FORMALIZED MATHEMATICS Volume 6, Number 4, 1997 University of Białystok

Introduction to the Homotopy Theory

Adam Grabowski University of Białystok

Summary. The paper introduces some preliminary notions concerning the homotopy theory according to [15]: paths and arcwise connected to topological spaces. The basic operations on paths (addition and reversing) are defined. In the last section the predicate: P, Q are homotopic is defined. We also showed some properties of the product of two topological spaces needed to prove reflexivity and symmetry of the above predicate.

MML Identifier: BORSUK 2.

The articles [27], [30], [26], [16], [10], [32], [7], [23], [13], [12], [25], [28], [24], [4], [1], [33], [11], [21], [31], [9], [19], [29], [17], [8], [34], [14], [6], [5], [22], [20], [2], [18], and [3] provide the notation and terminology for this paper.

1. Preliminaries

In this paper T, T_{1}, T_{2}, S denote non empty topological spaces.

The scheme FrCard deals with a non empty set A, a set B, a unary functor F yielding a set, and a unary predicate P, and states that:

\overline{\overline{\{F(w); w ranges over elements of A :w ∈ B ∧ P[w]\}}} \overline{\overline{B}} for all values of the parameters.

The following proposition is true

(1) Let f be a map from T_{1} into S and g be a map from T_{2} into S. Suppose that

(i) T_{1} is a subspace of T, (ii) T_{2} is a subspace of T, (iii) Ω_{(T_{1})} ∪ Ω_{(T_{2})} = Ω_{T} , (iv) T_{1} is compact, (v) T_{2} is compact, (vi) T is a T_{2} space, (vii) f is continuous, (viii) g is continuous, and (ix) for every set p such that p ∈ Ω_{(T_{1})} ∩ Ω_{(T_{2})} holds f(p) = g(p).

Then there exists a map h from T into S such that h = f+·g and h is continuous.

Let S, T be non empty topological spaces. One can verify that there exists a map from S into T which is continuous.

One can prove the following proposition

(2) For all non empty topological spaces S, T holds every continuous mapping from S into T is a continuous map from S into T.

Let T be a non empty topological structure. Note that id_{T} is open and continuous.

Let T be a non empty topological structure. Observe that there exists a map from T into T which is continuous and one-to-one.

We now state the proposition

(3) Let S, T be non empty topological spaces and f be a map from S into T. If f is a homeomorphism, then f^{−1} is open.

2. Paths and arcwise connected spaces

Let T be a topological structure and let a, b be points of T. Let us assume that there exists a map f from 𝕀 into T such that f is continuous and f(0) = a and f(1) = b. A map from 𝕀 into T is said to be a path from a to b if:

(Def. 1) It is continuous and it(0) = a and it(1) = b.

Next we state the proposition

(4) Let T be a non empty topological space and a be a point of T. Then there exists a map f from 𝕀 into T such that f is continuous and f(0) = a and f(1) = a.

Let T be a non empty topological space and let a be a point of T. Note that there exists a path from a to a which is continuous.

Let T be a topological structure. We say that T is arcwise connected if and only if:

(Def. 2) For all points a, b of T there exists a map f from 𝕀 into T such that f

is continuous and f(0) = a and f(1) = b. Let us observe that there exists a topological space which is arcwise connec-

ted and non empty.

Let T be an arcwise connected topological structure and let a, b be points of T. Let us note that the path from a to b can be characterized by the following (equivalent) condition:

(Def. 3) It is continuous and it(0) = a and it(1) = b.

Let T be an arcwise connected topological structure and let a, b be points of T. Note that every path from a to b is continuous.

Next we state the proposition

(5) For every non empty topological space G_{1} such that G_{1} is arcwise connected holds G_{1} is connected.

Let us mention that every non empty topological space which is arcwise connected is also connected.

3. Basic operations on paths

Let T be a non empty topological space, let a, b, c be points of T, let P be a path from a to b, and let Q be a path from b to c. Let us assume that there exist maps f, g from 𝕀 into T such that f is continuous and f(0) = a and f(1) = b and g is continuous and g(0) = b and g(1) = c. The functor P + Q yielding a path from a to c is defined by the condition (Def. 4).

(Def. 4) Let t be a point of 𝕀 and t′ be a real number such that t = t′. Then (i) if 0 t′ and t′ \frac{1}{2}, then (P + Q)(t) = P (2 · t′), and (ii) if \frac{1}{2} t′ and t′ 1, then (P + Q)(t) = Q(2 · t′− 1). Let T be a non empty topological space and let a be a point of T. Note that

there exists a path from a to a which is constant.

One can prove the following two propositions:

(6) Let T be a non empty topological space, a be a point of T, and P be a constant path from a to a. Then P = 𝕀 ⟼→ a.

(7) Let T be a non empty topological space, a be a point of T, and P be a constant path from a to a. Then P + P = P.

Let T be a non empty topological space, let a be a point of T, and let P be a constant path from a to a. Observe that P + P is constant.

Let T be a non empty topological space, let a, b be points of T, and let P be a path from a to b. Let us assume that there exists a map f from 𝕀 into T such that f is continuous and f(0) = a and f(1) = b. The functor − P yields a path from b to a and is defined as follows:

(Def. 5) For every point t of 𝕀 and for every real number t′ such that t = t′ holds

(− P)(t) = P (1 − t′). The following proposition is true (8) Let T be a non empty topological space, a be a point of T, and P be a

constant path from a to a. Then − P = P. Let T be a non empty topological space, let a be a point of T, and let P be

a constant path from a to a. One can verify that − P is constant.

4. The product of two topological spaces

One can prove the following proposition

(9) Let X, Y be non empty topological spaces, A be a family of subsets of Y, and f be a map from X into Y. Then f^{−1} (⋃ A) = ⋃(f^{−1} (A)).

Let S_{1}, S_{2}, T_{1}, T_{2} be non empty topological spaces, let f be a map from S_{1} into S_{2}, and let g be a map from T_{1} into T_{2}. Then [: f, g:] is a map from [: S_{1}, T_{1} :] into [: S_{2}, T_{2} :].

Next we state three propositions:

(10) Let S_{1}, S_{2}, T_{1}, T_{2} be non empty topological spaces, f be a continuous map from S_{1} into T_{1}, g be a continuous map from S_{2} into T_{2}, and P_{1}, P_{2} be subsets of the carrier of [: T_{1}, T_{2} :]. If P_{2} ∈ BaseAppr(P_{1}), then [: f, g:]^{−1} (P_{2}) is open.

(11) Let S_{1}, S_{2}, T_{1}, T_{2} be non empty topological spaces, f be a continuous map from S_{1} into T_{1}, g be a continuous map from S_{2} into T_{2}, and P_{2} be a subset of the carrier of [: T_{1}, T_{2} :]. If P_{2} is open, then [: f, g:]^{−1} (P_{2}) is open.

(12) Let S_{1}, S_{2}, T_{1}, T_{2} be non empty topological spaces, f be a continuous map from S_{1} into T_{1}, and g be a continuous map from S_{2} into T_{2}. Then [: f, g:] is continuous.

Let us note that every topological structure which is empty is also T_{0}. Let T_{1}, T_{2} be discernible non empty topological spaces. One can check that

[: T_{1}, T_{2} :] is discernible.

We now state two propositions:

(13) For all T_{0}-spaces T_{1}, T_{2} holds [: T_{1}, T_{2} :] is a T_{0}-space.

(14) Let T_{1}, T_{2} be non empty topological spaces. Suppose T_{1} is a T_{1} space

and T_{2} is a T_{1} space. Then [: T_{1}, T_{2} :] is a T_{1} space. Let T_{1}, T_{2} be a T_{1} space non empty topological spaces. Observe that [: T_{1},

T_{2} :] is a T_{1} space.

Let T_{1}, T_{2} be T_{2} non empty topological spaces. Observe that [: T_{1}, T_{2} :] is T_{2}. Let us note that 𝕀 is compact and T_{2}. Let us mention that E_{T}^{2} is T_{2}. Let T be a non empty arcwise connected topological space, let a, b be points

of T, and let P, Q be paths from a to b. We say that P, Q are homotopic if and only if the condition (Def. 6) is satisfied.

introduction to the homotopy theory 453

(Def. 6) There exists a map f from [: 𝕀, 𝕀:] into T such that (i) f is continuous, and (ii) for every point s of 𝕀 holds f(s, 0) = P (s) and f(s, 1) = Q(s) and for

every point t of 𝕀 holds f(0, t) = a and f(1, t) = b. Let us notice that the predicate P, Q are homotopic is reflexive and symmetric.

References

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

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

[3] Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353–359, 1996.

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

[5] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.

[6] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

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

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

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

[15] Marvin J. Greenberg. Lectures on Algebraic Topology. W. A. Benjamin, Inc., 1973.

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

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

[18] Zbigniew Karno. On Kolmogorov topological spaces. Formalized Mathematics, 5(1):119– 124, 1996.

[19] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[20] Michał Muzalewski. Categories of groups. Formalized Mathematics, 2(4):563–571, 1991.

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

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

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

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

[25] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

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

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

454 adam grabowski

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

[30] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

[31] Toshihiko Watanabe. The Brouwer fixed point theorem for intervals. Formalized Mathematics, 3(1):85–88, 1992.

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

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

[34] Mariusz Żynel and Adam Guzowski. T_{0} topological spaces. Formalized Mathematics, 5(1):75–77, 1996.

Some Properties of Real Maps

Adam Grabowski^{1} University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. The main goal of the paper is to show logical equivalence of the two definitions of the open subset: one from [2] and the other from [23]. This has been used to show that the other two definitions are equivalent: the continuity of the map as in [20] and in [22]. We used this to show that continuous and one-to-one maps are monotone (see theorems 16 and 17 for details).

MML Identifier: JORDAN5A.

The terminology and notation used here are introduced in the following articles: [26], [13], [27], [28], [4], [5], [24], [22], [17], [18], [10], [3], [23], [6], [25], [29], [16], [14], [19], [11], [20], [8], [7], [9], [15], [21], [2], [1], and [12].

1. Preliminaries

One can prove the following four propositions:

(1) For all points p, q of E_{T}^{2} and for every subset P of E_{T}^{2} such that P is an arc from p to q holds P is compact.

(2) For every real number r holds 0 r and r 1 iff r ∈ the carrier of 𝕀.

(3) For all points p_{1}, p_{2} of E_{T}^{2} and for all real numbers r_{1}, r_{2} such that (1 − r_{1}) · p_{1} + r_{1} · p_{2} = (1 − r_{2}) · p_{1} + r_{2} · p_{2} holds r_{1} = r_{2} or p_{1} = p_{2}.

(4) Let p_{1}, p_{2} be points of E_{T}^{2}. Suppose p_{1} ≠ p_{2}. Then there exists a map f from 𝕀 into (E_{T}^{2})↾L(p_{1}, p_{2}) such that for every real number x such that x ∈ [0, 1] holds f(x) = (1 − x) · p_{1} + x · p_{2} and f is a homeomorphism and f(0) = p_{1} and f(1) = p_{2}.

1This paper was written while the author visited the Shinshu University in the winter of 1997.

One can verify that E_{T}^{2} is arcwise connected. One can check that there exists a subspace of E_{T}^{2} which is compact and non

empty.

The following proposition is true

(5) Let a, b be points of E_{T}^{2}, f be a path from a to b, P be a non empty compact subspace of E_{T}^{2}, and g be a map from 𝕀 into P. If f is one-to-one and g = f and Ω_{P} = rngf, then g is a homeomorphism.

2. Equivalence of analytical and topological definitions of continuity

We now state a number of propositions:

(6) Let X be a subset of ℝ. Then X ∈ the open set family of the metric space of real numbers if and only if X is open.

(7) Let f be a map from ℝ^{1} into ℝ^{1}, x be a point of ℝ^{1}, g be a partial function from ℝ to ℝ, and x_{1} be a real number. If f is continuous at x and f = g and x = x_{1}, then g is continuous in x_{1}.

(8) Let f be a continuous map from ℝ^{1} into ℝ^{1} and g be a partial function from ℝ to ℝ. If f = g, then g is continuous on ℝ.

(9) Let f be a continuous one-to-one map from ℝ^{1} into ℝ^{1}. Then

(i) for all points x, y of 𝕀 and for all real numbers p, q, f_{1}, f_{2} such that x = p and y = q and p < q and f_{1} = f(x) and f_{2} = f(y) holds f_{1} < f_{2}, or

(ii) for all points x, y of 𝕀 and for all real numbers p, q, f_{1}, f_{2} such that

x = p and y = q and p < q and f_{1} = f(x) and f_{2} = f(y) holds f_{1} > f_{2}.

(10) Let r, g_{1}, a, b be real numbers and x be an element of the carrier of [a, b]_{M}. If a b and x = r and g_{1} > 0 and ]r − g_{1}, r + g_{1}[ ⊆ [a, b], then ]r − g_{1}, r + g_{1}[ = Ball(x, g_{1}).

(11) Let a, b be real numbers and X be a subset of ℝ. Suppose a < b and a ∉ X and b ∉ X. If X ∈ the open set family of [a, b]_{M}, then X is open.

(12) For every open subset X of ℝ and for all real numbers a, b such that X ⊆ [a, b] holds a ∉ X and b ∉ X.

(13) Let a, b be real numbers, X be a subset of ℝ, and V be a subset of the carrier of [a, b]_{M}. Suppose a b and V = X. If X is open, then V ∈ the open set family of [a, b]_{M}.

(14) Let a, b, c, d, x_{1} be real numbers, f be a map from [a, b]_{T} into [c, d]_{T}, x be a point of [a, b]_{T}, and g be a partial function from ℝ to ℝ. Suppose a < b and c < d and f is continuous at x and f(a) = c and f(b) = d and f is one-to-one and f = g and x = x_{1}. Then g↾[a, b] is continuous in x_{1}.

(15) Let a, b, c, d be real numbers, f be a map from [a, b]_{T} into [c, d]_{T}, and g be a partial function from ℝ to ℝ. Suppose f is continuous and one-to-one and a < b and c < d and f = g and f(a) = c and f(b) = d. Then g is continuous on [a, b].

3. On the monotonicity of continuous maps

One can prove the following propositions:

(16) Let a, b, c, d be real numbers and f be a map from [a, b]_{T} into [c, d]_{T}. Suppose a < b and c < d and f is continuous and one-to-one and f(a) = c and f(b) = d. Let x, y be points of [a, b]_{T} and p, q, f_{1}, f_{2} be real numbers. If x = p and y = q and p < q and f_{1} = f(x) and f_{2} = f(y), then f_{1} < f_{2}.

(17) Let f be a continuous one-to-one map from 𝕀 into 𝕀. Suppose f(0) = 0 and f(1) = 1. Let x, y be points of 𝕀 and p, q, f_{1}, f_{2} be real numbers. If x = p and y = q and p < q and f_{1} = f(x) and f_{2} = f(y), then f_{1} < f_{2}.

(18) Let a, b, c, d be real numbers, f be a map from [a, b]_{T} into [c, d]_{T}, P be a non empty subset of [a, b]_{T}, and P_{1}, Q_{1} be subsets of ℝ^{1}. Suppose a < b and c < d and P_{1} = P and f is continuous and one-to-one and P_{1} is compact and f(a) = c and f(b) = d and f^{∘}P = Q_{1}. Then f(inf(Ω_{(P_{1})})) = inf(Ω_{(Q_{1})}).

(19) Let a, b, c, d be real numbers, f be a map from [a, b]_{T} into [c, d]_{T}, P, Q be non empty subsets of [a, b]_{T}, and P_{1}, Q_{1} be subsets of ℝ^{1}. Suppose a < b and c < d and P_{1} = P and Q_{1} = Q and f is continuous and oneto-one and P_{1} is compact and f(a) = c and f(b) = d and f^{∘}P = Q. Then f(sup(Ω_{(P_{1})})) = sup(Ω_{(Q_{1})}).

(20) For all real numbers a, b such that a b holds inf[a, b] = a and sup[a, b] = b.

(21) Let a, b, c, d, e, f, g, h be real numbers and F be a map from [a, b]_{T} into [c, d]_{T}. Suppose a < b and c < d and e < f and a e and f b and F is a homeomorphism and F (a) = c and F (b) = d and g = F (e) and h = F (f). Then F^{∘}[e, f] = [g, h].

(22) Let P, Q be subsets of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P meets Q and P ∩ Q is closed and P is an arc from p_{1} to p_{2}. Then there exists a point E_{1} of E_{T}^{2} such that

(i) E_{1} ∈ P ∩ Q, and (ii) there exists a map g from 𝕀 into (E_{T}^{2})↾P and there exists a real number

s_{2} such that g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{2}) = E_{1} and 0 s_{2} and s_{2} 1 and for every real number t such that 0 t and t < s_{2} holds g(t) ∉ Q.

458 adam grabowski and yatsuka nakamura

(23) Let P, Q be subsets of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P meets Q and P ∩ Q is closed and P is an arc from p_{1} to p_{2}. Then there exists a point E_{1} of E_{T}^{2} such that

(i) E_{1} ∈ P ∩ Q, and (ii) there exists a map g from 𝕀 into (E_{T}^{2})↾P and there exists a real number

s_{2} such that g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{2}) = E_{1} and 0 s_{2} and s_{2} 1 and for every real number t such that 1 t and t > s_{2} holds g(t) ∉ Q.

References

[1] Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353–359, 1996.

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

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

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

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

[10] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 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] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

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

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

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

[16] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[17] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[18] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

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

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

[21] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[22] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

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

[24] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

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

some properties of real maps 459

[26] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

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

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

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The Ordering of Points on a Curve. Part I

Adam Grabowski^{1} University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. Some auxiliary theorems needed to formalize the proof of the Jordan Curve Theorem according to [25] are proved.

MML Identifier: JORDAN5B.

The articles [26], [29], [13], [1], [22], [24], [31], [2], [4], [5], [11], [28], [20], [12], [16], [23], [9], [8], [27], [10], [30], [15], [17], [18], [14], [19], [21], [6], [7], and [3] provide the terminology and notation for this paper.

1. Preliminaries

The following propositions are true:

(1) For every natural number i_{1} such that 1 i_{1} holds i_{1} − ′1 < i_{1}.

(2) For all natural numbers i, k such that i + 1 k holds 1 k − ′i.

(3) For all natural numbers i, k such that 1 i and 1 k holds k− ′i+1 k.

(4) For every real number r such that r ∈ the carrier of 𝕀 holds 1 − r ∈ the carrier of 𝕀.

(5) For all points p, q, p_{1} of E_{T}^{2} such that p_{2} ≠ q_{2} and p_{1} ∈ L(p, q) holds if (p_{1})_{2} = p_{2}, then (p_{1})_{1} = p_{1}.

(6) For all points p, q, p_{1} of E_{T}^{2} such that p_{1} ≠ q_{1} and p_{1} ∈ L(p, q) holds if (p_{1})_{1} = p_{1}, then (p_{1})_{2} = p_{2}.

1This paper was written while the author visited the Shinshu University in the winter of 1997.

(7) Let f be a finite sequence of elements of E_{T}^{2}, P be a non empty subset of the carrier of E_{T}^{2}, F be a map from 𝕀 into (E_{T}^{2})↾P, and i be a natural number. Suppose 1 i and i + 1 lenf and f is a special sequence and P = L̃(f) and F is a homeomorphism and F (0) = π_{1}f and F (1) = π_{len f}f. Then there exist real numbers p_{1}, p_{2} such that p_{1} < p_{2} and 0 p_{1} and p_{1} 1 and 0 p_{2} and p_{2} 1 and L(f, i) = F^{∘}[p_{1}, p_{2}] and F (p_{1}) = π_{i}f and F (p_{2}) = π_{i+1}f.

(8) Let f be a finite sequence of elements of E_{T}^{2}, Q, R be non empty subsets of the carrier of E_{T}^{2}, F be a map from 𝕀 into (E_{T}^{2})↾Q, i be a natural number, and P be a non empty subset of 𝕀. Suppose that

(i) f is a special sequence, (ii) F is a homeomorphism, (iii) F (0) = π_{1}f, (iv) F (1) = π_{len f}f,

(v) 1 i, (vi) i + 1 lenf, (vii) F^{∘}P = L(f, i), (viii) Q = L̃(f), and (ix) R = L(f, i).

Then there exists a map G from 𝕀↾P into (E_{T}^{2})↾R such that G = F ↾P and G is a homeomorphism.

2. Some properties of real intervals

One can prove the following propositions:

(9) For all points p_{1}, p_{2}, p of E_{T}^{2} such that p_{1} ≠ p_{2} and p ∈ L(p_{1}, p_{2}) holds LE(p, p, p_{1}, p_{2}).

(10) For all points p, p_{1}, p_{2} of E_{T}^{2} such that p_{1} ≠ p_{2} and p ∈ L(p_{1}, p_{2}) holds LE(p_{1}, p, p_{1}, p_{2}).

(11) For all points p, p_{1}, p_{2} of E_{T}^{2} such that p ∈ L(p_{1}, p_{2}) and p_{1} ≠ p_{2} holds LE(p, p_{2}, p_{1}, p_{2}).

(12) For all points p_{1}, p_{2}, q_{1}, q_{2}, q_{3} of E_{T}^{2} such that p_{1} ≠ p_{2} and LE(q_{1}, q_{2}, p_{1}, p_{2}) and LE(q_{2}, q_{3}, p_{1}, p_{2}) holds LE(q_{1}, q_{3}, p_{1}, p_{2}).

(13) For all points p, q of E_{T}^{2} such that p ≠ q holds L(p, q) = {p_{1}; p_{1} ranges over points of E_{T}^{2}: LE(p, p_{1}, p, q) ∧ LE(p_{1}, q, p, q)}.

(14) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then P is an arc from p_{2} to p_{1}.

(15) Let f be a finite sequence of elements of E_{T}^{2}, P be a subset of the carrier of E_{T}^{2}, and i be a natural number. Suppose f is a special sequence and 1 i and i + 1 lenf and P = L(f, i). Then P is an arc from π_{i}f to π_{i+1}f.

3. Cutting off sequences

One can prove the following propositions:

(16) Let g_{1} be a finite sequence of elements of E_{T}^{2} and i be a natural number. Suppose 1 i and i leng_{1} and g_{1} is a special sequence. If π_{1}g_{1} ∈ L̃(mid(g_{1}, i, leng_{1})), then i = 1.

(17) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence and p = f(lenf), then ⇃ p, f = ⟨p, p⟩.

(18) Let f be a finite sequence of elements of E_{T}^{2} and k be a natural number. If 1 k and k lenf, then mid(f, k, k) = ⟨π_{k}f⟩.

(19) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence and p = f(1), then ⇂ f, p = ⟨p⟩.

(20) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence and p ∈ L̃(f), then L̃(⇂ f, p) ⊆ L̃(f).

(21) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f) and p ≠ f(lenf) and f is a special sequence, then Index(p, ⇃ p, f) = 1.

(22) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f) and f is a special sequence, then p ∈ L̃(⇃ p, f).

(23) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f) and f is a special sequence and p ≠ f(1), then p ∈ L̃(⇂ f, p).

(24) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f) and p ≠ f(lenf) and f is a special sequence, then ⇃⇂ p, f, p = ⟨p⟩.

(25) Let f be a finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. If p ∈ L̃(f) and q ∈ L̃(f) and p = f(lenf) and f is a special sequence, then p ∈ L̃(⇃ q, f).

(26) Let f be a finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. If p ∈ L̃(f) and q ∈ L̃(f) and f is a special sequence, then p ∈ L̃(⇃ q, f) or q ∈ L̃(⇃ p, f).

(27) Let f be a finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. Suppose p ∈ L̃(f) and q ∈ L̃(f) and p ≠ f(lenf) or q ≠ f(lenf) and f is a special sequence. Then L̃(⇃⇂ p, f, q) ⊆ L̃(f).

(28) Let f be a non constant standard special circular sequence and i, j be natural numbers. Suppose 1 i and j lenthe Go-board of f and i < j. Then L((the Go-board of f)_{1,width the Go-board of f}, (the Go-board of f)_{i,width the Go-board of f})∩L((the Go-board of f)_{j,width the Go-board of f}, (the Go-board of f)_{len the Go-board of f, width the Go-board of f}) = ∅.

(29) Let f be a non constant standard special circular sequence and i, j be natural numbers. Suppose 1 i and j widththe Go-board of f and i < j. Then L((the Go-board of f)_{len the Go-board of f, 1}, (the Go-board of f)_{len the Go-board of f, i}) ∩ L((the Go-board of f)_{len the Go-board of f, j}, (the Go-board of f)_{len the Go-board of f, width the Go-board of f}) = ∅.

(30) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence, then ⇃ π_{1}f, f = f.

(31) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence, then ⇂ f, π_{len f}f = f.

(32) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f) and f is a special sequence and p ≠ f(lenf), then p ∈ L(π_{Index(p,f)}f, π_{Index(p,f)+1}f).

(33) Let f be a finite sequence of elements of E_{T}^{2}, p be a point of E_{T}^{2}, and i be a natural number. If f is a special sequence, then if π_{1}f ∈ L(f, i), then i = 1.

(34) Let f be a non constant standard special circular sequence, j be a natural number, and P be a subset of the carrier of E_{T}^{2}. Suppose 1 j and j widththe Go-board of f and P = L((the Go-board of f)_{1,j}, (the Goboard of f)_{len the Go-board of f, j}). Then P is a special polygonal arc joining (the Go-board of f)_{1,j} and (the Go-board of f)_{len the Go-board of f, j}.

