is defined by:

(Def. 10) For every point x of X holds (transl(a, X))(x) = a + x.

The following propositions are true:

(34) Let X be a non empty real linear topological structure, a be a point of X, and V be a subset of X. Then (transl(a, X))^{∘}V = a + V.

(35) For every non empty linear topological space X and for every point a of X holds rng transl(a, X) = Ω_{X}.

(36) For every non empty linear topological space X and for every point a of

X holds (transl(a, X))^{−1} = transl(−a, X). Let X be a non empty linear topological space and let a be a point of X.

Note that transl(a, X) is homeomorphism.

Let X be a non empty linear topological space, let E be an open subset of X, and let x be a point of X. Note that x + E is open.

Let X be a non empty linear topological space, let E be an open subset of X, and let x be a point of X. Observe that x + E is open.

Let X be a non empty linear topological space, let E be an open subset of X, and let K be a subset of X. Observe that K + E is open.

Let X be a non empty linear topological space, let D be a closed subset of X, and let x be a point of X. Note that x + D is closed.

We now state several propositions:

(37) For every non empty linear topological space X and for all subsets V_{1}, V_{2}, V of X such that V_{1} + V_{2} ⊆ V holds Int V_{1} + Int V_{2} ⊆ Int V.

(38) For every non empty linear topological space X and for every point x of X and for every subset V of X holds x + Int V = Int(x + V).

(39) For every non empty linear topological space X and for every point x of X and for every subset V of X holds x + \overline{V} = \overline{x + V} .

(40) Let X be a non empty linear topological space, x, v be points of X, and V be a neighbourhood of x. Then v + V is a neighbourhood of v + x.

(41) Let X be a non empty linear topological space, x be a point of X, and

V be a neighbourhood of x. Then −x + V is a neighbourhood of 0_{X}. Let X be a non empty real linear topological structure. A local base of X

is a generalized basis of 0_{X}.

Let X be a non empty real linear topological structure. We say that X is locally-convex if and only if:

(Def. 11) There exists a local base of X which is convex-membered.

Let X be a non empty linear topological space and let E be a subset of X. We say that E is bounded if and only if:

(Def. 12) For every neighbourhood V of 0_{X} there exists s such that s > 0 and for

every t such that t > s holds E ⊆ t · V. Let X be a non empty linear topological space. Note that ∅_{X} is bounded. Let X be a non empty linear topological space. Observe that there exists a

subset of X which is bounded.

The following propositions are true:

(42) For every non empty linear topological space X and for all bounded subsets V_{1}, V_{2} of X holds V_{1} ∪ V_{2} is bounded.

(43) Let X be a non empty linear topological space, P be a bounded subset of X, and Q be a subset of X. If Q ⊆ P, then Q is bounded.

(44) Let X be a non empty linear topological space and F be a family of subsets of X. Suppose F is finite and F = {P : P ranges over bounded subsets of X}. Then ⋃ F is bounded.

(45) Let X be a non empty linear topological space and P be a family of subsets of X. Suppose P = {U : U ranges over neighbourhoods of 0_{X}}. Then P is a local base of X.

(46) Let X be a non empty linear topological space, O be a local base of X, and P be a family of subsets of X. Suppose P = {a + U; a ranges over points of X, U ranges over subsets of X: U ∈ O}. Then P is a generalized basis of X.

Let X be a non empty real linear topological structure and let r be a real number. The functor r • X yielding a map from X into X is defined as follows:

(Def. 13) For every point x of X holds (r • X)(x) = r · x.

The following propositions are true:

(47) Let X be a non empty real linear topological structure, V be a subset of X, and r be a non zero real number. Then (r • X)^{∘}V = r · V.

(48) For every non empty linear topological space X and for every non zero real number r holds rng(r • X) = Ω_{X}.

(49) For every non empty linear topological space X and for every non zero

real number r holds (r • X)^{−1} = r^{−1} • X. Let X be a non empty linear topological space and let r be a non zero real

number. One can check that r • X is homeomorphism.

Next we state several propositions:

(50) Let X be a non empty linear topological space, V be an open subset of X, and r be a non zero real number. Then r · V is open.

(51) Let X be a non empty linear topological space, V be a closed subset of X, and r be a non zero real number. Then r · V is closed.

(52) Let X be a non empty linear topological space, V be a subset of X, and r be a non zero real number. Then r · Int V = Int(r · V).

(53) Let X be a non empty linear topological space, A be a subset of X, and r be a non zero real number. Then r · \overline{A} = \overline{r · A}.

(54) Let X be a non empty linear topological space and A be a subset of X. If X is a T_{1} space, then 0 · \overline{A} = \overline{0 · A}.

(55) Let X be a non empty linear topological space, x be a point of X, V be a neighbourhood of x, and r be a non zero real number. Then r · V is a neighbourhood of r · x.

(56) Let X be a non empty linear topological space, V be a neighbourhood of 0_{X}, and r be a non zero real number. Then r · V is a neighbourhood of 0_{X}.

Let X be a non empty linear topological space, let V be a bounded subset of X, and let r be a real number. Observe that r · V is bounded.

We now state four propositions:

(57) Let X be a non empty linear topological space and W be a neighbourhood of 0_{X}. Then there exists an open neighbourhood U of 0_{X} such that U is symmetric and U + U ⊆ W.

(58) Let X be a non empty linear topological space, K be a compact subset of X, and C be a closed subset of X. Suppose K misses C. Then there exists a neighbourhood V of 0_{X} such that K + V misses C + V.

(59) Let X be a non empty linear topological space, B be a local base of X, and V be a neighbourhood of 0_{X}. Then there exists a neighbourhood W of 0_{X} such that W ∈ B and \overline{W} ⊆ V.

(60) Let X be a non empty linear topological space and V be a neighbourhood

of 0_{X}. Then there exists a neighbourhood W of 0_{X} such that \overline{W} ⊆ V. Let us observe that every non empty linear topological space which is T_{1} is

also Hausdorff.

We now state three propositions:

(61) Let X be a non empty linear topological space and A be a subset of X. Then \overline{A} = ⋂\{A + V : V ranges over neighbourhoods of 0_{X}\}.

(62) For every non empty linear topological space X and for all subsets A, B of X holds Int A + Int B ⊆ Int(A + B).

(63) For every non empty linear topological space X and for all subsets A, B

of X holds \overline{A} + \overline{B} ⊆ \overline{A + B}. Let X be a non empty linear topological space and let C be a convex subset

of X. Note that \overline{C} is convex.

Let X be a non empty linear topological space and let C be a convex subset of X. Note that Int C is convex.

Let X be a non empty linear topological space and let B be a circled subset of X. One can check that \overline{B} is circled.

One can prove the following proposition

(64) Let X be a non empty linear topological space and B be a circled subset

of X. If 0_{X} ∈ Int B, then Int B is circled. Let X be a non empty linear topological space and let E be a bounded

subset of X. Note that \overline{E} is bounded.

The following propositions are true:

(65) Let X be a non empty linear topological space and U be a neighbourhood of 0_{X}. Then there exists a neighbourhood W of 0_{X} such that W is circled and W ⊆ U.

(66) Let X be a non empty linear topological space and U be a neighbourhood of 0_{X}. Suppose U is convex. Then there exists a neighbourhood W of 0_{X} such that W is circled and convex and W ⊆ U.

(67) For every non empty linear topological space X holds there exists a local base of X which is circled-membered.

introduction to real linear topological … 107

(68) For every non empty linear topological space X such that X is locallyconvex holds there exists a local base of X which is convex-membered.

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

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

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

[14] Artur Korniłowicz. Introduction to meet-continuous topological lattices. Formalized Mathematics, 7(2):279–283, 1998.

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

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

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

[18] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 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] Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213–225, 1997.

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

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

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108 czesław byliński

Some Properties of Rectangles on the Plane^{1}

Artur Korniłowicz University of Białystok Yasunari Shidama Shinshu University Nagano

MML Identifier: TOPREALA.

The terminology and notation used in this paper have been introduced in the following articles: [25], [9], [28], [2], [29], [5], [30], [8], [6], [16], [3], [23], [24], [27], [1], [4], [7], [22], [17], [21], [20], [26], [13], [10], [19], [31], [14], [12], [11], [18], and [15].

1. Real Numbers

We adopt the following rules: i is an integer and a, b, r, s are real numbers. The following propositions are true:

(1) frac(r + i) = frac r.

(2) If r ≤ a and a < ⌊r⌋ + 1, then ⌊a⌋ = ⌊r⌋.

(3) If r ≤ a and a < ⌊r⌋ + 1, then frac r ≤ frac a.

(4) If r < a and a < ⌊r⌋ + 1, then frac r < frac a.

(5) If a ≥ ⌊r⌋ + 1 and a ≤ r + 1, then ⌊a⌋ = ⌊r⌋ + 1.

(6) If a ≥ ⌊r⌋ + 1 and a < r + 1, then frac a < frac r.

(7) If r ≤ a and a < r + 1 and r ≤ b and b < r + 1 and frac a = frac b, then a = b.

1The paper was written during the first author’s post-doctoral fellowship granted by the Shinshu University, Japan.

2. Subsets of ℝ

Let r be a real number and let s be a positive real number. One can verify the following observations:

∗ ]r, r + s[ is non empty,

∗ [r, r + s[ is non empty,

∗ ]r, r + s] is non empty,

∗ [r, r + s] is non empty,

∗ ]r − s, r[ is non empty,

∗ [r − s, r[ is non empty,

∗ ]r − s, r] is non empty, and

∗ [r − s, r] is non empty. Let r be a non positive real number and let s be a positive real number.

One can verify the following observations:

∗ ]r, s[ is non empty,

∗ [r, s[ is non empty,

∗ ]r, s] is non empty, and

∗ [r, s] is non empty. Let r be a negative real number and let s be a non negative real number.

One can check the following observations:

∗ ]r, s[ is non empty,

∗ [r, s[ is non empty,

∗ ]r, s] is non empty, and

∗ [r, s] is non empty. We now state a number of propositions:

(8) If r ≤ a and b < s, then [a, b] ⊆ [r, s[.

(9) If r < a and b ≤ s, then [a, b] ⊆ ]r, s].

(10) If r < a and b < s, then [a, b] ⊆ ]r, s[.

(11) If r ≤ a and b ≤ s, then [a, b[⊆ [r, s].

(12) If r ≤ a and b ≤ s, then [a, b[⊆ [r, s[.

(13) If r < a and b ≤ s, then [a, b[⊆ ]r, s].

(14) If r < a and b ≤ s, then [a, b[⊆ ]r, s[.

(15) If r ≤ a and b ≤ s, then ]a, b] ⊆ [r, s].

(16) If r ≤ a and b < s, then ]a, b] ⊆ [r, s[.

(17) If r ≤ a and b ≤ s, then ]a, b] ⊆ ]r, s].

(18) If r ≤ a and b < s, then ]a, b] ⊆ ]r, s[.

(19) If r ≤ a and b ≤ s, then ]a, b[ ⊆ [r, s].

(20) If r ≤ a and b ≤ s, then ]a, b[ ⊆ [r, s[.

(21) If r ≤ a and b ≤ s, then ]a, b[ ⊆ ]r, s].

3. Functions

The following propositions are true:

(22) For every function f and for all sets x, X such that x ∈ dom f and f(x) ∈ f^{∘}X and f is one-to-one holds x ∈ X.

(23) For every finite sequence f and for every natural number i such that i + 1 ∈ dom f holds i ∈ dom f or i = 0.

(24) For all sets x, y, X, Y and for every function f such that x ≠ y and f ∈ ∏[x ⟼ X, y ⟼ Y] holds f(x) ∈ X and f(y) ∈ Y.

(25) For all sets a, b holds ⟨a, b⟩ = [1 ⟼ a, 2 ⟼ b].

4. General Topology

Let us note that there exists a topological space which is constituted finite sequences, non empty, and strict.

Let T be a constituted finite sequences topological space. Note that every subspace of T is constituted finite sequences.

One can prove the following proposition

(26) Let T be a non empty topological space, Z be a non empty subspace of T, t be a point of T, z be a point of Z, N be an open neighbourhood of t, and M be a subset of Z. If t = z and M = N ∩ Ω_{Z}, then M is an open neighbourhood of z.

Let us note that every topological space which is empty is also discrete and anti-discrete.

Let X be a discrete topological space and let Y be a topological space. Note that every map from X into Y is continuous.

The following proposition is true

(27) Let X be a topological space, Y be a topological structure, and f be a

map from X into Y. If f is empty, then f is continuous. Let X be a topological space and let Y be a topological structure. Observe

that every map from X into Y which is empty is also continuous.

One can prove the following propositions:

(28) Let X be a topological structure, Y be a non empty topological structure, and Z be a non empty subspace of Y. Then every map from X into Z is a map from X into Y.

(29) Let S, T be non empty topological spaces, X be a subset of S, Y be a subset of T, f be a continuous map from S into T, and g be a map from S↾X into T ↾Y. If g = f↾X, then g is continuous.

(30) Let S, T be non empty topological spaces, Z be a non empty subspace of T, f be a map from S into T, and g be a map from S into Z. If f = g and f is open, then g is open.

(31) Let S, T be non empty topological spaces, S_{1} be a subset of S, T_{1} be a subset of T, f be a map from S into T, and g be a map from S↾S_{1} into T ↾T_{1}. If g = f↾S_{1} and g is onto and f is open and one-to-one, then g is open.

(32) Let X, Y, Z be non empty topological spaces, f be a map from X into Y, and g be a map from Y into Z. If f is open and g is open, then g · f is open.

(33) Let X, Y be topological spaces, Z be an open subspace of Y, f be a map from X into Y, and g be a map from X into Z. If f = g and g is open, then f is open.

(34) Let S, T be non empty topological spaces and f be a map from S into T. Suppose f is one-to-one and onto. Then f is continuous if and only if f^{−1} is open.

(35) Let S, T be non empty topological spaces and f be a map from S into T. Suppose f is one-to-one and onto. Then f is open if and only if f^{−1} is continuous.

(36) Let S be a topological space and T be a non empty topological space. Then S and T are homeomorphic if and only if the topological structure of S and the topological structure of T are homeomorphic.

(37) Let S, T be non empty topological spaces and f be a map from S into T. Suppose f is one-to-one, onto, continuous, and open. Then f is a homeomorphism.

5. ℝ^{1}

One can prove the following propositions:

(38) For every partial function f from ℝ to ℝ such that f = ℝ ⟼ r holds f is continuous on ℝ.

(39) Let f, f_{1}, f_{2} be partial functions from ℝ to ℝ. Suppose that dom f = dom f_{1}∪dom f_{2} and dom f_{1} is open and dom f_{2} is open and f_{1} is continuous on dom f_{1} and f_{2} is continuous on dom f_{2} and for every set z such that z ∈ dom f_{1} holds f(z) = f_{1}(z) and for every set z such that z ∈ dom f_{2} holds f(z) = f_{2}(z). Then f is continuous on dom f.

(40) Let x be a point of ℝ^{1}, N be a subset of ℝ, and M be a subset of ℝ^{1}. Suppose M = N. If N is a neighbourhood of x, then M is a neighbourhood of x.

(41) For every point x of ℝ^{1} and for every neighbourhood M of x there exists a neighbourhood N of x such that N ⊆ M.

(42) Let f be a map from ℝ^{1} into ℝ^{1}, g be a partial function from ℝ to ℝ, and x be a point of ℝ^{1}. If f = g and g is continuous in x, then f is continuous at x.

(43) Let f be a map from ℝ^{1} into ℝ^{1} and g be a function from ℝ into ℝ. If f = g and g is continuous on ℝ, then f is continuous.

(44) If a ≤ r and s ≤ b, then [r, s] is a closed subset of [a, b]_{T}.

(45) If a ≤ r and s ≤ b, then ]r, s[ is an open subset of [a, b]_{T}.

(46) If a ≤ b and a ≤ r, then ]r, b] is an open subset of [a, b]_{T}.

(47) If a ≤ b and r ≤ b, then [a, r[ is an open subset of [a, b]_{T}.

(48) If a ≤ b and r ≤ s, then the carrier of [: [a, b]_{T}, [r, s]_{T} :] = [: [a, b], [r, s] :].

6. E_{T}^{2}

Next we state four propositions:

(49) [a, b] = [1 ⟼ a, 2 ⟼ b].

(50) [a, b](1) = a and [a, b](2) = b.

(51) ClosedInsideOfRectangle(a, b, r, s) = ∏[1 ⟼ [a, b], 2 ⟼ [r, s]].

(52) If a ≤ b and r ≤ s, then [a, r] ∈ ClosedInsideOfRectangle(a, b, r, s). Let a, b, c, d be real numbers. The functor Trectangle(a, b, c, d) yielding a

subspace of E_{T}^{2} is defined by: (Def. 1) Trectangle(a, b, c, d) = (E_{T}^{2})↾ ClosedInsideOfRectangle(a, b, c, d).

The following propositions are true:

(53) The carrier of Trectangle(a, b, r, s) = ClosedInsideOfRectangle(a, b, r, s).

(54) If a ≤ b and r ≤ s, then Trectangle(a, b, r, s) is non empty.

Let a, c be non positive real numbers and let b, d be non negative real numbers. Observe that Trectangle(a, b, c, d) is non empty.

The map R2Homeo from [: ℝ^{1}, ℝ^{1} :] into E_{T}^{2} is defined by: (Def. 2) For all real numbers x, y holds R2Homeo(⟨⟨x, y⟩⟩) = ⟨x, y⟩.

Next we state several propositions:

(55) For all subsets A, B of ℝ holds R2Homeo^{∘}[: A, B:] = ∏[1 ⟼ A, 2 ⟼ B].

(56) R2Homeo is a homeomorphism.

114 artur korniłowicz and yasunari shidama

(57) If a ≤ b and r ≤ s, then R2Homeo ↾the carrier of [: [a, b]_{T}, [r, s]_{T} :] is a map from [: [a, b]_{T}, [r, s]_{T} :] into Trectangle(a, b, r, s).

(58) Suppose a ≤ b and r ≤ s. Let h be a map from [: [a, b]_{T}, [r, s]_{T} :] into Trectangle(a, b, r, s). If h = R2Homeo ↾the carrier of [: [a, b]_{T}, [r, s]_{T} :], then h is a homeomorphism.

(59) If a ≤ b and r ≤ s, then [: [a, b]_{T}, [r, s]_{T} :] and Trectangle(a, b, r, s) are homeomorphic.

(60) If a ≤ b and r ≤ s, then for every subset A of [a, b]_{T} and for every subset B of [r, s]_{T} holds ∏[1 ⟼ A, 2 ⟼ B] is a subset of Trectangle(a, b, r, s).

(61) Suppose a ≤ b and r ≤ s. Let A be an open subset of [a, b]_{T} and B be an open subset of [r, s]_{T}. Then ∏[1 ⟼ A, 2 ⟼ B] is an open subset of Trectangle(a, b, r, s).

(62) Suppose a ≤ b and r ≤ s. Let A be a closed subset of [a, b]_{T} and B be a closed subset of [r, s]_{T}. Then ∏[1 ⟼ A, 2 ⟼ B] is a closed subset of Trectangle(a, b, r, s).

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some properties of rectangles on the plane 115

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116 artur korniłowicz and yasunari shidama

Some Properties of Circles on the Plane^{1}

Artur Korniłowicz University of Bialystok Yasunari Shidama Shinshu University Nagano

MML Identifier: TOPREALB.

The articles [30], [34], [1], [5], [35], [7], [6], [23], [29], [17], [4], [33], [2], [27], [24], [26], [31], [9], [25], [37], [12], [18], [11], [10], [28], [3], [14], [36], [15], [32], [13], [16], [20], [19], [21], [8], and [22] provide the terminology and notation for this paper.