(35) Let f be a non constant standard special circular sequence, j be a natural number, and P be a subset of the carrier of E_{T}^{2}. Suppose 1 j and j lenthe Go-board of f and P = L((the Go-board of f)_{j,1}, (the Goboard of f)_{j,width the Go-board of f}). Then P is a special polygonal arc joining (the Go-board of f)_{j,1} and (the Go-board of f)_{j,width the Go-board of f}.

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the ordering of points on a curve. part i 465

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466 adam grabowski and yatsuka nakamura

The Ordering of Points on a Curve. Part II

Adam Grabowski^{1} University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. The proof of the Jordan Curve Theorem according to [14] is continued. The notions of the first and last point of a oriented arc are introduced as well as ordering of points on a curve in E_{T}^{2} .

MML Identifier: JORDAN5C.

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

1. First and last point of a curve

One can prove the following proposition

(1) Let P, Q be subsets of the carrier of E_{T}^{2}, p_{1}, p_{2}, q_{1} be points of E_{T}^{2}, f be a map from 𝕀 into (E_{T}^{2})↾P, and s_{1} be a real number. Suppose that

(i) P is an arc from p_{1} to p_{2}, (ii) q_{1} ∈ P, (iii) q_{1} ∈ Q, (iv) f(s_{1}) = q_{1},

(v) f is a homeomorphism, (vi) f(0) = p_{1}, (vii) f(1) = p_{2}, (viii) 0 s_{1}, (ix) s_{1} 1, and (x) for every real number t such that 0 t and t < s_{1} holds f(t) ∉ Q.

1This paper was written while the author visited the Shinshu University in the winter of 1997.

Let g be a map from 𝕀 into (E_{T}^{2})↾P and s_{2} be a real number. Suppose g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{2}) = q_{1} and 0 s_{2} and s_{2} 1. Let t be a real number. If 0 t and t < s_{2}, then g(t) ∉ Q.

Let P, Q be subsets of the carrier of E_{T}^{2} and let p_{1}, p_{2} be points of E_{T}^{2}. Let us assume that P meets Q and P ∩ Q is closed and P is an arc from p_{1} to p_{2}. The functor FPoint(P, p_{1}, p_{2}, Q) yielding a point of E_{T}^{2} is defined by the conditions (Def. 1).

(Def. 1)(i) FPoint(P, p_{1}, p_{2}, Q) ∈ P ∩ Q, and

(ii) for every map g from 𝕀 into (E_{T}^{2})↾P and for every real number s_{2} such that g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{2}) = FPoint(P, p_{1}, p_{2}, Q) and 0 s_{2} and s_{2} 1 and for every real number t such that 0 t and t < s_{2} holds g(t) ∉ Q.

One can prove the following three propositions:

(2) Let P, Q be subsets of the carrier of E_{T}^{2} and p, p_{1}, p_{2} be points of E_{T}^{2}. If p ∈ P and P is an arc from p_{1} to p_{2} and Q = {p}, then FPoint(P, p_{1}, p_{2}, Q) = p.

(3) Let P be a non empty subset of the carrier of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and p_{1}, p_{2} be points of E_{T}^{2}. If p_{1} ∈ Q and P ∩ Q is closed and P is an arc from p_{1} to p_{2}, then FPoint(P, p_{1}, p_{2}, Q) = p_{1}.

(4) Let P, Q be subsets of the carrier of E_{T}^{2}, p_{1}, p_{2}, q_{1} be points of E_{T}^{2}, f be a map from 𝕀 into (E_{T}^{2})↾P, and s_{1} be a real number. Suppose that

(i) P is an arc from p_{1} to p_{2}, (ii) q_{1} ∈ P, (iii) q_{1} ∈ Q, (iv) f(s_{1}) = q_{1},

(v) f is a homeomorphism, (vi) f(0) = p_{1}, (vii) f(1) = p_{2}, (viii) 0 s_{1}, (ix) s_{1} 1, and

(x) for every real number t such that 1 t and t > s_{1} holds f(t) ∉ Q. Let g be a map from 𝕀 into (E_{T}^{2})↾P and s_{2} be a real number. Suppose g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{2}) = q_{1} and 0 s_{2} and s_{2} 1. Let t be a real number. If 1 t and t > s_{2}, then g(t) ∉ Q.

Let P, Q be subsets of the carrier of E_{T}^{2} and let p_{1}, p_{2} be points of E_{T}^{2}. Let us assume that P meets Q and P ∩ Q is closed and P is an arc from p_{1} to p_{2}. The functor LPoint(P, p_{1}, p_{2}, Q) yielding a point of E_{T}^{2} is defined by the conditions (Def. 2).

(Def. 2)(i) LPoint(P, p_{1}, p_{2}, Q) ∈ P ∩ Q, and

(ii) for every map g from 𝕀 into (E_{T}^{2})↾P and for every real number s_{2} such that g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{2}) = LPoint(P, p_{1}, p_{2}, Q) and 0 s_{2} and s_{2} 1 and for every real number t such that 1 t and t > s_{2} holds g(t) ∉ Q.

One can prove the following propositions:

(5) Let P, Q be subsets of the carrier of E_{T}^{2} and p, p_{1}, p_{2} be points of E_{T}^{2}. If p ∈ P and P is an arc from p_{1} to p_{2} and Q = {p}, then LPoint(P, p_{1}, p_{2}, Q) = p.

(6) Let P be a non empty subset of the carrier of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and p_{1}, p_{2} be points of E_{T}^{2}. If p_{2} ∈ Q and P ∩ Q is closed and P is an arc from p_{1} to p_{2}, then LPoint(P, p_{1}, p_{2}, Q) = p_{2}.

(7) Let P be a non empty subset of the carrier of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P ⊆ Q and P is closed and an arc from p_{1} to p_{2}. Then FPoint(P, p_{1}, p_{2}, Q) = p_{1} and LPoint(P, p_{1}, p_{2}, Q) = p_{2}.

2. The ordering of points on a curve

Let P be a subset of the carrier of E_{T}^{2} and let p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. We say that LE q_{1}, q_{2}, P, p_{1}, p_{2} if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) q_{1} ∈ P, (ii) q_{2} ∈ P, and (iii) for every map g from 𝕀 into (E_{T}^{2})↾P and for all real numbers s_{1}, s_{2} such

that g is a homeomorphism and g(0) = p_{1} and g(1) = p_{2} and g(s_{1}) = q_{1} and 0 s_{1} and s_{1} 1 and g(s_{2}) = q_{2} and 0 s_{2} and s_{2} 1 holds s_{1} s_{2}.

The following propositions are true:

(8) Let P be a non empty subset of the carrier of E_{T}^{2}, p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}, g be a map from 𝕀 into (E_{T}^{2})↾P, and s_{1}, s_{2} be real numbers. Suppose that

(i) P is an arc from p_{1} to p_{2}, (ii) g is a homeomorphism, (iii) g(0) = p_{1}, (iv) g(1) = p_{2},

(v) g(s_{1}) = q_{1}, (vi) 0 s_{1}, (vii) s_{1} 1, (viii) g(s_{2}) = q_{2}, (ix) 0 s_{2},

(x) s_{2} 1, and (xi) s_{1} s_{2}.

Then LE q_{1}, q_{2}, P, p_{1}, p_{2}.

(9) Let P be a subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and q_{1} ∈ P, then LE q_{1}, q_{1}, P, p_{1}, p_{2}.

(10) Let P be a subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and q_{1} ∈ P. Then LE p_{1}, q_{1}, P, p_{1}, p_{2} and LE q_{1}, p_{2}, P, p_{1}, p_{2}.

(11) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then LE p_{1}, p_{2}, P, p_{1}, p_{2}.

(12) Let P be a non empty 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 LE q_{1}, q_{2}, P, p_{1}, p_{2} and LE q_{2}, q_{1}, P, p_{1}, p_{2}. Then q_{1} = q_{2}.

(13) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2}, q_{3} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and LE q_{1}, q_{2}, P, p_{1}, p_{2} and LE q_{2}, q_{3}, P, p_{1}, p_{2}. Then LE q_{1}, q_{3}, P, p_{1}, p_{2}.

(14) 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 q_{1} ∈ P and q_{2} ∈ P and q_{1} ≠ q_{2}. Then LE q_{1}, q_{2}, P, p_{1}, p_{2} and not LE q_{2}, q_{1}, P, p_{1}, p_{2} or LE q_{2}, q_{1}, P, p_{1}, p_{2} and not LE q_{1}, q_{2}, P, p_{1}, p_{2}.

3. Some properties of the ordering of points on a curve

We now state a number of propositions:

(15) Let f be a finite sequence of elements of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and q be a point of E_{T}^{2}. Suppose f is a special sequence and L̃(f)∩Q is closed and q ∈ L̃(f) and q ∈ Q. Then LE FPoint(L̃(f), π_{1}f, π_{len f}f, Q), q, L̃(f), π_{1}f, π_{len f}f.

(16) Let f be a finite sequence of elements of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and q be a point of E_{T}^{2}. Suppose f is a special sequence and L̃(f)∩Q is closed and q ∈ L̃(f) and q ∈ Q. Then LE q, LPoint(L̃(f), π_{1}f, π_{len f}f, Q), L̃(f), π_{1}f, π_{len f}f.

(17) For all points q_{1}, q_{2}, p_{1}, p_{2} of E_{T}^{2} such that p_{1} ≠ p_{2} holds if LE q_{1}, q_{2}, L(p_{1}, p_{2}), p_{1}, p_{2}, then LE(q_{1}, q_{2}, p_{1}, p_{2}).

(18) Let P, Q be subsets of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and P ∩ Q ≠ ∅ and P ∩ Q is closed. Then FPoint(P, p_{1}, p_{2}, Q) = LPoint(P, p_{2}, p_{1}, Q) and LPoint(P, p_{1}, p_{2}, Q) = FPoint(P, p_{2}, p_{1}, Q).

(19) Let f be a finite sequence of elements of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and i be a natural number. Suppose L̃(f) meets Q and Q is closed and f is a special sequence and 1 i and i + 1 lenf and FPoint(L̃(f), π_{1}f, π_{len f}f, Q) ∈ L(f, i). Then FPoint(L̃(f), π_{1}f, π_{len f}f, Q) = FPoint(L(f, i), π_{i}f, π_{i+1}f, Q).

(20) Let f be a finite sequence of elements of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and i be a natural number. Suppose L̃(f) meets Q and Q is closed and f is a special sequence and 1 i and i + 1 lenf and LPoint(L̃(f), π_{1}f, π_{len f}f, Q) ∈ L(f, i). Then LPoint(L̃(f), π_{1}f, π_{len f}f, Q) = LPoint(L(f, i), π_{i}f, π_{i+1}f, Q).

(21) Let f be a finite sequence of elements of E_{T}^{2} and i be a natural number. Suppose 1 i and i + 1 lenf and f is a special sequence and FPoint(L̃(f), π_{1}f, π_{len f}f, L(f, i)) ∈ L(f, i). Then FPoint(L̃(f), π_{1}f, π_{len f}f, L(f, i)) = π_{i}f.

(22) Let f be a finite sequence of elements of E_{T}^{2} and i be a natural number. Suppose 1 i and i + 1 lenf and f is a special sequence and LPoint(L̃(f), π_{1}f, π_{len f}f, L(f, i)) ∈ L(f, i). Then LPoint(L̃(f), π_{1}f, π_{len f}f, L(f, i)) = π_{i+1}f.

(23) Let f be a finite sequence of elements of E_{T}^{2} and i be a natural number. Suppose f is a special sequence and 1 i and i + 1 lenf. Then LE π_{i}f, π_{i+1}f, L̃(f), π_{1}f, π_{len f}f.

(24) Let f be a finite sequence of elements of E_{T}^{2} and i, k be natural numbers. Suppose f is a special sequence and 1 i and i + k + 1 lenf. Then LE π_{i}f, π_{i+k}f, L̃(f), π_{1}f, π_{len f}f.

(25) Let f be a finite sequence of elements of E_{T}^{2}, q be a point of E_{T}^{2}, and i be a natural number. Suppose f is a special sequence and 1 i and i + 1 lenf and q ∈ L(f, i). Then LE π_{i}f, q, L̃(f), π_{1}f, π_{len f}f.

(26) Let f be a finite sequence of elements of E_{T}^{2}, q be a point of E_{T}^{2}, and i be a natural number. Suppose f is a special sequence and 1 i and i + 1 lenf and q ∈ L(f, i). Then LE q, π_{i+1}f, L̃(f), π_{1}f, π_{len f}f.

(27) Let f be a finite sequence of elements of E_{T}^{2}, Q be a subset of the carrier

of E_{T}^{2}, q be a point of E_{T}^{2}, and i, j be natural numbers. Suppose that (i) L̃(f) meets Q, (ii) f is a special sequence, (iii) Q is closed, (iv) FPoint(L̃(f), π_{1}f, π_{len f}f, Q) ∈ L(f, i),

(v) 1 i, (vi) i + 1 lenf, (vii) q ∈ L(f, j), (viii) 1 j, (ix) j + 1 lenf,

(x) q ∈ Q, and (xi) FPoint(L̃(f), π_{1}f, π_{len f}f, Q) ≠ q.

472 adam grabowski and yatsuka nakamura

Then i j and if i = j, then LE(FPoint(L̃(f), π_{1}f, π_{len f}f, Q), q, π_{i}f, π_{i+1}f).

(28) Let f be a finite sequence of elements of E_{T}^{2}, Q be a subset of the carrier

of E_{T}^{2}, q be a point of E_{T}^{2}, and i, j be natural numbers. Suppose that (i) L̃(f) meets Q, (ii) f is a special sequence, (iii) Q is closed, (iv) LPoint(L̃(f), π_{1}f, π_{len f}f, Q) ∈ L(f, i),

(v) 1 i, (vi) i + 1 lenf, (vii) q ∈ L(f, j), (viii) 1 j, (ix) j + 1 lenf,

(x) q ∈ Q, and (xi) LPoint(L̃(f), π_{1}f, π_{len f}f, Q) ≠ q.

Then i j and if i = j, then LE(q, LPoint(L̃(f), π_{1}f, π_{len f}f, Q), π_{i}f, π_{i+1}f).

(29) Let f be a finite sequence of elements of E_{T}^{2}, q_{1}, q_{2} be points of E_{T}^{2}, and i be a natural number. Suppose q_{1} ∈ L(f, i) and q_{2} ∈ L(f, i) and f is a special sequence and 1 i and i + 1 lenf. If LE q_{1}, q_{2}, L̃(f), π_{1}f, π_{len f}f, then LE q_{1}, q_{2}, L(f, i), π_{i}f, π_{i+1}f.

(30) Let f be a finite sequence of elements of E_{T}^{2} and q_{1}, q_{2} be points of E_{T}^{2}. Suppose q_{1} ∈ L̃(f) and q_{2} ∈ L̃(f) and f is a special sequence and q_{1} ≠ q_{2}. Then LE q_{1}, q_{2}, L̃(f), π_{1}f, π_{len f}f if and only if for all natural numbers i, j such that q_{1} ∈ L(f, i) and q_{2} ∈ L(f, j) and 1 i and i + 1 lenf and 1 j and j +1 lenf holds i j and if i = j, then LE(q_{1}, q_{2}, π_{i}f, π_{i+1}f).

References

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

[8] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

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

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

[14] 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.

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

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

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

[18] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

[19] Toshihiko Watanabe. The Brouwer fixed point theorem for intervals. Formalized Mathematics, 3(1):85–88, 1992.

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On the Categories Without Uniqueness of cod and dom .

Some Properties of the Morphisms and the Functors

Artur Korniłowicz University of Białystok

MML Identifier: ALTCAT 4.

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

1. Preliminaries

In this paper C denotes a category and o_{1}, o_{2}, o_{3} denote objects of C. Let C be a non empty category structure with units and let o be an object

of C. Observe that ⟨o, o⟩ is non empty.

The following propositions are true:

(1) Let v be a morphism from o_{1} to o_{2}, u be a morphism from o_{1} to o_{3}, and f be a morphism from o_{2} to o_{3}. If u = f · v and f^{−1} · f = id_{(o_{2})} and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{3}⟩ ≠ ∅ and ⟨o_{3}, o_{2}⟩ ≠ ∅, then v = f^{−1} · u.

(2) Let v be a morphism from o_{2} to o_{3}, u be a morphism from o_{1} to o_{3}, and f be a morphism from o_{1} to o_{2}. If u = v · f and f · f^{−1} = id_{(o_{2})} and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and ⟨o_{2}, o_{3}⟩ ≠ ∅, then v = u · f^{−1} .

(3) For every morphism m from o_{1} to o_{2} such that ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and m is iso holds m^{−1} is iso.

(4) For every non empty category structure C with units and for every object o of C holds id_{o} is epi and mono.

Let C be a non empty category structure with units and let o be an object of C. One can verify that id_{o} is epi mono retraction and coretraction.

Let C be a category and let o be an object of C. Note that id_{o} is iso. We now state two propositions:

(5) Let f be a morphism from o_{1} to o_{2} and g, h be morphisms from o_{2} to o_{1}. If h · f = id_{(o_{1})} and f · g = id_{(o_{2})} and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅, then g = h.

(6) Suppose that for all objects o_{1}, o_{2} of C holds every morphism from o_{1} to o_{2} is coretraction. Let a, b be objects of C and g be a morphism from a to b. If ⟨a, b⟩ ≠ ∅ and ⟨b, a⟩ ≠ ∅, then g is iso.

2. Some properties of the initial and terminal objects

The following propositions are true:

(7) For all morphisms m, m′ from o_{1} to o_{2} such that m is zero and m′ is zero and there exists an object of C which is zero holds m = m′.

(8) Let C be a non empty category structure, O, A be objects of C, and M be a morphism from O to A. If O is terminal, then M is mono.

(9) Let C be a non empty category structure, O, A be objects of C, and M be a morphism from A to O. If O is initial, then M is epi.

(10) If o_{2} is terminal and o_{1}, o_{2} are iso , then o_{1} is terminal.

(11) If o_{1} is initial and o_{1}, o_{2} are iso , then o_{2} is initial.

(12) If o_{1} is initial and o_{2} is terminal and ⟨o_{2}, o_{1}⟩ ≠ ∅, then o_{2} is initial and o_{1} is terminal.

3. The properties of the functors

One can prove the following propositions:

(13) Let A, B be transitive non empty category structures with units, F be a contravariant functor from A to B, and a be an object of A. Then F (id_{a}) = id_{F (a)} .

(14) Let C_{1}, C_{2} be non empty category structures and F be a precontravariant functor structure from C_{1} to C_{2}. Then F is full if and only if for all objects o_{1}, o_{2} of C_{1} holds Morph-Map_{F} (o_{2}, o_{1}) is onto.

(15) Let C_{1}, C_{2} be non empty category structures and F be a precontravariant functor structure from C_{1} to C_{2}. Then F is faithful if and only if for all objects o_{1}, o_{2} of C_{1} holds Morph-Map_{F} (o_{2}, o_{1}) is one-to-one.

(16) Let C_{1}, C_{2} be non empty category structures, F be a precovariant functor structure from C_{1} to C_{2}, o_{1}, o_{2} be objects of C_{1}, and F_{1} be a morphism from F (o_{1}) to F (o_{2}). Suppose ⟨o_{1}, o_{2}⟩ ≠ ∅ and F is full and feasible. Then there exists a morphism m from o_{1} to o_{2} such that F_{1} = F (m).

(17) Let C_{1}, C_{2} be non empty category structures, F be a precontravariant functor structure from C_{1} to C_{2}, o_{1}, o_{2} be objects of C_{1}, and F_{1} be a morphism from F (o_{2}) to F (o_{1}). Suppose ⟨o_{1}, o_{2}⟩ ≠ ∅ and F is full and feasible. Then there exists a morphism m from o_{1} to o_{2} such that F_{1} = F (m).

(18) Let A, B be transitive non empty category structures with units, F be a covariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. If ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and a is retraction, then F (a) is retraction.

(19) Let A, B be transitive non empty category structures with units, F be a covariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. If ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and a is coretraction, then F (a) is coretraction.

(20) Let A, B be categories, F be a covariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. If ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and a is iso, then F (a) is iso.

(21) Let A, B be categories, F be a covariant functor from A to B, and o_{1}, o_{2} be objects of A. If o_{1}, o_{2} are iso , then F (o_{1}), F (o_{2}) are iso .

(22) Let A, B be transitive non empty category structures with units, F be a contravariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. If ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and a is retraction, then F (a) is coretraction.

(23) Let A, B be transitive non empty category structures with units, F be a contravariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. If ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and a is coretraction, then F (a) is retraction.

(24) Let A, B be categories, F be a contravariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. If ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and a is iso, then F (a) is iso.

(25) Let A, B be categories, F be a contravariant functor from A to B, and o_{1}, o_{2} be objects of A. If o_{1}, o_{2} are iso , then F (o_{2}), F (o_{1}) are iso .

(26) Let A, B be transitive non empty category structures with units, F be a covariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (a) is retraction. Then a is retraction.

(27) Let A, B be transitive non empty category structures with units, F be a covariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (a) is coretraction. Then a is coretraction.

(28) Let A, B be categories, F be a covariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (a) is iso. Then a is iso.

(29) Let A, B be categories, F be a covariant functor from A to B, and o_{1}, o_{2} be objects of A. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (o_{1}), F (o_{2}) are iso . Then o_{1}, o_{2} are iso .

(30) Let A, B be transitive non empty category structures with units, F be a contravariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (a) is retraction. Then a is coretraction.

(31) Let A, B be transitive non empty category structures with units, F be a contravariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (a) is coretraction. Then a is retraction.

(32) Let A, B be categories, F be a contravariant functor from A to B, o_{1}, o_{2} be objects of A, and a be a morphism from o_{1} to o_{2}. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (a) is iso. Then a is iso.

(33) Let A, B be categories, F be a contravariant functor from A to B, and o_{1}, o_{2} be objects of A. Suppose F is full and faithful and ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and F (o_{2}), F (o_{1}) are iso . Then o_{1}, o_{2} are iso .

4. The subcategories of the morphisms

We now state two propositions:

(34) Let C be a category structure and D be a substructure of C. Suppose the carrier of C = the carrier of D and the arrows of C = the arrows of D. Then D is full.

(35) Let C be a non empty category structure with units and D be a substructure of C. Suppose the carrier of C = the carrier of D and the arrows of C = the arrows of D. Then D is full and id-inheriting.

Let C be a category. Observe that there exists a subcategory of C which is full, non empty, and strict.

Next we state several propositions:

(36) For every non empty subcategory B of C holds every non empty subcategory of B is a non empty subcategory of C.

(37) Let C be a non empty transitive category structure, D be a non empty transitive substructure of C, o_{1}, o_{2} be objects of C, p_{1}, p_{2} be objects of D, m be a morphism from o_{1} to o_{2}, and n be a morphism from p_{1} to p_{2} such that p_{1} = o_{1} and p_{2} = o_{2} and m = n and ⟨p_{1}, p_{2}⟩ ≠ ∅. Then

(i) if m is mono, then n is mono, and (ii) if m is epi, then n is epi.

(38) Let D be a non empty subcategory of C, o_{1}, o_{2} be objects of C, p_{1}, p_{2} be objects of D, m be a morphism from o_{1} to o_{2}, m_{1} be a morphism from o_{2} to o_{1}, n be a morphism from p_{1} to p_{2}, and n_{1} be a morphism from p_{2} to p_{1} such that p_{1} = o_{1} and p_{2} = o_{2} and m = n and m_{1} = n_{1} and ⟨p_{1}, p_{2}⟩ ≠ ∅ and ⟨p_{2}, p_{1}⟩ ≠ ∅. Then

(i) m is left inverse of m_{1} iff n is left inverse of n_{1}, and (ii) m is right inverse of m_{1} iff n is right inverse of n_{1}.

(39) Let D be a full non empty subcategory of C, o_{1}, o_{2} be objects of C, p_{1}, p_{2} be objects of D, m be a morphism from o_{1} to o_{2}, and n be a morphism from p_{1} to p_{2} such that p_{1} = o_{1} and p_{2} = o_{2} and m = n and ⟨p_{1}, p_{2}⟩ ≠ ∅ and ⟨p_{2}, p_{1}⟩ ≠ ∅. Then

(i) if m is retraction, then n is retraction, (ii) if m is coretraction, then n is coretraction, and (iii) if m is iso, then n is iso.

(40) Let D be a non empty subcategory of C, o_{1}, o_{2} be objects of C, p_{1}, p_{2} be objects of D, m be a morphism from o_{1} to o_{2}, and n be a morphism from p_{1} to p_{2} such that p_{1} = o_{1} and p_{2} = o_{2} and m = n and ⟨p_{1}, p_{2}⟩ ≠ ∅ and ⟨p_{2}, p_{1}⟩ ≠ ∅. Then

(i) if n is retraction, then m is retraction, (ii) if n is coretraction, then m is coretraction, and (iii) if n is iso, then m is iso. Let C be a category. The functor AllMonoC yields a strict non empty trans-

itive substructure of C and is defined by the conditions (Def. 1). (Def. 1)(i) The carrier of AllMonoC = the carrier of C,

(ii) the arrows of AllMonoC ˙⊆ the arrows of C, and (iii) for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds

m ∈ (the arrows of AllMonoC)(o_{1}, o_{2}) iff ⟨o_{1}, o_{2}⟩ ≠ ∅ and m is mono. Let C be a category. Note that AllMonoC is id-inheriting. Let C be a category. The functor AllEpiC yields a strict non empty transitive

substructure of C and is defined by the conditions (Def. 2). (Def. 2)(i) The carrier of AllEpiC = the carrier of C,

(ii) the arrows of AllEpiC ˙⊆ the arrows of C, and (iii) for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds

m ∈ (the arrows of AllEpiC)(o_{1}, o_{2}) iff ⟨o_{1}, o_{2}⟩ ≠ ∅ and m is epi.

Let C be a category. Observe that AllEpiC is id-inheriting. Let C be a category. The functor AllRetrC yielding a strict non empty

transitive substructure of C is defined by the conditions (Def. 3). (Def. 3)(i) The carrier of AllRetrC = the carrier of C,

(ii) the arrows of AllRetrC ˙⊆ the arrows of C, and (iii) for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds

m ∈ (the arrows of AllRetrC)(o_{1}, o_{2}) iff ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and m is retraction.

Let C be a category. One can check that AllRetrC is id-inheriting. Let C be a category. The functor AllCoretrC yielding a strict non empty

transitive substructure of C is defined by the conditions (Def. 4). (Def. 4)(i) The carrier of AllCoretrC = the carrier of C,

(ii) the arrows of AllCoretrC ˙⊆ the arrows of C, and (iii) for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds

m ∈ (the arrows of AllCoretrC)(o_{1}, o_{2}) iff ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and m is coretraction.

Let C be a category. One can verify that AllCoretrC is id-inheriting. Let C be a category. The functor AllIsoC yields a strict non empty transitive

substructure of C and is defined by the conditions (Def. 5). (Def. 5)(i) The carrier of AllIsoC = the carrier of C,

(ii) the arrows of AllIsoC ˙⊆ the arrows of C, and (iii) for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds

m ∈ (the arrows of AllIsoC)(o_{1}, o_{2}) iff ⟨o_{1}, o_{2}⟩ ≠ ∅ and ⟨o_{2}, o_{1}⟩ ≠ ∅ and m is iso.

Let C be a category. Note that AllIsoC is id-inheriting. Next we state a number of propositions:

(41) AllIsoC is a non empty subcategory of AllRetrC.

(42) AllIsoC is a non empty subcategory of AllCoretrC.

(43) AllCoretrC is a non empty subcategory of AllMonoC.

(44) AllRetrC is a non empty subcategory of AllEpiC.

(45) If for all objects o_{1}, o_{2} of C holds every morphism from o_{1} to o_{2} is mono, then the category structure of C = AllMonoC.

(46) If for all objects o_{1}, o_{2} of C holds every morphism from o_{1} to o_{2} is epi, then the category structure of C = AllEpiC.

(47) Suppose that for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds m is retraction and ⟨o_{2}, o_{1}⟩ ≠ ∅. Then the category structure of C = AllRetrC.

(48) Suppose that for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds m is coretraction and ⟨o_{2}, o_{1}⟩ ≠ ∅. Then the category structure of C = AllCoretrC.

on the categories without uniqueness of … 481

(49) Suppose that for all objects o_{1}, o_{2} of C and for every morphism m from o_{1} to o_{2} holds m is iso and ⟨o_{2}, o_{1}⟩ ≠ ∅. Then the category structure of C = AllIsoC.

(50) For all objects o_{1}, o_{2} of AllMonoC and for every morphism m from o_{1} to o_{2} such that ⟨o_{1}, o_{2}⟩ ≠ ∅ holds m is mono.

(51) For all objects o_{1}, o_{2} of AllEpiC and for every morphism m from o_{1} to o_{2} such that ⟨o_{1}, o_{2}⟩ ≠ ∅ holds m is epi.

(52) For all objects o_{1}, o_{2} of AllIsoC and for every morphism m from o_{1} to o_{2} such that ⟨o_{1}, o_{2}⟩ ≠ ∅ holds m is iso and m^{−1} ∈ ⟨o_{2}, o_{1}⟩.

(53) AllMonoAllMonoC = AllMonoC.

(54) AllEpiAllEpiC = AllEpiC.

(55) AllIsoAllIsoC = AllIsoC.

(56) AllIsoAllMonoC = AllIsoC.

(57) AllIsoAllEpiC = AllIsoC.

(58) AllIsoAllRetrC = AllIsoC.

(59) AllIsoAllCoretrC = AllIsoC.

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[5] Małgorzata Korolkiewicz. Homomorphisms of many sorted algebras. Formalized Mathematics, 5(1):61–65, 1996.

[6] Beata Madras. Basic properties of objects and morphisms. Formalized Mathematics, 6(3):329–334, 1997.

[7] Michał Muzalewski and Wojciech Skaba. Three-argument operations and four-argument operations. Formalized Mathematics, 2(2):221–224, 1991.

[8] Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.

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

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

[11] Andrzej Trybulec. Many-sorted sets. Formalized Mathematics, 4(1):15–22, 1993.

[12] Andrzej Trybulec. Categories without uniqueness of cod and dom. Formalized Mathematics, 5(2):259–267, 1996.

[13] Andrzej Trybulec. Examples of category structures. Formalized Mathematics, 5(4):493– 500, 1996.

[14] Andrzej Trybulec. Functors for alternative categories. Formalized Mathematics, 5(4):595– 608, 1996.

[15] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37–42, 1996.

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

482 artur korniłowicz

The loop and Times Macroinstruction for

SCM _{FSA}

Noriko Asamoto Ochanomizu University Tokyo

Summary. We implement two macroinstructions loop and Times which iterate macroinstructions of SCM _{FSA} . In a loop macroinstruction it jumps to the head when the original macroinstruction stops, in a Times macroinstruction it behaves as if the original macroinstrucion repeats n times.

MML Identifier: SCMFSA8C.

The articles [22], [29], [16], [8], [12], [30], [13], [14], [11], [7], [9], [28], [15], [17], [23], [20], [21], [27], [24], [25], [1], [10], [19], [26], [5], [6], [4], [2], [3], and [18] provide the terminology and notation for this paper.

1. Preliminaries

Let s be a state of SCM _{FSA} and let P be an initial finite partial state of SCM _{FSA}. We say that P is pseudo-closed on s if and only if the condition (Def. 1) is satisfied.

(Def. 1) There exists a natural number k such that IC_{(Computation (s+·(P +·Start-At (insloc(0)))))(k)} = insloc(cardProgramPart(P)) and for every natural number n such that n < k holds IC_{(Computation (s+·(P +·Start-At (insloc(0)))))(n)} ∈ domP.

Let P be an initial finite partial state of SCM _{FSA}. We say that P is pseudoparaclosed if and only if:

(Def. 2) For every state s of SCM _{FSA} holds P is pseudo-closed on s.

Let us note that there exists a macro instruction which is pseudo-paraclosed. Let s be a state of SCM _{FSA} and let P be an initial finite partial state of

SCM _{FSA}. Let us assume that P is pseudo-closed on s.

The functor pseudo − LifeSpan (s, P) yielding a natural number is defined as follows:

(Def. 3) IC_{(Computation (s+·(P +·Start-At (insloc(0)))))(pseudo−LifeSpan (s,P))} = insloc(cardProgramPart(P)) and for every natural number n such that IC_{(Computation (s+·(P +·Start-At (insloc(0)))))(n)} ∉ domP holds pseudo − LifeSpan (s, P) n.

We now state a number of propositions:

(1) Let s be a state of SCM _{FSA} and P be an initial finite partial state of SCM _{FSA}. Suppose P is pseudo-closed on s. Let n be a natural number. If n < pseudo − LifeSpan (s, P), then IC_{(Computation (s+·(P +·Start-At (insloc(0)))))(n)} ∈ domP and CurInstr((Computation(s+·(P +·Start-At(insloc(0)))))(n)) ≠ halt_{SCM _{FSA}} .

(2) Let s be a state of SCM _{FSA} and P be an initial finite partial state of SCM _{FSA}. Suppose P is pseudo-closed on s. Let k be a natural number. Suppose that for every natural number n such that n k holds IC_{(Computation (s+·(P +·Start-At (insloc(0)))))(n)} ∈ domP. Then k < pseudo − LifeSpan (s, P).

(3) Let s be a state of SCM _{FSA} and I, J be macro instructions. Suppose I is pseudo-closed on s. Let k be a natural number. Suppose k pseudo − LifeSpan (s, I). Then (Computation(s+·(I+·Start-At(insloc (0)))))(k) and (Computation(s+·((I;J)+·Start-At(insloc(0)))))(k) are equal outside the instruction locations of SCM _{FSA}.

(4) Let s be a state of SCM _{FSA} and I be a macro instruction. If I is closed on s and halting on s, then Directed(I) is pseudo-closed on s.

(5) Let s be a state of SCM _{FSA} and I be a macro instruction. If I is closed on s and halting on s, then pseudo − LifeSpan (s, Directed(I)) = LifeSpan(s+·(I+·Start-At(insloc(0)))) + 1.

(6) For every function f and for every set x such that x ∈ domf holds f+·(x⟼→ . f(x)) = f.

(7) For every instruction-location l of SCM _{FSA} holds l + 0 = l.

(8) For every instruction i of SCM _{FSA} holds IncAddr(i, 0) = i.

(9) For every programmed finite partial state P of SCM _{FSA} holds ProgramPart(Relocated(P, 0)) = P.

(10) For all finite partial states P, Q of SCM _{FSA} such that P ⊆ Q holds ProgramPart(P) ⊆ ProgramPart(Q).

(11) For all programmed finite partial states P, Q of SCM _{FSA} and for every natural number k such that P ⊆ Q holds Shift(P, k) ⊆ Shift(Q, k).

(12) For all finite partial states P, Q of SCM _{FSA} and for every natural number k such that P ⊆ Q holds ProgramPart(Relocated(P, k)) ⊆ ProgramPart(Relocated(Q, k)).

(13) Let I, J be macro instructions and k be a natural number. Suppose cardI k and k < cardI + cardJ. Let i be an instruction of SCM _{FSA}. If i = J(insloc(k − ′cardI)), then (I;J)(insloc(k)) = IncAddr(i, cardI).

(14) For every state s of SCM _{FSA} such that s(intloc(0)) = 1 and IC_{s} = insloc(0) holds Initialize(s) = s.

(15) For every state s of SCM _{FSA} holds Initialize(Initialize(s)) = Initialize(s).

(16) For every state s of SCM _{FSA} and for every macro instruction I holds s+·(Initialized(I)+·Start-At(insloc(0))) = Initialize(s)+·(I+·Start-At(insloc(0))).

(17) For every state s of SCM _{FSA} and for every macro instruction I holds IExec(I, s) = IExec(I, Initialize(s)).

(18) For every state s of SCM _{FSA} and for every macro instruction I such that s(intloc(0)) = 1 holds s+·(I+·Start-At(insloc(0))) = s+·Initialized(I).

(19) For every macro instruction I holds I+·Start-At(insloc(0)) ⊆ Initialized(I).

(20) For every instruction-location l of SCM _{FSA} and for every macro instruction I holds l ∈ domI iff l ∈ domInitialized(I).

(21) For every state s of SCM _{FSA} and for every macro instruction I holds Initialized(I) is closed on s iff I is closed on Initialize(s).

(22) For every state s of SCM _{FSA} and for every macro instruction I holds Initialized(I) is halting on s iff I is halting on Initialize(s).

(23) For every macro instruction I such that for every state s of SCM _{FSA} holds I is halting on Initialize(s) holds Initialized(I) is halting.

(24) For every macro instruction I such that for every state s of SCM _{FSA} holds Initialized(I) is halting on s holds Initialized(I) is halting.

(25) For every macro instruction I holds ProgramPart(Initialized(I)) = I.

(26) Let s be a state of SCM _{FSA}, I be a macro instruction, l be an instruction-location of SCM _{FSA}, and x be a set. If x ∈ domI, then I(x) = (s+·(I+·Start-At(l)))(x).

(27) For every state s of SCM _{FSA} such that s(intloc(0)) = 1 holds Initialize(s)↾(Int-Locations∪ FinSeq-Locations) = s↾(Int-Locations∪ FinSeq-Locations).

(28) Let s be a state of SCM _{FSA}, I be a macro instruction, a be an integer location, and l be an instruction-location of SCM _{FSA}. Then (s+·(I+·Start-At(l)))(a) = s(a).

(29) For every programmed finite partial state I of SCM _{FSA} and for every instruction-location l of SCM _{FSA} holds IC_{SCM _{FSA}} ∈ dom(I+·Start-At(l)).

(30) For every programmed finite partial state I of SCM _{FSA} and for every instruction-location l of SCM _{FSA} holds (I+·Start-At(l))(IC_{SCM _{FSA}} ) = l.

(31) Let s be a state of SCM _{FSA}, P be a finite partial state of SCM _{FSA}, and l be an instruction-location of SCM _{FSA}. Then IC_{s+·(P +·Start-At (l))} = l.

(32) For every state s of SCM _{FSA} and for every instruction i of SCM _{FSA} such that InsCode(i) ∈ {0, 6, 7, 8} holds Exec(i, s)↾(Int-Locations∪ FinSeq-Locations) = s↾(Int-Locations∪ FinSeq-Locations).

(33) Let s_{1}, s_{2} be states of SCM _{FSA}. Suppose that (i) s_{1}(intloc(0)) = s_{2}(intloc(0)), (ii) for every read-write integer location a holds s_{1}(a) = s_{2}(a), and (iii) for every finite sequence location f holds s_{1}(f) = s_{2}(f).

Then s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations ∪ FinSeq-Locations).

(34) For every state s of SCM _{FSA} and for every programmed finite partial state P of SCM _{FSA} holds (s+·P)↾(Int-Locations∪ FinSeq-Locations) = s↾(Int-Locations∪ FinSeq-Locations).

(35) For all states s, s_{3} of SCM _{FSA} holds (s+·s_{3}↾the instruction locations of SCM _{FSA})↾(Int-Locations∪ FinSeq-Locations) = s↾(Int-Locations∪ FinSeq-Locations).

(36) For every state s of SCM _{FSA} holds Initialize(s)↾the instruction locations of SCM _{FSA} = s↾the instruction locations of SCM _{FSA}.

(37) Let s, s_{3} be states of SCM _{FSA} and I be a macro instruction. Then (s_{3}+·s↾the instruction locations of SCM _{FSA})↾(Int-Locations ∪ FinSeq-Locations) = s_{3}↾(Int-Locations∪ FinSeq-Locations).

(38) For every state s of SCM _{FSA} holds IExec(Stop_{SCM_{FSA}} , s) = Initialize(s)+·Start-At(insloc(0)).

(39) For every state s of SCM _{FSA} and for every macro instruction I such that I is closed on s holds insloc(0) ∈ domI.

(40) For every state s of SCM _{FSA} and for every paraclosed macro instruction I holds insloc(0) ∈ domI.

(41) For every instruction i of SCM _{FSA} holds rngMacro(i) = {i, halt_{SCM _{FSA}} }.

(42) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s_{1} and I+·Start-At(insloc(0)) ⊆ s_{1}. Let n be a natural number. Suppose ProgramPart(Relocated(I, n)) ⊆ s_{2} and IC_{(s_{2})} = insloc(n) and s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations). Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} + n = IC_{(Computation (s_{2}))(i)} and IncAddr(CurInstr((Computation(s_{1}))(i)), n) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾(Int-Locations∪ FinSeq-Locations) = (Computation(s_{2}))(i)↾(Int-Locations∪ FinSeq-Locations).

(43) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s_{1} and I+·Start-At(insloc(0)) ⊆ s_{1} and I+·Start-At(insloc(0)) ⊆ s_{2} and s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations). Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾(Int-Locations∪ FinSeq-Locations) = (Computation(s_{2}))(i)↾(Int-Locations∪ FinSeq-Locations).

(44) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s_{1} and halting on s_{1} and I+·Start-At(insloc(0)) ⊆ s_{1} and I+·Start-At(insloc(0)) ⊆ s_{2} and s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations). Then LifeSpan(s_{1}) = LifeSpan(s_{2}).

(45) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose

that

(i) s_{1}(intloc(0)) = 1, (ii) I is closed on s_{1} and halting on s_{1}, (iii) for every read-write integer location a holds s_{1}(a) = s_{2}(a), and (iv) for every finite sequence location f holds s_{1}(f) = s_{2}(f).

Then IExec(I, s_{1})↾(Int-Locations∪ FinSeq-Locations) = IExec(I, s_{2})↾(Int-Locations∪ FinSeq-Locations).

(46) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose s_{1}(intloc(0)) = 1 and I is closed on s_{1} and halting on s_{1} and s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations). Then IExec(I, s_{1})↾(Int-Locations∪ FinSeq-Locations) = IExec(I, s_{2})↾(Int-Locations∪ FinSeq-Locations).

Let I be a macro instruction. Observe that Initialized(I) is initial. One can prove the following propositions:

(47) Let s be a state of SCM _{FSA} and I be a macro instruction. Then Initialized(I) is pseudo-closed on s if and only if I is pseudo-closed on Initialize(s).

(48) For every state s of SCM _{FSA} and for every macro instruction I such that I is pseudo-closed on Initialize(s) holds pseudo − LifeSpan (s, Initialized(I)) = pseudo − LifeSpan (Initialize(s), I).

(49) For every state s of SCM _{FSA} and for every macro instruction I such that Initialized(I) is pseudo-closed on s holds pseudo − LifeSpan (s, Initialized(I)) = pseudo − LifeSpan (Initialize(s), I).

(50) Let s be a state of SCM _{FSA} and I be an initial finite partial state of SCM _{FSA}. Suppose I is pseudo-closed on s. Then I is pseudoclosed on s+·(I+·Start-At(insloc(0))) and pseudo − LifeSpan (s, I) = pseudo − LifeSpan (s+·(I+·Start-At(insloc(0))), I).

(51) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose I+·Start-At(insloc(0)) ⊆ s_{1} and I is pseudo-closed on s_{1}. Let n be a natural number. Suppose ProgramPart(Relocated(I, n)) ⊆ s_{2} and IC_{(s_{2})} = insloc(n) and s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations). Then

(i) for every natural number i such that i < pseudo − LifeSpan (s_{1}, I) holds IncAddr(CurInstr((Computation(s_{1}))(i)), n) = CurInstr((Computation(s_{2}))(i)), and

(ii) for every natural number i such that i pseudo − LifeSpan (s_{1}, I) holds IC_{(Computation (s_{1}))(i)} + n = IC_{(Computation (s_{2}))(i)} and (Computation(s_{1}))(i)↾(Int-Locations∪ FinSeq-Locations) = (Computation(s_{2}))(i)↾(Int-Locations∪ FinSeq-Locations).

(52) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations). If I is pseudo-closed on s_{1}, then I is pseudo-closed on s_{2}.

(53) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose s(intloc(0)) = 1. Then I is pseudo-closed on s if and only if I is pseudoclosed on Initialize(s).

(54) Let a be an integer location and I, J be macro instructions. Then insloc(0) ∈ domif = 0(a, I, J) and insloc(1) ∈ domif = 0(a, I, J) and insloc(0) ∈ domif > 0(a, I, J) and insloc(1) ∈ domif > 0(a, I, J).

(55) Let a be an integer location and I, J be macro instructions. Then (if = 0(a, I, J))(insloc(0)) = if a = 0 goto insloc(cardJ + 3) and (if = 0(a, I, J))(insloc(1)) = goto insloc(2) and (if > 0(a, I, J))(insloc(0)) = if a > 0 goto insloc(cardJ + 3) and (if > 0(a, I, J))(insloc(1)) = goto insloc(2).

(56) Let a be an integer location, I, J be macro instructions, and n be a natural number. If n < cardI + cardJ + 3, then insloc(n) ∈ domif = 0(a, I, J) and (if = 0(a, I, J))(insloc(n)) ≠ halt_{SCM _{FSA}} .

(57) Let a be an integer location, I, J be macro instructions, and n be a natural number. If n < cardI + cardJ + 3, then insloc(n) ∈ domif > 0(a, I, J) and (if > 0(a, I, J))(insloc(n)) ≠ halt_{SCM _{FSA}} .

(58) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose

Directed(I) is pseudo-closed on s. Then

(i) I;Stop_{SCM_{FSA}} is closed on s, (ii) I;Stop_{SCM_{FSA}} is halting on s, (iii) LifeSpan(s+·((I;Stop_{SCM_{FSA}} )+·Start-At(insloc(0)))) =

pseudo − LifeSpan (s, Directed(I)), (iv) for every natural number n such that

n < pseudo − LifeSpan (s, Directed(I)) holds IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(n)} = IC_{(Computation (s+·((I;Stop_{_{SCM} _{FSA}} )+·Start-At (insloc(0)))))(n)}, and

(v) for every natural number n such that n pseudo − LifeSpan (s, Directed(I)) holds (Computation(s+·(I+·Start-At(insloc(0)))))(n)↾D = (Computation(s+·((I;Stop_{SCM_{FSA}} )+·Start-At(insloc(0)))))(n)↾D.

(59) Let s be a state of SCM _{FSA} and I be a macro instruction. If Directed(I) is pseudo-closed on s, then Result(s+·((I;Stop_{SCM_{FSA}} )+·Start-At(insloc(0))))↾D = (Computation(s+·(I+·Start-At(insloc(0))))) (pseudo − LifeSpan (s, Directed(I)))↾D.

(60) Let s be a state of SCM _{FSA} and I be a macro instruction. If s(intloc(0)) = 1 and Directed(I) is pseudo-closed on s, then IExec(I;Stop_{SCM_{FSA}} , s)↾D = (Computation(s+·(I+·Start-At(insloc(0))))) (pseudo − LifeSpan (s, Directed(I)))↾D.

(61) For all macro instructions I, J and for every integer location a holds (if = 0(a, I, J))(insloc(cardI + cardJ + 3)) = halt_{SCM _{FSA}} .

(62) For all macro instructions I, J and for every integer location a holds (if > 0(a, I, J))(insloc(cardI + cardJ + 3)) = halt_{SCM _{FSA}} .

(63) For all macro instructions I, J and for every integer location a holds (if = 0(a, I, J))(insloc(cardJ + 2)) = goto insloc(cardI + cardJ + 3).

(64) For all macro instructions I, J and for every integer location a holds (if > 0(a, I, J))(insloc(cardJ + 2)) = goto insloc(cardI + cardJ + 3).

(65) For every macro instruction J and for every integer location a holds (if = 0(a, Goto(insloc(2)), J))(insloc(cardJ + 3)) = goto insloc(cardJ + 5).

(66) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(a) = 0 and Directed(I) is pseudoclosed on s. Then if = 0(a, I, J) is halting on s and if = 0(a, I, J) is closed on s and LifeSpan(s+·(if = 0(a, I, J)+·Start-At(insloc(0)))) = LifeSpan(s+·((I;Stop_{SCM_{FSA}} )+·Start-At(insloc(0)))) + 1.

(67) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(intloc(0)) = 1 and s(a) = 0 and Directed(I) is pseudo-closed on s. Then IExec(if = 0(a, I, J), s)↾(Int-Locations∪ FinSeq-Locations) = IExec(I;Stop_{SCM_{FSA}} , s) ↾(Int-Locations∪ FinSeq-Locations).

(68) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(a) > 0 and Directed(I) is pseudoclosed on s. Then if > 0(a, I, J) is halting on s and if > 0(a, I, J) is closed on s and LifeSpan(s+·(if > 0(a, I, J)+·Start-At(insloc(0)))) = LifeSpan(s+·((I;Stop_{SCM_{FSA}} )+·Start-At(insloc(0)))) + 1.

(69) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(intloc(0)) = 1 and s(a) > 0 and Directed(I) is pseudo-closed on s. Then IExec(if > 0(a, I, J), s)↾(Int-Locations∪ FinSeq-Locations) = IExec(I;Stop_{SCM_{FSA}} , s) ↾(Int-Locations∪ FinSeq-Locations).

(70) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(a) ≠ 0 and Directed(J) is pseudoclosed on s. Then if = 0(a, I, J) is halting on s and if = 0(a, I, J) is closed on s and LifeSpan(s+·(if = 0(a, I, J)+·Start-At(insloc(0)))) = LifeSpan(s+·((J;Stop_{SCM_{FSA}} )+·Start-At(insloc(0)))) + 3.

(71) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(intloc(0)) = 1 and s(a) ≠ 0 and Directed(J) is pseudo-closed on s. Then IExec(if = 0(a, I, J), s)↾(Int-Locations∪ FinSeq-Locations) = IExec(J;Stop_{SCM_{FSA}} , s) ↾(Int-Locations∪ FinSeq-Locations).

(72) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(a) 0 and Directed(J) is pseudoclosed on s. Then if > 0(a, I, J) is halting on s and if > 0(a, I, J) is closed on s and LifeSpan(s+·(if > 0(a, I, J)+·Start-At(insloc(0)))) = LifeSpan(s+·((J;Stop_{SCM_{FSA}} )+·Start-At(insloc(0)))) + 3.

(73) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose s(intloc(0)) = 1 and s(a) 0 and Directed(J) is pseudo-closed on s. Then IExec(if > 0(a, I, J), s)↾(Int-Locations∪ FinSeq-Locations) = IExec(J;Stop_{SCM_{FSA}} , s) ↾(Int-Locations∪ FinSeq-Locations).

(74) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose Directed(I) is pseudo-closed on s and Directed(J) is pseudo-closed on s. Then if = 0(a, I, J) is closed on s and if = 0(a, I, J) is halting on s.

(75) Let s be a state of SCM _{FSA}, I, J be macro instructions, and a be a read-write integer location. Suppose Directed(I) is pseudo-closed on s and Directed(J) is pseudo-closed on s. Then if > 0(a, I, J) is closed on s and if > 0(a, I, J) is halting on s.