1. Preliminaries

For simplicity, we follow the rules: n is a natural number, i is an integer, a, b, r are real numbers, and x is a point of E_{T}^{n}.

One can check the following observations:

∗ ]0, 1[ is non empty,

∗ [−1, 1] is non empty, and

∗ ]\frac{1}{2}, \frac{3}{2}[ is non empty. One can verify the following observations:

∗ the function sin is continuous,

∗ the function cos is continuous,

∗ the function arcsin is continuous, and

∗ the function arccos is continuous. Next we state two propositions:

(1) sin(a · r + b) = ((the function sin) · AffineMap(a, b))(r).

(2) cos(a · r + b) = ((the function cos) · AffineMap(a, b))(r).

1The paper was written during the first author’s post-doctoral fellowship granted by the Shinshu University, Japan.

Let a be a non zero real number and let b be a real number. Note that AffineMap(a, b) is onto and one-to-one.

Let a, b be real numbers. The functor IntIntervals(a, b) is defined as follows: (Def. 1) IntIntervals(a, b) = {]a + n, b + n[ : n ranges over elements of ℤ}.

One can prove the following proposition

(3) For every set x holds x ∈ IntIntervals(a, b) iff there exists an element n of ℤ such that x = ]a + n, b + n[.

Let a, b be real numbers. Observe that IntIntervals(a, b) is non empty. Next we state the proposition

(4) If b − a ≤ 1, then IntIntervals(a, b) is mutually-disjoint. Let a, b be real numbers. Then IntIntervals(a, b) is a family of subsets of

ℝ^{1}.

Let a, b be real numbers. Then IntIntervals(a, b) is an open family of subsets of ℝ^{1}.

2. Correspondence between ℝ and ℝ^{1}

Let r be a real number. The functor R^{1}r yielding a point of ℝ^{1} is defined by:

(Def. 2) R^{1}r = r.

Let A be a subset of ℝ. The functor R^{1}A yielding a subset of ℝ^{1} is defined by:

(Def. 3) R^{1}A = A.

Let A be a non empty subset of ℝ. Observe that R^{1}A is non empty. Let A be an open subset of ℝ. Note that R^{1}A is open. Let A be a closed subset of ℝ. Observe that R^{1}A is closed. Let A be an open subset of ℝ. Observe that ℝ^{1}↾R^{1}A is open. Let A be a closed subset of ℝ. One can verify that ℝ^{1}↾R^{1}A is closed. Let f be a partial function from ℝ to ℝ. The functor R^{1}f yielding a map

from ℝ^{1}↾R^{1} dom f into ℝ^{1}↾R^{1} rng f is defined as follows: (Def. 4) R^{1}f = f.

Let f be a partial function from ℝ to ℝ. One can check that R^{1}f is onto. Let f be an one-to-one partial function from ℝ to ℝ. Observe that R^{1}f is

one-to-one.

One can prove the following four propositions:

(5) ℝ^{1}↾R^{1}(Ω_{ℝ}) = ℝ^{1}.

(6) For every partial function f from ℝ to ℝ such that dom f = ℝ holds ℝ^{1}↾R^{1} dom f = ℝ^{1}.

(7) Every function f from ℝ into ℝ is a map from ℝ^{1} into ℝ^{1}↾R^{1} rng f.

(8) Every function from ℝ into ℝ is a map from ℝ^{1} into ℝ^{1}. Let f be a continuous partial function from ℝ to ℝ. Note that R^{1}f is

continuous.

Let a be a non zero real number and let b be a real number. One can verify that R^{1} AffineMap(a, b) is open.

3. Circles

Let S be a subspace of E_{T}^{2}. We say that S satisfies conditions of simple closed curve if and only if:

(Def. 5) The carrier of S is a simple closed curve.

Let us note that every subspace of E_{T}^{2} which satisfies conditions of simple closed curve is also non empty, arcwise connected, and compact.

Let r be a positive real number and let x be a point of E_{T}^{2}. Observe that Sphere(x, r) satisfies conditions of simple closed curve.

Let n be a natural number, let x be a point of E_{T}^{n}, and let r be a real number. The functor Tcircle(x, r) yielding a subspace of E_{T}^{n} is defined by:

(Def. 6) Tcircle(x, r) = (E_{T}^{n})↾ Sphere(x, r).

Let n be a non empty natural number, let x be a point of E_{T}^{n}, and let r be a non negative real number. Note that Tcircle(x, r) is strict and non empty.

One can prove the following proposition

(9) The carrier of Tcircle(x, r) = Sphere(x, r). Let n be a natural number, let x be a point of E_{T}^{n}, and let r be an empty

real number. Note that Tcircle(x, r) is trivial.

Next we state the proposition

(10) Tcircle(0_{E_{T}^{2}}, r) is a subspace of Trectangle(−r, r, −r, r).

Let x be a point of E_{T}^{2} and let r be a positive real number. One can verify that Tcircle(x, r) satisfies conditions of simple closed curve.

Let us mention that there exists a subspace of E_{T}^{2} which is strict and satisfies conditions of simple closed curve.

Next we state the proposition

(11) For all subspaces S, T of E_{T}^{2} satisfying conditions of simple closed curve

holds S and T are homeomorphic. Let n be a natural number. The functor TopUnitCircle n yields a subspace

of E_{T}^{n} and is defined by: (Def. 7) TopUnitCircle n = Tcircle(0_{E_{T}^{n}}, 1).

Let n be a non empty natural number. Note that TopUnitCircle n is non empty.

We now state several propositions:

(12) For every non empty natural number n and for every point x of E_{T}^{n} such that x is a point of TopUnitCircle n holds |x| = 1.

(13) For every point x of E_{T}^{2} such that x is a point of TopUnitCircle 2 holds −1 ≤ x_{1} and x_{1} ≤ 1 and −1 ≤ x_{2} and x_{2} ≤ 1.

(14) For every point x of E_{T}^{2} such that x is a point of TopUnitCircle 2 and x_{1} = 1 holds x_{2} = 0.

(15) For every point x of E_{T}^{2} such that x is a point of TopUnitCircle 2 and x_{1} = −1 holds x_{2} = 0.

(16) For every point x of E_{T}^{2} such that x is a point of TopUnitCircle 2 and x_{2} = 1 holds x_{1} = 0.

(17) For every point x of E_{T}^{2} such that x is a point of TopUnitCircle 2 and

x_{2} = −1 holds x_{1} = 0. The following propositions are true:

(18) TopUnitCircle 2 is a subspace of Trectangle(−1, 1, −1, 1).

(19) Let n be a non empty natural number, r be a positive real number, x be a point of E_{T}^{n}, and f be a map from TopUnitCircle n into Tcircle(x, r). Suppose that for every point a of TopUnitCircle n and for every point b of E_{T}^{n} such that a = b holds f(a) = r · b + x. Then f is a homeomorphism.

Let us observe that TopUnitCircle 2 satisfies conditions of simple closed curve.

One can prove the following proposition

(20) Let n be a non empty natural number, r, s be positive real numbers, and x, y be points of E_{T}^{n}. Then Tcircle(x, r) and Tcircle(y, s) are homeomorphic.

Let x be a point of E_{T}^{2} and let r be a non negative real number. Observe that Tcircle(x, r) is arcwise connected.

The point c[10] of TopUnitCircle 2 is defined as follows: (Def. 8) c[10] = [1, 0].

The point c[−10] of TopUnitCircle 2 is defined as follows: (Def. 9) c[−10] = [−1, 0].

Next we state the proposition

(21) c[10] ≠ c[−10].

Let p be a point of TopUnitCircle 2. The functor TopOpenUnitCircle p yielding a strict subspace of TopUnitCircle 2 is defined by:

(Def. 10) The carrier of TopOpenUnitCircle p = (the carrier of TopUnitCircle 2) \

{p}. Let p be a point of TopUnitCircle 2. Note that TopOpenUnitCircle p is non

empty.

We now state several propositions:

(22) For every point p of TopUnitCircle 2 holds p is not a point of TopOpenUnitCircle p.

(23) For every point p of TopUnitCircle 2 holds TopOpenUnitCircle p = TopUnitCircle 2↾(Ω_{TopUnitCircle 2} \ {p}).

(24) For all points p, q of TopUnitCircle 2 such that p ≠ q holds q is a point of TopOpenUnitCircle p.

(25) For every point p of E_{T}^{2} such that p is a point of TopOpenUnitCircle c[10] and p_{2} = 0 holds p = c[−10].

(26) For every point p of E_{T}^{2} such that p is a point of TopOpenUnitCircle c[−10]

and p_{2} = 0 holds p = c[10]. Next we state three propositions:

(27) Let p be a point of TopUnitCircle 2 and x be a point of E_{T}^{2}. If x is a point of TopOpenUnitCircle p, then −1 ≤ x_{1} and x_{1} ≤ 1 and −1 ≤ x_{2} and x_{2} ≤ 1.

(28) For every point x of E_{T}^{2} such that x is a point of TopOpenUnitCircle c[10] holds −1 ≤ x_{1} and x_{1} < 1.

(29) For every point x of E_{T}^{2} such that x is a point of TopOpenUnitCircle c[−10]

holds −1 < x_{1} and x_{1} ≤ 1. Let p be a point of TopUnitCircle 2. Note that TopOpenUnitCircle p is open. We now state two propositions:

(30) For every point p of TopUnitCircle 2 holds TopOpenUnitCircle p and I(01) are homeomorphic.

(31) For all points p, q of TopUnitCircle 2 holds TopOpenUnitCircle p and TopOpenUnitCircle q are homeomorphic.

4. Correspondence between the Real Line and Circles

The map CircleMap from ℝ^{1} into TopUnitCircle 2 is defined by: (Def. 11) For every real number x holds CircleMap(x) = [cos(2·π ·x), sin(2·π ·x)].

Next we state several propositions:

(32) CircleMap(r) = CircleMap(r + i).

(33) CircleMap(i) = c[10].

(34) CircleMap^{−1}({c[10]}) = ℤ.

(35) If frac r = \frac{1}{2}, then CircleMap(r) = [−1, 0].

(36) If frac r = \frac{1}{4}, then CircleMap(r) = [0, 1].

(37) If frac r = \frac{3}{4}, then CircleMap(r) = [0, −1].

(38) For all integers i, j holds CircleMap(r) = [((the function cos) · AffineMap(2·π, 2·π·i))(r), ((the function sin) · AffineMap(2·π, 2·π·j))(r)].

Let us note that CircleMap is continuous. The following proposition is true

(39) For every subset B of ℝ^{1} and for every map f from ℝ^{1}↾B into TopUnitCircle 2 such that [0, 1[⊆ B and f = CircleMap ↾B holds f is onto.

Let us observe that CircleMap is onto. Let r be a real number. One can verify that CircleMap ↾[r, r + 1[ is one-to-

one.

Let r be a real number. One can verify that CircleMap ↾]r, r + 1[ is one-to-one.

One can prove the following two propositions:

(40) If b − a ≤ 1, then for every set d such that d ∈ IntIntervals(a, b) holds CircleMap ↾d is one-to-one.

(41) For every set d such that d ∈ IntIntervals(a, b) holds CircleMap^{∘} d =

CircleMap^{∘} ⋃ IntIntervals(a, b). Let r be a point of ℝ^{1}. The functor CircleMap r yielding a map from

ℝ^{1}↾R^{1}]r, r + 1[ into TopOpenUnitCircle CircleMap(r) is defined by: (Def. 12) CircleMap r = CircleMap ↾]r, r + 1[.

One can prove the following proposition

(42) CircleMap R^{1}(a+i) = CircleMap R^{1}a·(AffineMap(1, −i)↾]a+i, a+i+1[). Let r be a point of ℝ^{1}. One can check that CircleMap r is one-to-one, onto,

and continuous.

The map Circle2IntervalR from TopOpenUnitCircle c[10] into ℝ^{1}↾R^{1}]0, 1[ is defined by the condition (Def. 13).

(Def. 13) Let p be a point of TopOpenUnitCircle c[10]. Then there exist real numbers x, y such that p = [x, y] and if y ≥ 0, then Circle2IntervalR(p) = \frac{arccos x}{2·π} and if y ≤ 0, then Circle2IntervalR(p) = 1 − \frac{arccos x}{2·π}.

The map Circle2IntervalL from TopOpenUnitCircle c[−10] into ℝ^{1}↾R^{1}]\frac{1}{2}, \frac{3}{2}[ is defined by the condition (Def. 14).

(Def. 14) Let p be a point of TopOpenUnitCircle c[−10]. Then there exist real numbers x, y such that p = [x, y] and if y ≥ 0, then Circle2IntervalL(p) = 1 + \frac{arccos x}{2·π} and if y ≤ 0, then Circle2IntervalL(p) = 1 − \frac{arccos x}{2·π}.

We now state two propositions:

(43) (CircleMap R^{1}0)^{−1} = Circle2IntervalR .

(44) (CircleMap R^{1}(\frac{1}{2}))^{−1} = Circle2IntervalL .

Let us observe that Circle2IntervalR is one-to-one, onto, and continuous and Circle2IntervalL is one-to-one, onto, and continuous.

Let i be an integer. Observe that CircleMap R^{1}i is open and CircleMap R^{1}(\frac{1}{2} + i) is open.

some properties of circles on the plane 123

Let us observe that Circle2IntervalR is open and Circle2IntervalL is open. Next we state several propositions:

(45) CircleMap R^{1}0 is a homeomorphism.

(46) CircleMap R^{1}(\frac{1}{2}) is a homeomorphism.

(47) Circle2IntervalR is a homeomorphism.

(48) Circle2IntervalL is a homeomorphism.

(49) There exists a family F of subsets of TopUnitCircle 2 such that (i) F = \{CircleMap^{∘}]0, 1[, CircleMap^{∘}]\frac{1}{2}, \frac{3}{2}[\}, (ii) F is a cover of TopUnitCircle 2 and open, and (iii) for every subset U of TopUnitCircle 2 holds if U = CircleMap^{∘}]0, 1[,

then ⋃ IntIntervals(0, 1) = CircleMap^{−1}(U) and for every subset d of ℝ^{1} such that d ∈ IntIntervals(0, 1) and for every map f from ℝ^{1}↾d into TopUnitCircle 2↾U such that f = CircleMap ↾d holds f is a homeomorphism and if U = CircleMap^{∘}]\frac{1}{2}, \frac{3}{2}[, then ⋃ IntIntervals(\frac{1}{2}, \frac{3}{2}) = CircleMap^{−1}(U) and for every subset d of ℝ^{1} such that d ∈ IntIntervals(\frac{1}{2}, \frac{3}{2}) and for every map f from ℝ^{1}↾d into TopUnitCircle 2↾U such that f = CircleMap ↾d holds f is a homeomorphism.

References

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

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

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

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

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

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

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[13] Mariusz Giero. Hierarchies and classifications of sets. Formalized Mathematics, 9(4):865– 869, 2001.

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

[15] Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55–59, 1999.

[16] Adam Grabowski. On the decompositions of intervals and simple closed curves. Formalized Mathematics, 10(3):145–151, 2002.

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

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On the Characterization of Collineations of the Segre Product of Strongly Connected

Partial Linear Spaces^{1}

Adam Naumowicz University of Białystok

Summary. In this paper we characterize the automorphisms (collineations) of the Segre product of partial linear spaces. In particular, we show that if all components of the product are strongly connected, then every collineation is determined by a set of isomorphisms between its components. The formalization follows the ideas presented in the Journal of Geometry paper [16] by Naumowicz and Prażmowski.

MML Identifier: PENCIL 3.

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

1. Preliminaries

The following propositions are true:

(1) Let S be a non empty non void topological structure, f be a collineation of S, and p, q be points of S. If p, q are collinear, then f(p), f(q) are collinear.

(2) Let I be a non empty set, i be an element of I, and A be a non-Trivial-yielding 1-sorted yielding many sorted set indexed by I. Then A(i) is non trivial.

1This work has been partially supported by the KBN grant 4 T11C 039 24 and the FP6 IST grant TYPES No. 510096.

(3) Let S be a non void identifying close blocks topological structure such that S is strongly connected. Then S has no isolated points.

(4) Let S be a non empty non void identifying close blocks topological structure. If S is strongly connected, then S is connected.

(5) Let S be a non void non degenerated topological structure and L be a block of S. Then there exists a point x of S such that x ∉ L.

(6) Let I be a non empty set and A be a nonempty TopStruct-yielding many sorted set indexed by I. Then every point of SegreProduct A is an element of the support of A.

(7) Let I be a non empty set, A be a 1-sorted yielding many sorted set indexed by I, and x be an element of I. Then (the support of A)(x) = Ω_{A(x)}.

(8) Let I be a non empty set, i be an element of I, and A be a non-Trivial-yielding TopStruct-yielding many sorted set indexed by I. Then there exists a Segre-like non trivial-yielding many sorted subset L indexed by the support of A such that index(L) = i and ∏ L is a Segre coset of A.

(9) Let I be a non empty set, i be an element of I, A be a non-Trivial-yielding TopStruct-yielding many sorted set indexed by I, and p be a point of SegreProduct A. Then there exists a Segre-like non trivial-yielding many sorted subset L indexed by the support of A such that index(L) = i and ∏ L is a Segre coset of A and p ∈ ∏ L.

(10) Let I be a non empty set, A be a non-Trivial-yielding TopStruct-yielding many sorted set indexed by I, and b be a Segre-like non trivial-yielding many sorted subset indexed by the support of A. If ∏ b is a Segre coset of A, then b(index(b)) = Ω_{A(index(b))}.

(11) Let I be a non empty set, A be a non-Trivial-yielding TopStruct-yielding many sorted set indexed by I, and L_{1}, L_{2} be Segre-like non trivial-yielding many sorted subsets indexed by the support of A. Suppose ∏ L_{1} is a Segre coset of A and ∏ L_{2} is a Segre coset of A and index(L_{1}) = index(L_{2}) and ∏ L_{1} meets ∏ L_{2}. Then L_{1} = L_{2}.

(12) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, L be a Segre-like non trivial-yielding many sorted subset indexed by the support of A, and B be a block of A(index(L)). Then ∏(L +· (index(L), B)) is a block of SegreProduct A.

(13) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, i be an element of I, p be a point of A(i), and L be a Segre-like non trivial-yielding many sorted subset indexed by the support of A. Suppose i ≠ index(L). Then L +· (i, {p}) is a Segre-like non trivial-yielding many sorted subset indexed by the support of A.

(14) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, i be an element of I, S be a subset of A(i), and L be a Segre-like non trivial-yielding many sorted subset indexed by the support of A. Then ∏(L +· (i, S)) is a subset of SegreProduct A.

(15) Let I be a non empty set, P be a many sorted set indexed by I, and i be an element of I. Then {P }(i) is non empty and trivial.

(16) Let I be a non empty set, i be an element of I, A be a PLS-yielding many sorted set indexed by I, B be a block of A(i), and P be an element of the support of A. Then ∏({P } +· (i, B)) is a block of SegreProduct A.

(17) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, and p, q be points of SegreProduct A. Suppose p ≠ q. Then p, q are collinear if and only if for all many sorted sets p_{1}, q_{1} indexed by I such that p_{1} = p and q_{1} = q there exists an element i of I such that for all points a, b of A(i) such that a = p_{1}(i) and b = q_{1}(i) holds a ≠ b and a, b are collinear and for every element j of I such that j ≠ i holds p_{1}(j) = q_{1}(j).

(18) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, b be a Segre-like non trivial-yielding many sorted subset indexed by the support of A, and x be a point of A(index(b)). Then there exists a many sorted set p indexed by I such that p ∈ ∏ b and {(p +· (index(b), x) qua set) } = ∏(b +· (index(b), {x})).

Let I be a finite non empty set and let b_{1}, b_{2} be many sorted sets indexed by I. The functor b_{1}′(b_{2}) yields a natural number and is defined by:

(Def. 1) b_{1}′(b_{2}) = \overline{\overline{\{i; i ranges over elements of I: b_{1}(i) ≠ b_{2}(i)\}}}.