(76) Let I be a macro instruction and a be an integer location. If I does not destroy a, then Directed(I) does not destroy a.

(77) Let i be an instruction of SCM _{FSA} and a be an integer location. If i does not destroy a, then Macro(i) does not destroy a.

(78) For every integer location a holds halt_{SCM _{FSA}} does not refer a.

(79) For all integer locations a, b, c such that a ≠ b holds AddTo(c, b) does not refer a.

(80) Let i be an instruction of SCM _{FSA} and a be an integer location. If i does not refer a, then Macro(i) does not refer a.

(81) Let I, J be macro instructions and a be an integer location. Suppose I does not destroy a and J does not destroy a. Then I;J does not destroy a.

(82) Let J be a macro instruction, i be an instruction of SCM _{FSA}, and a be an integer location. Suppose i does not destroy a and J does not destroy a. Then i;J does not destroy a.

(83) Let I be a macro instruction, j be an instruction of SCM _{FSA}, and a be an integer location. Suppose I does not destroy a and j does not destroy a. Then I;j does not destroy a.

(84) Let i, j be instructions of SCM _{FSA} and a be an integer location. Suppose i does not destroy a and j does not destroy a. Then i;j does not destroy a.

(85) For every integer location a holds Stop_{SCM_{FSA}} does not destroy a.

(86) For every integer location a and for every instruction-location l of SCM _{FSA} holds Goto(l) does not destroy a.

(87) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose I is

halting on Initialize(s). Then

(i) for every read-write integer location a holds (IExec(I, s))(a) = (Computation(Initialize(s)+·(I+·Start-At(insloc(0))))) (LifeSpan(Initialize(s)+·(I+·Start-At(insloc(0)))))(a), and

(ii) for every finite sequence location f holds (IExec(I, s))(f) = (Computation(Initialize(s)+·(I+·Start-At(insloc(0))))) (LifeSpan(Initialize(s)+·(I+·Start-At(insloc(0)))))(f).

(88) Let s be a state of SCM _{FSA}, I be a parahalting macro instruction, and a be a read-write integer location. Then (IExec(I, s))(a) = (Computation(Initialize(s)+·(I+·Start-At(insloc(0))))) (LifeSpan(Initialize(s)+·(I+·Start-At(insloc(0)))))(a).

(89) Let s be a state of SCM _{FSA}, I be a macro instruction, a be an integer location, and k be a natural number. Suppose I is closed on Initialize(s) and halting on Initialize(s) and I does not destroy a. Then (IExec(I, s))(a) = (Computation(Initialize(s)+·(I+·Start-At(insloc(0)))))(k)(a).

(90) Let s be a state of SCM _{FSA}, I be a parahalting macro instruction, a be an integer location, and k be a natural number. If I does not destroy a, then (IExec(I, s))(a) = (Computation(Initialize(s)+·(I+·Start-At(insloc(0)))))(k)(a).

(91) Let s be a state of SCM _{FSA}, I be a parahalting macro instruction, and a be an integer location. If I does not destroy a, then (IExec(I, s))(a) = (Initialize(s))(a).

(92) Let s be a state of SCM _{FSA} and I be a keeping 0 macro instruction. Suppose I is halting on Initialize(s). Then (IExec(I, s))(intloc(0)) = 1 and for every natural number k holds (Computation(Initialize(s)+·(I+·Start-At (insloc(0)))))(k)(intloc(0)) = 1.

(93) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be an integer location. Suppose I does not destroy a. Let k be a natural number. If IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(k)} ∈ domI, then (Computation(s+·(I+·Start-At(insloc(0)))))(k + 1)(a) = (Computation(s+·(I+·Start-At(insloc(0)))))(k)(a).

(94) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be an integer location. Suppose I does not destroy a. Let m be a natural number. Suppose that for every natural number n such that n < m holds IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(n)} ∈ domI. Let n be a natural number. If n m, then (Computation(s+·(I+·Start-At(insloc(0)))))(n)(a) = s(a).

(95) Let s be a state of SCM _{FSA}, I be a good macro instruction, and m be a natural number. Suppose that for every natural number n such that n < m holds IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(n)} ∈ domI. Let n be a natural number. If n m, then (Computation(s+·(I+·Start-At (insloc(0)))))(n)(intloc(0)) = s(intloc(0)).

(96) Let s be a state of SCM _{FSA} and I be a good macro instruction. Suppose I is halting on Initialize(s) and closed on Initialize(s). Then (IExec(I, s))(intloc(0)) = 1 and for every natural number k holds (Computation(Initialize(s)+·(I+·Start-At(insloc(0)))))(k)(intloc(0)) = 1.

(97) Let s be a state of SCM _{FSA} and I be a good macro instruction. Suppose I is closed on s. Let k be a natural number. Then (Computation(s+·(I+·Start-At(insloc(0)))))(k)(intloc(0)) = s(intloc(0)).

(98) Let s be a state of SCM _{FSA}, I be a keeping 0 parahalting macro instruction, and a be a read-write integer location. Suppose I does not destroy a. Then (Computation(Initialize(s)+·((I;SubFrom(a, intloc(0)))+·Start-At (insloc(0)))))(LifeSpan(Initialize(s)+·((I;SubFrom(a, intloc(0)))+·Start-At (insloc(0)))))(a) = s(a) − 1.

(99) For every instruction i of SCM _{FSA} such that i does not destroy intloc(0) holds Macro(i) is good.

(100) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s_{1} and halting on s_{1} and s_{1}↾D = s_{2}↾D. Let k be a natural number. Then

(i) (Computation(s_{1}+·(I+·Start-At(insloc(0)))))(k) and (Computation(s_{2}+·(I+·Start-At(insloc(0)))))(k) are equal outside the instruction locations of SCM _{FSA}, and

(ii) CurInstr((Computation(s_{1}+·(I+·Start-At(insloc(0)))))(k)) =

CurInstr((Computation(s_{2}+·(I+·Start-At(insloc(0)))))(k)).

(101) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s_{1} and halting on s_{1} and s_{1}↾D = s_{2}↾D. Then LifeSpan(s_{1}+·(I+·Start-At(insloc(0)))) = LifeSpan(s_{2}+·(I+·Start-At(insloc(0)))) and Result(s_{1}+·(I+·Start-At(insloc(0)))) and Result(s_{2}+·(I+·Start-At (insloc(0)))) are equal outside the instruction locations of SCM _{FSA}.

(102) Let N be a non empty set with non empty elements, S be a steadyprogrammed von Neumann definite AMI over N, and s be a state of S. Suppose s is halting. Then there exists a natural number k such that s halts at IC_{(Computation (s))(k)}.

(103) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a macro instruction. Suppose

that

(i) I is closed on s_{1} and halting on s_{1}, (ii) I+·Start-At(insloc(0)) ⊆ s_{1}, (iii) I+·Start-At(insloc(0)) ⊆ s_{2}, and (iv) there exists a natural number k such that (Computation(s_{1}))(k) and

s_{2} are equal outside the instruction locations of SCM _{FSA}. Then Result(s_{1}) and Result(s_{2}) are equal outside the instruction locations of SCM _{FSA}.

2. The loop Macroinstruction

Let I be a macro instruction and let k be a natural number. One can verify that IncAddr(I, k) is initial and programmed.

Let I be a macro instruction. The functor loopI yields a halt-free macro instruction and is defined by:

(Def. 4) loopI = (id_{the instructions of SCM _{FSA}} +·(halt_{SCM _{FSA}} ⟼→ . goto insloc(0))) · I.

Next we state two propositions:

(104) For every macro instruction I holds loopI = Directed(I, insloc(0)).

(105) Let I be a macro instruction and a be an integer location. If I does not destroy a, then loopI does not destroy a.

Let I be a good macro instruction. One can verify that loopI is good. The following propositions are true:

(106) For every macro instruction I holds domloopI = domI.

(107) For every macro instruction I holds halt_{SCM _{FSA}} ∉ rngloopI.

(108) For every macro instruction I and for every set x such that x ∈ domI holds if I(x) ≠ halt_{SCM _{FSA}} , then (loopI)(x) = I(x).

(109) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s and halting on s. Let m be a natural number. Suppose m LifeSpan(s+·(I+·Start-At(insloc(0)))). Then (Computation(s+·(I+·Start-At(insloc(0)))))(m) and (Computation(s+·(loopI+·Start-At(insloc(0)))))(m) are equal outside the instruction locations of SCM _{FSA}.

(110) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s and halting on s. Let m be a natural number. If m < LifeSpan(s+·(I+·Start-At(insloc(0)))), then CurInstr((Computation(s+·(I+·Start-At(insloc(0)))))(m)) = CurInstr((Computation(s+·(loopI+·Start-At(insloc(0)))))(m)).

(111) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose I is closed on s and halting on s. Let m be a natural number. If m LifeSpan(s+·(I+·Start-At(insloc(0)))), then CurInstr((Computation(s+·(loopI+·Start-At(insloc(0)))))(m)) ≠ halt_{SCM _{FSA}} .

(112) Let s be a state of SCM _{FSA} and I be a macro instruction. If I is closed on s and halting on s, then CurInstr((Computation(s+·(loopI+·Start-At (insloc(0)))))(LifeSpan(s+·(I+·Start-At(insloc(0)))))) = goto insloc(0).

(113) Let s be a state of SCM _{FSA} and I be a paraclosed macro instruction. Suppose I+·Start-At(insloc(0)) ⊆ s and s is halting. Let m be a natural number. Suppose m LifeSpan(s). Then (Computation(s))(m) and (Computation(s+·loopI))(m) are equal outside the instruction locations of SCM _{FSA}.

(114) Let s be a state of SCM _{FSA} and I be a parahalting macro instruction. Suppose Initialized(I) ⊆ s. Let k be a natural number. If k LifeSpan(s), then CurInstr((Computation(s+·loopI))(k)) ≠ halt_{SCM _{FSA}} .

3. The Times Macroinstruction

Let a be an integer location and let I be a macro instruction. The functor Times(a, I) yields a macro instruction and is defined by:

(Def. 5) Times(a, I) = if > 0(a, loopif = 0(a, Goto(insloc(2)), I;SubFrom

(a, intloc(0))), Stop_{SCM_{FSA}} ). The following propositions are true:

(115) For every good macro instruction I and for every read-write integer location a holds if = 0(a, Goto(insloc(2)), I;SubFrom(a, intloc(0))) is good.

(116) For all macro instructions I, J and for every integer location a holds (if = 0(a, Goto(insloc(2)), I;SubFrom(a, intloc(0)))) (insloc(card(I;SubFrom(a, intloc(0))) + 3)) = goto insloc(card(I;SubFrom(a, intloc(0))) + 5).

(117) Let s be a state of SCM _{FSA}, I be a good parahalting macro instruction, and a be a read-write integer location. Suppose I does not destroy a and s(intloc(0)) = 1 and s(a) > 0. Then loopif = 0(a, Goto(insloc(2)), I;SubFrom(a, intloc(0))) is pseudo-closed on s.

(118) Let s be a state of SCM _{FSA}, I be a good parahalting macro instruction, and a be a read-write integer location. Suppose I does not destroy a and s(a) > 0. Then Initialized(loopif = 0(a, Goto(insloc(2)), I;SubFrom(a, intloc(0)))) is pseudo-closed on s.

(119) Let s be a state of SCM _{FSA}, I be a good parahalting macro instruction, and a be a read-write integer location. Suppose I does not destroy a and s(intloc(0)) = 1. Then Times(a, I) is closed on s and Times(a, I) is halting on s.

(120) Let I be a good parahalting macro instruction and a be a read-write integer location. If I does not destroy a, then Initialized(Times(a, I)) is halting.

(121) Let I, J be macro instructions and a, c be integer locations. Suppose I does not destroy c and J does not destroy c. Then if = 0(a, I, J) does not destroy c and if > 0(a, I, J) does not destroy c.

(122) Let s be a state of SCM _{FSA}, I be a good parahalting macro instruction, and a be a read-write integer location. Suppose I does not destroy a and s(intloc(0)) = 1 and s(a) > 0. Then there exists a state s_{2} of SCM _{FSA} and there exists a natural number k such that

(i) s_{2} = s+·(loopif = 0(a, Goto(insloc(2)), I;SubFrom(a, intloc(0))) +·Start-At(insloc(0))),

(ii) k = LifeSpan(s+·(if = 0(a, Goto(insloc(2)), I;SubFrom(a, intloc(0)))

+·Start-At(insloc(0)))) + 1, (iii) (Computation(s_{2}))(k)(a) = s(a) − 1, (iv) (Computation(s_{2}))(k)(intloc(0)) = 1,

(v) for every read-write integer location b such that b ≠ a holds (Computation(s_{2}))(k)(b) = (IExec(I, s))(b), 496 noriko asamoto

(vi) for every finite sequence location f holds (Computation(s_{2}))(k)(f) =

(IExec(I, s))(f), (vii) IC_{(Computation (s_{2}))(k)} = insloc(0), and (viii) for every natural number n such that n k holds

IC_{(Computation (s_{2}))(n)} ∈ domloopif = 0(a, Goto(insloc(2)), I;SubFrom (a, intloc(0))).

(123) Let s be a state of SCM _{FSA}, I be a good parahalting macro instruction, and a be a read-write integer location. If s(intloc(0)) = 1 and s(a) 0, then IExec(Times(a, I), s)↾(Int-Locations∪ FinSeq-Locations) = s↾(Int-Locations∪ FinSeq-Locations).

(124) Let s be a state of SCM _{FSA}, I be a good parahalting macro instruction, and a be a read-write integer location. Suppose I does not destroy a and s(a) > 0. Then (IExec(I;SubFrom(a, intloc(0)), s))(a) = s(a) − 1 and IExec(Times(a, I), s)↾(Int-Locations∪ FinSeq-Locations) = IExec(Times(a, I), IExec(I;SubFrom(a, intloc(0)), s))↾(Int-Locations ∪ FinSeq-Locations).

4. An example

One can prove the following proposition

(125) Let s be a state of SCM _{FSA} and a, b, c be read-write integer locations. If a ≠ b and a ≠ c and b ≠ c and s(a) 0, then (IExec(Times(a, Macro(AddTo(b, c))), s))(b) = s(b) + s(c) · s(a).

References

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[10] Grzegorz Bancerek and Piotr Rudnicki. Development of terminology for scm. Formalized Mathematics, 4(1):61–67, 1993.

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

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[20] Yasushi Tanaka. On the decomposition of the states of SCM. Formalized Mathematics, 5(1):1–8, 1996.

[21] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

[23] Andrzej Trybulec and Yatsuka Nakamura. Some remarks on the simple concrete model of computer. Formalized Mathematics, 4(1):51–56, 1993.

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[26] Andrzej Trybulec, Yatsuka Nakamura, and Noriko Asamoto. On the compositions of macro instructions. Part I. Formalized Mathematics, 6(1):21–27, 1997.

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498 noriko asamoto

Algebraic and Arithmetic Lattices. Part II^{1}

Robert Milewski University of Białystok

Summary. The article is a translation of [13, pp. 89–92]

MML Identifier: WAYBEL15.

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

1. Preliminaries

The following propositions are true:

(1) Let R be a relational structure and S be a full relational substructure of R. Then every full relational substructure of S is a full relational substructure of R.

(2) Let X, Y, Z be non empty 1-sorted structures, f be a map from X into Y, and g be a map from Y into Z. If f is onto and g is onto, then g · f is onto.

(3) For every non empty 1-sorted structure X and for every subset Y of the carrier of X holds (id_{X})^{∘}Y = Y.

(4) For every set X and for every element a of 2_{⊆}^{X} holds ↑a = {Y; Y ranges over subsets of X: a ⊆ Y }.

(5) Let L be an upper-bounded non empty antisymmetric relational structure and a be an element of L. If ⊤_{L} a, then a = ⊤_{L}.

1This work has been supported by KBN Grant 8 T11C 018 12.

(6) Let S, T be non empty posets, g be a map from S into T, and d be a map from T into S. If g is onto and ⟨⟨g, d⟩⟩ is Galois, then T and Imd are isomorphic.

(7) Let L_{1}, L_{2}, L_{3} be non empty posets, g_{1} be a map from L_{1} into L_{2}, g_{2} be a map from L_{2} into L_{3}, d_{1} be a map from L_{2} into L_{1}, and d_{2} be a map from L_{3} into L_{2}. If ⟨⟨g_{1}, d_{1}⟩⟩ is Galois and ⟨⟨g_{2}, d_{2}⟩⟩ is Galois, then ⟨⟨g_{2} · g_{1}, d_{1} · d_{2}⟩⟩ is Galois.

(8) Let L_{1}, L_{2} be non empty posets, f be a map from L_{1} into L_{2}, and f_{1} be a map from L_{2} into L_{1}. Suppose f_{1} = (f qua function) ^{−1} and f is isomorphic. Then ⟨⟨f, f_{1}⟩⟩ is Galois and ⟨⟨f_{1}, f⟩⟩ is Galois.

(9) For every set X holds 2_{⊆}^{X} is arithmetic. Next we state four propositions:

(10) Let L_{1}, L_{2} be up-complete non empty posets and f be a map from L_{1} into L_{2}. If f is isomorphic, then for every element x of L_{1} holds f^{∘}↓↓x = ↓↓f(x).

(11) For all non empty posets L_{1}, L_{2} such that L_{1} and L_{2} are isomorphic and L_{1} is continuous holds L_{2} is continuous.

(12) Let L_{1}, L_{2} be lattices. Suppose L_{1} and L_{2} are isomorphic and L_{1} is lower-bounded and arithmetic. Then L_{2} is arithmetic.

(13) Let L_{1}, L_{2}, L_{3} be non empty posets, f be a map from L_{1} into L_{2}, and g be a map from L_{2} into L_{3}. Suppose f is directed-sups-preserving and g is directed-sups-preserving. Then g · f is directed-sups-preserving.

2. Maps Preserving Sup’s and Inf’s

One can prove the following propositions:

(14) Let L_{1}, L_{2} be non empty relational structures, f be a map from L_{1} into L_{2}, and X be a subset of Imf. Then (f_{∘})^{∘}X = X.

(15) Let X be a set and c be a map from 2_{⊆}^{X} into 2_{⊆}^{X}. Suppose c is idempotent and directed-sups-preserving. Then c_{∘} is directed-sups-preserving.

(16) Let L be a continuous complete lattice and p be a kernel map from L into L. If p is directed-sups-preserving, then Imp is a continuous lattice.

(17) Let L be a continuous complete lattice and p be a projection map from L into L. If p is directed-sups-preserving, then Imp is a continuous lattice.

(18) Let L be a lower-bounded lattice. Then L is continuous if and only if there exists an arithmetic lower-bounded lattice A such that there exists a map from A into L which is onto, infs-preserving, and directed-sups-preserving.

(19) Let L be a lower-bounded lattice. Then L is continuous if and only if there exists an algebraic lower-bounded lattice A such that there exists a map from A into L which is onto, infs-preserving, and directed-sups-preserving.

(20) Let L be a lower-bounded lattice. Then L is continuous if and only if there exists a set X and there exists a projection map p from 2_{⊆}^{X} into 2_{⊆}^{X} such that p is directed-sups-preserving and L and Imp are isomorphic.

3. Atoms Elements

Next we state two propositions:

(21) For every non empty relational structure L and for every element x of L holds x ∈ PRIME(L^{op}) iff x is co-prime.

(22) Let L be a sup-semilattice and a be an element of L. Then a is co-prime if and only if for all elements x, y of L such that a x ⊔ y holds a x or a y.

Let L be a non empty relational structure and let a be an element of L. We say that a is an atom if and only if:

(Def. 1) ⊥_{L} < a and for every element b of L such that ⊥_{L} < b and b a holds

b = a. Let L be a non empty relational structure. The functor ATOM(L) yielding

a subset of L is defined by: (Def. 2) For every element x of L holds x ∈ ATOM(L) iff x is atom.

The following proposition is true

(23) For every Boolean lattice L and for every element a of L holds a is atom

iff a is co-prime and a ≠ ⊥_{L}. Let L be a Boolean lattice. Observe that every element of L which is atom

is also co-prime.

Next we state several propositions:

(24) For every Boolean lattice L holds ATOM(L) = PRIME(L^{op}) \ {⊥_{L}}.

(25) For every Boolean lattice L and for all elements x, a of L such that a is atom holds a x iff a ¬̸x.

(26) Let L be a complete Boolean lattice, X be a subset of L, and x be an element of L. Then x ⊓ supX = ⨆_{L}{x ⊓ y; y ranges over elements of L: y ∈ X}.

(27) Let L be a lower-bounded antisymmetric non empty relational structure with g.l.b.’s and x, y be elements of L. If x is atom and y is atom and x ≠ y, then x ⊓ y = ⊥_{L}.

(28) Let L be a complete Boolean lattice, x be an element of L, and A be a subset of L. If A ⊆ ATOM(L), then x ∈ A iff x is atom and x supA.

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(29) Let L be a complete Boolean lattice and X, Y be subsets of L. If X ⊆ ATOM(L) and Y ⊆ ATOM(L), then X ⊆ Y iff supX supY.

4. More on the Boolean Lattice

One can prove the following propositions:

(30) For every Boolean lattice L holds L is arithmetic iff there exists a set X such that L and 2_{⊆}^{X} are isomorphic.

(31) For every Boolean lattice L holds L is arithmetic iff L is algebraic.

(32) For every Boolean lattice L holds L is arithmetic iff L is continuous.

(33) For every Boolean lattice L holds L is arithmetic iff L is continuous and L^{op} is continuous.

(34) For every Boolean lattice L holds L is arithmetic iff L is completelydistributive.

(35) Let L be a Boolean lattice. Then L is arithmetic if and only if the following conditions are satisfied:

(i) L is complete, and (ii) for every element x of L there exists a subset X of L such that X ⊆

ATOM(L) and x = supX.

References

[1] Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.

[2] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719–725, 1991.

[3] Grzegorz Bancerek. Quantales. Formalized Mathematics, 5(1):85–91, 1996.

[4] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81–91, 1997.

[5] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.

[6] Grzegorz Bancerek. Duality in relation structures. Formalized Mathematics, 6(2):227–232, 1997.

[7] Grzegorz Bancerek. The “way-below” relation. Formalized Mathematics, 6(1):169–176, 1997.

[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. Galois connections. Formalized Mathematics, 6(1):131–143, 1997.

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

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

[13] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.

[14] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117–121, 1997.

[15] Beata Madras. Irreducible and prime elements. Formalized Mathematics, 6(2):233–239, 1997.

[16] Robert Milewski. Algebraic lattices. Formalized Mathematics, 6(2):249–254, 1997.

algebraic and arithmetic lattices. part … 503

[17] Michał Muzalewski. Categories of groups. Formalized Mathematics, 2(4):563–571, 1991.

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

[19] Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.

[20] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

[21] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

[23] Mariusz Żynel. The equational characterization of continuous lattices. Formalized Mathematics, 6(2):199–205, 1997.

[24] Mariusz Żynel and Czesław Byliński. Properties of relational structures, posets, lattices and maps. Formalized Mathematics, 6(1):123–130, 1997.

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Projections in n-Dimensional Euclidean Space to Each Coordinates

Roman Matuszewski^{1} University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. In the n-dimensional Euclidean space E_{T}^{n}, a projection operator to each coordinate is defined. It is proven that such an operator is linear. Moreover, it is continuous as a mapping from E_{T}^{n} to R^{1}, the carrier of which is a set of all reals. If n is 1, the projection becomes a homeomorphism, which means that E_{T}^{1} is homeomorphic to R^{1}.

MML Identifier: JORDAN2B.

The notation and terminology used in this paper are introduced in the following articles: [30], [35], [34], [20], [1], [37], [33], [27], [12], [29], [11], [26], [23], [36], [2], [8], [9], [5], [32], [3], [18], [17], [25], [15], [10], [14], [31], [16], [19], [22], [7], [24], [13], [21], [4], [6], and [28].

1. Projections

For simplicity, we use the following convention: a, b, s, s_{1}, r, r_{1}, r_{2} denote real numbers, n, i denote natural numbers, X denotes a non empty topological space, p, p_{1}, p_{2}, q denote points of E_{T}^{n}, P denotes a subset of the carrier of E_{T}^{n}, and f denotes a map from E_{T}^{n} into ℝ^{1}.

Let n, i be natural numbers and let p be an element of the carrier of E_{T}^{n}. The functor Proj(p, i) yielding a real number is defined as follows:

(Def. 1) For every finite sequence g of elements of ℝ such that g = p holds Proj(p, i) = π_{i}g.

1The work was done, while the author stayed at Nagano in the fall of 1996.

The following propositions are true:

(1) For every i there exists a map f from E_{T}^{n} into ℝ^{1} such that for every element p of the carrier of E_{T}^{n} holds f(p) = Proj(p, i).

(2) For every i such that i ∈ Segn holds ⟨\underbrace{0, …, 0}_{n}⟩(i) = 0.

(3) For every i such that i ∈ Segn holds Proj(0_{E_{T}^{n}}, i) = 0.

(4) For all r, p, i such that i ∈ Segn holds Proj(r · p, i) = r · Proj(p, i).

(5) For all p, i such that i ∈ Segn holds Proj(− p, i) = − Proj(p, i).

(6) For all p_{1}, p_{2}, i such that i ∈ Segn holds Proj(p_{1} + p_{2}, i) = Proj(p_{1}, i) + Proj(p_{2}, i).

(7) For all p_{1}, p_{2}, i such that i ∈ Segn holds Proj(p_{1} − p_{2}, i) = Proj(p_{1}, i) − Proj(p_{2}, i).