One can prove the following proposition

(19) Let I be a finite non empty set, b_{1}, b_{2} be many sorted sets indexed by I, and i be an element of I. If b_{1}(i) ≠ b_{2}(i), then b_{1}′(b_{2}) = b_{1}′(b_{2} +· (i, b_{1}(i))) + 1.

2. The Adherence of Segre Cosets

Let I be a non empty set, let A be a PLS-yielding many sorted set indexed by I, and let B_{1}, B_{2} be Segre cosets of A. The predicate B_{1}||B_{2} is defined as follows:

(Def. 2) For every point x of SegreProduct A such that x ∈ B_{1} there exists a

point y of SegreProduct A such that y ∈ B_{2} and x, y are collinear. Next we state several propositions:

(20) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, and B_{1}, B_{2} be Segre cosets of A. Suppose B_{1}||B_{2}. Let f be a collineation of SegreProduct A and C_{1}, C_{2} be Segre cosets of A. If C_{1} = f^{∘}B_{1} and C_{2} = f^{∘}B_{2}, then C_{1}||C_{2}.

(21) Let I be a non empty set, A be a PLS-yielding many sorted set indexed by I, and B_{1}, B_{2} be Segre cosets of A. Suppose B_{1} misses B_{2}. Then B_{1}||B_{2} if and only if for all Segre-like non trivial-yielding many sorted subsets b_{1}, b_{2} indexed by the support of A such that B_{1} = ∏ b_{1} and B_{2} = ∏ b_{2} holds index(b_{1}) = index(b_{2}) and there exists an element r of I such that r ≠ index(b_{1}) and for every element i of I such that i ≠ r holds b_{1}(i) = b_{2}(i) and for all points c_{1}, c_{2} of A(r) such that b_{1}(r) = {c_{1}} and b_{2}(r) = {c_{2}} holds c_{1}, c_{2} are collinear.

(22) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is connected. Let i be an element of I, p be a point of A(i), and b_{1}, b_{2} be Segre-like non trivial-yielding many sorted subsets indexed by the support of A. Suppose ∏ b_{1} is a Segre coset of A and ∏ b_{2} is a Segre coset of A and b_{1} = b_{2} +· (i, {p}) and p ∉ b_{2}(i). Then there exists a finite sequence D of elements of 2^{the carrier of SegreProduct A} such that

(i) D(1) = ∏ b_{1}, (ii) D(len D) = ∏ b_{2}, (iii) for every natural number i such that i ∈ dom D holds D(i) is a Segre

coset of A, and (iv) for every natural number i such that 1 ≤ i and i < len D and for all

Segre cosets D_{1}, D_{2} of A such that D_{1} = D(i) and D_{2} = D(i + 1) holds D_{1} misses D_{2} and D_{1}||D_{2}.

(23) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is connected. Let B_{1}, B_{2} be Segre cosets of A. Suppose B_{1} misses B_{2}. Let b_{1}, b_{2} be Segrelike non trivial-yielding many sorted subsets indexed by the support of A. Suppose B_{1} = ∏ b_{1} and B_{2} = ∏ b_{2}. Then index(b_{1}) = index(b_{2}) if and only if there exists a finite sequence D of elements of 2^{the carrier of SegreProduct A} such that D(1) = B_{1} and D(len D) = B_{2} and for every natural number i such that i ∈ dom D holds D(i) is a Segre coset of A and for every natural number i such that 1 ≤ i and i < len D and for all Segre cosets D_{1}, D_{2} of A such that D_{1} = D(i) and D_{2} = D(i + 1) holds D_{1} misses D_{2} and D_{1}||D_{2}.

(24) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A, B_{1}, B_{2} be Segre cosets of A, and b_{1}, b_{2}, b_{3}, b_{4} be Segre-like non trivial-yielding many sorted subsets indexed by the support of A. If B_{1} = ∏ b_{1} and B_{2} = ∏ b_{2} and f^{∘}B_{1} = ∏ b_{3} and f^{∘}B_{2} = ∏ b_{4}, then if index(b_{1}) = index(b_{2}), then index(b_{3}) = index(b_{4}).

(25) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A. Then there exists a permutation s of I such that for all elements i, j of I holds s(i) = j if and only if for every Segre coset B_{1} of A and for all Segre-like non trivialyielding many sorted subsets b_{1}, b_{2} indexed by the support of A such that B_{1} = ∏ b_{1} and f^{∘}B_{1} = ∏ b_{2} holds if index(b_{1}) = i, then index(b_{2}) = j.

Let I be a finite non empty set and let A be a PLS-yielding many sorted set indexed by I. Let us assume that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A. The functor IndPerm(f) yields a permutation of I and is defined by the condition (Def. 3).

(Def. 3) Let i, j be elements of I. Then (IndPerm(f))(i) = j if and only if for every Segre coset B_{1} of A and for all Segre-like non trivial-yielding many sorted subsets b_{1}, b_{2} indexed by the support of A such that B_{1} = ∏ b_{1} and f^{∘}B_{1} = ∏ b_{2} holds if index(b_{1}) = i, then index(b_{2}) = j.

3. Canonical Embeddings and Automorphisms of Segre Product

Let I be a finite non empty set and let A be a PLS-yielding many sorted set indexed by I. Let us assume that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A and let b_{1} be a Segre-like non trivial-yielding many sorted subset indexed by the support of A. Let us assume that ∏ b_{1} is a Segre coset of A. The functor CanEmb(f, b_{1}) yields a map from A(index(b_{1})) into A((IndPerm(f))(index(b_{1}))) and is defined by the conditions (Def. 4).

(Def. 4)(i) CanEmb(f, b_{1}) is isomorphic, and

(ii) for every many sorted set a indexed by I such that a is a point of SegreProduct A and a ∈ ∏ b_{1} and for every many sorted set b indexed by I such that b = f(a) holds b((IndPerm(f))(index(b_{1}))) = (CanEmb(f, b_{1}))(a(index(b_{1}))).

Next we state two propositions:

(26) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A and B_{1}, B_{2} be Segre cosets of A. Suppose B_{1} misses B_{2} and B_{1}||B_{2}. Let b_{1}, b_{2} be Segre-like non trivial-yielding many sorted subsets indexed by the support of A. If ∏ b_{1} = B_{1} and ∏ b_{2} = B_{2}, then CanEmb(f, b_{1}) = CanEmb(f, b_{2}).

(27) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A and b_{1}, b_{2} be Segrelike non trivial-yielding many sorted subsets indexed by the support of A.

130 adam naumowicz

Suppose ∏ b_{1} is a Segre coset of A and ∏ b_{2} is a Segre coset of A and index(b_{1}) = index(b_{2}). Then CanEmb(f, b_{1}) = CanEmb(f, b_{2}).

Let I be a finite non empty set and let A be a PLS-yielding many sorted set indexed by I. Let us assume that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A and let i be an element of I. The functor CanEmb(f, i) yields a map from A(i) into A((IndPerm(f))(i)) and is defined by the condition (Def. 5).

(Def. 5) Let b be a Segre-like non trivial-yielding many sorted subset indexed by the support of A. If ∏ b is a Segre coset of A and index(b) = i, then CanEmb(f, i) = CanEmb(f, b).

Next we state the proposition

(28) Let I be a finite non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let f be a collineation of SegreProduct A. Then there exists a permutation s of I and there exists a function yielding many sorted set B indexed by I such that for every element i of I holds

(i) B(i) is a map from A(i) into A(s(i)), (ii) for every map m from A(i) into A(s(i)) such that m = B(i) holds m is

isomorphic, and (iii) for every point p of SegreProduct A and for every many sorted set a

indexed by I such that a = p and for every many sorted set b indexed by I such that b = f(p) and for every map m from A(i) into A(s(i)) such that m = B(i) holds b(s(i)) = m(a(i)).

References

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

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

[3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

[4] Grzegorz Bancerek. The “way-below” relation. Formalized Mathematics, 6(1):169–176, 1997.

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

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

[7] Józef Białas. Properties of fields. Formalized Mathematics, 1(5):807–812, 1990.

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

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

[10] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

[12] Artur Korniłowicz. Some basic properties of many sorted sets. Formalized Mathematics, 5(3):395–399, 1996.

[13] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103–108, 1993.

[14] Adam Naumowicz. On cosets in Segre’s product of partial linear spaces. Formalized Mathematics, 9(4):795–800, 2001.

on the characterization of collineations of … 131

[15] Adam Naumowicz. On Segre’s product of partial line spaces. Formalized Mathematics, 9(2):383–390, 2001.

[16] Adam Naumowicz and Krzysztof Prażmowski. On Segre’s product of partial line spaces and spaces of pencils. Journal of Geometry, 71(1):128–143, 2001.

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

[18] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[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] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15–22, 1993.

[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.

132 adam naumowicz

Spaces of Pencils, Grassmann Spaces, and Generalized Veronese Spaces^{1}

Adam Naumowicz University of Białystok

Summary. In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of k-pencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definition presented in the paper [8] by Naumowicz and Prażmowski. For all spaces defined, we state the conditions under which they are not degenerated to a single line.

MML Identifier: PENCIL 4.

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

1. Spaces of k-Pencils

One can prove the following propositions:

(1) For all natural numbers k, n such that 1 ≤ k and k < n and 3 ≤ n holds k + 1 < n or 2 ≤ k.

(2) For every finite set X and for every natural number n such that n ≤ card X there exists a subset Y of X such that card Y = n.

(3) For every field F and for every vector space V over F holds every subspace of V is a subspace of Ω_{V}.

(4) For every field F and for every vector space V over F holds every subspace of Ω_{V} is a subspace of V.

1This work has been partially supported by the KBN grant 4 T11C 039 24OB and the FP6 IST grant TYPES No. 510096.

(5) For every field F and for every vector space V over F and for every subspace W of V holds Ω_{W} is a subspace of V.

(6) Let F be a field and V, W be vector spaces over F. If Ω_{W} is a subspace of V, then W is a subspace of V.

Let F be a field, let V be a vector space over F, and let W_{1}, W_{2} be subspaces of V. The functor segment(W_{1}, W_{2}) yielding a subset of Subspaces V is defined by:

(Def. 1)(i) For every strict subspace W of V holds W ∈ segment(W_{1}, W_{2}) iff W_{1}

is a subspace of W and W is a subspace of W_{2} if W_{1} is a subspace of W_{2}, (ii) segment(W_{1}, W_{2}) = ∅, otherwise. We now state the proposition

(7) Let F be a field, V be a vector space over F, and W_{1}, W_{2}, W_{3}, W_{4} be subspaces of V. Suppose W_{1} is a subspace of W_{2} and W_{3} is a subspace of W_{4} and Ω_{(W_{1})} = Ω_{(W_{3})} and Ω_{(W_{2})} = Ω_{(W_{4})}. Then segment(W_{1}, W_{2}) = segment(W_{3}, W_{4}).

Let F be a field, let V be a vector space over F, and let W_{1}, W_{2} be subspaces of V. The functor pencil(W_{1}, W_{2}) yielding a subset of Subspaces V is defined by:

(Def. 2) pencil(W_{1}, W_{2}) = segment(W_{1}, W_{2}) \ {Ω_{(W_{1})}, Ω_{(W_{2})}}.

Next we state the proposition

(8) Let F be a field, V be a vector space over F, and W_{1}, W_{2}, W_{3}, W_{4} be subspaces of V. Suppose W_{1} is a subspace of W_{2} and W_{3} is a subspace of W_{4} and Ω_{(W_{1})} = Ω_{(W_{3})} and Ω_{(W_{2})} = Ω_{(W_{4})}. Then pencil(W_{1}, W_{2}) = pencil(W_{3}, W_{4}).

Let F be a field, let V be a finite dimensional vector space over F, let W_{1}, W_{2} be subspaces of V, and let k be a natural number. The functor pencil(W_{1}, W_{2}, k) yielding a subset of Sub_{k}(V) is defined by:

(Def. 3) pencil(W_{1}, W_{2}, k) = pencil(W_{1}, W_{2}) ∩ Sub_{k}(V).

We now state two propositions:

(9) Let F be a field, V be a finite dimensional vector space over F, k be a natural number, and W_{1}, W_{2}, W be subspaces of V. If W ∈ pencil(W_{1}, W_{2}, k), then W_{1} is a subspace of W and W is a subspace of W_{2}.

(10) Let F be a field, V be a finite dimensional vector space over F, k be a natural number, and W_{1}, W_{2}, W_{3}, W_{4} be subspaces of V. Suppose W_{1} is a subspace of W_{2} and W_{3} is a subspace of W_{4} and Ω_{(W_{1})} = Ω_{(W_{3})} and Ω_{(W_{2})} = Ω_{(W_{4})}. Then pencil(W_{1}, W_{2}, k) = pencil(W_{3}, W_{4}, k).

Let F be a field, let V be a finite dimensional vector space over F, and let k be a natural number. k pencils of V yields a family of subsets of Sub_{k}(V) and is defined by the condition (Def. 4).

(Def. 4) Let X be a set. Then X ∈ k pencils of V if and only if there exist subspaces W_{1}, W_{2} of V such that W_{1} is a subspace of W_{2} and dim(W_{1}) + 1 = k and dim(W_{2}) = k + 1 and X = pencil(W_{1}, W_{2}, k).

We now state several propositions:

(11) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If 1 ≤ k and k < dim(V), then k pencils of V is non empty.

(12) Let F be a field, V be a finite dimensional vector space over F, W_{1}, W_{2}, P, Q be subspaces of V, and k be a natural number. Suppose 1 ≤ k and k < dim(V) and dim(W_{1}) + 1 = k and dim(W_{2}) = k + 1 and P ∈ pencil(W_{1}, W_{2}, k) and Q ∈ pencil(W_{1}, W_{2}, k) and P ≠ Q. Then P ∩ Q = Ω_{(W_{1})} and P + Q = Ω_{(W_{2})}.

(13) Let F be a field, V be a finite dimensional vector space over F, and v be a vector of V. If v ≠ 0_{V} , then dim(Lin({v})) = 1.

(14) Let F be a field, V be a finite dimensional vector space over F, W be a subspace of V, and v be a vector of V. If v ∉ W, then dim(W +Lin({v})) = dim(W) + 1.

(15) Let F be a field, V be a finite dimensional vector space over F, W be a subspace of V, and v, u be vectors of V. Suppose v ∉ W and u ∉ W and v ≠ u and {v, u} is linearly independent and W ∩ Lin({v, u}) = 0_{V} . Then dim(W + Lin({v, u})) = dim(W) + 2.

(16) Let F be a field, V be a finite dimensional vector space over F, and W_{1}, W_{2} be subspaces of V. Suppose W_{1} is a subspace of W_{2}. Let k be a natural number. Suppose 1 ≤ k and k < dim(V) and dim(W_{1}) + 1 = k and dim(W_{2}) = k + 1. Let v be a vector of V. If v ∈ W_{2} and v ∉ W_{1}, then W_{1} + Lin({v}) ∈ pencil(W_{1}, W_{2}, k).

(17) Let F be a field, V be a finite dimensional vector space over F, and W_{1}, W_{2} be subspaces of V. Suppose W_{1} is a subspace of W_{2}. Let k be a natural number. If 1 ≤ k and k < dim(V) and dim(W_{1}) + 1 = k and dim(W_{2}) = k + 1, then pencil(W_{1}, W_{2}, k) is non trivial.

Let F be a field, let V be a finite dimensional vector space over F, and let k be a natural number. The functor PencilSpace(V, k) yielding a strict topological structure is defined by:

(Def. 5) PencilSpace(V, k) = ⟨Sub_{k}(V), k pencils of V ⟩.

Next we state several propositions:

(18) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If k ≤ dim(V), then PencilSpace(V, k) is non empty.

(19) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If 1 ≤ k and k < dim(V), then PencilSpace(V, k) is non void.

(20) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If 1 ≤ k and k < dim(V) and 3 ≤ dim(V), then PencilSpace(V, k) is non degenerated.

(21) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If 1 ≤ k and k < dim(V), then PencilSpace(V, k) has non trivial blocks.

(22) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If 1 ≤ k and k < dim(V), then PencilSpace(V, k) is identifying close blocks.

(23) Let F be a field, V be a finite dimensional vector space over F, and k be a natural number. If 1 ≤ k and k < dim(V) and 3 ≤ dim(V), then PencilSpace(V, k) is a PLS.

2. Grassmann Spaces

Let F be a field, let V be a finite dimensional vector space over F, and let m, n be natural numbers. The functor SubspaceSet(V, m, n) yields a family of subsets of Sub_{m}(V) and is defined by:

(Def. 6) For every set X holds X ∈ SubspaceSet(V, m, n) iff there exists a sub-

space W of V such that dim(W) = n and X = Sub_{m}(W). One can prove the following propositions:

(24) Let F be a field, V be a finite dimensional vector space over F, and m, n be natural numbers. If n ≤ dim(V), then SubspaceSet(V, m, n) is non empty.

(25) Let F be a field and W_{1}, W_{2} be finite dimensional vector spaces over F. If Ω_{(W_{1})} = Ω_{(W_{2})}, then for every natural number m holds Sub_{m}(W_{1}) = Sub_{m}(W_{2}).

(26) Let F be a field, V be a finite dimensional vector space over F, W be a subspace of V, and m be a natural number. If 1 ≤ m and m ≤ dim(V) and Sub_{m}(V) ⊆ Sub_{m}(W), then Ω_{V} = Ω_{W} .

Let F be a field, let V be a finite dimensional vector space over F, and let m, n be natural numbers. The functor GrassmannSpace(V, m, n) yields a strict topological structure and is defined as follows:

(Def. 7) GrassmannSpace(V, m, n) = ⟨Sub_{m}(V), SubspaceSet(V, m, n)⟩.

We now state several propositions:

(27) Let F be a field, V be a finite dimensional vector space over F, and m, n be natural numbers. If m ≤ dim(V), then GrassmannSpace(V, m, n) is non empty.

(28) Let F be a field, V be a finite dimensional vector space over F, and m, n be natural numbers. If n ≤ dim(V), then GrassmannSpace(V, m, n) is non void.

(29) Let F be a field, V be a finite dimensional vector space over F, and m, n be natural numbers. If 1 ≤ m and m < n and n < dim(V), then GrassmannSpace(V, m, n) is non degenerated.

(30) Let F be a field, V be a finite dimensional vector space over F, and m, n be natural numbers. If 1 ≤ m and m < n and n ≤ dim(V), then GrassmannSpace(V, m, n) has non trivial blocks.

(31) Let F be a field, V be a finite dimensional vector space over F, and m be a natural number. If 1 ≤ m and m + 1 ≤ dim(V), then GrassmannSpace(V, m, m + 1) is identifying close blocks.

(32) Let F be a field, V be a finite dimensional vector space over F, and m be a natural number. If 1 ≤ m and m + 1 < dim(V), then GrassmannSpace(V, m, m + 1) is a PLS.

3. Veronese Spaces

Let X be a set. The functor PairSet X is defined as follows: (Def. 8) For every set z holds z ∈ PairSet X iff there exist sets x, y such that

x ∈ X and y ∈ X and z = {x, y}. Let X be a non empty set. One can verify that PairSet X is non empty. Let t, X be sets. The functor PairSet(t, X) is defined as follows:

(Def. 9) For every set z holds z ∈ PairSet(t, X) iff there exists a set y such that

y ∈ X and z = {t, y}. Let t be a set and let X be a non empty set. One can verify that PairSet(t, X)

is non empty.

Let t be a set and let X be a non trivial set. One can verify that PairSet(t, X) is non trivial.

Let X be a set and let L be a family of subsets of X. The functor PairSetFamily L yields a family of subsets of PairSet X and is defined as follows:

(Def. 10) For every set S holds S ∈ PairSetFamily L iff there exists a set t and

there exists a subset l of X such that t ∈ X and l ∈ L and S = PairSet(t, l). Let X be a non empty set and let L be a non empty family of subsets of X.

Note that PairSetFamily L is non empty.

Let S be a topological structure. The functor VeroneseSpace S yielding a strict topological structure is defined by:

(Def. 11) VeroneseSpace S = ⟨PairSet (the carrier of S),PairSetFamily (the topology of S)⟩.

138 adam naumowicz

Let S be a non empty topological structure. One can verify that VeroneseSpace S is non empty.

Let S be a non empty non void topological structure. One can check that VeroneseSpace S is non void.

Let S be a non empty non void non degenerated topological structure. Note that VeroneseSpace S is non degenerated.

Let S be a non empty non void topological structure with non trivial blocks. One can check that VeroneseSpace S has non trivial blocks.

Let S be an identifying close blocks topological structure. Note that VeroneseSpace S is identifying close blocks.

Let S be a PLS. Then VeroneseSpace S is a strict PLS.

References

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

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

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

[4] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

[6] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[7] Adam Naumowicz. On Segre’s product of partial line spaces. Formalized Mathematics, 9(2):383–390, 2001.

[8] Adam Naumowicz and Krzysztof Prażmowski. The geometry of generalized Veronese spaces. Results in Mathematics, 45:115–136, 2004.

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

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

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

[12] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.

[13] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871–876, 1990.

[14] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.

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

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

[17] Mariusz Żynel. The Steinitz theorem and the dimension of a vector space. Formalized Mathematics, 5(3):423–428, 1996.

Received November 8, 2004

On the Boundary and Derivative of a Set^{1}

Adam Grabowski University of Białystok

Summary. This is the first Mizar article in a series aiming at a complete formalization of the textbook “General Topology” by Engelking [7]. We cover the first part of Section 1.3, by defining such notions as a derivative of a subset A of a topological space (usually denoted by A^{d}, but Der A in our notation), the derivative and the boundary of families of subsets, points of accumulation and isolated points. We also introduce dense-in-itself, perfect and scattered topological spaces and formulate the notion of the density of a space. Some basic properties are given as well as selected exercises from [7].

MML Identifier: TOPGEN 1.

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

1. Preliminaries

Let T be a set, let A be a subset of T, and let B be a set. Then A \ B is a subset of T.

The following three propositions are true:

(1) For every 1-sorted structure T and for all subsets A, B of T holds A meets B^{c} iff A \ B ≠ ∅.

(2) For every 1-sorted structure T holds T is countable iff Ω_{T} is countable.

(3) For every 1-sorted structure T holds T is countable iff \overline{\overline{Ω_{T}}} ≤ ℵ_{0}. Let T be a finite 1-sorted structure. Note that Ω_{T} is finite. Let us note that every 1-sorted structure which is finite is also countable.

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

Let us observe that there exists a 1-sorted structure which is countable and non empty and there exists a topological space which is countable and non empty.

Let T be a countable 1-sorted structure. Observe that Ω_{T} is countable. Let us observe that there exists a topological space which is T_{1} and non

empty.

2. Boundary of a Subset

Next we state two propositions:

(4) For every topological structure T and for every subset A of T holds A ∪ Ω_{T} = Ω_{T} .

(5) For every topological space T and for every subset A of T holds Int A = \overline{A^{c}}^{c}.

Let T be a topological space and let F be a family of subsets of T. The functor Fr F yielding a family of subsets of T is defined by:

(Def. 1) For every subset A of T holds A ∈ Fr F iff there exists a subset B of T

such that A = Fr B and B ∈ F. The following propositions are true:

(6) For every topological space T and for every family F of subsets of T such that F = ∅ holds Fr F = ∅.

(7) Let T be a topological space, F be a family of subsets of T, and A be a subset of T. If F = {A}, then Fr F = {Fr A}.

(8) For every topological space T and for all families F, G of subsets of T such that F ⊆ G holds Fr F ⊆ Fr G.

(9) For every topological space T and for all families F, G of subsets of T holds Fr(F ∪ G) = Fr F ∪ Fr G.

(10) For every topological structure T and for every subset A of T holds Fr A = \overline{A} \ Int A.

(11) Let T be a non empty topological space, A be a subset of T, and p be a point of T. Then p ∈ Fr A if and only if for every subset U of T such that U is open and p ∈ U holds A meets U and U \ A ≠ ∅.

(12) Let T be a non empty topological space, A be a subset of T, and p be a point of T. Then p ∈ Fr A if and only if for every basis B of p and for every subset U of T such that U ∈ B holds A meets U and U \ A ≠ ∅.

(13) Let T be a non empty topological space, A be a subset of T, and p be a point of T. Then p ∈ Fr A if and only if there exists a basis B of p such that for every subset U of T such that U ∈ B holds A meets U and U \ A ≠ ∅.

(14) For every topological space T and for all subsets A, B of T holds Fr(A ∩ B) ⊆ \overline{A} ∩ Fr B ∪ Fr A ∩ \overline{B}.

(15) For every topological space T and for every subset A of T holds the carrier of T = Int A ∪ Fr A ∪ Int(A^{c}).

(16) For every topological space T and for every subset A of T holds A is open and closed iff Fr A = ∅.

3. Accumulation Points and Derivative of a Set

Let T be a topological structure, let A be a subset of T, and let x be a set. We say that x is an accumulation point of A if and only if:

(Def. 2) x ∈ \overline{A \ \{x\}}.

We now state the proposition

(17) Let T be a topological space, A be a subset of T, and x be a set. If x is

an accumulation point of A, then x is a point of T. Let T be a topological structure and let A be a subset of T. The functor

Der A yielding a subset of T is defined by: (Def. 3) For every set x such that x ∈ the carrier of T holds x ∈ Der A iff x is an

accumulation point of A. Next we state four propositions:

(18) Let T be a non empty topological space, A be a subset of T, and x be a set. Then x ∈ Der A if and only if x is an accumulation point of A.

(19) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ Der A if and only if for every open subset U of T such that x ∈ U there exists a point y of T such that y ∈ A ∩ U and x ≠ y.

(20) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ Der A if and only if for every basis B of x and for every subset U of T such that U ∈ B there exists a point y of T such that y ∈ A ∩ U and x ≠ y.

(21) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ Der A if and only if there exists a basis B of x such that for every subset U of T such that U ∈ B there exists a point y of T such that y ∈ A ∩ U and x ≠ y.

4. Isolated Points

Let T be a topological space, let A be a subset of T, and let x be a set. We say that x is isolated in A if and only if:

(Def. 4) x ∈ A and x is not an accumulation point of A.

The following three propositions are true:

(22) Let T be a non empty topological space, A be a subset of T, and p be a set. Then p ∈ A \ Der A if and only if p is isolated in A.

(23) Let T be a non empty topological space, A be a subset of T, and p be a point of T. Then p is an accumulation point of A if and only if for every open subset U of T such that p ∈ U there exists a point q of T such that q ≠ p and q ∈ A and q ∈ U.

(24) Let T be a non empty topological space, A be a subset of T, and p be a point of T. Then p is isolated in A if and only if there exists an open subset G of T such that G ∩ A = {p}.

Let T be a topological space and let p be a point of T. We say that p is isolated if and only if:

(Def. 5) p is isolated in Ω_{T}.

Next we state the proposition

(25) For every non empty topological space T and for every point p of T holds p is isolated iff {p} is open.

5. Derivative of a Subset-Family

Let T be a topological space and let F be a family of subsets of T. The functor Der F yielding a family of subsets of T is defined by:

(Def. 6) For every subset A of T holds A ∈ Der F iff there exists a subset B of T

such that A = Der B and B ∈ F. For simplicity, we follow the rules: T is a non empty topological space, A,

B are subsets of T, F, G are families of subsets of T, and x is a set.

One can prove the following propositions:

(26) If F = ∅, then Der F = ∅.

(27) If F = {A}, then Der F = {Der A}.

(28) If F ⊆ G, then Der F ⊆ Der G.

(29) Der(F ∪ G) = Der F ∪ Der G.

(30) For every non empty topological space T and for every subset A of T holds Der A ⊆ \overline{A}.

(31) For every topological space T and for every subset A of T holds \overline{A} = A ∪ Der A.

(32) For every non empty topological space T and for all subsets A, B of T such that A ⊆ B holds Der A ⊆ Der B.

(33) For every non empty topological space T and for all subsets A, B of T holds Der(A ∪ B) = Der A ∪ Der B.

(34) For every non empty topological space T and for every subset A of T such that T is T_{1} holds Der Der A ⊆ Der A.

(35) For every topological space T and for every subset A of T such that T

is T_{1} holds \overline{Der A} = Der A. Let T be a T_{1} non empty topological space and let A be a subset of T.

Observe that Der A is closed.

One can prove the following two propositions:

(36) For every non empty topological space T and for every family F of subsets of T holds ⋃ Der F ⊆ Der ⋃ F.

(37) If A ⊆ B and x is an accumulation point of A, then x is an accumulation point of B.

6. Density-in-itself

Let T be a topological space and let A be a subset of T. We say that A is dense-in-itself if and only if:

(Def. 7) A ⊆ Der A.

Let T be a non empty topological space. We say that T is dense-in-itself if and only if:

(Def. 8) Ω_{T} is dense-in-itself.

Next we state the proposition

(38) If T is T_{1} and A is dense-in-itself, then \overline{A} is dense-in-itself.

Let T be a topological space and let F be a family of subsets of T. We say that F is dense-in-itself if and only if:

(Def. 9) For every subset A of T such that A ∈ F holds A is dense-in-itself.

The following propositions are true:

(39) For every family F of subsets of T such that F is dense-in-itself holds ⋃ F ⊆ ⋃ Der F.

(40) If F is dense-in-itself, then ⋃ F is dense-in-itself.

(41) Fr(∅_{T}) = ∅.

Let T be a topological space and let A be an open closed subset of T. Note that Fr A is empty.

Let T be a non empty non discrete topological space. Note that there exists a subset of T which is non open and there exists a subset of T which is non closed.

Let T be a non empty non discrete topological space and let A be a non open subset of T. Observe that Fr A is non empty.

Let T be a non empty non discrete topological space and let A be a non closed subset of T. One can check that Fr A is non empty.

7. Perfect Sets

Let T be a topological space and let A be a subset of T. We say that A is perfect if and only if:

(Def. 10) A is closed and dense-in-itself.

Let T be a topological space. One can check that every subset of T which is perfect is also closed and dense-in-itself and every subset of T which is closed and dense-in-itself is also perfect.

We now state three propositions:

(42) For every topological space T holds Der(∅_{T}) = ∅_{T} .

(43) For every topological space T and for every subset A of T holds A is perfect iff Der A = A.

(44) For every topological space T holds ∅_{T} is perfect.

Let T be a topological space. Note that every subset of T which is empty is also perfect.

Let T be a topological space. Observe that there exists a subset of T which is perfect.

8. Scattered Subsets

Let T be a topological space and let A be a subset of T. We say that A is scattered if and only if:

(Def. 11) It is not true that there exists a subset B of T such that B is non empty

and B ⊆ A and B is dense-in-itself. Let T be a non empty topological space. Observe that every subset of T

which is non empty and scattered is also non dense-in-itself and every subset of T which is dense-in-itself and non empty is also non scattered.

The following proposition is true

(45) For every topological space T holds ∅_{T} is scattered.

Let T be a topological space. Note that every subset of T which is empty is also scattered.

One can prove the following proposition

(46) Let T be a non empty topological space. Suppose T is T_{1}. Then there exist subsets A, B of T such that A ∪ B = Ω_{T} and A misses B and A is perfect and B is scattered.

Let T be a discrete topological space and let A be a subset of T. Observe that Fr A is empty.

Let T be a discrete topological space. Observe that every subset of T is closed and open.

The following proposition is true (47) For every discrete topological space T and for every subset A of T holds

Der A = ∅. Let T be a discrete non empty topological space and let A be a subset of T.

Note that Der A is empty.

One can prove the following proposition

(48) For every discrete non empty topological space T and for every subset A of T such that A is dense holds A = Ω_{T} .

9. Density of a Topological Space and Separable Spaces

Let T be a topological space. The functor density T yielding a cardinal number is defined by:

(Def. 12) There exists a subset A of T such that A is dense and density T = \overline{\overline{A}}

and for every subset B of T such that B is dense holds density T ≤ \overline{\overline{B}} . Let T be a topological space. We say that T is separable if and only if:

(Def. 13) density T ≤ ℵ_{0}.

One can prove the following proposition

(49) Every countable topological space is separable.

Let us observe that every topological space which is countable is also separable.

10. Exercises

The following propositions are true:

(50) For every subset A of ℝ^{1} such that A = ℚ holds A^{c} = 𝕀ℚ.

(51) For every subset A of ℝ^{1} such that A = 𝕀ℚ holds A^{c} = ℚ.

(52) For every subset A of ℝ^{1} such that A = ℚ holds Int A = ∅.

(53) For every subset A of ℝ^{1} such that A = 𝕀ℚ holds Int A = ∅.

(54) For every subset A of ℝ^{1} such that A = ℚ holds A is dense.

(55) For every subset A of ℝ^{1} such that A = 𝕀ℚ holds A is dense.

(56) For every subset A of ℝ^{1} such that A = ℚ holds A is boundary.

(57) For every subset A of ℝ^{1} such that A = 𝕀ℚ holds A is boundary.

(58) For every subset A of ℝ^{1} such that A = ℝ holds A is non boundary.

(59) There exist subsets A, B of ℝ^{1} such that A is boundary and B is boundary and A ∪ B is non boundary.

146 adam grabowski

References

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

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

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

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

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

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

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

[8] Adam Grabowski. On the subcontinua of a real line. Formalized Mathematics, 11(3):313– 322, 2003.

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

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

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

[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. Baire spaces, Sober spaces. Formalized Mathematics, 6(2):289–294, 1997.

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

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

Received November 8, 2004

Construction of Gröbner Bases: Avoiding S-Polynomials – Buchberger’s

First Criterium^{1}

Christoph Schwarzweller University of Gdańsk

Summary. We continue the formalization of Groebner bases following the book “Groebner Bases – A Computational Approach to Commutative Algebra” by Becker and Weispfenning. Here we prove Buchberger’s first criterium on avoiding S-polynomials: S-polynomials for polynomials with disjoint head terms need not be considered when constructing Groebner bases. In the course of formalizing this theorem we also introduced the splitting of a polynomial in an upper and a lower polynomial containing the greater resp. smaller terms of the original polynomial with respect to a given term order.

MML Identifier: GROEB 3.

The terminology and notation used in this paper have been introduced in the following articles: [24], [28], [29], [31], [1], [3], [12], [2], [8], [30], [9], [10], [17], [25], [16], [26], [11], [7], [5], [15], [13], [19], [27], [6], [4], [14], [23], [20], [22], [21], and [18].

1. Preliminaries

One can prove the following propositions:

(1) For every set X and for all bags b_{1}, b_{2} of X holds \frac{b_{1}+b_{2}}{b_{2}} = b_{1}.

(2) Let n be an ordinal number, T be an admissible term order of n, and b_{1}, b_{2}, b_{3} be bags of n. If b_{1} ≤_{T} b_{2}, then b_{1} + b_{3} ≤_{T} b_{2} + b_{3}.

(3) Let n be an ordinal number, T be a term order of n, and b_{1}, b_{2}, b_{3} be bags of n. If b_{1} ≤_{T} b_{2} and b_{2} <_{T} b_{3}, then b_{1} <_{T} b_{3}.

1This work has been partially supported by grant BW 5100-5-0147-4.

(4) Let n be an ordinal number, T be an admissible term order of n, and b_{1}, b_{2}, b_{3} be bags of n. If b_{1} <_{T} b_{2}, then b_{1} + b_{3} <_{T} b_{2} + b_{3}.

(5) Let n be an ordinal number, T be an admissible term order of n, and b_{1}, b_{2}, b_{3}, b_{4} be bags of n. If b_{1} <_{T} b_{2} and b_{3} ≤_{T} b_{4}, then b_{1} + b_{3} <_{T} b_{2} + b_{4}.

(6) Let n be an ordinal number, T be an admissible term order of n, and b_{1}, b_{2}, b_{3}, b_{4} be bags of n. If b_{1} ≤_{T} b_{2} and b_{3} <_{T} b_{4}, then b_{1} + b_{3} <_{T} b_{2} + b_{4}.

2. More on Polynomials

One can prove the following propositions:

(7) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, and m_{1}, m_{2} be non-zero monomials of n, L. Then term m_{1} ∗ m_{2} = term m_{1} + term m_{2}.

(8) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, m be a non-zero monomial of n, L, and b be a bag of n. Then b ∈ Support p if and only if term m + b ∈ Support(m ∗ p).

(9) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, and m be a non-zero monomial of n, L. Then Support(m ∗ p) = {term m + b; b ranges over elements of Bags n : b ∈ Support p}.

(10) Let n be an ordinal number, L be an add-associative right complementable left zeroed right zeroed unital distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, and m be a non-zero monomial of n, L. Then card Support p = card Support(m ∗ p).

(11) Let n be an ordinal number, T be a connected term order of n, and L be an add-associative right complementable right zeroed non trivial loop structure. Then Red(0_{n}L, T) = 0_{n}L.

(12) Let n be an ordinal number, L be an Abelian add-associative right zeroed right complementable commutative unital distributive non trivial double loop structure, and p, q be polynomials of n, L. If p−q = 0_{n}L, then p = q.

(13) Let X be a set and L be an add-associative right zeroed right complementable non empty loop structure. Then −0_{X}L = 0_{X}L.

(14) Let X be a set, L be an add-associative right zeroed right complementable non empty loop structure, and f be a series of X, L. Then 0_{X}L − f = −f.

(15) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial double loop structure, and p be a polynomial of n, L. Then p−Red(p, T) = HM(p, T).

Let n be an ordinal number, let L be an add-associative right complementable right zeroed non empty loop structure, and let p be a polynomial of n, L. Observe that Support p is finite.

Let n be an ordinal number, let L be a right zeroed add-associative right complementable unital distributive non trivial double loop structure, and let p, q be polynomials of n, L. Then {p, q} is a non empty finite subset of Polynom-Ring(n, L).

3. Restriction and Splitting of Polynomials

Let X be a set, let L be a non empty zero structure, let s be a series of X, L, and let Y be a subset of Bags X. The functor s↾Y yields a series of X, L and is defined as follows:

(Def. 1) s↾Y = s+·(Support s \ Y ⟼ 0_{L}).

Let n be an ordinal number, let L be a non empty zero structure, let p be a polynomial of n, L, and let Y be a subset of Bags n. Note that p↾Y is finite-Support.

Next we state three propositions:

(16) Let X be a set, L be a non empty zero structure, s be a series of X, L, and Y be a subset of Bags X. Then Support(s↾Y) = Support s ∩ Y and for every bag b of X such that b ∈ Support(s↾Y) holds (s↾Y)(b) = s(b).

(17) Let X be a set, L be a non empty zero structure, s be a series of X, L, and Y be a subset of Bags X. Then Support(s↾Y) ⊆ Support s.

(18) For every set X and for every non empty zero structure L and for every

series s of X, L holds s↾ Support s = s and s↾∅_{Bags X} = 0_{X}L. Let n be an ordinal number, let T be a connected term order of n, let L be

an add-associative right zeroed right complementable non empty loop structure, let p be a polynomial of n, L, and let i be a natural number. Let us assume that i ≤ card Support p. The functor UpperSupport(p, T, i) yielding a finite subset of Bags n is defined by the conditions (Def. 2).