(8) len⟨\underbrace{0, …, 0}_{n}⟩ = n.

(9) For every i such that i n holds ⟨\underbrace{0, …, 0}_{n}⟩↾i = ⟨\underbrace{0, …, 0}_{i}⟩.

(10) For every i holds ⟨\underbrace{0, …, 0}_{n}⟩_{⇂i} = ⟨\underbrace{0, …, 0}_{n− ′i}⟩.

(11) For every i holds ∑⟨\underbrace{0, …, 0}_{i}⟩ = 0.

(12) For every finite sequence w and for all r, i holds len(w +·(i, r)) = lenw.

(13) For every finite sequence w of elements of ℝ and for all r, i such that i ∈ Seglenw holds w +·(i, r) = (w↾i − ′1)^{⌢} ⟨r⟩^{⌢} (w_{⇂i}).

(14) For all i, r such that i ∈ Segn holds ∑(⟨\underbrace{0, …, 0}_{n}⟩ +·(i, r)) = r.

(15) For every element q of R^{n} and for all p, i such that i ∈ Segn and q = p holds Proj(p, i) |q| and (Proj(p, i))^{2} |q|^{2}.

2. Continuity of Projections

Next we state several propositions:

(16) For all s_{1}, P, i such that P = {p : s_{1} > Proj(p, i)} and i ∈ Segn holds P is open.

(17) For all s_{1}, P, i such that P = {p : s_{1} < Proj(p, i)} and i ∈ Segn holds P is open.

(18) Let P be a subset of the carrier of E_{T}^{n}, a, b be real numbers, and given i. Suppose P = {p; p ranges over elements of the carrier of E_{T}^{n}: a < Proj(p, i) ∧ Proj(p, i) < b} and i ∈ Segn. Then P is open.

(19) Let a, b be real numbers, f be a map from E_{T}^{n} into ℝ^{1}, and given i. Suppose that for every element p of the carrier of E_{T}^{n} holds f(p) = Proj(p, i). Then f^{−1} ({s : a < s ∧ s < b}) = {p; p ranges over elements of the carrier of E_{T}^{n}: a < Proj(p, i) ∧ Proj(p, i) < b}.

(20) Let M be a metric space and f be a map from X into M_{top}. Suppose that for every real number r and for every element u of the carrier of M and for every subset P of the carrier of M_{top} such that r > 0 and P = Ball(u, r) holds f^{−1} (P) is open. Then f is continuous.

(21) Let u be a point of the metric space of real numbers and r, u_{1} be real numbers. If u_{1} = u and r > 0, then Ball(u, r) = {s : u_{1} − r < s ∧ s < u_{1} + r}.

(22) Let f be a map from E_{T}^{n} into ℝ^{1} and given i. Suppose i ∈ Segn and for every element p of the carrier of E_{T}^{n} holds f(p) = Proj(p, i). Then f is continuous.

3. 1-dimensional and 2-dimensional Cases

The following three propositions are true:

(23) For every s holds |⟨s⟩| = ⟨|s|⟩.

(24) For every element p of the carrier of E_{T}^{1} there exists r such that p = ⟨r⟩.

(25) For every element w of the carrier of E^{1} there exists r such that w = ⟨r⟩.

Let us consider r. The functor |[r]| yields a point of E_{T}^{1} and is defined by: (Def. 2) |[r]| = ⟨r⟩.

The following propositions are true:

(26) For all r, s holds s · |[r]| = |[s · r]|.

(27) For all r_{1}, r_{2} holds |[r_{1} + r_{2}]| = |[r_{1}]| + |[r_{2}]|.

(28) |[0]| = 0_{E_{T}^{1}}.

(29) For all r_{1}, r_{2} such that |[r_{1}]| = |[r_{2}]| holds r_{1} = r_{2}.

(30) For every subset P of the carrier of ℝ^{1} and for every real number b such that P = {s : s < b} holds P is open.

(31) For every subset P of the carrier of ℝ^{1} and for every real number a such that P = {s : a < s} holds P is open.

(32) For every subset P of the carrier of ℝ^{1} and for all real numbers a, b such that P = {s : a < s ∧ s < b} holds P is open.

(33) For every point u of E^{1} and for all real numbers r, u_{1} such that ⟨u_{1}⟩ = u and r > 0 holds Ball(u, r) = {⟨s⟩ : u_{1} − r < s ∧ s < u_{1} + r}.

(34) Let f be a map from E_{T}^{1} into ℝ^{1}. Suppose that for every element p of the carrier of E_{T}^{1} holds f(p) = Proj(p, 1). Then f is a homeomorphism.

508 roman matuszewski and yatsuka nakamura

(35) For every element p of the carrier of E_{T}^{2} holds Proj(p, 1) = p_{1} and Proj(p, 2) = p_{2}.

(36) For every element p of the carrier of E_{T}^{2} holds Proj(p, 1) = (proj1)(p) and Proj(p, 2) = (proj2)(p).

References

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

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

[3] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.

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

[5] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[6] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

[7] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 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. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[11] Czesław Byliński. Semigroup operations on finite subsets. Formalized Mathematics, 1(4):651–656, 1990.

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

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

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

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

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

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

[18] 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.

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

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

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

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

[23] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

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

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

[26] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

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

projections in n-dimensional euclidean … 509

[28] Agnieszka Sakowicz, Jarosław Gryko, and Adam Grabowski. Sequences in E_{T}^{N}. Formalized Mathematics, 5(1):93–96, 1996.

[29] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

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

[32] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

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

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

[35] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

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

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Intermediate Value Theorem and Thickness of Simple Closed Curves

Yatsuka Nakamura Shinshu University Nagano

Andrzej Trybulec University of Białystok

Summary. Various types of the intermediate value theorem ([25]) are proved. For their special cases, the Bolzano theorem is also proved. Using such a theorem, it is shown that if a curve is a simple closed curve, then it is not horizontally degenerated, neither is it vertically degenerated.

MML Identifier: TOPREAL5.

The articles [29], [33], [28], [16], [1], [27], [34], [6], [7], [4], [8], [32], [22], [35], [11], [10], [24], [2], [5], [31], [17], [3], [12], [13], [14], [15], [18], [19], [21], [26], [23], [30], [9], and [20] provide the notation and terminology for this paper.

1. Intermediate Value Theorems and Bolzano Theorem

For simplicity, we adopt the following convention: a, b, c, d, r_{1}, r_{2}, r_{3}, r, r_{4}, s_{1}, s_{2} are real numbers, p, q are points of E_{T}^{2}, P is a subset of the carrier of E_{T}^{2}, and X, Y, Z are non empty topological spaces.

Next we state a number of propositions:

(1) For all a, b, c holds c ∈ [a, b] iff a c and c b.

(2) Let f be a continuous mapping from X into Y and g be a continuous mapping from Y into Z. Then g · f is a continuous mapping from X into Z.

(3) Let A, B be subsets of the carrier of X. If A is open and B is open and A ∩ B = ∅_{X}, then A and B are separated.

(4) Let A, B_{1}, B_{2} be subsets of the carrier of X. Suppose B_{1} is open and B_{2} is open and B_{1} ∩ A ≠ ∅ and B_{2} ∩ A ≠ ∅ and A ⊆ B_{1} ∪ B_{2} and B_{1} ∩ B_{2} = ∅. Then A is not connected.

(5) Let f be a continuous mapping from X into Y and A be a subset of the carrier of X. If A is connected and A ≠ ∅, then f^{∘}A is connected.

(6) For all r_{1}, r_{2} such that r_{1} r_{2} holds Ω_{[(r_{1}), r_{2}]_{T}} is connected.

(7) For every subset A of the carrier of ℝ^{1} and for every a such that A = {r : a < r} holds A is open.

(8) For every subset A of the carrier of ℝ^{1} and for every a such that A = {r : a > r} holds A is open.

(9) Let A be a subset of the carrier of ℝ^{1} and given a. Suppose a ∉ A and there exist b, d such that b ∈ A and d ∈ A and b < a and a < d. Then A is not connected.

(10) Let X be a non empty topological space, x_{1}, x_{2} be points of X, a, b, d be real numbers, and f be a continuous mapping from X into ℝ^{1}. Suppose X is connected and f(x_{1}) = a and f(x_{2}) = b and a d and d b. Then there exists a point x_{3} of X such that f(x_{3}) = d.

(11) Let X be a non empty topological space, x_{1}, x_{2} be points of X, B be a subset of the carrier of X, a, b, d be real numbers, and f be a continuous mapping from X into ℝ^{1}. Suppose B is connected and f(x_{1}) = a and f(x_{2}) = b and a d and d b and x_{1} ∈ B and x_{2} ∈ B. Then there exists a point x_{3} of X such that x_{3} ∈ B and f(x_{3}) = d.

(12) Let given r_{1}, r_{2}, a, b. Suppose r_{1} < r_{2}. Let f be a continuous mapping from [(r_{1}), r_{2}]_{T} into ℝ^{1} and given d. Suppose f(r_{1}) = a and f(r_{2}) = b and a < d and d < b. Then there exists r_{3} such that f(r_{3}) = d and r_{1} < r_{3} and r_{3} < r_{2}.

(13) Let given r_{1}, r_{2}, a, b. Suppose r_{1} < r_{2}. Let f be a continuous mapping from [(r_{1}), r_{2}]_{T} into ℝ^{1} and given d. Suppose f(r_{1}) = a and f(r_{2}) = b and a > d and d > b. Then there exists r_{3} such that f(r_{3}) = d and r_{1} < r_{3} and r_{3} < r_{2}.

(14) Let r_{1}, r_{2} be real numbers, g be a continuous mapping from [(r_{1}), r_{2}]_{T} into ℝ^{1}, and given s_{1}, s_{2}. Suppose r_{1} < r_{2} and s_{1} · s_{2} < 0 and s_{1} = g(r_{1}) and s_{2} = g(r_{2}). Then there exists r_{4} such that g(r_{4}) = 0 and r_{1} < r_{4} and r_{4} < r_{2}.

(15) Let g be a map from 𝕀 into ℝ^{1} and given s_{1}, s_{2}. Suppose g is continuous and g(0) ≠ g(1) and s_{1} = g(0) and s_{2} = g(1). Then there exists r_{4} such that 0 < r_{4} and r_{4} < 1 and g(r_{4}) = \frac{s_{1}+s_{2}}{2} .

2. Simple Closed Curves Are Not Flat

Next we state a number of propositions:

(16) For every map f from E_{T}^{2} into ℝ^{1} such that f = proj1 holds f is continuous.

(17) For every map f from E_{T}^{2} into ℝ^{1} such that f = proj2 holds f is continuous.

(18) Let P be a non empty subset of the carrier of E_{T}^{2} and f be a map from 𝕀 into (E_{T}^{2})↾P. Suppose f is continuous. Then there exists a map g from 𝕀 into ℝ^{1} such that g is continuous and for all r, q such that r ∈ the carrier of 𝕀 and q = f(r) holds q_{1} = g(r).

(19) Let P be a non empty subset of the carrier of E_{T}^{2} and f be a map from 𝕀 into (E_{T}^{2})↾P. Suppose f is continuous. Then there exists a map g from 𝕀 into ℝ^{1} such that g is continuous and for all r, q such that r ∈ the carrier of 𝕀 and q = f(r) holds q_{2} = g(r).

(20) Let P be a non empty subset of the carrier of E_{T}^{2}. Suppose P is simple closed curve. Then it is not true that there exists r such that for every p such that p ∈ P holds p_{2} = r.

(21) Let P be a non empty subset of the carrier of E_{T}^{2}. Suppose P is simple closed curve. Then it is not true that there exists r such that for every p such that p ∈ P holds p_{1} = r.

(22) For every compact non empty subset C of E_{T}^{2} such that C is a simple closed curve holds N-boundC > S-boundC.

(23) For every compact non empty subset C of E_{T}^{2} such that C is a simple closed curve holds E-boundC > W-boundC.

(24) For every compact non empty subset P of E_{T}^{2} such that P is a simple closed curve holds S-minP ≠ N-maxP.

(25) For every compact non empty subset P of E_{T}^{2} such that P is a simple closed curve holds W-minP ≠ E-maxP.

References

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

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

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

[4] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

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

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

514 yatsuka nakamura and andrzej trybulec

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

[8] 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.

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

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

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

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

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

[14] 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.

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

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

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

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

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The Jónson’s Theorem

Jarosław Gryko University of Białystok

MML Identifier: LATTICE5.

The papers [30], [16], [34], [36], [35], [13], [14], [6], [33], [29], [21], [26], [2], [18], [23], [3], [4], [1], [31], [28], [22], [15], [19], [24], [27], [32], [25], [20], [10], [12], [5], [17], [37], [7], [11], [8], [9], and [38] provide the notation and terminology for this paper.

1. Preliminaries

The scheme RecChoice deals with a set A and a ternary predicate P, and states that:

There exists a function f such that domf = ℕ and f(0) = A and for every element n of ℕ holds P[n, f(n), f(n + 1)]

provided the following condition is satisfied:

• For every natural number n and for every set x there exists a set y such that P[n, x, y].

One can prove the following propositions:

(1) For every function f and for every function yielding function F such that f = ⋃ rngF holds domf = ⋃ rng(dom_{κ} F (κ)).

(2) For all non empty sets A, B holds [: ⋃ A, ⋃ B:] = ⋃{[: a, b:]; a ranges over elements of A, b ranges over elements of B: a ∈ A ∧ b ∈ B}.

(3) For every non empty set A such that A is ⊆-linear holds [: ⋃ A, ⋃ A:] = ⋃{[: a, a:]; a ranges over elements of A: a ∈ A}.

2. An equivalence lattice of a set

In the sequel X is a non empty set. Let A be a non empty set. The functor EqRelPoset(A) yielding a poset is

defined as follows: (Def. 1) EqRelPoset(A) = Poset(EqRelLatt(A)).

Let A be a non empty set. One can check that EqRelPoset(A) is non empty and has g.l.b.’s and l.u.b.’s.

One can prove the following propositions:

(4) Let A be a non empty set and x be a set. Then x ∈ the carrier of EqRelPoset(A) if and only if x is an equivalence relation of A.

(5) For every non empty set A and for all elements x, y of the carrier of EqRelLatt(A) holds x ⊑ y iff x ⊆ y.

(6) For every non empty set A and for all elements a, b of EqRelPoset(A) holds a b iff a ⊆ b.

(7) For every lattice L and for all elements a, b of Poset(L) holds a⊓b = ^{·}a⊓^{·}b.

(8) For every non empty set A and for all elements a, b of EqRelPoset(A) holds a ⊓ b = a ∩ b.

(9) For every lattice L and for all elements a, b of Poset(L) holds a⊔b = ^{·}a⊔^{·}b.

(10) Let A be a non empty set, a, b be elements of EqRelPoset(A), and E_{1}, E_{2} be equivalence relations of A. If a = E_{1} and b = E_{2}, then a ⊔ b = E_{1} ⊔ E_{2}.

(11) Let L be a lattice, X be a set, and b be an element of L. Then b X if

and only if b X ∩ the carrier of L. Let L be a non empty relational structure. Let us observe that L is complete

if and only if the condition (Def. 2) is satisfied. (Def. 2) Let X be a subset of L. Then there exists an element a of L such that

a X and for every element b of L such that b X holds b a. Let A be a non empty set. Note that EqRelPoset(A) is complete.

3. A type of a sublattice of equivalence lattice of a set

Let L_{1}, L_{2} be lattices. One can check that there exists a map from L_{1} into L_{2} which is meet-preserving and join-preserving.

Let L_{1}, L_{2} be lattices. A homomorphism from L_{1} to L_{2} is a meet-preserving join-preserving map from L_{1} into L_{2}.

Let L be a lattice. One can check that there exists a relational substructure of L which is meet-inheriting, join-inheriting, and strict.

Let L_{1}, L_{2} be lattices and let f be a homomorphism from L_{1} to L_{2}. Then Imf is a strict full sublattice of L_{2}.

We follow the rules: e, e_{1}, e_{2} denote equivalence relations of X and x, y denote sets.

Let us consider X, let f be a non empty finite sequence of elements of X, let us consider x, y, and let R be a binary relation. We say that x and y are joint by f and R if and only if:

(Def. 3) f(1) = x and f(lenf) = y and for every natural number i such that

1 i and i < lenf holds ⟨⟨f(i), f(i + 1)⟩⟩∈ R. One can prove the following propositions:

(12) Let x be a set, o be a natural number, R be a binary relation, and f be a non empty finite sequence of elements of X. If R is reflexive in X and f = o ↦ x, then x and x are joint by f and R.

(13) Let x, y, z be sets, R be a binary relation, and f, g be non empty finite sequences of elements of X. Suppose R is reflexive in X and x and y are joint by f and R and y and z are joint by g and R. Then there exists a non empty finite sequence h of elements of X such that h = f ^{⌢} g and x and z are joint by h and R.

(14) Let x, y be sets, R be a binary relation, and n, m be natural numbers.

Suppose that

(i) n m, (ii) R is reflexive in X, and (iii) there exists a non empty finite sequence f of elements of X such that

lenf = n and x and y are joint by f and R. Then there exists a non empty finite sequence h of elements of X such that lenh = m and x and y are joint by h and R.

Let us consider X and let Y be a sublattice of EqRelPoset(X). Let us assume that there exists e such that e ∈ the carrier of Y e ≠ id_{X}. And let us assume that there exists a natural number o such that for all e_{1}, e_{2}, x, y such that e_{1} ∈ the carrier of Y and e_{2} ∈ the carrier of Y and ⟨⟨x, y⟩⟩∈ e_{1} ⊔ e_{2} there exists a non empty finite sequence F of elements of X such that lenF = o and x and y are joint by F and e_{1} ∪ e_{2}. The type of Y is a natural number and is defined by the conditions (Def. 4).

(Def. 4)(i) For all e_{1}, e_{2}, x, y such that e_{1} ∈ the carrier of Y and e_{2} ∈ the carrier of Y and ⟨⟨x, y⟩⟩ ∈ e_{1} ⊔ e_{2} there exists a non empty finite sequence F of elements of X such that lenF = (the type of Y) + 2 and x and y are joint by F and e_{1} ∪ e_{2}, and

(ii) there exist e_{1}, e_{2}, x, y such that e_{1} ∈ the carrier of Y and e_{2} ∈ the carrier of Y and ⟨⟨x, y⟩⟩∈ e_{1} ⊔ e_{2} and it is not true that there exists a non empty finite sequence F of elements of X such that lenF = (the type of Y) + 1 and x and y are joint by F and e_{1} ∪ e_{2}.

One can prove the following proposition

(15) Let Y be a sublattice of EqRelPoset(X) and n be a natural number.

Suppose that

(i) there exists e such that e ∈ the carrier of Y and e ≠ id_{X}, and (ii) for all e_{1}, e_{2}, x, y such that e_{1} ∈ the carrier of Y and e_{2} ∈ the carrier

of Y and ⟨⟨x, y⟩⟩ ∈ e_{1} ⊔ e_{2} there exists a non empty finite sequence F of elements of X such that lenF = n + 2 and x and y are joint by F and e_{1} ∪ e_{2}. Then the type of Y n.

4. A meet-representation of a lattice

In the sequel A is a non empty set and L is a lower-bounded lattice.

Let us consider A, L. (Def. 5) A function from [: A, A:] into the carrier of L is said to be a bifunction

from A into L. Let us consider A, L, let f be a bifunction from A into L, and let x, y be

elements of A. Then f(x, y) is an element of L.

Let us consider A, L and let f be a bifunction from A into L. We say that f is symmetric if and only if:

(Def. 6) For all elements x, y of A holds f(x, y) = f(y, x).

We say that f is zeroed if and only if: (Def. 7) For every element x of A holds f(x, x) = ⊥_{L}.

We say that f satisfies triangle inequality if and only if: (Def. 8) For all elements x, y, z of A holds f(x, y) ⊔ f(y, z) f(x, z).

Let us consider A, L. Observe that there exists a bifunction from A into L which is symmetric and zeroed and satisfies triangle inequality.

Let us consider A, L. A distance function of A, L is a symmetric zeroed bifunction from A into L satisfying triangle inequality.

Let us consider A, L and let d be a distance function of A, L. The functor α(d) yielding a map from L into EqRelPoset(A) is defined by the condition (Def. 9).

(Def. 9) Let e be an element of L. Then there exists an equivalence relation E of A such that E = (α(d))(e) and for all elements x, y of A holds ⟨⟨x, y⟩⟩∈ E iff d(x, y) e.

The following two propositions are true:

(16) For every distance function d of A, L holds α(d) is meet-preserving.

(17) For every distance function d of A, L such that d is onto holds α(d) is one-to-one.

5. Jónson’s theorem

Let A be a set. The functor A^{∗} is defined as follows: (Def. 10) A^{∗} = A ∪ {{A}, {{A}}, {{{A}}}}.

Let A be a set. One can verify that A^{∗} is non empty. Let us consider A, L, let d be a bifunction from A into L, and let q be an

element of [: A, A, the carrier of L, the carrier of L:]. The functor d_{q}^{∗} yields a bifunction from A^{∗} into L and is defined by the conditions (Def. 11).

(Def. 11)(i) For all elements u, v of A holds d_{q}^{∗}(u, v) = d(u, v), (ii) d_{q}^{∗}({A}, {A}) = ⊥_{L}, (iii) d_{q}^{∗}({{A}}, {{A}}) = ⊥_{L}, (iv) d_{q}^{∗}({{{A}}}, {{{A}}}) = ⊥_{L},

(v) d_{q}^{∗}({{A}}, {{{A}}}) = q_{3}, (vi) d_{q}^{∗}({{{A}}}, {{A}}) = q_{3}, (vii) d_{q}^{∗}({A}, {{A}}) = q_{4}, (viii) d_{q}^{∗}({{A}}, {A}) = q_{4}, (ix) d_{q}^{∗}({A}, {{{A}}}) = q_{3} ⊔ q_{4},

(x) d_{q}^{∗}({{{A}}}, {A}) = q_{3} ⊔ q_{4}, and (xi) for every element u of A holds d_{q}^{∗}(u, {A}) = d(u, q_{1}) ⊔ q_{3} and d_{q}^{∗}({A},

u) = d(u, q_{1}) ⊔ q_{3} and d_{q}^{∗}(u, {{A}}) = d(u, q_{1}) ⊔ q_{3} ⊔ q_{4} and d_{q}^{∗}({{A}},

u) = d(u, q_{1})⊔q_{3} ⊔q_{4} and d_{q}^{∗}(u, {{{A}}}) = d(u, q_{2})⊔q_{4} and d_{q}^{∗}({{{A}}},

u) = d(u, q_{2}) ⊔ q_{4}. Next we state several propositions:

(18) Let d be a bifunction from A into L. Suppose d is zeroed. Let q be an element of [: A, A, the carrier of L, the carrier of L:]. Then d_{q}^{∗} is zeroed.

(19) Let d be a bifunction from A into L. Suppose d is symmetric. Let q be an element of [: A, A, the carrier of L, the carrier of L:]. Then d_{q}^{∗} is symmetric.

(20) Let d be a bifunction from A into L. Suppose d is symmetric and satisfies triangle inequality. Let q be an element of [: A, A, the carrier of L, the carrier of L:]. If d(q_{1}, q_{2}) q_{3} ⊔ q_{4}, then d_{q}^{∗} satisfies triangle inequality.

(21) For every set A holds A ⊆ A^{∗}.

(22) Let d be a bifunction from A into L and q be an element of [: A, A, the

carrier of L, the carrier of L:]. Then d ⊆ d_{q}^{∗}. Let us consider A, L and let d be a bifunction from A into L. The functor

DistEsti(d) yields a cardinal number and is defined as follows: (Def. 12) DistEsti(d) ≈ {⟨⟨x, y, a, b⟩⟩; x ranges over elements of A, y ranges over

elements of A, a ranges over elements of L, b ranges over elements of L: d(x, y) a ⊔ b}.

We now state the proposition

(23) For every distance function d of A, L holds DistEsti(d) ≠ ∅. In the sequel T denotes a transfinite sequence and O, O_{1}, O_{2} denote ordinal

numbers.

Let us consider A and let us consider O. The functor ConsecutiveSet(A, O) is defined by the condition (Def. 13).

(Def. 13) There exists a transfinite sequence L_{0} such that (i) ConsecutiveSet(A, O) = lastL_{0}, (ii) domL_{0} = succO, (iii) L_{0}(∅) = A, (iv) for every ordinal number C and for every set z such that succC ∈ succO

and z = L_{0}(C) holds L_{0}(succC) = z^{∗}, and

(v) for every ordinal number C and for every transfinite sequence L_{1} such that C ∈ succO and C ≠ ∅ and C is a limit ordinal number and L_{1} = L_{0}↾C holds L_{0}(C) = ⋃ rngL_{1}.

We now state three propositions:

(24) ConsecutiveSet(A, ∅) = A.

(25) ConsecutiveSet(A, succO) = (ConsecutiveSet(A, O))^{∗}.

(26) Suppose O ≠ ∅ and O is a limit ordinal number and domT = O and for every ordinal number O_{1} such that O_{1} ∈ O holds T (O_{1}) = ConsecutiveSet(A, O_{1}). Then ConsecutiveSet(A, O) = ⋃ rngT.

Let us consider A and let us consider O. Note that ConsecutiveSet(A, O) is non empty.

One can prove the following proposition

(27) A ⊆ ConsecutiveSet(A, O).

Let us consider A, L and let d be a bifunction from A into L. A transfinite sequence of elements of [: A, A, the carrier of L, the carrier of L:] is said to be a sequence of quadruples of d if it satisfies the conditions (Def. 14).