(Def. 2)(i) UpperSupport(p, T, i) ⊆ Support p, (ii) card UpperSupport(p, T, i) = i, and (iii) for all bags b, b′ of n such that b ∈ UpperSupport(p, T, i) and b′ ∈

Support p and b ≤_{T} b′ holds b′ ∈ UpperSupport(p, T, i). Let n be an ordinal number, let T be a connected term order of n, let L be

an add-associative right zeroed right complementable non empty loop structure, let p be a polynomial of n, L, and let i be a natural number. The functor LowerSupport(p, T, i) yielding a finite subset of Bags n is defined by:

(Def. 3) LowerSupport(p, T, i) = Support p \ UpperSupport(p, T, i).

We now state several propositions:

(19) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i ≤ card Support p, then UpperSupport(p, T, i) ∪ LowerSupport(p, T, i) = Support p and UpperSupport(p, T, i) ∩ LowerSupport(p, T, i) = ∅.

(20) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Let b, b′ be bags of n. If b ∈ UpperSupport(p, T, i) and b′ ∈ LowerSupport(p, T, i), then b′ <_{T} b.

(21) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, and p be a polynomial of n, L. Then UpperSupport(p, T, 0) = ∅ and LowerSupport(p, T, 0) = Support p.

(22) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, and p be a polynomial of n, L. Then UpperSupport(p, T, card Support p) = Support p and LowerSupport(p, T, card Support p) = ∅.

(23) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non trivial loop structure, p be a non-zero polynomial of n, L, and i be a natural number. If 1 ≤ i and i ≤ card Support p, then HT(p, T) ∈ UpperSupport(p, T, i).

(24) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Then LowerSupport(p, T, i) ⊆ Support p and card LowerSupport(p, T, i) = card Support p − i and for all bags b, b′ of n such that b ∈ LowerSupport(p, T, i) and b′ ∈ Support p and b′ ≤_{T} b holds b′ ∈ LowerSupport(p, T, i).

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right zeroed right complementable non empty loop structure, let p be a polynomial of n, L, and let i be a natural number. The functor Up(p, T, i) yields a polynomial of n, L and is defined by:

(Def. 4) Up(p, T, i) = p↾ UpperSupport(p, T, i).

The functor Low(p, T, i) yielding a polynomial of n, L is defined by: (Def. 5) Low(p, T, i) = p↾ LowerSupport(p, T, i).

One can prove the following propositions:

(25) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i ≤ card Support p, then Support Up(p, T, i) = UpperSupport(p, T, i) and Support Low(p, T, i) = LowerSupport(p, T, i).

(26) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i ≤ card Support p, then Support Up(p, T, i) ⊆ Support p and Support Low(p, T, i) ⊆ Support p.

(27) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed non trivial loop structure, p be a polynomial of n, L, and i be a natural number. If 1 ≤ i and i ≤ card Support p, then Support Low(p, T, i) ⊆ Support Red(p, T).

(28) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Let b be a bag of n. If b ∈ Support p, then b ∈ Support Up(p, T, i) or b ∈ Support Low(p, T, i) but b ∉ Support Up(p, T, i) ∩ Support Low(p, T, i).

(29) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Let b, b′ be bags of n. If b ∈ Support Low(p, T, i) and b′ ∈ Support Up(p, T, i), then b <_{T} b′.

(30) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If 1 ≤ i and i ≤ card Support p, then HT(p, T) ∈ Support Up(p, T, i).

(31) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Let b be a bag of n. If b ∈ Support Low(p, T, i), then (Low(p, T, i))(b) = p(b) and (Up(p, T, i))(b) = 0_{L}.

(32) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Let b be a bag of n. If b ∈ Support Up(p, T, i), then (Up(p, T, i))(b) = p(b) and (Low(p, T, i))(b) = 0_{L}.

(33) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i ≤ card Support p, then Up(p, T, i) + Low(p, T, i) = p.

(34) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, and p be a polynomial of n, L. Then Up(p, T, 0) = 0_{n}L and Low(p, T, 0) = p.

(35) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable Abelian non empty double loop structure, and p be a polynomial of n, L. Then Up(p, T, card Support p) = p and Low(p, T, card Support p) = 0_{n}L.

(36) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable Abelian non trivial double loop structure, and p be a non-zero polynomial of n, L. Then Up(p, T, 1) = HM(p, T) and Low(p, T, 1) = Red(p, T).

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right zeroed right complementable non trivial loop structure, and let p be a non-zero polynomial of n, L. Observe that Up(p, T, 0) is monomial-like.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right zeroed right complementable Abelian non trivial double loop structure, and let p be a non-zero polynomial of n, L. Note that Up(p, T, 1) is non-zero and monomial-like and Low(p, T, card Support p) is monomial-like.

The following propositions are true:

(37) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non trivial loop structure, p be a polynomial of n, L, and j be a natural number. If j = card Support p − 1, then Low(p, T, j) is a non-zero monomial of n, L.

(38) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i < card Support p, then HT(Low(p, T, i + 1), T) ≤_{T} HT(Low(p, T, i), T).

(39) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If 0 < i and i < card Support p, then HT(Low(p, T, i), T) <_{T} HT(p, T).

(40) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, m be a non-zero monomial of n, L, and i be a natural number. Suppose i ≤ card Support p. Let b be a bag of n. Then term m+b ∈ Support Low(m∗p, T, i) if and only if b ∈ Support Low(p, T, i).

(41) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i < card Support p, then Support Low(p, T, i + 1) ⊆ Support Low(p, T, i).

(42) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right zeroed right complementable non empty loop structure, p be a polynomial of n, L, and i be a natural number. If i < card Support p, then Support Low(p, T, i) \ Support Low(p, T, i + 1) = {HT(Low(p, T, i), T)}.

(43) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right zeroed right complementable non trivial loop structure, p be a polynomial of n, L, and i be a natural number. If i < card Support p, then Low(p, T, i + 1) = Red(Low(p, T, i), T).

(44) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed unital distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, m be a non-zero monomial of n, L, and i be a natural number. If i ≤ card Support p, then Low(m ∗ p, T, i) = m ∗ Low(p, T, i).

4. More on Polynomial Reduction

Next we state several propositions:

(45) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and f, g, p be polynomials of n, L. If f reduces to g, p, T, then −f reduces to −g, p, T.

(46) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and f, f_{1}, g, p be polynomials of n, L. Suppose f reduces to f_{1}, {p}, T and for every bag b_{1} of n such that b_{1} ∈ Support g holds HT(p, T) ∤ b_{1}. Then f + g reduces to f_{1} + g, {p}, T.

(47) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and f, g be non-zero polynomials of n, L. Then f ∗ g reduces to Red(f, T) ∗ g, {g}, T.

(48) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, f, g be non-zero polynomials of n, L, and p be a polynomial of n, L. If p(HT(f ∗ g, T)) = 0_{L}, then f ∗ g + p reduces to Red(f, T) ∗ g + p, {g}, T.

(49) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and f, g be polynomials of n, L. If PolyRedRel(P, T) reduces f to g, then PolyRedRel(P, T) reduces −f to −g.

(50) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and f, f_{1}, g, p be polynomials of n, L. Suppose PolyRedRel({p}, T) reduces f to f_{1} and for every bag b_{1} of n such that b_{1} ∈ Support g holds HT(p, T) ∤ b_{1}. Then PolyRedRel({p}, T) reduces f +g to f_{1} + g.

(51) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and f, g be non-zero polynomials of n, L. Then PolyRedRel({g}, T) reduces f ∗ g to 0_{n}L.

5. The Criterium

We now state several propositions:

(52) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive field-like non trivial double loop structure, and p_{1}, p_{2} be polynomials of n, L. Suppose HT(p_{1}, T), HT(p_{2}, T) are disjoint. Let b_{1}, b_{2} be bags of n. If b_{1} ∈ Support Red(p_{1}, T) and b_{2} ∈ Support Red(p_{2}, T), then HT(p_{1}, T) + b_{2} ≠ HT(p_{2}, T) + b_{1}.

(53) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and p_{1}, p_{2} be polynomials of n, L. If HT(p_{1}, T), HT(p_{2}, T) are disjoint, then S-Poly(p_{1}, p_{2}, T) = HM(p_{2}, T)∗Red(p_{1}, T)−HM(p_{1}, T)∗ Red(p_{2}, T).

(54) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and p_{1}, p_{2} be polynomials of n, L. If HT(p_{1}, T), HT(p_{2}, T) are disjoint, then S-Poly(p_{1}, p_{2}, T) = Red(p_{1}, T) ∗ p_{2} − Red(p_{2}, T) ∗ p_{1}.

(55) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and p_{1}, p_{2} be non-zero polynomials of n, L. Suppose HT(p_{1}, T), HT(p_{2}, T) are disjoint and Red(p_{1}, T) is non-zero and Red(p_{2}, T) is non-zero. Then PolyRedRel({p_{1}}, T) reduces HM(p_{2}, T) ∗ Red(p_{1}, T) − HM(p_{1}, T) ∗ Red(p_{2}, T) to p_{2} ∗ Red(p_{1}, T).

(56) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non trivial double loop structure, and p_{1}, p_{2} be polynomials of n, L. If HT(p_{1}, T), HT(p_{2}, T) are disjoint, then PolyRedRel({p_{1}, p_{2}}, T) reduces S-Poly(p_{1}, p_{2}, T) to 0_{n}L.

(57) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non degenerated non empty double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose G is a Groebner basis wrt T. Let g_{1}, g_{2} be polynomials of n, L. Suppose g_{1} ∈ G and g_{2} ∈ G and HT(g_{1}, T), HT(g_{2}, T) are not disjoint. Then PolyRedRel(G, T) reduces S-Poly(g_{1}, g_{2}, T) to 0_{n}L.

(58) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non degenerated non trivial double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose 0_{n}L ∉ G. Suppose that for all polynomials g_{1}, g_{2} of n, L such that g_{1} ∈ G and g_{2} ∈ G and HT(g_{1}, T), HT(g_{2}, T) are not disjoint holds PolyRedRel(G, T) reduces S-Poly(g_{1}, g_{2}, T) to 0_{n}L. Let g_{1}, g_{2}, h be polynomials of n, L. Suppose g_{1} ∈ G and g_{2} ∈ G and HT(g_{1}, T), HT(g_{2}, T) are not disjoint and h is a normal form of S-Poly(g_{1}, g_{2}, T) w.r.t. PolyRedRel(G, T). Then h = 0_{n}L.

(59) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative left unital right unital distributive Abelian field-like non degenerated non empty double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose 0_{n}L ∉ G. Suppose that for all polynomials g_{1}, g_{2}, h of n, L such that g_{1} ∈ G and g_{2} ∈ G and HT(g_{1}, T), HT(g_{2}, T) are not disjoint and h is a normal form of S-Poly(g_{1}, g_{2}, T) w.r.t. PolyRedRel(G, T) holds h = 0_{n}L. Then G is a Groebner basis wrt T.

156 christoph schwarzweller

References

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

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

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

[4] Grzegorz Bancerek. Reduction relations. Formalized Mathematics, 5(4):469–478, 1996.

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

[6] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.

[7] Józef Białas. Properties of fields. Formalized Mathematics, 1(5):807–812, 1990.

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

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

[10] 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.

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

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

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[16] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

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A Theory of Matrices of Complex Elements

Wenpai Chang Shinshu University

Nagano Hiroshi Yamazaki Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. A concept of “Matrix of Complex” is defined here. Addition, subtraction, scalar multiplication and product are introduced using correspondent definitions of “Matrix of Field”. Many equations for such operations consist of a case of “Matrix of Field”. A calculation method of product of matrices is shown using a finite sequence of Complex in the last theorem.

MML Identifier: MATRIX 5.

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

The following two propositions are true:

(1) 1 = 1_{ℂ_{F}}.

(2) 0_{ℂ_{F}} = 0. Let A be a matrix over ℂ. The functor A_{ℂ_{F}} yields a matrix over ℂ_{F} and is

defined by: (Def. 1) A_{ℂ_{F}} = A.

Let A be a matrix over ℂ_{F}. The functor A_{ℂ} yielding a matrix over ℂ is defined by:

(Def. 2) A_{ℂ} = A. We now state four propositions:

(3) For all matrices A, B over ℂ such that A_{ℂ_{F}} = B_{ℂ_{F}} holds A = B.

(4) For all matrices A, B over ℂ_{F} such that A_{ℂ} = B_{ℂ} holds A = B.

(5) For every matrix A over ℂ holds A = (A_{ℂ_{F}})_{ℂ}.

(6) For every matrix A over ℂ_{F} holds A = (A_{ℂ})_{ℂ_{F}}. Let A, B be matrices over ℂ. The functor A + B yielding a matrix over ℂ

is defined as follows:

(Def. 3) A + B = (A_{ℂ_{F}} + B_{ℂ_{F}})_{ℂ}.

Let A be a matrix over ℂ. The functor −A yielding a matrix over ℂ is defined as follows:

(Def. 4) −A = (−A_{ℂ_{F}})_{ℂ}.

Let A, B be matrices over ℂ. The functor A − B yields a matrix over ℂ and is defined as follows:

(Def. 5) A − B = (A_{ℂ_{F}} − B_{ℂ_{F}})_{ℂ}.

Let A, B be matrices over ℂ. The functor A · B yielding a matrix over ℂ is defined as follows:

(Def. 6) A · B = (A_{ℂ_{F}} · B_{ℂ_{F}})_{ℂ}.

Let x be a complex number and let A be a matrix over ℂ. The functor x · A yielding a matrix over ℂ is defined as follows:

(Def. 7) For every element e_{1} of ℂ_{F} such that e_{1} = x holds x · A = (e_{1} · A_{ℂ_{F}})_{ℂ}. One can prove the following propositions:

(7) For every matrix A over ℂ holds len A = len(A_{ℂ_{F}}) and width A = width(A_{ℂ_{F}}).

(8) For every matrix A over ℂ_{F} holds len A = len(A_{ℂ}) and width A = width(A_{ℂ}).

(9) For every matrix M over ℂ such that len M > 0 holds −−M = M.

(10) For every field K and for every matrix M over K holds 1_{K} · M = M.

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

(12) For every field K and for all elements a, b of K and for every matrix M over K holds a · (b · M) = (a · b) · M.

(13) For every field K and for all elements a, b of K and for every matrix M over K holds (a + b) · M = a · M + b · M.

(14) For every matrix M over ℂ holds M + M = 2 · M.

(15) For every matrix M over ℂ holds M + M + M = 3 · M.

Let n, m be natural numbers. The functor ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{n×m} yields a

matrix over ℂ and is defined by:

(Def. 8) ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{n×m} = (( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ_{F}}^{n×m})_{ℂ}.

One can prove the following propositions:

a theory of matrices of complex elements 159

(16) For all natural numbers n, m holds ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{n×m}_{ℂ_{F}} =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ_{F}}^{n×m}.

(17) For every matrix M over ℂ such that len M > 0 holds M + −M =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M)×(width M)}.

(18) For every matrix M over ℂ such that len M > 0 holds M − M =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M)×(width M)}.

(19) For all matrices M_{1}, M_{2}, M_{3} over ℂ such that len M_{1} = len M_{2} and len M_{2} = len M_{3} and width M_{1} = width M_{2} and width M_{2} = width M_{3} and len M_{1} > 0 and M_{1} + M_{3} = M_{2} + M_{3} holds M_{1} = M_{2}.

(20) For all matrices M_{1}, M_{2} over ℂ such that len M_{2} > 0 holds M_{1} −−M_{2} = M_{1} + M_{2}.

(21) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 and M_{1} = M_{1} + M_{2} holds

M_{2} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{1})}.

(22) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 and M_{1} − M_{2} =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{1})} holds M_{1} = M_{2}.

(23) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 and M_{1} + M_{2} =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{1})} holds M_{2} = −M_{1}.

(24) For all natural numbers n, m such that n > 0 holds

−( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{n×m} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{n×m}.

(25) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 and M_{2} − M_{1} = M_{2} holds

M_{1} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{1})}.

(26) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 holds M_{1} = M_{1} − (M_{2} − M_{2}).

(27) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 holds −(M_{1} + M_{2}) = −M_{1} + −M_{2}.

(28) For all matrices M_{1}, M_{2} over ℂ such that len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0 holds M_{1} − (M_{1} − M_{2}) = M_{2}.

(29) For all matrices M_{1}, M_{2}, M_{3} over ℂ such that len M_{1} = len M_{2} and len M_{2} = len M_{3} and width M_{1} = width M_{2} and width M_{2} = width M_{3} and len M_{1} > 0 and M_{1} − M_{3} = M_{2} − M_{3} holds M_{1} = M_{2}.

(30) For all matrices M_{1}, M_{2}, M_{3} over ℂ such that len M_{1} = len M_{2} and len M_{2} = len M_{3} and width M_{1} = width M_{2} and width M_{2} = width M_{3} and len M_{1} > 0 and M_{3} − M_{1} = M_{3} − M_{2} holds M_{1} = M_{2}.

(31) For all matrices M_{1}, M_{2}, M_{3} over ℂ such that len M_{2} = len M_{3} and width M_{2} = width M_{3} and width M_{1} = len M_{2} and len M_{1} > 0 and len M_{2} > 0 holds M_{1} · (M_{2} + M_{3}) = M_{1} · M_{2} + M_{1} · M_{3}.

(32) For all matrices M_{1}, M_{2}, M_{3} over ℂ such that len M_{2} = len M_{3} and width M_{2} = width M_{3} and len M_{1} = width M_{2} and len M_{2} > 0 and len M_{1} > 0 holds (M_{2} + M_{3}) · M_{1} = M_{2} · M_{1} + M_{3} · M_{1}.

(33) 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 M_{1} + M_{2} = M_{2} + M_{1}.

(34) For all matrices M_{1}, M_{2}, M_{3} over ℂ such that len M_{1} = len M_{2} and len M_{1} = len M_{3} and width M_{1} = width M_{2} and width M_{1} = width M_{3} holds (M_{1} + M_{2}) + M_{3} = M_{1} + (M_{2} + M_{3}).

(35) For every matrix M over ℂ such that len M > 0 holds M +

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M)×(width M)} = M.

(36) Let K be a field, b be an element of K, and M_{1}, M_{2} be matrices over K. If len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0, then b · (M_{1} + M_{2}) = b · M_{1} + b · M_{2}.

(37) Let M_{1}, M_{2} be matrices over ℂ and a be a complex number. If len M_{1} = len M_{2} and width M_{1} = width M_{2} and len M_{1} > 0, then a · (M_{1} + M_{2}) = a · M_{1} + a · M_{2}.

(38) For every field K and for all matrices M_{1}, M_{2} over K such that width M_{1} = len M_{2} and len M_{1} > 0 and len M_{2} > 0 holds

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(len M_{1})×(width M_{1})} · M_{2} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(len M_{1})×(width M_{2})}.

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

len M_{1} > 0 and len M_{2} > 0 holds ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{1})} · M_{2} =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{2})}.

(40) For every field K and for every matrix M_{1} over K such that len M_{1} > 0

holds 0_{K} · M_{1} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{(len M_{1})×(width M_{1})}.

(41) For every matrix M_{1} over ℂ such that len M_{1} > 0 holds 0 · M_{1} =

( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{ℂ}^{(len M_{1})×(width M_{1})}.

Let s be a finite sequence of elements of ℂ and let k be a natural number. Then s(k) is an element of ℂ.

We now state the proposition

(42) Let i, j be natural numbers and M_{1}, M_{2} be matrices over ℂ. Suppose len M_{1} > 0 and len M_{2} > 0 and width M_{1} = len M_{2} and 1 ≤ i and i ≤ len M_{1} and 1 ≤ j and j ≤ width M_{2}. Then there exists a finite sequence s of elements of ℂ such that len s = len M_{2} and s(1) = (M_{1}∘(i, 1))·(M_{2}∘(1, j)) and for every natural number k such that 1 ≤ k and k < len M_{2} holds s(k + 1) = s(k) + (M_{1} ∘ (i, k + 1)) · (M_{2} ∘ (k + 1, j)).

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

162 wenpai chang et al.

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

[5] Library Committee. Binary operations on numbers. To appear in Formalized Mathematics.

[6] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

[7] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[8] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2):265–269, 2001.

[9] Yatsuka Nakamura and Hiroshi Yamazaki. Calculation of matrices of field elements. Part I. Formalized Mathematics, 11(4):385–391, 2003.

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

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

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

[15] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1–8, 1993.

On the Characteristic and Weight of a Topological Space^{1}

Grzegorz Bancerek Białystok Technical University

Summary. We continue Mizar formalization of General Topology according to the book [13] by Engelking. In the article the formalization of Section 1.1 is completed. Namely, the paper includes the formalization of theorems on correspondence of the basis and basis in a point, definitions of the character of a point and a topological space, a neighborhood system, and the weight of a topological space. The formalization is tested with almost discrete topological spaces with infinity.

MML Identifier: TOPGEN 2.

The notation and terminology used here are introduced in the following articles: [22], [26], [21], [16], [27], [9], [28], [10], [7], [3], [18], [5], [4], [12], [24], [1], [2], [25], [17], [29], [11], [14], [8], [19], [20], [23], [6], and [15].

1. Characteristic of Topological Spaces

One can prove the following propositions:

(1) Let T be a non empty topological space, B be a basis of T, and x be an element of T. Then {U; U ranges over subsets of T: x ∈ U ∧ U ∈ B} is a basis of x.

(2) Let T be a non empty topological space and B be a many sorted set indexed by T. Suppose that for every element x of T holds B(x) is a basis of x. Then ⋃ B is a basis of T.

Let T be a non empty topological structure and let x be an element of T. The functor Chi(x, T) yielding a cardinal number is defined as follows:

1This work has been partially supported by the KBN grant 4 T11C 039 24.

(Def. 1) There exists a basis B of x such that Chi(x, T) = \overline{\overline{B}} and for every basis

B of x holds Chi(x, T) ≤ \overline{\overline{B}} . One can prove the following proposition

(3) Let X be a set. Suppose that for every set a such that a ∈ X holds a is a cardinal number. Then ⋃ X is a cardinal number.

Let T be a non empty topological structure. The functor Chi T yields a cardinal number and is defined by the conditions (Def. 2).

(Def. 2)(i) For every point x of T holds Chi(x, T) ≤ Chi T, and

(ii) for every cardinal number M such that for every point x of T holds

Chi(x, T) ≤ M holds Chi T ≤ M. The following three propositions are true:

(4) For every non empty topological structure T holds Chi T = ⋃{Chi(x, T) : x ranges over points of T }.

(5) For every non empty topological structure T and for every point x of T holds Chi(x, T) ≤ Chi T.

(6) For every non empty topological space T holds T is first-countable iff Chi T ≤ ℵ_{0}.

2. Neighborhood Systems

Let T be a non empty topological space. A many sorted set indexed by T is said to be a neighborhood system of T if:

(Def. 3) For every element x of T holds it(x) is a basis of x.

Let T be a non empty topological space and let N be a neighborhood system of T. Then ⋃ N is a basis of T. Let x be a point of T. Then N(x) is a basis of x.

We now state several propositions:

(7) Let T be a non empty topological space, N be a neighborhood system of T, and x be an element of T. Then N(x) is non empty and for every set U such that U ∈ N(x) holds x ∈ U.

(8) Let T be a non empty topological space, x, y be points of T, B_{1} be a basis of x, B_{2} be a basis of y, and U be a set. If x ∈ U and U ∈ B_{2}, then there exists an open subset V of T such that V ∈ B_{1} and V ⊆ U.

(9) Let T be a non empty topological space, x be a point of T, B be a basis of x, and U_{1}, U_{2} be sets. If U_{1} ∈ B and U_{2} ∈ B, then there exists an open subset V of T such that V ∈ B and V ⊆ U_{1} ∩ U_{2}.

(10) Let T be a non empty topological space, A be a subset of T, and x be an element of T. Then x ∈ \overline{A} if and only if for every basis B of x and for every set U such that U ∈ B holds U meets A.

(11) Let T be a non empty topological space, A be a subset of T, and x be an element of T. Then x ∈ \overline{A} if and only if there exists a basis B of x such that for every set U such that U ∈ B holds U meets A.

Let T be a topological space. Note that there exists a family of subsets of T which is open and non empty.

3. Weights of Topological Spaces

Next we state the proposition

(12) Let T be a non empty topological space and S be an open family of subsets of T. Then there exists an open family G of subsets of T such that G ⊆ S and ⋃ G = ⋃ S and \overline{\overline{G}} ≤ weight T.

Let T be a topological structure. We say that T is finite-weight if and only if:

(Def. 4) weight T is finite.

Let T be a topological structure. We introduce T is infinite-weight as an antonym of T is finite-weight.

Let us mention that every topological structure which is finite is also finiteweight and every topological structure which is infinite-weight is also infinite.

Let us note that there exists a topological space which is finite and non empty.

The following propositions are true:

(13) For every set X holds \overline{\overline{SmallestPartition(X)}} = \overline{\overline{X}} .

(14) Let T be a discrete non empty topological structure. Then SmallestPartition(the carrier of T) is a basis of T and for every basis B of T holds SmallestPartition(the carrier of T) ⊆ B.

(15) For every discrete non empty topological structure T holds weight T =

\overline{\overline{the carrier of T}} . One can verify that there exists a topological space which is infinite-weight. Next we state several propositions:

(16) Let T be an infinite-weight topological space and B be a basis of T. Then there exists a basis B_{1} of T such that B_{1} ⊆ B and \overline{\overline{B_{1}}} = weight T.

(17) Let T be a finite-weight non empty topological space. Then there exists a basis B_{0} of T and there exists a function f from the carrier of T into the topology of T such that B_{0} = rng f and for every point x of T holds x ∈ f(x) and for every open subset U of T such that x ∈ U holds f(x) ⊆ U.

(18) Let T be a topological space, B_{0}, B be bases of T, and f be a function from the carrier of T into the topology of T. Suppose B_{0} = rng f and for every point x of T holds x ∈ f(x) and for every open subset U of T such that x ∈ U holds f(x) ⊆ U. Then B_{0} ⊆ B.

(19) Let T be a topological space, B_{0} be a basis of T, and f be a function from the carrier of T into the topology of T. Suppose B_{0} = rng f and for every point x of T holds x ∈ f(x) and for every open subset U of T such that x ∈ U holds f(x) ⊆ U. Then weight T = \overline{\overline{B_{0}}} .

(20) For every non empty topological space T and for every basis B of T there exists a basis B_{1} of T such that B_{1} ⊆ B and \overline{\overline{B_{1}}} = weight T.

4. Example of Almost Discrete Topological Space with Infinity

Let X, x_{0} be sets. The functor DiscrWithInfin(X, x_{0}) yielding a strict topological structure is defined by the conditions (Def. 5).

(Def. 5)(i) The carrier of DiscrWithInfin(X, x_{0}) = X, and

(ii) the topology of DiscrWithInfin(X, x_{0}) = {U; U ranges over subsets of

X: x_{0} ∉ U} ∪ {F^{c}; F ranges over subsets of X: F is finite}. Let X, x_{0} be sets. Observe that DiscrWithInfin(X, x_{0}) is topological space-

like.

Let X be a non empty set and let x_{0} be a set. One can verify that DiscrWithInfin(X, x_{0}) is non empty.

Next we state a number of propositions:

(21) For all sets X, x_{0} and for every subset A of DiscrWithInfin(X, x_{0}) holds A is open iff x_{0} ∉ A or A^{c} is finite.

(22) For all sets X, x_{0} and for every subset A of DiscrWithInfin(X, x_{0}) holds A is closed iff if x_{0} ∈ X, then x_{0} ∈ A or A is finite.

(23) For all sets X, x_{0}, x such that x ∈ X holds {x} is a closed subset of DiscrWithInfin(X, x_{0}).

(24) For all sets X, x_{0}, x such that x ∈ X and x ≠ x_{0} holds {x} is an open subset of DiscrWithInfin(X, x_{0}).

(25) For all sets X, x_{0} such that X is infinite and for every subset U of DiscrWithInfin(X, x_{0}) such that U = {x_{0}} holds U is not open.

(26) For all sets X, x_{0} and for every subset A of DiscrWithInfin(X, x_{0}) such that A is finite holds \overline{A} = A.

(27) Let T be a non empty topological space and A be a subset of T. Suppose A is not closed. Let a be a point of T. If A∪\{a\} is closed, then \overline{A} = A∪\{a\}.

(28) For all sets X, x_{0} such that x_{0} ∈ X and for every subset A of DiscrWithInfin(X, x_{0}) such that A is infinite holds \overline{A} = A ∪ \{x_{0}\}.

(29) For all sets X, x_{0} and for every subset A of DiscrWithInfin(X, x_{0}) such that A^{c} is finite holds Int A = A.

(30) For all sets X, x_{0} such that x_{0} ∈ X and for every subset A of DiscrWithInfin(X, x_{0}) such that A^{c} is infinite holds Int A = A \ {x_{0}}.

(31) For all sets X, x_{0} there exists a basis B_{0} of DiscrWithInfin(X, x_{0}) such that B_{0} = (SmallestPartition(X) \ {{x_{0}}}) ∪ {F^{c}; F ranges over subsets of X: F is finite}.

In the sequel Z denotes an infinite set. The following proposition is true

(32) \overline{\overline{Fin Z}} = \overline{\overline{Z}} .

In the sequel F is a subset of Z. One can prove the following propositions:

(33) \overline{\overline{\{F^{c} : F is finite\}}} = \overline{\overline{Z}} .

(34) Let X be an infinite set, x_{0} be a set, and B_{0} be a basis of DiscrWithInfin(X, x_{0}). If B_{0} = (SmallestPartition(X) \ {{x_{0}}}) ∪ {F^{c}; F ranges over subsets of X: F is finite\}, then \overline{\overline{B_{0}}} = \overline{\overline{X}} .

(35) For every infinite set X and for every set x_{0} and for every basis B of DiscrWithInfin(X, x_{0}) holds \overline{\overline{X}} ≤ \overline{\overline{B}} .

(36) For every infinite set X and for every set x_{0} holds weight DiscrWithInfin(X, x_{0}) = \overline{\overline{X}} .

(37) Let X be a non empty set and x_{0} be a set. Then there exists a prebasis B_{0} of DiscrWithInfin(X, x_{0}) such that B_{0} = (SmallestPartition(X)\{{x_{0}}})∪ {{x}^{c} : x ranges over elements of X}.

5. Exercises

Next we state four propositions:

(38) Let T be a topological space, F be a family of subsets of T, and I be a non empty family of subsets of F. Suppose that for every set G such that G ∈ I holds F \ G is finite. Then \overline{⋃ F} = ⋃ clf F ∪ ⋂\{\overline{⋃ G}; G ranges over families of subsets of T: G ∈ I}.

(39) Let T be a topological space and F be a family of subsets of T. Then \overline{⋃ F} = ⋃ clf F ∪ ⋂\{\overline{⋃ G}; G ranges over families of subsets of T: G ⊆ F ∧ F \ G is finite}.

(40) Let X be a set and O be a family of subsets of 2^{X}. Suppose that for every family o of subsets of X such that o ∈ O holds ⟨X, o⟩ is a topological space. Then there exists a family B of subsets of X such that

(i) B = Intersect(O), (ii) ⟨X, B⟩ is a topological space, (iii) for every family o of subsets of X such that o ∈ O holds ⟨X, o⟩ is a

topological extension of ⟨X, B⟩, and (iv) for every topological space T such that the carrier of T = X and for

every family o of subsets of X such that o ∈ O holds ⟨X, o⟩ is a topological extension of T holds ⟨X, B⟩ is a topological extension of T.

168 grzegorz bancerek

(41) Let X be a set and O be a family of subsets of 2^{X}. Then there exists a family B of subsets of X such that

(i) B = UniCl(FinMeetCl(⋃ O)), (ii) ⟨X, B⟩ is a topological space, (iii) for every family o of subsets of X such that o ∈ O holds ⟨X, B⟩ is a

topological extension of ⟨X, o⟩, and (iv) for every topological space T such that the carrier of T = X and for

every family o of subsets of X such that o ∈ O holds T is a topological extension of ⟨X, o⟩ holds T is a topological extension of ⟨X, B⟩.

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170 grzegorz bancerek

On Constructing Topological Spaces and Sorgenfrey Line^{1}

Grzegorz Bancerek Białystok Technical University

Summary. We continue Mizar formalization of General Topology according to the book [19] by Engelking. In the article the formalization of Section 1.2 is almost completed. Namely, we formalize theorems on introduction of topologies by bases, neighborhood systems, closed sets, closure operator, and interior operator. The Sorgenfrey line is defined by a basis. It is proved that the weight of it is continuum. Other techniques are used to demonstrate introduction of discrete and anti-discrete topologies.

MML Identifier: TOPGEN 3.

The terminology and notation used in this paper have been introduced in the following articles: [39], [17], [45], [30], [18], [38], [43], [46], [47], [15], [16], [10], [6], [7], [3], [5], [13], [20], [2], [8], [1], [14], [4], [42], [27], [44], [23], [37], [35], [11], [25], [24], [32], [33], [34], [29], [40], [26], [31], [48], [21], [22], [36], [12], [41], [28], and [9].

1. Introducing Topology

In this paper a is a set.

Let X be a set and let B be a family of subsets of X. We say that B is point-filtered if and only if:

(Def. 1) For all sets x, U_{1}, U_{2} such that U_{1} ∈ B and U_{2} ∈ B and x ∈ U_{1} ∩ U_{2}

there exists a subset U of X such that U ∈ B and x ∈ U and U ⊆ U_{1} ∩U_{2}. We now state four propositions:

1This work has been partially supported by the KBN grant 4 T11C 039 24.

(1) Let X be a set and B be a family of subsets of X. Then B is covering if and only if for every set x such that x ∈ X there exists a subset U of X such that U ∈ B and x ∈ U.

(2) Let X be a set and B be a family of subsets of X. Suppose B is pointfiltered and covering. Let T be a topological structure. Suppose the carrier of T = X and the topology of T = UniCl(B). Then T is a topological space and B is a basis of T.

(3) Let X be a set and B be a non-empty many sorted set indexed by X. Suppose that

(i) rng B ⊆ 2^{2^{X}} , (ii) for all sets x, U such that x ∈ X and U ∈ B(x) holds x ∈ U, (iii) for all sets x, y, U such that x ∈ U and U ∈ B(y) and y ∈ X there

exists a set V such that V ∈ B(x) and V ⊆ U, and (iv) for all sets x, U_{1}, U_{2} such that x ∈ X and U_{1} ∈ B(x) and U_{2} ∈ B(x)

there exists a set U such that U ∈ B(x) and U ⊆ U_{1} ∩ U_{2}. Then there exists a family P of subsets of X such that

(v) P = ⋃ B, and (vi) for every topological structure T such that the carrier of T = X and the

topology of T = UniCl(P) holds T is a topological space and for every non empty topological space T′ such that T′ = T holds B is a neighborhood system of T′.

(4) Let X be a set and F be a family of subsets of X. Suppose that

(i) ∅ ∈ F, (ii) X ∈ F, (iii) for all sets A, B such that A ∈ F and B ∈ F holds A ∪ B ∈ F, and (iv) for every family G of subsets of X such that G ⊆ F holds Intersect(G) ∈

F. Let T be a topological structure. Suppose the carrier of T = X and the topology of T = F^{c}. Then T is a topological space and for every subset A of T holds A is closed iff A ∈ F.

The scheme TopDefByClosedPred deals with a set A and a unary predicate P, and states that:

There exists a strict topological space T such that the carrier of T = A and for every subset A of T holds A is closed iff P[A]

provided the following conditions are satisfied: • P[∅] and P[A],

• For all sets A, B such that P[A] and P[B] holds P[A ∪ B], and

• For every family G of subsets of A such that for every set A such that A ∈ G holds P[A] holds P[Intersect(G)].

We now state two propositions:

(5) Let T_{1}, T_{2} be topological spaces. Suppose that for every set A holds A is an open subset of T_{1} iff A is an open subset of T_{2}. Then the topological structure of T_{1} = the topological structure of T_{2}.

(6) Let T_{1}, T_{2} be topological spaces. Suppose that for every set A holds A is a closed subset of T_{1} iff A is a closed subset of T_{2}. Then the topological structure of T_{1} = the topological structure of T_{2}.

Let X, Y be sets and let r be a subset of [: X, 2^{Y} :]. Then rng r is a family of subsets of Y.

We now state the proposition

(7) Let X be a set and c be a function from 2^{X} into 2^{X}. Suppose that (i) c(∅) = ∅, (ii) for every subset A of X holds A ⊆ c(A), (iii) for all subsets A, B of X holds c(A ∪ B) = c(A) ∪ c(B), and (iv) for every subset A of X holds c(c(A)) = c(A).

Let T be a topological structure. Suppose the carrier of T = X and the topology of T = (rng c)^{c}. Then T is a topological space and for every subset A of T holds \overline{A} = c(A).

The scheme TopDefByClosureOp deals with a set A and a unary functor F yielding a set, and states that:

There exists a strict topological space T such that the carrier of T = A and for every subset A of T holds \overline{A} = F(A)

provided the parameters satisfy the following conditions: • F(∅) = ∅,

• For every subset A of A holds A ⊆ F(A) and F(A) ⊆ A,

• For all subsets A, B of A holds F(A ∪ B) = F(A) ∪ F(B), and

• For every subset A of A holds F(F(A)) = F(A). We now state two propositions:

(8) Let T_{1}, T_{2} be topological spaces. Suppose that

(i) the carrier of T_{1} = the carrier of T_{2}, and (ii) for every subset A_{1} of T_{1} and for every subset A_{2} of T_{2} such that

A_{1} = A_{2} holds \overline{A_{1}} = \overline{A_{2}}. Then the topology of T_{1} = the topology of T_{2}.

(9) Let X be a set and i be a function from 2^{X} into 2^{X}. Suppose that

(i) i(X) = X, (ii) for every subset A of X holds i(A) ⊆ A, (iii) for all subsets A, B of X holds i(A ∩ B) = i(A) ∩ i(B), and (iv) for every subset A of X holds i(i(A)) = i(A).

Let T be a topological structure. Suppose the carrier of T = X and the topology of T = rng i. Then T is a topological space and for every subset A of T holds Int A = i(A).

The scheme TopDefByInteriorOp deals with a set A and a unary functor F yielding a set, and states that:

There exists a strict topological space T such that the carrier of T = A and for every subset A of T holds Int A = F(A)

provided the following conditions are met:

• F(A) = A,

• For every subset A of A holds F(A) ⊆ A,

• For all subsets A, B of A holds F(A ∩ B) = F(A) ∩ F(B), and

• For every subset A of A holds F(F(A)) = F(A).

Next we state the proposition

(10) Let T_{1}, T_{2} be topological spaces. Suppose that (i) the carrier of T_{1} = the carrier of T_{2}, and (ii) for every subset A_{1} of T_{1} and for every subset A_{2} of T_{2} such that

A_{1} = A_{2} holds Int A_{1} = Int A_{2}.

Then the topology of T_{1} = the topology of T_{2}.

2. Sorgenfrey Line

In the sequel x, q denote elements of ℝ.

The strict non empty topological space Sorgenfrey line is defined by the conditions (Def. 2).

(Def. 2)(i) The carrier of Sorgenfrey line = ℝ, and

(ii) there exists a family B of subsets of ℝ such that the topology of Sor-

genfrey line = UniCl(B) and B = {[x, q[: x < q ∧ q is rational}. We now state several propositions:

(11) For all real numbers x, y holds [x, y[ is an open subset of Sorgenfrey line.