(Def. 14)(i) domit is a cardinal number, (ii) it is one-to-one, and (iii) rngit = {⟨⟨x, y, a, b⟩⟩; x ranges over elements of A, y ranges over elements

of A, a ranges over elements of L, b ranges over elements of L: d(x, y) a ⊔ b}.

Let us consider A, L, let d be a bifunction from A into L, let q be a sequence of quadruples of d, and let us consider O. Let us assume that O ∈ domq. The functor Quadr(q, O) yielding an element of [: ConsecutiveSet(A, O), ConsecutiveSet(A, O), the carrier of L, the carrier of L:] is defined as follows:

(Def. 15) Quadr(q, O) = q(O).

One can prove the following proposition (28) Let d be a bifunction from A into L and q be a sequence of quadruples

of d. Then O ∈ DistEsti(d) if and only if O ∈ domq. Let us consider A, L and let z be a set. Let us assume that z is a bifunction

from A into L. The functor BiFun(z, A, L) yields a bifunction from A into L and is defined as follows:

(Def. 16) BiFun(z, A, L) = z.

Let us consider A, L, let d be a bifunction from A into L, let q be a sequence of quadruples of d, and let us consider O. The functor ConsecutiveDelta(q, O) is defined by the condition (Def. 17).

(Def. 17) There exists a transfinite sequence L_{0} such that (i) ConsecutiveDelta(q, O) = lastL_{0}, (ii) domL_{0} = succO, (iii) L_{0}(∅) = d, (iv) for every ordinal number C and for every set z such

that succC ∈ succO and z = L_{0}(C) holds L_{0}(succC) = (BiFun(z, ConsecutiveSet(A, C), L))_{Quadr(q,C)}^{∗}, and

(v) for every ordinal number C and for every transfinite sequence L_{1} such that C ∈ succO and C ≠ ∅ and C is a limit ordinal number and L_{1} = L_{0}↾C holds L_{0}(C) = ⋃ rngL_{1}.

Next we state four propositions:

(29) For every bifunction d from A into L and for every sequence q of quadruples of d holds ConsecutiveDelta(q, ∅) = d.

(30) For every bifunction d from A into L and for every sequence q of quadruples of d holds ConsecutiveDelta(q, succO) = (BiFun(ConsecutiveDelta(q, O), ConsecutiveSet(A, O), L))_{Quadr(q,O)}^{∗}.

(31) Let d be a bifunction from A into L and q be a sequence of quadruples of d. Suppose O ≠ ∅ and O is a limit ordinal number and domT = O and for every ordinal number O_{1} such that O_{1} ∈ O holds T (O_{1}) = ConsecutiveDelta(q, O_{1}). Then ConsecutiveDelta(q, O) = ⋃ rngT.

(32) If O_{1} ⊆ O_{2}, then ConsecutiveSet(A, O_{1}) ⊆ ConsecutiveSet(A, O_{2}). Let O be a non empty ordinal number. Note that every element of O is

ordinal-like.

Next we state the proposition

(33) Let d be a bifunction from A into L and q be a sequence of quadruples of d. Then ConsecutiveDelta(q, O) is a bifunction from ConsecutiveSet(A, O) into L.

Let us consider A, L, let d be a bifunction from A into L, let q be a sequence of quadruples of d, and let us consider O. Then ConsecutiveDelta(q, O) is a bifunction from ConsecutiveSet(A, O) into L.

Next we state several propositions:

(34) For every bifunction d from A into L and for every sequence q of quadruples of d holds d ⊆ ConsecutiveDelta(q, O).

(35) For every bifunction d from A into L and for every sequence q of quadruples of d such that O_{1} ⊆ O_{2} holds ConsecutiveDelta(q, O_{1}) ⊆ ConsecutiveDelta(q, O_{2}).

(36) Let d be a bifunction from A into L. Suppose d is zeroed. Let q be a sequence of quadruples of d. Then ConsecutiveDelta(q, O) is zeroed.

(37) Let d be a bifunction from A into L. Suppose d is symmetric. Let q be a sequence of quadruples of d. Then ConsecutiveDelta(q, O) is symmetric.

(38) Let d be a bifunction from A into L. Suppose d is symmetric and satisfies triangle inequality. Let q be a sequence of quadruples of d. If O ⊆ DistEsti(d), then ConsecutiveDelta(q, O) satisfies triangle inequality.

(39) Let d be a distance function of A, L and q be a sequence of quadruples of d. If O ⊆ DistEsti(d), then ConsecutiveDelta(q, O) is a distance function of ConsecutiveSet(A, O), L.

Let us consider A, L and let d be a bifunction from A into L. The functor NextSet(d) is defined as follows:

(Def. 18) NextSet(d) = ConsecutiveSet(A, DistEsti(d)).

Let us consider A, L and let d be a bifunction from A into L. One can check that NextSet(d) is non empty.

Let us consider A, L, let d be a bifunction from A into L, and let q be a sequence of quadruples of d. The functor NextDelta(q) is defined as follows:

(Def. 19) NextDelta(q) = ConsecutiveDelta(q, DistEsti(d)).

Let us consider A, L, let d be a distance function of A, L, and let q be a sequence of quadruples of d. Then NextDelta(q) is a distance function of NextSet(d), L.

Let us consider A, L, let d be a distance function of A, L, let A_{1} be a non empty set, and let d_{1} be a distance function of A_{1}, L. We say that (A_{1}, d_{1}) is extension of (A, d) if and only if:

(Def. 20) There exists a sequence q of quadruples of d such that A_{1} = NextSet(d)

and d_{1} = NextDelta(q). The following proposition is true

(40) Let d be a distance function of A, L, A_{1} be a non empty set, and d_{1} be a distance function of A_{1}, L. Suppose (A_{1}, d_{1}) is extension of (A, d). Let x, y be elements of A and a, b be elements of L. Suppose d(x, y) a ⊔ b. Then there exist elements z_{1}, z_{2}, z_{3} of A_{1} such that d_{1}(x, z_{1}) = a and d_{1}(z_{2}, z_{3}) = a and d_{1}(z_{1}, z_{2}) = b and d_{1}(z_{3}, y) = b.

Let us consider A, L and let d be a distance function of A, L. A function is called an extension sequence of (A, d) if it satisfies the conditions (Def. 21).

(Def. 21)(i) domit = ℕ,

the jónson’s theorem 523

(ii) it(0) = ⟨⟨A, d⟩⟩, and (iii) for every natural number n there exists a non empty set A′ and there

exists a distance function d′ of A′, L and there exists a non empty set A_{1} and there exists a distance function d_{1} of A_{1}, L such that (A_{1}, d_{1}) is extension of (A′, d′) and it(n) = ⟨⟨A′, d′⟩⟩ and it(n + 1) = ⟨⟨A_{1}, d_{1}⟩⟩.

Next we state two propositions:

(41) Let d be a distance function of A, L, S be an extension sequence of (A, d), and k, l be natural numbers. If k l, then S(k)_{1} ⊆ S(l)_{1}.

(42) Let d be a distance function of A, L, S be an extension sequence of (A, d), and k, l be natural numbers. If k l, then S(k)_{2} ⊆ S(l)_{2}.

Let us consider L. The functor δ_{0}(L) yields a distance function of the carrier of L, L and is defined by:

(Def. 22) For all elements x, y of the carrier of L holds if x ≠ y, then (δ_{0}(L))(x, y) = x ⊔ y and if x = y, then (δ_{0}(L))(x, y) = ⊥_{L}.

We now state two propositions:

(43) δ_{0}(L) is onto.

(44) There exists a non empty set A and there exists a homomorphism f from L to EqRelPoset(A) such that f is one-to-one and the type of Imf 3.

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

[25] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

[27] Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.

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

[29] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

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

[32] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

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[37] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

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Lebesgue’s Covering Lemma, Uniform Continuity and Segmentation of Arcs

Yatsuka Nakamura Shinshu University Nagano

Andrzej Trybulec University of Białystok

Summary. For mappings from a metric space to a metric space, a notion of uniform continuity is defined. If we introduce natural topologies to the metric spaces, a uniformly continuous function becomes continuous. On the other hand, if the domain is compact, a continuous function is uniformly continuous. For this proof, Lebesgue’s covering lemma is also proved. An arc, which is homeomorphic to [0,1], can be devided into small segments, as small as one wishes.

MML Identifier: UNIFORM1.

The notation and terminology used in this paper have been introduced in the following articles: [35], [41], [40], [34], [28], [23], [1], [43], [38], [27], [39], [31], [11], [33], [10], [30], [26], [42], [2], [7], [8], [4], [19], [20], [18], [29], [15], [9], [14], [36], [17], [21], [16], [6], [22], [13], [24], [3], [5], [32], [12], [25], and [37].

1. Lebesgue’s Covering Lemma

We adopt the following rules: s, s_{1}, s_{2}, t, r, r_{1}, r_{2} are real numbers, n, m are natural numbers, and X, Y are non empty metric spaces.

The following two propositions are true:

(1) t − r − (t − s) = − r + s and t − r − (t − s) = s − r.

(2) For every r such that r > 0 there exists a natural number n such that n > 0 and \frac{1}{n} < r.

Let X, Y be non empty metric structures and let f be a map from X into Y. We say that f is uniformly continuous if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let given r. Suppose 0 < r. Then there exists s such that 0 < s and for all elements u_{1}, u_{2} of the carrier of X such that ρ(u_{1}, u_{2}) < s holds ρ(f_{u_{1}}, f_{u_{2}}) < r.

Next we state several propositions:

(3) Let X be a non empty topological space, M be a metric space, and f be a map from X into M_{top}. Suppose f is continuous. Let r be a real number, u be an element of the carrier of M, and P be a subset of the carrier of M_{top}. If P = Ball(u, r), then f^{−1} (P) is open.

(4) Let N, M be metric spaces and f be a map from N_{top} into M_{top}. Suppose that for every real number r and for every element u of the carrier of N and for every element u_{1} of the carrier of M such that r > 0 and u_{1} = f(u) there exists s such that s > 0 and for every element w of the carrier of N and for every element w_{1} of the carrier of M such that w_{1} = f(w) and ρ(u, w) < s holds ρ(u_{1}, w_{1}) < r. Then f is continuous.

(5) Let N be a metric space, M be a non empty metric space, and f be a map from N_{top} into M_{top}. Suppose f is continuous. Let r be a real number, u be an element of the carrier of N, and u_{1} be an element of the carrier of M. Suppose r > 0 and u_{1} = f(u). Then there exists s such that

(i) s > 0, and (ii) for every element w of the carrier of N and for every element w_{1} of the

carrier of M such that w_{1} = f(w) and ρ(u, w) < s holds ρ(u_{1}, w_{1}) < r.

(6) Let N, M be non empty metric spaces, f be a map from N into M, and g be a map from N_{top} into M_{top}. If f = g and f is uniformly continuous, then g is continuous.

(7) Let N be a non empty metric space and G be a family of subsets of N_{top}. Suppose G is a cover of N_{top} and open and N_{top} is compact. Then there exists r such that

(i) r > 0, and (ii) for all elements w_{1}, w_{2} of the carrier of N such that ρ(w_{1}, w_{2}) < r there

exists a subset G_{1} of the carrier of N_{top} such that w_{1} ∈ G_{1} and w_{2} ∈ G_{1} and G_{1} ∈ G.

2. Uniformity of Continuous Functions on Compact Spaces

Next we state three propositions:

(8) Let N, M be non empty metric spaces, f be a map from N into M, and g be a map from N_{top} into M_{top}. Suppose g = f and N_{top} is compact and g is continuous. Then f is uniformly continuous.

(9) Let g be a map from 𝕀 into E_{T}^{n} and f be a map from [0, 1]_{M} into E^{n}. If g is continuous and f = g, then f is uniformly continuous.

(10) Let P be a subset of the carrier of E_{T}^{n}, Q be a non empty subset of the carrier of E^{n}, g be a map from 𝕀 into (E_{T}^{n})↾P, and f be a map from [0, 1]_{M} into E^{n}↾Q. If P = Q and g is continuous and f = g, then f is uniformly continuous.

3. Segmentation of Arcs

We now state four propositions:

(11) For every map g from 𝕀 into E_{T}^{n} there exists a map f from [0, 1]_{M} into E^{n} such that f = g.

(12) For every r such that r 0 holds ⌈r⌉ 0 and ⌊r⌋ 0 and ⌈r⌉ is a natural number and ⌊r⌋ is a natural number.

(13) For all r, s holds |r − s| = |s − r|.

(14) For all r_{1}, r_{2}, s_{1}, s_{2} such that r_{1} ∈ [s_{1}, s_{2}] and r_{2} ∈ [s_{1}, s_{2}] holds |r_{1}−r_{2}|

s_{2} − s_{1}. Let I_{1} be a finite sequence of elements of ℝ. We say that I_{1} is decreasing if

and only if: (Def. 2) For all n, m such that n ∈ domI_{1} and m ∈ domI_{1} and n < m holds

I_{1}(n) > I_{1}(m). We now state the proposition

(15) Let e be a real number, g be a map from 𝕀 into E_{T}^{n}, and p_{1}, p_{2} be elements of E_{T}^{n}. Suppose e > 0 and g is continuous and one-to-one and g(0) = p_{1} and g(1) = p_{2}. Then there exists a finite sequence h of elements of ℝ such that

(i) h(1) = 1, (ii) h(lenh) = 0, (iii) 5 lenh, (iv) rngh ⊆ the carrier of 𝕀,

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

and for every subset W of the carrier of E^{n} such that 1 i and i < lenh and Q = [π_{i+1}h, π_{i}h] and W = g^{∘}Q holds ØW < e.

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530 yatsuka nakamura and andrzej trybulec

On the Rectangular Finite Sequences of the Points of the Plane

Andrzej Trybulec University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. The article deals with a rather technical concept – rectangular sequences of the points of the plane. We mean by that a finite sequence consisting of five elements, that is circular, i.e. the first element and the fifth one of it are equal, and such that the polygon determined by it is a non degenerated rectangle, with sides parallel to axes. The main result is that for the rectangle determined by such a sequence the left and the right component of the complement of it are different and disjoint.

MML Identifier: SPRECT 1.

The terminology and notation used in this paper are introduced in the following papers: [29], [35], [34], [28], [7], [36], [13], [2], [25], [1], [27], [32], [5], [6], [3], [33], [31], [17], [16], [14], [15], [4], [26], [24], [37], [10], [23], [11], [12], [21], [18], [19], [22], [30], [20], [8], and [9].

1. General preliminaries

One can prove the following proposition

(1) For every trivial set A and for every set B such that B ⊆ A holds B is trivial.

One can verify that every function which is non constant is also non trivial.

Let us observe that every function which is trivial is also constant.

One can prove the following proposition

(2) For every function f such that rngf is trivial holds f is constant.

Let f be a constant function. One can verify that rngf is trivial. Let us observe that there exists a finite sequence which is non empty and

constant.

We now state three propositions:

(3) For all finite sequences f, g such that f^{⌢}g is constant holds f is constant and g is constant.

(4) For all sets x, y such that ⟨x, y⟩ is constant holds x = y.

(5) For all sets x, y, z such that ⟨x, y, z⟩ is constant holds x = y and y = z and z = x.

2. Preliminaries (general topology)

One can prove the following four propositions:

(6) Let G_{1} be a non empty topological space, A be a subset of the carrier of G_{1}, and B be a non empty subset of the carrier of G_{1}. If A is a component of B, then A ≠ ∅.

(7) Let G_{1} be a non empty topological space, A be a subset of the carrier of G_{1}, and B be a non empty subset of the carrier of G_{1}. If A is a component of B, then A ⊆ B.

(8) Let T be a non empty topological space, A be a non empty subset of the carrier of T, and B_{1}, B_{2}, C be subsets of the carrier of T. Suppose B_{1} is a component of A and B_{2} is a component of A and C is a component of A and B_{1} ∪ B_{2} = A. Then C = B_{1} or C = B_{2}.

(9) Let T be a non empty topological space, A be a non empty subset of the carrier of T, and B_{1}, B_{2}, C_{1}, C_{2} be subsets of the carrier of T. Suppose B_{1} is a component of A and B_{2} is a component of A and C_{1} is a component of A and C_{2} is a component of A and B_{1} ∪B_{2} = A and C_{1} ∪C_{2} = A. Then {B_{1}, B_{2}} = {C_{1}, C_{2}}.

3. Preliminaries (the topology of the plane)

We follow the rules: C, C_{1}, C_{2} are non empty compact subsets of E_{T}^{2} and p, q are points of E_{T}^{2}.

Next we state the proposition

(10) For all points p, q, r of E_{T}^{2} holds L̃(⟨p, q, r⟩) = L(p, q) ∪ L(q, r). Let n be a natural number and let f be a non trivial finite sequence of

elements of E_{T}^{n}. Observe that L̃(f) is non empty.

Let f be a finite sequence of elements of E_{T}^{2}. Note that L̃(f) is compact. We now state two propositions:

(11) For all subsets A, B of the carrier of E_{T}^{2} such that A ⊆ B and B is horizontal holds A is horizontal.

(12) For all subsets A, B of the carrier of E_{T}^{2} such that A ⊆ B and B is

vertical holds A is vertical. Let us observe that □_{E^{2}} is special polygonal. One can check that □_{E^{2}} is non horizontal and non vertical. One can check that there exists a subset of E_{T}^{2} which is non vertical, non

horizontal, non empty, and compact.

4. Special points of a compact non empty subset of the plane

The following propositions are true:

(13) N-minC ∈ C and N-maxC ∈ C.

(14) S-minC ∈ C and S-maxC ∈ C.

(15) W-minC ∈ C and W-maxC ∈ C.

(16) E-minC ∈ C and E-maxC ∈ C.

(17) C is vertical iff W-boundC = E-boundC.

(18) C is horizontal iff S-boundC = N-boundC.

(19) For every C such that NW-cornerC = NE-cornerC holds C is vertical.

(20) For every C such that SW-cornerC = SE-cornerC holds C is vertical.

(21) For every C such that NW-cornerC = SW-cornerC holds C is horizontal.

(22) For every C such that NE-cornerC = SE-cornerC holds C is horizontal. In the sequel t, r_{1}, r_{2}, s_{1}, s_{2} are real numbers. The following propositions are true:

(23) W-boundC E-boundC.

(24) S-boundC N-boundC.

(25) L(SE-cornerC, NE-cornerC) = {p : p_{1} = E-boundC ∧ p_{2} N-boundC ∧ p_{2} S-boundC}.

(26) L(SW-cornerC, SE-cornerC) = {p : p_{1} E-boundC ∧ p_{1} W-boundC ∧ p_{2} = S-boundC}.

(27) L(NW-cornerC, NE-cornerC) = {p : p_{1} E-boundC ∧ p_{1} W-boundC ∧ p_{2} = N-boundC}.

(28) L(SW-cornerC, NW-cornerC) = {p : p_{1} = W-boundC ∧ p_{2} N-boundC ∧ p_{2} S-boundC}.

(29) L(SW-cornerC, NW-cornerC) ∩ L(NW-cornerC, NE-cornerC) = {NW-cornerC}.

(30) L(NW-cornerC, NE-cornerC) ∩ L(NE-cornerC, SE-cornerC) = {NE-cornerC}.

(31) L(SE-cornerC, NE-cornerC) ∩ L(SW-cornerC, SE-cornerC) = {SE-cornerC}.

(32) L(NW-cornerC, SW-cornerC) ∩ L(SW-cornerC, SE-cornerC) = {SW-cornerC}.

5. Subsets of the plane that are neither vertical nor horizontal

In the sequel D is a non vertical non horizontal non empty compact subset of E_{T}^{2}.

The following propositions are true:

(33) W-boundD < E-boundD.

(34) S-boundD < N-boundD.

(35) L(SW-cornerD, NW-cornerD) ∩ L(SE-cornerD, NE-cornerD) = ∅.

(36) L(SW-cornerD, SE-cornerD) ∩ L(NW-cornerD, NE-cornerD) = ∅.

6. A special sequence related to a compact non empty subset of the plane

Let us consider C. The functor SpStSeqC yielding a finite sequence of elements of E_{T}^{2} is defined as follows:

(Def. 1) SpStSeqC = ⟨NW-cornerC, NE-cornerC, SE-cornerC⟩^{⌢} ⟨SW-cornerC,

NW-cornerC⟩. The following propositions are true:

(37) π_{1} SpStSeqC = NW-cornerC.

(38) π_{2} SpStSeqC = NE-cornerC.

(39) π_{3} SpStSeqC = SE-cornerC.

(40) π_{4} SpStSeqC = SW-cornerC.

(41) π_{5} SpStSeqC = NW-cornerC.

(42) lenSpStSeqC = 5.

(43) L̃(SpStSeqC) = L(NW-cornerC, NE-cornerC) ∪ L(NE-cornerC, SE-cornerC) ∪ (L(SE-cornerC, SW-cornerC) ∪ L(SW-cornerC, NW-cornerC)).

Let D be a non vertical non empty compact subset of E_{T}^{2}. Note that SpStSeqD is non constant.

Let D be a non horizontal non empty compact subset of E_{T}^{2}. Note that SpStSeqD is non constant.

Let us consider D. One can check that SpStSeqD is special unfolded circular s.c.c. and standard.

Next we state four propositions:

(44) L̃(SpStSeqD) = [. W-boundD, E-boundD, S-boundD, N-boundD.].

(45) Let T be a non empty topological space, X be a non empty subset of T, and f be a real map of T. Then rng(f ↾ X) = f^{∘}X.

(46) Let T be a non empty topological space, X be a non empty compact subset of T, and f be a continuous real map of T. Then f^{∘}X is lower bounded.

(47) Let T be a non empty topological space, X be a non empty compact subset of T, and f be a continuous real map of T. Then f^{∘}X is upper bounded.

Let us observe that there exists a subset of ℝ which is non empty, upper bounded, and lower bounded.

We now state a number of propositions:

(48) W-boundC = inf((proj1)^{∘}C).

(49) S-boundC = inf((proj2)^{∘}C).

(50) N-boundC = sup((proj2)^{∘}C).

(51) E-boundC = sup((proj1)^{∘}C).

(52) For all non empty lower bounded subsets A, B of ℝ holds inf(A ∪ B) = min(inf A, inf B).

(53) For all non empty upper bounded subsets A, B of ℝ holds sup(A ∪ B) = max(supA, supB).

(54) If C = C_{1} ∪ C_{2}, then W-boundC = min(W-boundC_{1}, W-boundC_{2}).

(55) If C = C_{1} ∪ C_{2}, then S-boundC = min(S-boundC_{1}, S-boundC_{2}).

(56) If C = C_{1} ∪ C_{2}, then N-boundC = max(N-boundC_{1}, N-boundC_{2}).

(57) If C = C_{1} ∪ C_{2}, then E-boundC = max(E-boundC_{1}, E-boundC_{2}). Let us consider p, q. One can check that L(p, q) is compact. One can verify that ∅_{ℝ} is bounded. Next we state the proposition

(58) s_{1} ∈ [r_{1}, r_{2}] iff r_{1} s_{1} and s_{1} r_{2}.

Let us consider r_{1}, r_{2}. One can check that [r_{1}, r_{2}] is bounded.

Let us observe that every subset of ℝ which is bounded is also lower bounded and upper bounded and every subset of ℝ which is lower bounded and upper bounded is also bounded.

The following propositions are true:

(59) If r_{1} r_{2}, then t ∈ [r_{1}, r_{2}] iff there exists s_{1} such that 0 s_{1} and s_{1} 1 and t = s_{1} · r_{1} + (1 − s_{1}) · r_{2}.

(60) If p_{1} q_{1}, then (proj1)^{∘}L(p, q) = [p_{1}, q_{1}].

(61) If p_{2} q_{2}, then (proj2)^{∘}L(p, q) = [p_{2}, q_{2}].

(62) If p_{1} q_{1}, then W-boundL(p, q) = p_{1}.

(63) If p_{2} q_{2}, then S-boundL(p, q) = p_{2}.

(64) If p_{2} q_{2}, then N-boundL(p, q) = q_{2}.

(65) If p_{1} q_{1}, then E-boundL(p, q) = q_{1}.

(66) W-bound L̃(SpStSeqD) = W-boundD.

(67) S-bound L̃(SpStSeqD) = S-boundD.

(68) N-bound L̃(SpStSeqD) = N-boundD.

(69) E-bound L̃(SpStSeqD) = E-boundD.

(70) NW-corner L̃(SpStSeqD) = NW-cornerD.

(71) NE-corner L̃(SpStSeqD) = NE-cornerD.

(72) SW-corner L̃(SpStSeqD) = SW-cornerD.

(73) SE-corner L̃(SpStSeqD) = SE-cornerD.

(74) W-most L̃(SpStSeqD) = L(SW-cornerD, NW-cornerD).

(75) N-most L̃(SpStSeqD) = L(NW-cornerD, NE-cornerD).

(76) S-most L̃(SpStSeqD) = L(SW-cornerD, SE-cornerD).

(77) E-most L̃(SpStSeqD) = L(SE-cornerD, NE-cornerD).

(78) (proj2)^{∘}L(SW-cornerD, NW-cornerD) = [S-boundD, N-boundD].

(79) (proj1)^{∘}L(NW-cornerD, NE-cornerD) = [W-boundD, E-boundD].

(80) (proj2)^{∘}L(NE-cornerD, SE-cornerD) = [S-boundD, N-boundD].

(81) (proj1)^{∘}L(SE-cornerD, SW-cornerD) = [W-boundD, E-boundD].

(82) W-min L̃(SpStSeqD) = SW-cornerD.