(12) For all real numbers x, y holds ]x, y[ is an open subset of Sorgenfrey line.

(13) For every real number x holds ]−∞, x[ is an open subset of Sorgenfrey line.

(14) For every real number x holds ]x, +∞[ is an open subset of Sorgenfrey line.

(15) For every real number x holds [x, +∞[ is an open subset of Sorgenfrey line.

(16) \overline{\overline{ℤ}} = ℵ_{0}.

(17) \overline{\overline{ℚ}} = ℵ_{0}.

(18) Let A be a set. Suppose A is mutually-disjoint and for every a such that a ∈ A there exist real numbers x, y such that x < y and ]x, y[ ⊆ a. Then A is countable.

Let X be a set and let x be a real number. We say that x is local minimum of X if and only if:

(Def. 3) x ∈ X and there exists a real number y such that y < x and ]y, x[ misses

X. In the sequel x is an element of ℝ. One can prove the following proposition

(19) For every subset U of ℝ holds {x : x is local minimum of U} is countable. One can check the following observations:

∗ ℤ is infinite,

∗ ℚ is infinite, and

∗ ℝ is infinite. Let X be an infinite set. Note that 2^{X} is infinite.

Let M be an aleph. Observe that 2^{M} is infinite. The infinite cardinal number 𝔠 is defined by:

(Def. 4) 𝔠 = \overline{\overline{ℝ}}.

In the sequel x, q are elements of ℝ. One can prove the following proposition

(20) \overline{\overline{\{[x, q[: x < q ∧ q is rational\}}} = 𝔠.

Let X be an infinite set. Observe that there exists a subset of X which is infinite.

Let X be a set and let r be a real number. The functor X-powers(r) yields a function from ℕ into ℝ and is defined by:

(Def. 5) For every natural number i holds i ∈ X and (X-powers(r))(i) = r^{i} or

i ∉ X and (X-powers(r))(i) = 0. Next we state the proposition

(21) For every set X and for every real number r such that 0 < r and r < 1

holds X-powers(r) is summable. In the sequel r denotes a real number, X denotes a set, and n denotes an

element of ℕ.

The following propositions are true:

(22) If 0 < r and r < 1, then ∑((r^{κ})_{κ∈ℕ} ↑ n) = \frac{r^{n}}{1−r} .

(23) ∑(((\frac{1}{2})^{κ})_{κ∈ℕ} ↑ (n + 1)) = (\frac{1}{2})^{n}.

(24) If 0 < r and r < 1, then ∑(X-powers(r)) ≤ ∑((r^{κ})_{κ∈ℕ}).

(25) ∑((X-powers(\frac{1}{2})) ↑ (n + 1)) ≤ (\frac{1}{2})^{n}.

(26) For every infinite subset X of ℕ and for every natural number i holds (∑_{α=0}^{κ}(X-powers(\frac{1}{2}))(α))_{κ∈ℕ}(i) < ∑(X-powers(\frac{1}{2})).

(27) For all infinite subsets X, Y of ℕ such that ∑(X-powers(\frac{1}{2})) = ∑(Y -powers(\frac{1}{2})) holds X = Y.

(28) If X is countable, then Fin X is countable.

(29) 𝔠 = 2^{ℵ_{0}}.

(30) ℵ_{0} < 𝔠.

(31) For every family A of subsets of ℝ such that \overline{\overline{A}} < 𝔠 holds \overline{\overline{\{x : ⋁_{U : set} (U ∈ UniCl(A) ∧ x is local minimum of U)\}}} < 𝔠.

(32) Let X be a family of subsets of ℝ. Suppose \overline{\overline{X}} < 𝔠. Then there exists a real number x and there exists a rational number q such that x < q and [x, q[∉ UniCl(X).

(33) weight Sorgenfrey line = 𝔠.

3. Example: closed = finite

Let X be a set. The functor ClFinTop(X) yielding a strict topological space is defined by:

(Def. 6) The carrier of ClFinTop(X) = X and for every subset F of ClFinTop(X)

holds F is closed iff F is finite or F = X. The following two propositions are true:

(34) For every set X and for every subset A of ClFinTop(X) holds A is open iff A = ∅ or A^{c} is finite.

(35) For every infinite set X and for every subset A of X such that A is finite

holds A^{c} is infinite. Let X be a non empty set. Note that ClFinTop(X) is non empty. The following proposition is true

(36) For every infinite set X and for all non empty open subsets U, V of ClFinTop(X) holds U meets V.

4. Example: one point closure

Let X, x_{0} be sets. The functor x_{0}-PointClTop(X) yielding a strict topological space is defined as follows:

(Def. 7) The carrier of x_{0}-PointClTop(X) = X and for every subset A of

x_{0}-PointClTop(X) holds \overline{A} = (A = ∅ → A, A ∪ \{x_{0}\} ∩ X). Let X be a non empty set and let x_{0} be a set. One can check that

x_{0}-PointClTop(X) is non empty.

We now state two propositions:

(37) For every non empty set X and for every element x_{0} of X and for every non empty subset A of x_{0}-PointClTop(X) holds \overline{A} = A ∪ \{x_{0}\}.

(38) Let X be a non empty set, x_{0} be an element of X, and A be a non empty subset of x_{0}-PointClTop(X). Then A is closed if and only if x_{0} ∈ A.

Let X be a non empty set and let A be a proper subset of X. Observe that A^{c} is non empty.

The following propositions are true:

(39) Let X be a non empty set, x_{0} be an element of X, and A be a proper subset of x_{0}-PointClTop(X). Then A is open if and only if x_{0} ∉ A.

(40) For all sets X, x_{0}, x such that x_{0} ∈ X holds {x} is a closed subset of x_{0}-PointClTop(X) iff x = x_{0}.

(41) For all sets X, x_{0}, x such that {x_{0}} ⊂ X holds {x} is an open subset of x_{0}-PointClTop(X) iff x ∈ X and x ≠ x_{0}.

5. Example: discrete on subset

Let X, X_{0} be sets. The functor X_{0}-DiscreteTop(X) yielding a strict topological space is defined as follows:

(Def. 8) The carrier of X_{0}-DiscreteTop(X) = X and for every subset A of

X_{0}-DiscreteTop(X) holds Int A = (A = X → A, A ∩ X_{0}). Let X be a non empty set and let X_{0} be a set. One can check that

X_{0}-DiscreteTop(X) is non empty.

We now state several propositions:

(42) For every non empty set X and for every set X_{0} and for every proper subset A of X_{0}-DiscreteTop(X) holds Int A = A ∩ X_{0}.

(43) For every non empty set X and for every set X_{0} and for every proper subset A of X_{0}-DiscreteTop(X) holds A is open iff A ⊆ X_{0}.

(44) For every set X and for every subset X_{0} of X holds the topology of X_{0}-DiscreteTop(X) = {X} ∪ 2^{X_{0}}.

(45) For every set X holds ADTS(X) = ∅-DiscreteTop(X).

(46) For every set X holds {X}_{top} = X-DiscreteTop(X).

References

[1] Grzegorz Bancerek. Arithmetic of non negative rational numbers. To appear in Formalized Mathematics.

[2] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

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

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

[5] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

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

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

[9] Grzegorz Bancerek. On the characteristic and weight of a topological space. Formalized Mathematics, 13(1):163–169, 2005.

178 grzegorz bancerek

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

[11] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.

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

[13] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

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

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

[17] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

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

[20] Mariusz Giero. Hierarchies and classifications of sets. Formalized Mathematics, 9(4):865– 869, 2001.

[21] Zbigniew Karno. On discrete and almost discrete topological spaces. Formalized Mathematics, 3(2):305–310, 1992.

[22] Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125–135, 1993.

[23] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.

[24] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

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

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

[27] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

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

[29] Yatsuka Nakamura. Half open intervals in real numbers. Formalized Mathematics, 10(1):21–22, 2002.

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

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

[32] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

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

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

[35] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

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

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

[38] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

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

[40] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15–22, 1993.

[41] Andrzej Trybulec. Baire spaces, Sober spaces. Formalized Mathematics, 6(2):289–294, 1997.

[42] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

on constructing topological spaces and … 179

[43] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

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

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

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

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

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

180 grzegorz bancerek

On the Real Valued Functions^{1}

Artur Korniłowicz University of Bialystok

MML Identifier: PARTFUN3.

The terminology and notation used here have been introduced in the following articles: [9], [12], [1], [10], [11], [13], [14], [2], [3], [4], [6], [5], [8], and [7].

Let r be a real number. Observe that \frac{r}{r} is non negative. Let r be a real number. Observe that r · r is non negative and r · r^{−1} is non

negative.

Let r be a non negative real number. One can check that \sqrt{r} is non negative. Let r be a positive real number. Observe that \sqrt{r} is positive. We now state the proposition

(1) For every function f and for every set A such that f is one-to-one and A ⊆ dom(f^{−1}) holds f^{∘}(f^{−1})^{∘}A = A.

Let f be a non-empty function. One can verify that f^{−1}({0}) is empty. Let R be a binary relation. We say that R is positive yielding if and only if:

(Def. 1) For every real number r such that r ∈ rng R holds 0 < r.

We say that R is negative yielding if and only if: (Def. 2) For every real number r such that r ∈ rng R holds 0 > r.

We say that R is non-positive yielding if and only if: (Def. 3) For every real number r such that r ∈ rng R holds 0 ≥ r.

We say that R is non-negative yielding if and only if: (Def. 4) For every real number r such that r ∈ rng R holds 0 ≤ r.

Let X be a set and let r be a positive real number. Observe that X ⟼ r is positive yielding.

Let X be a set and let r be a negative real number. Note that X ⟼ r is negative yielding.

1The paper was written during the author’s post-doctoral fellowship granted by the Shinshu University, Japan.

Let X be a set and let r be a non positive real number. Note that X ⟼ r is non-positive yielding.

Let X be a set and let r be a non negative real number. Observe that X ⟼ r is non-negative yielding.

Let X be a non empty set. Note that X ⟼ 0 is non non-empty.

Let us observe that every binary relation which is positive yielding is also non-negative yielding and non-empty and every binary relation which is negative yielding is also non-positive yielding and non-empty.

Let X be a set. One can check that there exists a function from X into ℝ which is negative yielding and there exists a function from X into ℝ which is positive yielding.

One can check that there exists a function which is non-empty and realyielding.

We now state two propositions:

(2) For every non-empty real-yielding function f holds dom(\frac{1}{f}) = dom f.

(3) Let X be a non empty set, f be a partial function from X to ℝ, and g be a non-empty partial function from X to ℝ. Then dom(\frac{f}{g}) = dom f ∩dom g.

Let X be a set and let f, g be non-positive yielding partial functions from X to ℝ. Observe that f + g is non-positive yielding.

Let X be a set and let f, g be non-negative yielding partial functions from X to ℝ. Note that f + g is non-negative yielding.

Let X be a set, let f be a positive yielding partial function from X to ℝ, and let g be a non-negative yielding partial function from X to ℝ. Observe that f + g is positive yielding.

Let X be a set, let f be a non-negative yielding partial function from X to ℝ, and let g be a positive yielding partial function from X to ℝ. One can verify that f + g is positive yielding.

Let X be a set, let f be a non-positive yielding partial function from X to ℝ, and let g be a negative yielding partial function from X to ℝ. Note that f + g is negative yielding.

Let X be a set, let f be a negative yielding partial function from X to ℝ, and let g be a non-positive yielding partial function from X to ℝ. Note that f + g is negative yielding.

Let X be a set, let f be a non-negative yielding partial function from X to ℝ, and let g be a non-positive yielding partial function from X to ℝ. Note that f − g is non-negative yielding.

Let X be a set, let f be a non-positive yielding partial function from X to ℝ, and let g be a non-negative yielding partial function from X to ℝ. Observe that f − g is non-positive yielding.

Let X be a set, let f be a positive yielding partial function from X to ℝ, and let g be a non-positive yielding partial function from X to ℝ. One can check that f − g is positive yielding.

Let X be a set, let f be a non-positive yielding partial function from X to ℝ, and let g be a positive yielding partial function from X to ℝ. Observe that f − g is negative yielding.

Let X be a set, let f be a negative yielding partial function from X to ℝ, and let g be a non-negative yielding partial function from X to ℝ. Note that f − g is negative yielding.

Let X be a set, let f be a non-negative yielding partial function from X to ℝ, and let g be a negative yielding partial function from X to ℝ. One can verify that f − g is positive yielding.

Let X be a set and let f, g be non-positive yielding partial functions from X to ℝ. One can verify that f g is non-negative yielding.

Let X be a set and let f, g be non-negative yielding partial functions from X to ℝ. Note that f g is non-negative yielding.

Let X be a set, let f be a non-positive yielding partial function from X to ℝ, and let g be a non-negative yielding partial function from X to ℝ. One can verify that f g is non-positive yielding.

Let X be a set, let f be a non-negative yielding partial function from X to ℝ, and let g be a non-positive yielding partial function from X to ℝ. Observe that f g is non-positive yielding.

Let X be a set, let f be a positive yielding partial function from X to ℝ, and let g be a negative yielding partial function from X to ℝ. Note that f g is negative yielding.

Let X be a set, let f be a negative yielding partial function from X to ℝ, and let g be a positive yielding partial function from X to ℝ. One can verify that f g is negative yielding.

Let X be a set and let f, g be positive yielding partial functions from X to ℝ. One can verify that f g is positive yielding.

Let X be a set and let f, g be negative yielding partial functions from X to ℝ. One can check that f g is positive yielding.

Let X be a set and let f, g be non-empty partial functions from X to ℝ. Observe that f g is non-empty.

Let X be a set and let f be a partial function from X to ℝ. Note that f f is non-negative yielding.

Let X be a set, let r be a non positive real number, and let f be a non-positive yielding partial function from X to ℝ. One can verify that r f is non-negative yielding.

Let X be a set, let r be a non negative real number, and let f be a nonnegative yielding partial function from X to ℝ. Observe that r f is non-negative yielding.

Let X be a set, let r be a non positive real number, and let f be a nonnegative yielding partial function from X to ℝ. One can verify that r f is non-positive yielding.

Let X be a set, let r be a non negative real number, and let f be a nonpositive yielding partial function from X to ℝ. One can verify that r f is nonpositive yielding.

Let X be a set, let r be a positive real number, and let f be a negative yielding partial function from X to ℝ. Note that r f is negative yielding.

Let X be a set, let r be a negative real number, and let f be a positive yielding partial function from X to ℝ. One can check that r f is negative yielding.

Let X be a set, let r be a positive real number, and let f be a positive yielding partial function from X to ℝ. One can verify that r f is positive yielding.

Let X be a set, let r be a negative real number, and let f be a negative yielding partial function from X to ℝ. Note that r f is positive yielding.

Let X be a set, let r be a non zero real number, and let f be a non-empty partial function from X to ℝ. Observe that r f is non-empty.

Let X be a non empty set and let f, g be non-positive yielding partial functions from X to ℝ. Note that \frac{f}{g} is non-negative yielding.

Let X be a non empty set and let f, g be non-negative yielding partial functions from X to ℝ. Observe that \frac{f}{g} is non-negative yielding.

Let X be a non empty set, let f be a non-positive yielding partial function from X to ℝ, and let g be a non-negative yielding partial function from X to ℝ. Note that \frac{f}{g} is non-positive yielding.

Let X be a non empty set, let f be a non-negative yielding partial function from X to ℝ, and let g be a non-positive yielding partial function from X to ℝ. Note that \frac{f}{g} is non-positive yielding.

Let X be a non empty set, let f be a positive yielding partial function from X to ℝ, and let g be a negative yielding partial function from X to ℝ. One can verify that \frac{f}{g} is negative yielding.

Let X be a non empty set, let f be a negative yielding partial function from X to ℝ, and let g be a positive yielding partial function from X to ℝ. Observe that \frac{f}{g} is negative yielding.

Let X be a non empty set and let f, g be positive yielding partial functions from X to ℝ. One can check that \frac{f}{g} is positive yielding.

Let X be a non empty set and let f, g be negative yielding partial functions from X to ℝ. One can check that \frac{f}{g} is positive yielding.

Let X be a non empty set and let f be a partial function from X to ℝ. Observe that \frac{f}{f} is non-negative yielding.

Let X be a non empty set and let f, g be non-empty partial functions from X to ℝ. One can verify that \frac{f}{g} is non-empty.

Let X be a set and let f be a non-positive yielding function from X into ℝ. One can verify that Inv f is non-positive yielding.

Let X be a set and let f be a non-negative yielding function from X into ℝ. Observe that Inv f is non-negative yielding.

Let X be a set and let f be a positive yielding function from X into ℝ. One can verify that Inv f is positive yielding.

Let X be a set and let f be a negative yielding function from X into ℝ. Note that Inv f is negative yielding.

Let X be a set and let f be a non-empty function from X into ℝ. Note that Inv f is non-empty.

Let X be a set and let f be a non-empty function from X into ℝ. One can verify that −f is non-empty.

Let X be a set and let f be a non-positive yielding function from X into ℝ. Observe that −f is non-negative yielding.

Let X be a set and let f be a non-negative yielding function from X into ℝ. One can check that −f is non-positive yielding.

Let X be a set and let f be a positive yielding function from X into ℝ. Observe that −f is negative yielding.

Let X be a set and let f be a negative yielding function from X into ℝ. Observe that −f is positive yielding.

Let X be a set and let f be a function from X into ℝ. Note that |f| is non-negative yielding.

Let X be a set and let f be a non-empty function from X into ℝ. One can check that |f| is positive yielding.

Let X be a non empty set and let f be a non-positive yielding function from X into ℝ. Observe that \frac{1}{f} is non-positive yielding.

Let X be a non empty set and let f be a non-negative yielding function from X into ℝ. Note that \frac{1}{f} is non-negative yielding.

Let X be a non empty set and let f be a positive yielding function from X into ℝ. One can check that \frac{1}{f} is positive yielding.

Let X be a non empty set and let f be a negative yielding function from X into ℝ. Note that \frac{1}{f} is negative yielding.

Let X be a non empty set and let f be a non-empty function from X into ℝ. One can check that \frac{1}{f} is non-empty.

Let f be a real-yielding function. The functor \sqrt{f} yields a function and is defined as follows:

(Def. 5) dom \sqrt{f} = dom f and for every set x such that x ∈ dom \sqrt{f} holds

\sqrt{f}(x) = \sqrt{f(x)}. Let f be a real-yielding function. Observe that \sqrt{f} is real-yielding. Let C be a set, let D be a real-membered set, and let f be a partial function

from C to D. Then \sqrt{f} is a partial function from C to ℝ.

Let X be a set and let f be a non-negative yielding function from X into ℝ. One can check that \sqrt{f} is non-negative yielding.

Let X be a set and let f be a positive yielding function from X into ℝ. Note that \sqrt{f} is positive yielding.

Let X be a set and let f, g be functions from X into ℝ. Then f + g is a function from X into ℝ. Then f − g is a function from X into ℝ. Then f g is a function from X into ℝ.

Let X be a set and let f be a function from X into ℝ. Then −f is a function from X into ℝ. Then |f| is a function from X into ℝ. Then \sqrt{f} is a function from X into ℝ.

Let X be a set, let f be a function from X into ℝ, and let r be a real number. Then r f is a function from X into ℝ.

Let X be a set and let f be a non-empty function from X into ℝ. Then \frac{1}{f} is a function from X into ℝ.

Let X be a non empty set, let f be a function from X into ℝ, and let g be a non-empty function from X into ℝ. Then \frac{f}{g} is a function from X into ℝ.

In the sequel T is a non empty topological space, f, g are continuous real maps of T, and r is a real number.

Let us consider T, f, g. Then f + g is a continuous real map of T. Then f − g is a continuous real map of T. Then f g is a continuous real map of T.

Let us consider T, f. Then −f is a continuous real map of T.

Let us consider T, f. Then |f| is a continuous real map of T.

Let us consider T. Observe that there exists a real map of T which is positive yielding and continuous and there exists a real map of T which is negative yielding and continuous.

Let us consider T and let f be a non-negative yielding continuous real map of T. Then \sqrt{f} is a continuous real map of T.

Let us consider T, f, r. Then r f is a continuous real map of T.

Let us consider T and let f be a non-empty continuous real map of T. Then \frac{1}{f} is a continuous real map of T.

Let us consider T, f and let g be a non-empty continuous real map of T. Then \frac{f}{g} is a continuous real map of T.

References

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

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

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

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

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

on the real valued functions 187

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

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

[10] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341– 347, 2003.

[11] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 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.

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

188 artur korniłowicz

Formalization of Ortholattices via Orthoposets

Adam Grabowski^{1} University of Białystok Markus Moschner University of Vienna

Summary. There are two approaches to lattices used in the Mizar Mathematical Library: on the one hand, these structures are based on the set with two binary operations (with an equational characterization as in [17]). On the other hand, we may look at them as at relational structures (posets – see [12]). As the main result of this article we can state that the Mizar formalization enables us to use both approaches simultaneously (Section 3). This is especially useful because most of lemmas on ortholattices in the literature are stated in the poset setting, so we cannot use equational theorem provers in a straightforward way. We give also short equational characterization of lattices via four axioms (as it was done in [7] with the help of the Otter prover). Some corresponding results about ortholattices are also formalized.

MML Identifier: ROBBINS3.

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

1. Another Short Axiomatization of Lattices

Let L be a non empty ⊔-semi lattice structure. We say that L is quasi-join-associative if and only if:

(Def. 1) For all elements x, y, z of L holds x ⊔ (y ⊔ z) = y ⊔ (x ⊔ z).

Let L be a non empty ⊓-semi lattice structure. We say that L is quasi-meet-associative if and only if:

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

(Def. 2) For all elements x, y, z of L holds x ⊓ (y ⊓ z) = y ⊓ (x ⊓ z).

Let L be a non empty lattice structure. We say that L is quasi-meet-absorbing if and only if:

(Def. 3) For all elements x, y of L holds x ⊔ (x ⊓ y) = x.

One can prove the following propositions:

(1) Let L be a non empty lattice structure. Suppose L is quasimeet-associative, quasi-join-associative, quasi-meet-absorbing, and joinabsorbing. Then L is meet-idempotent and join-idempotent.

(2) Let L be a non empty lattice structure. Suppose L is quasimeet-associative, quasi-join-associative, quasi-meet-absorbing, and joinabsorbing. Then L is meet-commutative and join-commutative.

(3) Let L be a non empty lattice structure. Suppose L is quasimeet-associative, quasi-join-associative, quasi-meet-absorbing, and joinabsorbing. Then L is meet-absorbing.

(4) Let L be a non empty lattice structure. Suppose L is quasimeet-associative, quasi-join-associative, quasi-meet-absorbing, and joinabsorbing. Then L is meet-associative and join-associative.

(5) Let L be a non empty lattice structure. Then L is lattice-like if and only if L is quasi-meet-associative, quasi-join-associative, quasi-meet-absorbing, and join-absorbing.

One can verify that every non empty lattice structure which is lattice-like is also quasi-meet-associative, quasi-join-associative, meet-absorbing, and joinabsorbing and every non empty lattice structure which is quasi-meet-associative, quasi-join-associative, quasi-meet-absorbing, and join-absorbing is also latticelike.

2. Orthoposets

Let us note that every PartialOrdered non empty orthorelational structure which is OrderInvolutive is also Dneg.

The following propositions are true:

(6) For every Dneg non empty orthorelational structure L and for every element x of L holds (x^{c})^{c} = x.

(7) Let O be an OrderInvolutive PartialOrdered non empty orthorelational structure and x, y be elements of O. If x ≤ y, then y^{c} ≤ x^{c}.

Let us note that there exists a PreOrthoPoset which is strict and has g.l.b.’s and l.u.b.’s.

Let L be a non empty ⊔-semi lattice structure and let x, y be elements of L. We introduce x\underline{⊔}y as a synonym of x ⊔ y.

Let L be a non empty ⊓-semi lattice structure and let x, y be elements of L. We introduce x\overline{⊓}y as a synonym of x ⊓ y.

Let L be a non empty relational structure and let x, y be elements of L. We introduce x ⊓_{≤} y as a synonym of x ⊓ y. We introduce x ⊔_{≤} y as a synonym of x ⊔ y.

3. Merging Relational Structures and Lattice Structures Together

We introduce ⊔-relational semilattice structures which are extensions of ⊔- semi lattice structure and relational structure and are systems

⟨ a carrier, a join operation, an internal relation ⟩, where the carrier is a set, the join operation is a binary operation on the carrier, and the internal relation is a binary relation on the carrier.

We introduce ⊓-relational semilattice structures which are extensions of ⊓- semi lattice structure and relational structure and are systems

⟨ a carrier, a meet operation, an internal relation ⟩, where the carrier is a set, the meet operation is a binary operation on the carrier, and the internal relation is a binary relation on the carrier.

We introduce relational lattice structures which are extensions of ⊓-relational semilattice structure, ⊔-relational semilattice structure, and lattice structure and are systems

⟨ a carrier, a join operation, a meet operation, an internal relation ⟩, where the carrier is a set, the join operation and the meet operation are binary operations on the carrier, and the internal relation is a binary relation on the carrier.

The relational lattice structure TrivLattRelStr is defined as follows: (Def. 4) TrivLattRelStr = ⟨{∅}, op_{2}, op_{2}, id_{{∅}}⟩.

Let us note that TrivLattRelStr is non empty and trivial.

One can check the following observations:

∗ there exists a ⊔-relational semilattice structure which is non empty,

∗ there exists a ⊓-relational semilattice structure which is non empty, and ∗ there exists a relational lattice structure which is non empty. One can prove the following proposition

(8) Let R be a non empty relational structure. Suppose that

(i) the internal relation of R is reflexive in the carrier of R, and (ii) the internal relation of R is antisymmetric and transitive.

Then R is reflexive, antisymmetric, and transitive. Let us mention that TrivLattRelStr is reflexive.

Let us note that there exists a relational lattice structure which is antisymmetric, reflexive, and transitive and has l.u.b.’s and g.l.b.’s.

One can verify that TrivLattRelStr is quasi-meet-absorbing. One can verify that there exists a non empty relational lattice structure

which is lattice-like.

Let L be a lattice. Then LattRel(L) is an order in the carrier of L.

4. Binary Approach to Ortholattices

We consider relational ortholattice structures as extensions of relational lattice structure, ortholattice structure, and orthorelational structure as systems

⟨ a carrier, a join operation, a meet operation, an internal relation, a complement operation ⟩, where the carrier is a set, the join operation and the meet operation are binary operations on the carrier, the internal relation is a binary relation on the carrier, and the complement operation is a unary operation on the carrier.

The relational ortholattice structure TrivCLRelStr is defined by: (Def. 5) TrivCLRelStr = ⟨{∅}, op_{2}, op_{2}, id_{{∅}}, op_{1}⟩.

Let L be a non empty ComplStr. We say that L is involutive if and only if: (Def. 6) For every element x of L holds (x^{c})^{c} = x.

Let L be a non empty complemented lattice structure. We say that L has top if and only if:

(Def. 7) For all elements x, y of L holds x\underline{⊔}x^{c} = y\underline{⊔}y^{c}. One can verify that TrivOrtLat is involutive and has top. One can verify that TrivCLRelStr is non empty and trivial. One can check that TrivCLRelStr is reflexive. Let us observe that TrivCLRelStr is involutive and has top. Let us observe that there exists a non empty ortholattice structure which is

involutive, de Morgan, and lattice-like and has top.

An ortholattice is an involutive de Morgan lattice-like non empty ortholattice structure with top.

5. Lemmas

Next we state a number of propositions:

(9) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is join-commutative. Then L is join-commutative.

(10) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is meet-commutative. Then L is meet-commutative.

(11) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is join-associative. Then L is join-associative.

(12) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is meet-associative. Then L is meet-associative.

(13) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is join-absorbing. Then L is join-absorbing.

(14) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is meet-absorbing. Then L is meet-absorbing.

(15) Let K, L be non empty lattice structures. Suppose the lattice structure of K = the lattice structure of L and K is lattice-like. Then L is latticelike.

(16) Let L_{1}, L_{2} be non empty ⊔-semi lattice structures. Suppose the upper semilattice structure of L_{1} = the upper semilattice structure of L_{2}. Let a_{1}, b_{1} be elements of L_{1} and a_{2}, b_{2} be elements of L_{2}. If a_{1} = a_{2} and b_{1} = b_{2}, then a_{1} ⊔ b_{1} = a_{2} ⊔ b_{2}.

(17) Let L_{1}, L_{2} be non empty ⊓-semi lattice structures. Suppose the lower semilattice structure of L_{1} = the lower semilattice structure of L_{2}. Let a_{1}, b_{1} be elements of L_{1} and a_{2}, b_{2} be elements of L_{2}. If a_{1} = a_{2} and b_{1} = b_{2}, then a_{1} ⊓ b_{1} = a_{2} ⊓ b_{2}.

(18) Let K, L be non empty ComplStr, x be an element of K, and y be an element of L. Suppose the complement operation of K = the complement operation of L and x = y. Then x^{c} = y^{c}.

(19) Let K, L be non empty complemented lattice structures such that the complemented lattice structure of K = the complemented lattice structure of L and K has top. Then L has top.

(20) Let K, L be non empty ortholattice structures. Suppose the ortholattice structure of K = the ortholattice structure of L and K is de Morgan. Then L is de Morgan.

(21) Let K, L be non empty ortholattice structures. Suppose the ortholattice structure of K = the ortholattice structure of L and K is involutive. Then L is involutive.

6. Structure Extensions

Let R be a relational structure. A relational lattice structure is said to be a relational augmentation of R if:

(Def. 8) The relational structure of it = the relational structure of R.

Let R be a lattice structure. A relational lattice structure is said to be a lattice augmentation of R if:

(Def. 9) The lattice structure of it = the lattice structure of R.

Let L be a non empty lattice structure. Observe that every lattice augmentation of L is non empty.

Let L be a meet-associative non empty lattice structure. Note that every lattice augmentation of L is meet-associative.

Let L be a join-associative non empty lattice structure. One can check that every lattice augmentation of L is join-associative.

Let L be a meet-commutative non empty lattice structure. One can verify that every lattice augmentation of L is meet-commutative.

Let L be a join-commutative non empty lattice structure. Note that every lattice augmentation of L is join-commutative.

Let L be a join-absorbing non empty lattice structure. One can check that every lattice augmentation of L is join-absorbing.

Let L be a meet-absorbing non empty lattice structure. Observe that every lattice augmentation of L is meet-absorbing.

Let L be a non empty ⊔-relational semilattice structure. We say that L is naturally sup-generated if and only if:

(Def. 10) For all elements x, y of L holds x ≤ y iff x\underline{⊔}y = y.

Let L be a non empty ⊓-relational semilattice structure. We say that L is naturally inf-generated if and only if:

(Def. 11) For all elements x, y of L holds x ≤ y iff x\overline{⊓}y = x.

Let L be a lattice. One can verify that there exists a lattice augmentation of L which is naturally sup-generated, naturally inf-generated, and lattice-like.

Let us mention that there exists a relational lattice structure which is trivial, non empty, and reflexive.

Let us mention that there exists a relational ortholattice structure which is trivial, non empty, and reflexive.

Let us note that there exists a orthorelational structure which is trivial, non empty, and reflexive.

One can check that every non empty ortholattice structure which is trivial is also involutive, de Morgan, and well-complemented and has top.

Let us note that every non empty reflexive orthorelational structure which is trivial is also OrderInvolutive, Pure, and PartialOrdered.

One can check that every non empty reflexive relational lattice structure which is trivial is also naturally sup-generated and naturally inf-generated.

Let us note that there exists a non empty relational ortholattice structure which is naturally sup-generated, naturally inf-generated, de Morgan, latticelike, OrderInvolutive, Pure, and PartialOrdered and has g.l.b.’s and l.u.b.’s.

Let us observe that there exists a non empty relational lattice structure which is naturally sup-generated, naturally inf-generated, and lattice-like and has g.l.b.’s and l.u.b.’s.

Next we state two propositions:

(22) Let L be a naturally sup-generated non empty relational lattice structure and x, y be elements of L. Then x ≤ y if and only if x ⊑ y.

(23) Let L be a naturally sup-generated lattice-like non empty relational lat-

tice structure. Then the relational structure of L = Poset(L). One can check that every non empty relational lattice structure which is

naturally sup-generated and lattice-like has also g.l.b.’s and l.u.b.’s.

7. Extending Orthocomplemented Lattice Structure

Let R be an ortholattice structure. A relational ortholattice structure is said to be a complemented lattice augmentation of R if:

(Def. 12) The ortholattice structure of it = the ortholattice structure of R.

Let L be a non empty ortholattice structure. One can check that every complemented lattice augmentation of L is non empty.

Let L be a meet-associative non empty ortholattice structure. Note that every complemented lattice augmentation of L is meet-associative.

Let L be a join-associative non empty ortholattice structure. One can verify that every complemented lattice augmentation of L is join-associative.

Let L be a meet-commutative non empty ortholattice structure. Observe that every complemented lattice augmentation of L is meet-commutative.

Let L be a join-commutative non empty ortholattice structure. Note that every complemented lattice augmentation of L is join-commutative.

Let L be a meet-absorbing non empty ortholattice structure. Note that every complemented lattice augmentation of L is meet-absorbing.

Let L be a join-absorbing non empty ortholattice structure. Note that every complemented lattice augmentation of L is join-absorbing.

Let L be a non empty ortholattice structure with top. Observe that every complemented lattice augmentation of L has top.

Let L be a non empty ortholattice. Note that there exists a complemented lattice augmentation of L which is naturally sup-generated, naturally inf-generated, and lattice-like.

Let us observe that there exists a non empty relational ortholattice structure which is involutive, de Morgan, lattice-like, naturally sup-generated, and wellcomplemented and has top.

Next we state the proposition (24) Let L be a PartialOrdered non empty orthorelational structure with

g.l.b.’s and l.u.b.’s and x, y be elements of L. If x ≤ y, then y = x ⊔_{≤} y and x = x ⊓_{≤} y.

Let L be a meet-commutative non empty ⊓-semi lattice structure and let a, b be elements of L. Let us observe that the functor a\overline{⊓}b is commutative.

Let L be a join-commutative non empty ⊔-semi lattice structure and let a, b be elements of L. Let us notice that the functor a\underline{⊔}b is commutative.

One can check that every non empty relational lattice structure which is meet-absorbing, join-absorbing, meet-commutative, and naturally supgenerated is also reflexive.

Let us observe that every non empty relational lattice structure which is join-associative and naturally sup-generated is also transitive.

One can check that every non empty relational lattice structure which is join-commutative and naturally sup-generated is also antisymmetric.

Next we state three propositions:

(25) Let L be a naturally sup-generated lattice-like non empty relational ortholattice structure with g.l.b.’s and l.u.b.’s and x, y be elements of L. Then x ⊔_{≤} y = x\underline{⊔}y.

(26) Let L be a naturally sup-generated lattice-like non empty relational ortholattice structure with g.l.b.’s and l.u.b.’s and x, y be elements of L. Then x ⊓_{≤} y = x\overline{⊓}y.

(27) Every naturally sup-generated naturally inf-generated lattice-like Order-Involutive PartialOrdered non empty relational ortholattice structure with g.l.b.’s and l.u.b.’s is de Morgan.

Let L be an ortholattice. Note that every complemented lattice augmentation of L is involutive.

Let L be an ortholattice. Observe that every complemented lattice augmentation of L is de Morgan.

The following two propositions are true:

(28) Let L be a non empty relational ortholattice structure. Suppose L is involutive, de Morgan, lattice-like, and naturally sup-generated and has top. Then L is Orthocomplemented and PartialOrdered.

(29) For every ortholattice L holds every naturally sup-generated comple-

mented lattice augmentation of L is Orthocomplemented. Let L be an ortholattice. Observe that every naturally sup-generated com-

plemented lattice augmentation of L is Orthocomplemented.

We now state the proposition

(30) Let L be a non empty ortholattice structure. Suppose L is Boolean,

well-complemented, and lattice-like. Then L is an ortholattice. Let us observe that every non empty ortholattice structure which is Boolean,

formalization of ortholattices … 197

well-complemented, and lattice-like is also involutive and de Morgan and has top.

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Index of MML Identifiers

BVFUNC26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 CALCUL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 CALCUL 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 GOEDELCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 GROEB 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 HENMODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 HOLDER 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 JORDAN21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 JORDAN22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 LP SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 MATRIX 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 MESFUNC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 NCFCONT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 PARTFUN3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 PENCIL 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 PENCIL 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 RLTOPSP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 ROBBINS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 SERIES 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 SIN COS6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 SUBLEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 SUBSTUT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 SUBSTUT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 TOPGEN 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 TOPGEN 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 TOPGEN 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 TOPREALA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 TOPREALB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Contents Formaliz. Math. 13 (1)

Partial Sum of Some Series

By Ming Liang and Yuzhong Ding . . . . . . . . . . . . . . . . . . . . . . 1

Substitution in First-Order Formulas: Elementary Properties

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 5

Coincidence Lemma and Substitution Lemma

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 17

Substitution in First-Order Formulas. Part II. The Construction

of First-Order Formulas

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 27

A Sequent Calculus for First-Order Logic

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 33

Consequences of the Sequent Calculus

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 41

Equivalences of Inconsistency and Henkin Models

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 45

Gödel’s Completeness Theorem

By Patrick Braselmann and Peter Koepke . . . . . . . . . . . . . 49

Propositional Calculus for Boolean Valued Functions. Part VIII

By Shunichi Kobayashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Hölder’s Inequality and Minkowski’s Inequality

By Yasumasa Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

The Banach Space l^{p}

By Yasumasa Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Lebesgue Integral of Simple Valued Function

By Yasunari Shidama and Noboru Endou . . . . . . . . . . . . . . . 67

Continued on inside back cover

Inverse Trigonometric Functions Arcsin and Arccos

By Artur Korniłowicz and Yasunari Shidama . . . . . . . . . . 73

On Some Points of a Simple Closed Curve

By Artur Korniłowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

On Some Points of a Simple Closed Curve. Part II

By Artur Korniłowicz and Adam Grabowski . . . . . . . . . . . 89

Uniform Continuity of Functions on Normed Complex Linear

Spaces

By Noboru Endou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Introduction to Real Linear Topological Spaces

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Some Properties of Rectangles on the Plane

By Artur Korniłowicz and Yasunari Shidama . . . . . . . . . . 109

Some Properties of Circles on the Plane

By Artur Korniłowicz and Yasunari Shidama . . . . . . . . . . 117

On the Characterization of Collineations of the Segre Product of

Strongly Connected Partial Linear Spaces

By Adam Naumowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Spaces of Pencils, Grassmann Spaces, and Generalized Veronese

Spaces

By Adam Naumowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

On the Boundary and Derivative of a Set

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Construction of Gröbner Bases: Avoiding S-Polynomials – Buch-

berger’s First Criterium

By Christoph Schwarzweller . . . . . . . . . . . . . . . . . . . . . . . . . . 147

A Theory of Matrices of Complex Elements

By Wenpai Chang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

On the Characteristic and Weight of a Topological Space

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

On Constructing Topological Spaces and Sorgenfrey Line

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

On the Real Valued Functions

By Artur Korniłowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Formalization of Ortholattices via Orthoposets

By Adam Grabowski and Markus Moschner . . . . . . . . . . . . 189

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Continued on inside back cover