(83) W-max L̃(SpStSeqD) = NW-cornerD.

(84) N-min L̃(SpStSeqD) = NW-cornerD.

(85) N-max L̃(SpStSeqD) = NE-cornerD.

(86) E-min L̃(SpStSeqD) = SE-cornerD.

(87) E-max L̃(SpStSeqD) = NE-cornerD.

(88) S-min L̃(SpStSeqD) = SW-cornerD.

(89) S-max L̃(SpStSeqD) = SE-cornerD.

7. Rectangular finite suequences of the points of the plane

Let f be a finite sequence of elements of E_{T}^{2}. We say that f is rectangular if and only if:

(Def. 2) There exists D such that f = SpStSeqD. Let us consider D. Note that SpStSeqD is rectangular. Let us mention that there exists a finite sequence of elements of E_{T}^{2} which is

rectangular.

In the sequel s denotes a rectangular finite sequence of elements of E_{T}^{2}. The following proposition is true

(90) lens = 5.

Let us note that every finite sequence of elements of E_{T}^{2} which is rectangular is also non constant.

One can verify that every non empty finite sequence of elements of E_{T}^{2} which is rectangular is also standard, special, unfolded, circular, and s.c.c..

In the sequel s is a rectangular finite sequence of elements of E_{T}^{2}. Next we state four propositions:

(91) π_{1}s = N-min L̃(s) and π_{1}s = W-max L̃(s).

(92) π_{2}s = N-max L̃(s) and π_{2}s = E-max L̃(s).

(93) π_{3}s = S-max L̃(s) and π_{3}s = E-min L̃(s).

(94) π_{4}s = S-min L̃(s) and π_{4}s = W-min L̃(s).

8. Jordan property

One can prove the following proposition

(95) If r_{1} < r_{2} and s_{1} < s_{2}, then [.r_{1}, r_{2}, s_{1}, s_{2}.] is Jordan.

Let f be a rectangular finite sequence of elements of E_{T}^{2}. Observe that L̃(f) is Jordan.

Let S be a subset of the carrier of E_{T}^{2}. Let us observe that S is Jordan if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) S^{c} ≠ ∅, and

(ii) there exist subsets A_{1}, A_{2} of the carrier of E_{T}^{2} such that S^{c} = A_{1} ∪ A_{2} and A_{1} misses A_{2} and \overline{A_{1}} \ A_{1} = \overline{A_{2}} \ A_{2} and A_{1} is a component of S^{c} and A_{2} is a component of S^{c}.

Next we state the proposition

(96) For every rectangular finite sequence f of elements of E_{T}^{2} holds LeftComp(f) misses RightComp(f).

538 andrzej trybulec and yatsuka nakamura

Let f be a non constant standard special circular sequence. One can verify that LeftComp(f) is non empty and RightComp(f) is non empty.

The following proposition is true

(97) For every rectangular finite sequence f of elements of E_{T}^{2} holds LeftComp(f) ≠ RightComp(f).

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

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

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

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

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

[8] Czesław Byliński and Yatsuka Nakamura. Special polygons. Formalized Mathematics, 5(2):247–252, 1996.

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

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

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

[12] 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.

[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] Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711–717, 1991.

[16] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[17] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

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

[19] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

[20] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons, part I. Formalized Mathematics, 5(1):97–102, 1996.

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

[22] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

[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] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

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

on the rectangular finite sequences of the … 539

[27] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[28] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

[30] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

[31] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5(3):317–322, 1996.

[32] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

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

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

[35] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

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

540 andrzej trybulec and yatsuka nakamura

On the Order on a Special Polygon

Andrzej Trybulec University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. The goal of the article is to determine the order of the special points defined in [10] on a special polygon. We restrict ourselves to the clockwise oriented finite sequences (the concept defined in this article) that start in N-min C (C being a compact non empty subset of the plane).

MML Identifier: SPRECT 2.

The papers [28], [33], [27], [7], [15], [29], [34], [1], [5], [6], [3], [32], [8], [30], [16], [17], [2], [25], [4], [19], [18], [26], [11], [12], [13], [14], [21], [20], [22], [9], [24], [23], [10], and [31] provide the terminology and notation for this paper.

1. Preliminaries

One can prove the following propositions:

(1) For all sets A, B, C, p such that A ∩ B ⊆ {p} and p ∈ C and C misses B holds A ∪ C misses B.

(2) For all sets A, B, C, p such that A ∩ C = {p} and p ∈ B and B ⊆ C holds A ∩ B = {p}.

(3) For all sets A, B such that for every set y such that y ∈ B holds A misses y holds A misses ⋃ B.

(4) For all sets A, B such that for all sets x, y such that x ∈ A and y ∈ B holds x misses y holds ⋃ A misses ⋃ B.

2. On the finite sequences

We adopt the following convention: i, j, k, m, n denote natural numbers, D denotes a non empty set, and f denotes a finite sequence of elements of D.

The following propositions are true:

(5) For all i, j, k such that i j and i ∈ domf and j ∈ domf and k ∈ dommid(f, i, j) holds (k + i) − ′1 ∈ domf.

(6) For all i, j, k such that i > j and i ∈ domf and j ∈ domf and k ∈ dommid(f, i, j) holds i − ′k + 1 ∈ domf.

(7) For all i, j, k such that i j and i ∈ domf and j ∈ domf and k ∈ dommid(f, i, j) holds π_{k} mid(f, i, j) = π_{(k+i)− ′1}f.

(8) For all i, j, k such that i > j and i ∈ domf and j ∈ domf and k ∈ dommid(f, i, j) holds π_{k} mid(f, i, j) = π_{i− ′k+1}f.

(9) If i ∈ domf and j ∈ domf, then lenmid(f, i, j) 1.

(10) If i ∈ domf and j ∈ domf and lenmid(f, i, j) = 1, then i = j.

(11) If i ∈ domf and j ∈ domf, then mid(f, i, j) is non empty.

(12) If i ∈ domf and j ∈ domf, then π_{1} mid(f, i, j) = π_{i}f.

(13) If i ∈ domf and j ∈ domf, then π_{len mid(f,i,j)} mid(f, i, j) = π_{j}f.

3. Compact subsets of the plane

In the sequel X denotes a non empty compact subset of E_{T}^{2}. One can prove the following four propositions:

(14) For every point p of E_{T}^{2} such that p ∈ X and p_{2} = N-boundX holds p ∈ N-mostX.

(15) For every point p of E_{T}^{2} such that p ∈ X and p_{2} = S-boundX holds p ∈ S-mostX.

(16) For every point p of E_{T}^{2} such that p ∈ X and p_{1} = W-boundX holds p ∈ W-mostX.

(17) For every point p of E_{T}^{2} such that p ∈ X and p_{1} = E-boundX holds p ∈ E-mostX.

4. Finite sequences on the plane

We now state several propositions:

(18) For every finite sequence f of elements of E_{T}^{2} such that 1 i and i j and j lenf holds L̃(mid(f, i, j)) = ⋃{L(f, k) : i k ∧ k < j}.

(19) For every finite sequence f of elements of E_{T}^{2} holds domX-coordinate(f) = domf.

(20) For every finite sequence f of elements of E_{T}^{2} holds domY-coordinate(f) = domf.

(21) For all points a, b, c of E_{T}^{2} such that b ∈ L(a, c) and a_{1} b_{1} and c_{1} b_{1} holds a = b or b = c or a_{1} = b_{1} and c_{1} = b_{1}.

(22) For all points a, b, c of E_{T}^{2} such that b ∈ L(a, c) and a_{2} b_{2} and c_{2} b_{2} holds a = b or b = c or a_{2} = b_{2} and c_{2} = b_{2}.

(23) For all points a, b, c of E_{T}^{2} such that b ∈ L(a, c) and a_{1} b_{1} and c_{1} b_{1} holds a = b or b = c or a_{1} = b_{1} and c_{1} = b_{1}.

(24) For all points a, b, c of E_{T}^{2} such that b ∈ L(a, c) and a_{2} b_{2} and c_{2} b_{2} holds a = b or b = c or a_{2} = b_{2} and c_{2} = b_{2}.

5. The area of a sequence

Let f be a non trivial finite sequence of elements of E_{T}^{2} and let g be a finite sequence of elements of E_{T}^{2}. We say that g is in the area of f if and only if:

(Def. 1) For every n such that n ∈ domg holds W-bound L̃(f) (π_{n}g)_{1} and (π_{n}g)_{1} E-bound L̃(f) and S-bound L̃(f) (π_{n}g)_{2} and (π_{n}g)_{2} N-bound L̃(f).

We now state several propositions:

(25) Every non trivial finite sequence f of elements of E_{T}^{2} is in the area of f.

(26) Let f be a non trivial finite sequence of elements of E_{T}^{2} and g be a finite sequence of elements of E_{T}^{2}. Suppose g is in the area of f. Let given i, j. If i ∈ domg and j ∈ domg, then mid(g, i, j) is in the area of f.

(27) Let f be a non trivial finite sequence of elements of E_{T}^{2} and given i, j. If i ∈ domf and j ∈ domf, then mid(f, i, j) is in the area of f.

(28) Let f be a non trivial finite sequence of elements of E_{T}^{2} and g, h be finite sequences of elements of E_{T}^{2}. Suppose g is in the area of f and h is in the area of f. Then g^{⌢} h is in the area of f.

(29) For every non trivial finite sequence f of elements of E_{T}^{2} holds ⟨NE-corner L̃(f)⟩ is in the area of f.

(30) For every non trivial finite sequence f of elements of E_{T}^{2} holds ⟨NW-corner L̃(f)⟩ is in the area of f.

(31) For every non trivial finite sequence f of elements of E_{T}^{2} holds ⟨SE-corner L̃(f)⟩ is in the area of f.

(32) For every non trivial finite sequence f of elements of E_{T}^{2} holds ⟨SW-corner L̃(f)⟩ is in the area of f.

6. Horizontal and vertical connections

Let f be a non trivial finite sequence of elements of E_{T}^{2} and let g be a finite sequence of elements of E_{T}^{2}. We say that g is a h.c. for f if and only if:

(Def. 2) g is in the area of f and (π_{1}g)_{1} = W-bound L̃(f) and (π_{len g}g)_{1} =

E-bound L̃(f). We say that g is a v.c. for f if and only if:

(Def. 3) g is in the area of f and (π_{1}g)_{2} = S-bound L̃(f) and (π_{len g}g)_{2} =

N-bound L̃(f). Next we state the proposition

(33) Let f be a non trivial finite sequence of elements of E_{T}^{2} and g, h be Ssequences in ℝ^{2}. If g is a h.c. for f and h is a v.c. for f, then L̃(g) meets L̃(h).

7. Orientation

Let f be a non trivial finite sequence of elements of E_{T}^{2}. We say that f is clockwise oriented if and only if:

(Def. 4) π_{2}f_{⟲}^{N-min L̃(f)} ∈ N-most L̃(f).

The following proposition is true

(34) Let f be a non constant standard special circular sequence. If π_{1}f =

N-min L̃(f), then f is clockwise oriented iff π_{2}f ∈ N-most L̃(f). Let us note that □_{E^{2}} is compact. We now state several propositions:

(35) N-bound□_{E^{2}} = 1.

(36) W-bound□_{E^{2}} = 0.

(37) E-bound□_{E^{2}} = 1.

(38) S-bound□_{E^{2}} = 0.

(39) N-most□_{E^{2}} = L([0, 1], [1, 1]).

(40) N-min□_{E^{2}} = [0, 1].

Let X be a non vertical non horizontal non empty compact subset of E_{T}^{2}. One can verify that SpStSeqX is clockwise oriented.

One can verify that there exists a non constant standard special circular sequence which is clockwise oriented.

One can prove the following propositions:

(41) Let f be a non constant standard special circular sequence and given i, j. Suppose i > j but 1 < j and i lenf or 1 j and i < lenf. Then mid(f, i, j) is a S-sequence in ℝ^{2}.

(42) Let f be a non constant standard special circular sequence and given i, j. Suppose i < j but 1 < i and j lenf or 1 i and j < lenf. Then mid(f, i, j) is a S-sequence in ℝ^{2}.

In the sequel f is a clockwise oriented non constant standard special circular sequence.

One can prove the following propositions:

(43) N-min L̃(f) ∈ rngf.

(44) N-max L̃(f) ∈ rngf.

(45) S-min L̃(f) ∈ rngf.

(46) S-max L̃(f) ∈ rngf.

(47) W-min L̃(f) ∈ rngf.

(48) W-max L̃(f) ∈ rngf.

(49) E-min L̃(f) ∈ rngf.

(50) E-max L̃(f) ∈ rngf.

(51) If 1 i and i j and j < m and m n and n lenf and 1 < i or n < lenf, then L̃(mid(f, i, j)) misses L̃(mid(f, m, n)).

(52) If 1 i and i j and j < m and m n and n lenf and 1 < i or n < lenf, then L̃(mid(f, i, j)) misses L̃(mid(f, n, m)).

(53) If 1 i and i j and j < m and m n and n lenf and 1 < i or n < lenf, then L̃(mid(f, j, i)) misses L̃(mid(f, n, m)).

(54) If 1 i and i j and j < m and m n and n lenf and 1 < i or n < lenf, then L̃(mid(f, j, i)) misses L̃(mid(f, m, n)).

(55) (N-min L̃(f))_{1} < (N-max L̃(f))_{1}.

(56) N-min L̃(f) ≠ N-max L̃(f).

(57) (E-min L̃(f))_{2} < (E-max L̃(f))_{2}.

(58) E-min L̃(f) ≠ E-max L̃(f).

(59) (S-min L̃(f))_{1} < (S-max L̃(f))_{1}.

(60) S-min L̃(f) ≠ S-max L̃(f).

(61) (W-min L̃(f))_{2} < (W-max L̃(f))_{2}.

(62) W-min L̃(f) ≠ W-max L̃(f).

(63) L(NW-corner L̃(f), N-min L̃(f)) misses L(N-max L̃(f), NE-corner L̃(f)).

(64) Let f be a S-sequence in ℝ^{2} and p be a point of E_{T}^{2}. Suppose p ≠ π_{1}f but p_{1} = (π_{1}f)_{1} or p_{2} = (π_{1}f)_{2} but L(p, π_{1}f) ∩ L̃(f) = {π_{1}f}. Then ⟨p⟩^{⌢} f is a S-sequence in ℝ^{2}.

(65) Let f be a S-sequence in ℝ^{2} and p be a point of E_{T}^{2}. Suppose p ≠ π_{len f}f but p_{1} = (π_{len f}f)_{1} or p_{2} = (π_{len f}f)_{2} but L(p, π_{len f}f) ∩ L̃(f) = {π_{len f}f}. Then f ^{⌢} ⟨p⟩ is a S-sequence in ℝ^{2}.

8. Appending corners

We now state several propositions:

(66) Let given i, j. Suppose i ∈ domf and j ∈ domf and mid(f, i, j) is a Ssequence in ℝ^{2} and π_{j}f = N-max L̃(f) and N-max L̃(f) ≠ NE-corner L̃(f). Then (mid(f, i, j))^{⌢} ⟨NE-corner L̃(f)⟩ is a S-sequence in ℝ^{2}.

(67) Let given i, j. Suppose i ∈ domf and j ∈ domf and mid(f, i, j) is a Ssequence in ℝ^{2} and π_{j}f = E-max L̃(f) and E-max L̃(f) ≠ NE-corner L̃(f). Then (mid(f, i, j))^{⌢} ⟨NE-corner L̃(f)⟩ is a S-sequence in ℝ^{2}.

(68) Let given i, j. Suppose i ∈ domf and j ∈ domf and mid(f, i, j) is a Ssequence in ℝ^{2} and π_{j}f = S-max L̃(f) and S-max L̃(f) ≠ SE-corner L̃(f). Then (mid(f, i, j))^{⌢} ⟨SE-corner L̃(f)⟩ is a S-sequence in ℝ^{2}.

(69) Let given i, j. Suppose i ∈ domf and j ∈ domf and mid(f, i, j) is a Ssequence in ℝ^{2} and π_{j}f = E-max L̃(f) and E-max L̃(f) ≠ NE-corner L̃(f). Then (mid(f, i, j))^{⌢} ⟨NE-corner L̃(f)⟩ is a S-sequence in ℝ^{2}.

(70) Let given i, j. Suppose i ∈ domf and j ∈ domf and mid(f, i, j) is a Ssequence in ℝ^{2} and π_{i}f = N-min L̃(f) and N-min L̃(f) ≠ NW-corner L̃(f). Then ⟨NW-corner L̃(f)⟩^{⌢} mid(f, i, j) is a S-sequence in ℝ^{2}.

(71) Let given i, j. Suppose i ∈ domf and j ∈ domf and mid(f, i, j) is a S-sequence in ℝ^{2} and π_{i}f = W-min L̃(f) and W-min L̃(f) ≠ SW-corner L̃(f). Then ⟨SW-corner L̃(f)⟩ ^{⌢} mid(f, i, j) is a S-sequence in ℝ^{2}.

Let f be a non constant standard special circular sequence. One can check that L̃(f) is simple closed curve. on the order on a special polygon 547

9. The order

We now state a number of propositions:

(72) If π_{1}f = N-min L̃(f), then (N-min L̃(f)) ↫ f < (N-max L̃(f)) ↫ f.

(73) If π_{1}f = N-min L̃(f), then (N-max L̃(f)) ↫ f > 1.

(74) If π_{1}f = N-min L̃(f) and N-max L̃(f) ≠ E-max L̃(f), then (N-max L̃(f)) ↫ f < (E-max L̃(f)) ↫ f.

(75) If π_{1}f = N-min L̃(f), then (E-max L̃(f)) ↫ f < (E-min L̃(f)) ↫ f.

(76) If π_{1}f = N-min L̃(f) and E-min L̃(f) ≠ S-max L̃(f), then (E-min L̃(f)) ↫ f < (S-max L̃(f)) ↫ f.

(77) If π_{1}f = N-min L̃(f), then (S-max L̃(f)) ↫ f < (S-min L̃(f)) ↫ f.

(78) If π_{1}f = N-min L̃(f) and S-min L̃(f) ≠ W-min L̃(f), then (S-min L̃(f)) ↫ f < (W-min L̃(f)) ↫ f.

(79) If π_{1}f = N-min L̃(f) and N-min L̃(f) ≠ W-max L̃(f), then (W-min L̃(f)) ↫ f < (W-max L̃(f)) ↫ f.

(80) If π_{1}f = N-min L̃(f), then (W-min L̃(f)) ↫ f < lenf.

(81) If π_{1}f = N-min L̃(f), then (W-max L̃(f)) ↫ f < lenf.

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

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

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

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

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

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

[9] Czesław Byliński and Yatsuka Nakamura. Special polygons. Formalized Mathematics, 5(2):247–252, 1996.

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

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

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

[13] 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.

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

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

[16] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

548 andrzej trybulec and yatsuka nakamura

[17] Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711–717, 1991.

[18] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[19] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[20] Jarosław Kotowicz and Yatsuka Nakamura. Go-board theorem. Formalized Mathematics, 3(1):125–129, 1992.

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

[22] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons, part I. Formalized Mathematics, 5(1):97–102, 1996.

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

[24] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

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

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

[27] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

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

[30] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5(3):317–322, 1996.

[31] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Formalized Mathematics, 6(4):531–539, 1997.

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

[33] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

The Euler’s Function

Yoshinori Fujisawa Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Summary. This article is concerned with the Euler’s function [10] that plays an important role in cryptograms. In the first section, we present some selected theorems on integers. Next, we define the Euler’s function. Finally, three theorems relating to the Euler’s function are proved. The third theorem concerns two relatively prime integers which make up the Euler’s function parameter. In the public key cryptography these two integer values are used as public and secret keys.

MML Identifier: EULER 1.

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

1. Preliminary

We follow the rules: a, b, c, k, l, m, n are natural numbers and i, j, x, y are integers.

The following propositions are true:

(1) k ∈ n iff k < n.

(2) n and n are relative prime iff n = 1.

(3) If k ≠ 0 and k < n and n is prime, then k and n are relative prime.

(4) n is prime and k ∈ {k_{1}; k_{1} ranges over natural numbers: n and k_{1} are relative prime ∧ k_{1} 1 ∧ k_{1} n} if and only if n is prime and k ∈ n and k ∉ {0}.

(5) For every finite set A and for every set x such that x ∈ A holds \overline{\overline{A \ \{x\}}} = \overline{\overline{A}} − \overline{\overline{\{x\}}}.

(6) If gcd(a, b) = 1, then for every c holds gcd(a · c, b · c) = c.

(7) If a ≠ 0 and b ≠ 0 and c ≠ 0 and gcd(a · c, b · c) = c, then a and b are relative prime.

(8) If gcd(a, b) = 1, then gcd(a + b, b) = 1.

(9) For every c holds gcd(a + b · c, b) = gcd(a, b).

(10) Suppose m and n are relative prime. Then there exists k such that (i) there exist integers i_{0}, j_{0} such that k = i_{0} · m + j_{0} · n and k > 0, and (ii) for every l such that there exist integers i, j such that l = i · m + j · n

and l > 0 holds k l.

(11) If m and n are relative prime, then for every k there exist i, j such that i · m + j · n = k.

(12) For all non empty finite sets A, B such that there exists a function from A into B which is one-to-one and onto holds \overline{\overline{A}} = \overline{\overline{B}} .

(13) For all integers i, k, n such that n ≠ 0 holds (i + k · n) modn = i modn.

(14) If a ≠ 0 and b ≠ 0 and c ≠ 0 and c | a · b and a and c are relative prime, then c | b.

(15) Suppose a ≠ 0 and b ≠ 0 and c ≠ 0 and a and c are relative prime and b and c are relative prime. Then a · b and c are relative prime.

(16) If x ≠ 0 and y ≠ 0 and i > 0, then i · x gcdi · y = i · (x gcdy).

(17) For every x such that a ≠ 0 and b ≠ 0 holds a + x · b gcdb = a gcdb.

2. Definition of Euler’s Function

Let n be a natural number. The functor Eulern yields a natural number and is defined as follows:

(Def. 1) Eulern = \overline{\overline{\{k; k ranges over natural numbers: n and k are}}

\overline{\overline{\{relative prime ∧ k 1 ∧ k n\}}} .

We now state several propositions:

(18) Euler1 = 1.

(19) Euler2 = 1.

(20) If n > 1, then Eulern n − 1.

(21) If n is prime, then Eulern = n − 1.

(22) If m > 1 and n > 1 and m and n are relative prime, then Eulerm · n = Eulerm · Eulern.

Acknowledgments

The authors wish to thank Professor A. Trybulec for all his advice on this article. the euler’s function 551

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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 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. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

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

[9] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[10] Teiji Takagi. Elementary Theory of Numbers. Kyoritsu Publishing Co., Ltd., second edition, 1995.

[11] Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1):137–144, 1996.

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

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

[14] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

552 yoshinori fujisawa and yasushi fuwa

While Macro Instructions of SCM _{FSA}

Jing-Chao Chen^{1} Shanghai Jiaotong University Shanghai

Summary. The article defines while macro instructions based on SCM _{FSA} . Some theorems about the generalized halting problems of while macro instructions are proved.

MML Identifier: SCMFSA 9.

The notation and terminology used in this paper are introduced in the following papers: [24], [32], [19], [8], [13], [33], [15], [16], [17], [12], [34], [7], [10], [14], [31], [18], [9], [20], [21], [25], [11], [23], [30], [29], [26], [27], [1], [28], [22], [5], [6], [4], [2], and [3].

The following propositions are true:

(1) For every macro instruction I and for every integer location a holds cardif = 0(a, I; Goto(insloc(0)), Stop_{SCM_{FSA}} ) = cardI + 6.

(2) For every macro instruction I and for every integer location a holds cardif > 0(a, I; Goto(insloc(0)), Stop_{SCM_{FSA}} ) = cardI + 6.

Let a be an integer location and let I be a macro instruction. The functor while = 0(a, I) yielding a macro instruction is defined as follows:

(Def. 1) while = 0(a, I) = if = 0(a, I; Goto(insloc(0)), Stop_{SCM_{FSA}} )+·(insloc

(cardI + 4)⟼→ . goto insloc(0)). The functor while > 0(a, I) yielding a macro instruction is defined by:

(Def. 2) while > 0(a, I) = if > 0(a, I; Goto(insloc(0)), Stop_{SCM_{FSA}} )+·(insloc

(cardI + 4)⟼→ . goto insloc(0)). The following proposition is true (3) For every macro instruction I and for every integer location a holds

1Part of the work was done while the author was visiting the Institute of Mathematics at the University of Białystok.

cardif = 0(a, Stop_{SCM_{FSA}} , if > 0(a, Stop_{SCM_{FSA}} , I; Goto(insloc(0)))) = cardI + 11.

Let a be an integer location and let I be a macro instruction. The functor while < 0(a, I) yields a macro instruction and is defined as follows:

(Def. 3) while < 0(a, I) = if = 0(a, Stop_{SCM_{FSA}} , if > 0(a, Stop_{SCM_{FSA}} , I; Goto

(insloc(0))))+·(insloc(cardI + 4)⟼→ . goto insloc(0)). Next we state a number of propositions:

(4) For every macro instruction I and for every integer location a holds cardwhile = 0(a, I) = cardI + 6.

(5) For every macro instruction I and for every integer location a holds cardwhile > 0(a, I) = cardI + 6.

(6) For every macro instruction I and for every integer location a holds cardwhile < 0(a, I) = cardI + 11.

(7) For every integer location a and for every instruction-location l of SCM _{FSA} holds if a = 0 goto l ≠ halt_{SCM _{FSA}} .

(8) For every integer location a and for every instruction-location l of SCM _{FSA} holds if a > 0 goto l ≠ halt_{SCM _{FSA}} .

(9) For every instruction-location l of SCM _{FSA} holds goto l ≠ halt_{SCM _{FSA}} .

(10) Let a be an integer location and I be a macro instruction. Then insloc(0) ∈ domwhile = 0(a, I) and insloc(1) ∈ domwhile = 0(a, I) and insloc(0) ∈ domwhile > 0(a, I) and insloc(1) ∈ domwhile > 0(a, I).

(11) Let a be an integer location and I be a macro instruction. Then (while = 0(a, I))(insloc(0)) = if a = 0 goto insloc(4) and (while = 0(a, I))(insloc(1)) = goto insloc(2) and (while > 0(a, I))(insloc(0)) = if a > 0 goto insloc(4) and (while > 0(a, I))(insloc(1)) = goto insloc(2).

(12) Let a be an integer location, I be a macro instruction, and k be a natural number. If k < 6, then insloc(k) ∈ domwhile = 0(a, I).

(13) Let a be an integer location, I be a macro instruction, and k be a natural number. If k < 6, then insloc(cardI + k) ∈ domwhile = 0(a, I).

(14) For every integer location a and for every macro instruction I holds (while = 0(a, I))(insloc(cardI + 5)) = halt_{SCM _{FSA}} .

(15) For every integer location a and for every macro instruction I holds (while = 0(a, I))(insloc(3)) = goto insloc(cardI + 5).

(16) For every integer location a and for every macro instruction I holds (while = 0(a, I))(insloc(2)) = goto insloc(3).

(17) Let a be an integer location, I be a macro instruction, and k be a natural number. If k < cardI + 6, then insloc(k) ∈ domwhile = 0(a, I).

(18) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a readwrite integer location. If s(a) ≠ 0, then while = 0(a, I) is halting on s and while = 0(a, I) is closed on s.

(19) Let a be an integer location, I be a macro instruction, s be a state of

SCM _{FSA}, and k be a natural number. Suppose that

(i) I is closed on s and halting on s, (ii) k < LifeSpan(s+·(I+·Start-At(insloc(0)))), (iii) IC_{(Computation (s+·(while=0(a,I)+·Start-At (insloc(0)))))(1+k)} =

IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(k)} + 4, and (iv) (Computation(s+·(while = 0(a, I)+·Start-At(insloc(0)))))(1 + k)

↾(Int-Locations∪ FinSeq-Locations) = (Computation(s+·(I+·Start-At (insloc(0)))))(k)↾(Int-Locations∪ FinSeq-Locations). Then IC_{(Computation (s+·(while=0(a,I)+·Start-At (insloc(0)))))(1+k+1)} = IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(k+1)}+4 and (Computation(s+·(while = 0(a, I)+·Start-At(insloc(0)))))(1 + k + 1)↾(Int-Locations ∪ FinSeq-Locations) = (Computation(s+·(I+·Start-At(insloc(0))))) (k + 1)↾(Int-Locations∪ FinSeq-Locations).

(20) Let a be an integer location, I be a macro instruction, and s be a state of SCM _{FSA}. Suppose I is closed on s and halting on s and

IC_{(Computation (s+·(while=0(a,I)+·Start-At (insloc(0)))))(1+LifeSpan (s+·(I+·Start-At}

(insloc(0))))) = IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(LifeSpan (s+·(I+·Start-At (insloc(0)))))}+4. Then CurInstr((Computation(s+·(while = 0(a, I)+·Start-At(insloc(0))))) (1 + LifeSpan(s+·(I+·Start-At(insloc(0)))))) = goto insloc(cardI + 4).

(21) For every integer location a and for every macro instruction I holds (while = 0(a, I))(insloc(cardI + 4)) = goto insloc(0).

(22) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. Suppose I is closed on s and halting on s and s(a) = 0. Then IC_{(Computation (s+·(while=0(a,I)+·Start-At (insloc(0)))))} (LifeSpan (s+·(I+·Start-At (insloc(0))))+3) = insloc(0) and for every natural number k such that k LifeSpan(s+·(I+·Start-At(insloc(0)))) + 3 holds IC_{(Computation (s+·(while=0(a,I)+·Start-At (insloc(0)))))(k)} ∈ domwhile = 0(a, I).

In the sequel s denotes a state of SCM _{FSA}, I denotes a macro instruction, and a denotes a read-write integer location.

Let us consider s, I, a. The functor StepW hile = 0(a, I, s) yields a function from ℕ into ∏ (the object kind of SCM _{FSA}) and is defined by the conditions (Def. 4).

(Def. 4)(i) (StepW hile = 0(a, I, s))(0) = s, and

(ii) for every natural number i and for every element x of ∏ (the object kind of SCM _{FSA}) such that x = (StepW hile = 0(a, I, s))(i) holds (StepW hile = 0(a, I, s))(i + 1) = (Computation(x+·(while = 0(a, I)+·s_{0})))(LifeSpan(x+·(I+·s_{0})) + 3).

In the sequel k, n are natural numbers. We now state three propositions:

(23) (StepW hile = 0(a, I, s))(0) = s.

(24) (StepW hile = 0(a, I, s))(k + 1) = (Computation((StepW hile = 0(a, I, s))(k)+·(while = 0(a, I)+·s_{0})))(LifeSpan((StepW hile = 0(a, I, s)) (k)+·(I+·s_{0})) + 3).

(25) (StepW hile = 0(a, I, s))(k + 1) = (StepW hile = 0(a, I, (StepW hile =

0(a, I, s))(k)))(1). The scheme MinIndex deals with a unary functor F yielding a natural num-

ber and a natural number A, and states that:

There exists k such that F(k) = 0 and for every n such that F(n) = 0 holds k n

provided the parameters meet the following conditions: • F(0) = A, and

• For every k holds F(k + 1) < F(k) or F(k) = 0. We now state a number of propositions:

(26) For all functions f, g holds f+·g+·g = f+·g.

(27) For all functions f, g, h and for every set D such that (f+·g)↾D = h↾D holds (h+·g)↾D = (f+·g)↾D.

(28) For all functions f, g, h and for every set D such that f↾D = h↾D holds (h+·g)↾D = (f+·g)↾D.

(29) For all states s_{1}, s_{2} of SCM _{FSA} such that IC_{(s_{1})} = IC_{(s_{2})} and s_{1}↾(Int-Locations∪ FinSeq-Locations) = s_{2}↾(Int-Locations∪ FinSeq-Locations) and s_{1}↾I_{1} = s_{2}↾I_{1} holds s_{1} = s_{2}.

(30) Let I be a macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Then (StepW hile = 0(a, I, s))(0 + 1) = (Computation(s+·(while = 0(a, I)+·s_{0})))(LifeSpan(s+·(I+·s_{0})) + 3).

(31) Let I be a macro instruction, a be a read-write integer location, s be a state of SCM _{FSA}, and k, n be natural numbers. Suppose IC_{(StepW hile=0(a,I,s))(k)} = insloc(0) and (StepW hile = 0(a, I, s))(k) = (Computation(s+·(while = 0(a, I)+·Start-At(insloc(0)))))(n). Then (StepW hile = 0(a, I, s))(k) = (StepW hile = 0(a, I, s))(k)+·(while = 0(a, I)+·Start-At(insloc(0))) and (StepW hile = 0(a, I, s))(k + 1) = (Computation(s+·(while = 0(a, I)+·Start-At(insloc(0)))))(n + (LifeSpan ((StepW hile = 0(a, I, s))(k)+·(I+·Start-At(insloc(0)))) + 3)).

(32) Let I be a macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Suppose that (i) for every natural number k holds I is closed on (StepW hile =

0(a, I, s))(k) and halting on (StepW hile = 0(a, I, s))(k), and (ii) there exists a function f from ∏ (the object kind of SCM _{FSA})

into ℕ such that for every natural number k holds f((StepW hile = 0(a, I, s))(k + 1)) < f((StepW hile = 0(a, I, s))(k)) or f((StepW hile = 0(a, I, s))(k)) = 0 but f((StepW hile = 0(a, I, s))(k)) = 0 iff (StepW hile = 0(a, I, s))(k)(a) ≠ 0. Then while = 0(a, I) is halting on s and while = 0(a, I) is closed on s.

(33) Let I be a parahalting macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Given a function f from ∏ (the object kind of SCM _{FSA}) into ℕ such that let k be a natural number. Then f((StepW hile = 0(a, I, s))(k + 1)) < f((StepW hile = 0(a, I, s))(k)) or f((StepW hile = 0(a, I, s))(k)) = 0 but f((StepW hile = 0(a, I, s))(k)) = 0 iff (StepW hile = 0(a, I, s))(k)(a) ≠ 0. Then while = 0(a, I) is halting on s and while = 0(a, I) is closed on s.

(34) Let I be a parahalting macro instruction and a be a read-write integer location. Given a function f from ∏ (the object kind of SCM _{FSA}) into ℕ such that let s be a state of SCM _{FSA}. Then f((StepW hile = 0(a, I, s))(1)) < f(s) or f(s) = 0 but f(s) = 0 iff s(a) ≠ 0. Then while = 0(a, I) is parahalting.

(35) For all instructions-locations l_{1}, l_{2} of SCM _{FSA} and for every integer location a holds l_{1}⟼→ . goto l_{2} does not destroy a.

(36) For every instruction i of SCM _{FSA} such that i does not destroy intloc(0)

holds Macro(i) is good. Let I, J be good macro instructions and let a be an integer location. Note

that if = 0(a, I, J) is good.

Let I be a good macro instruction and let a be an integer location. One can verify that while = 0(a, I) is good.

We now state a number of propositions:

(37) Let a be an integer location, I be a macro instruction, and k be a natural number. If k < 6, then insloc(k) ∈ domwhile > 0(a, I).

(38) Let a be an integer location, I be a macro instruction, and k be a natural number. If k < 6, then insloc(cardI + k) ∈ domwhile > 0(a, I).

(39) For every integer location a and for every macro instruction I holds (while > 0(a, I))(insloc(cardI + 5)) = halt_{SCM _{FSA}} .

(40) For every integer location a and for every macro instruction I holds (while > 0(a, I))(insloc(3)) = goto insloc(cardI + 5).

(41) For every integer location a and for every macro instruction I holds (while > 0(a, I))(insloc(2)) = goto insloc(3).

(42) Let a be an integer location, I be a macro instruction, and k be a natural

number. If k < cardI + 6, then insloc(k) ∈ domwhile > 0(a, I).

(43) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a readwrite integer location. If s(a) 0, then while > 0(a, I) is halting on s and while > 0(a, I) is closed on s.

(44) Let a be an integer location, I be a macro instruction, s be a state of

SCM _{FSA}, and k be a natural number. Suppose that

(i) I is closed on s and halting on s, (ii) k < LifeSpan(s+·(I+·Start-At(insloc(0)))), (iii) IC_{(Computation (s+·(while>0(a,I)+·Start-At (insloc(0)))))(1+k)} =

IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(k)} + 4, and (iv) (Computation(s+·(while > 0(a, I)+·Start-At(insloc(0)))))(1+k)↾D =

(Computation(s+·(I+·Start-At(insloc(0)))))(k)↾D. Then IC_{(Computation (s+·(while>0(a,I)+·Start-At (insloc(0)))))(1+k+1)} = IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(k+1)}+4 and (Computation(s+·(while > 0(a, I)+·Start-At(insloc(0)))))(1 + k + 1)↾D = (Computation(s+·(I+·Start-At(insloc(0)))))(k + 1)↾D.

(45) Let a be an integer location, I be a macro instruction, and s be a state of SCM _{FSA}. Suppose I is closed on s and halting on s and

IC_{(Computation (s+·(while>0(a,I)+·Start-At (insloc(0)))))(1+LifeSpan (s+·(I+·Start-At}

(insloc(0))))) = IC_{(Computation (s+·(I+·Start-At (insloc(0)))))(LifeSpan (s+·(I+·Start-At (insloc(0)))))}+4. Then CurInstr((Computation(s+·(while > 0(a, I)+·Start-At(insloc(0))))) (1 + LifeSpan(s+·(I+·Start-At(insloc(0)))))) = goto insloc(cardI + 4).

(46) For every integer location a and for every macro instruction I holds (while > 0(a, I))(insloc(cardI + 4)) = goto insloc(0).

(47) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. Suppose I is closed on s and halting on s and s(a) > 0. Then IC_{(Computation (s+·(while>0(a,I)+·Start-At (insloc(0)))))} (LifeSpan (s+·(I+·Start-At (insloc(0))))+3) = insloc(0) and for every natural number k such that k LifeSpan(s+·(I+·Start-At(insloc(0)))) + 3 holds IC_{(Computation (s+·(while>0(a,I)+·Start-At (insloc(0)))))(k)} ∈ domwhile > 0(a, I).

In the sequel s denotes a state of SCM _{FSA}, I denotes a macro instruction, and a denotes a read-write integer location.

Let us consider s, I, a. The functor StepW hile > 0(a, I, s) yielding a function from ℕ into ∏ (the object kind of SCM _{FSA}) is defined by the conditions (Def. 5).

(Def. 5)(i) (StepW hile > 0(a, I, s))(0) = s, and

(ii) for every natural number i and for every element x of ∏ (the object kind of SCM _{FSA}) such that x = (StepW hile > 0(a, I, s))(i) holds (StepW hile > 0(a, I, s))(i + 1) = (Computation(x+·(while > 0(a, I)+·s_{0})))(LifeSpan(x+·(I+·s_{0})) + 3).

One can prove the following propositions:

(48) (StepW hile > 0(a, I, s))(0) = s.

(49) (StepW hile > 0(a, I, s))(k + 1) = (Computation((StepW hile > 0(a, I, s))(k)+·(while > 0(a, I)+·s_{0})))(LifeSpan((StepW hile > 0(a, I, s)) (k)+·(I+·s_{0})) + 3).

(50) (StepW hile > 0(a, I, s))(k + 1) = (StepW hile > 0(a, I, (StepW hile > 0(a, I, s))(k)))(1).

(51) Let I be a macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Then (StepW hile > 0(a, I, s))(0 + 1) = (Computation(s+·(while > 0(a, I)+·s_{0})))(LifeSpan(s+·(I+·s_{0})) + 3).

(52) Let I be a macro instruction, a be a read-write integer location, s be a state of SCM _{FSA}, and k, n be natural numbers. Suppose IC_{(StepW hile>0(a,I,s))(k)} = insloc(0) and (StepW hile > 0(a, I, s))(k) = (Computation(s+·(while > 0(a, I)+·Start-At(insloc(0)))))(n). Then (StepW hile > 0(a, I, s))(k) = (StepW hile > 0(a, I, s))(k)+·(while > 0(a, I)+·Start-At(insloc(0))) and (StepW hile > 0(a, I, s))(k + 1) = (Computation(s+·(while > 0(a, I)+·Start-At(insloc(0)))))(n + (LifeSpan ((StepW hile > 0(a, I, s))(k)+·(I+·Start-At(insloc(0)))) + 3)).

(53) Let I be a macro instruction, a be a read-write integer location, and s

be a state of SCM _{FSA}. Suppose that

(i) for every natural number k holds I is closed on (StepW hile > 0(a, I, s))(k) and halting on (StepW hile > 0(a, I, s))(k), and

(ii) there exists a function f from ∏ (the object kind of SCM _{FSA}) into ℕ such that for every natural number k holds f((StepW hile > 0(a, I, s))(k + 1)) < f((StepW hile > 0(a, I, s))(k)) or f((StepW hile > 0(a, I, s))(k)) = 0 but f((StepW hile > 0(a, I, s))(k)) = 0 iff (StepW hile > 0(a, I, s))(k)(a) 0. Then while > 0(a, I) is halting on s and while > 0(a, I) is closed on s.

(54) Let I be a parahalting macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Given a function f from ∏ (the object kind of SCM _{FSA}) into ℕ such that let k be a natural number. Then f((StepW hile > 0(a, I, s))(k + 1)) < f((StepW hile > 0(a, I, s))(k)) or f((StepW hile > 0(a, I, s))(k)) = 0 but f((StepW hile > 0(a, I, s))(k)) = 0 iff (StepW hile > 0(a, I, s))(k)(a) 0. Then while > 0(a, I) is halting on s and while > 0(a, I) is closed on s.

(55) Let I be a parahalting macro instruction and a be a read-write integer location. Given a function f from ∏ (the object kind of SCM _{FSA}) into ℕ such that let s be a state of SCM _{FSA}. Then f((StepW hile > 0(a, I, s))(1)) < f(s) or f(s) = 0 but f(s) = 0 iff s(a) 0. Then while > 0(a, I) is parahalting.

560 jing-chao chen

Let I, J be good macro instructions and let a be an integer location. One can verify that if > 0(a, I, J) is good.

Let I be a good macro instruction and let a be an integer location. One can verify that while > 0(a, I) is good.

Acknowledgments

The author wishes to thank Prof. Andrzej Trybulec and Dr. Grzegorz Bancerek for their helpful comments and encouragement during his stay in Białystok.

References

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[3] Noriko Asamoto. Conditional branch macro instructions of SCM _{FSA} . Part II. Formalized Mathematics, 6(1):73–80, 1997.

[4] Noriko Asamoto. Constant assignment macro instructions of SCM _{FSA} . Part II. Formalized Mathematics, 6(1):59–63, 1997.

[5] Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, and Andrzej Trybulec. On the composition of macro instructions. Part II. Formalized Mathematics, 6(1):41–47, 1997.

[6] Noriko Asamoto, Yatsuka Nakamura, Piotr Rudnicki, and Andrzej Trybulec. On the composition of macro instructions. Part III. Formalized Mathematics, 6(1):53–57, 1997.

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

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

[9] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

[11] Grzegorz Bancerek and Piotr Rudnicki. Development of terminology for scm. Formalized Mathematics, 4(1):61–67, 1993.

[12] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.

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

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

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[17] 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.

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

[20] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Formalized Mathematics, 3(2):151–160, 1992.

[21] Yatsuka Nakamura and Andrzej Trybulec. On a mathematical model of programs. Formalized Mathematics, 3(2):241–250, 1992.

[22] Piotr Rudnicki and Andrzej Trybulec. Memory handling for SCM _{FSA} . Formalized Mathematics, 6(1):29–36, 1997.

[23] Yasushi Tanaka. On the decomposition of the states of SCM. Formalized Mathematics, 5(1):1–8, 1996.

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

[25] Andrzej Trybulec and Yatsuka Nakamura. Some remarks on the simple concrete model of computer. Formalized Mathematics, 4(1):51–56, 1993.

[26] Andrzej Trybulec and Yatsuka Nakamura. Modifying addresses of instructions of SCM _{FSA} . Formalized Mathematics, 5(4):571–576, 1996.

[27] Andrzej Trybulec and Yatsuka Nakamura. Relocability for SCM _{FSA} . Formalized Mathematics, 5(4):583–586, 1996.

[28] Andrzej Trybulec, Yatsuka Nakamura, and Noriko Asamoto. On the compositions of macro instructions. Part I. Formalized Mathematics, 6(1):21–27, 1997.

[29] Andrzej Trybulec, Yatsuka Nakamura, and Piotr Rudnicki. The SCM _{FSA} computer. Formalized Mathematics, 5(4):519–528, 1996.

[30] Andrzej Trybulec, Yatsuka Nakamura, and Piotr Rudnicki. An extension of scm. Formalized Mathematics, 5(4):507–512, 1996.

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

[32] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

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562 jing-chao chen

A Decomposition of a Simple Closed Curves and the Order of Their Points

Yatsuka Nakamura Shinshu University Nagano

Andrzej Trybulec University of Białystok

Summary. The goal of the article is to introduce an order on a simple closed curve. To do this, we fix two points on the curve and devide it into two arcs. We prove that such a decomposition is unique. Other auxiliary theorems about arcs are proven for preparation of the proof of the above.

MML Identifier: JORDAN6.

The papers [41], [46], [45], [40], [26], [1], [49], [44], [37], [12], [39], [10], [36], [32], [48], [2], [7], [8], [4], [20], [21], [34], [33], [29], [11], [43], [28], [19], [35], [16], [9], [15], [42], [18], [22], [17], [6], [23], [27], [3], [31], [5], [38], [13], [25], [47], [14], [30], and [24] provide the notation and terminology for this paper.

1. Middle Points of Arcs

For simplicity, we use the following convention: a, b, c, s, r are real numbers, n is a natural number, p, q are points of E_{T}^{2}, and P is a subset of the carrier of E_{T}^{2}.

The following propositions are true:

(1) If a = \frac{a+b}{2}, then a = b.

(2) If r s, then r \frac{r+s}{2} and \frac{r+s}{2} s.

(3) Let T_{1} be a non empty topological space, P be a subset of the carrier of T_{1}, A be a subset of the carrier of T_{1}↾P, and B be a subset of the carrier of T_{1}. If B is closed and A = B ∩ P, then A is closed.

(4) Let T_{1}, T_{2} be non empty topological spaces, P be a non empty subset of the carrier of T_{2}, and f be a map from T_{1} into T_{2}↾P. Then

(i) f is a map from T_{1} into T_{2}, and (ii) for every map f_{2} from T_{1} into T_{2} such that f_{2} = f and f is continuous

holds f_{2} is continuous.

(5) Let r be a real number and P be a subset of the carrier of E_{T}^{2}. If P = {p; p ranges over points of E_{T}^{2}: p_{1} r}, then P is closed.

(6) Let r be a real number and P be a subset of the carrier of E_{T}^{2}. If P = {p; p ranges over points of E_{T}^{2}: p_{1} r}, then P is closed.

(7) Let r be a real number and P be a subset of the carrier of E_{T}^{2}. If P = {p; p ranges over points of E_{T}^{2}: p_{1} = r}, then P is closed.

(8) Let r be a real number and P be a subset of the carrier of E_{T}^{2}. If P = {p; p ranges over points of E_{T}^{2}: p_{2} r}, then P is closed.

(9) Let r be a real number and P be a subset of the carrier of E_{T}^{2}. If P = {p; p ranges over points of E_{T}^{2}: p_{2} r}, then P is closed.

(10) Let r be a real number and P be a subset of the carrier of E_{T}^{2}. If P = {p; p ranges over points of E_{T}^{2}: p_{2} = r}, then P is closed.

(11) Let P be a non empty subset of the carrier of E_{T}^{n} and p_{1}, p_{2} be points of E_{T}^{n}. If P is an arc from p_{1} to p_{2}, then P is connected.

(12) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then P is closed.

(13) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2}. Then there exists a point q of E_{T}^{2} such that q ∈ P and q_{1} = \frac{(p_{1})_{1} +(p_{2})_{1}}{2} .

(14) Let P be a non empty subset of the carrier of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and Q = \{q : q_{1} = \frac{(p_{1})_{1} +(p_{2})_{1}}{2} \}. Then P meets Q and P ∩ Q is closed.

(15) Let P be a non empty subset of the carrier of E_{T}^{2}, Q be a subset of the carrier of E_{T}^{2}, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P is an arc from p_{1} to p_{2} and Q = \{q : q_{2} = \frac{(p_{1})_{2} +(p_{2})_{2}}{2} \}. Then P meets Q and P ∩ Q is closed.

Let P be a non empty subset of the carrier of E_{T}^{2} and let p_{1}, p_{2} be points of E_{T}^{2}. Let us assume that P is an arc from p_{1} to p_{2}. The functor xMiddle(P, p_{1}, p_{2}) yields a point of E_{T}^{2} and is defined as follows:

(Def. 1) For every subset Q of the carrier of E_{T}^{2} such that Q = {q : q_{1} =

\frac{(p_{1})_{1} +(p_{2})_{1}}{2} \} holds xMiddle(P, p_{1}, p_{2}) = FPoint(P, p_{1}, p_{2}, Q). Let P be a non empty subset of the carrier of E_{T}^{2} and let p_{1}, p_{2} be points of

E_{T}^{2}. Let us assume that P is an arc from p_{1} to p_{2}. The functor yMiddle(P, p_{1}, p_{2}) yields a point of E_{T}^{2} and is defined by:

(Def. 2) For every subset Q of the carrier of E_{T}^{2} such that Q = {q : q_{2} =

\frac{(p_{1})_{2} +(p_{2})_{2}}{2} \} holds yMiddle(P, p_{1}, p_{2}) = FPoint(P, p_{1}, p_{2}, Q). One can prove the following propositions:

(16) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then xMiddle(P, p_{1}, p_{2}) ∈ P and yMiddle(P, p_{1}, p_{2}) ∈ P.

(17) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then p_{1} = xMiddle(P, p_{1}, p_{2}) iff (p_{1})_{1} = (p_{2})_{1}.

(18) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then p_{1} = yMiddle(P, p_{1}, p_{2}) iff (p_{1})_{2} = (p_{2})_{2}.

2. Segments of Arcs

The following proposition is true

(19) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and LE q_{1}, q_{2}, P, p_{1}, p_{2}, then LE q_{2}, q_{1}, P, p_{2}, p_{1}.

Let P be a non empty subset of the carrier of E_{T}^{2} and let p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. The functor LSegment(P, p_{1}, p_{2}, q_{1}) yields a subset of the carrier of E_{T}^{2} and is defined by:

(Def. 3) LSegment(P, p_{1}, p_{2}, q_{1}) = {q : LE q, q_{1}, P, p_{1}, p_{2}}.

Let P be a non empty subset of the carrier of E_{T}^{2} and let p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. The functor RSegment(P, p_{1}, p_{2}, q_{1}) yielding a subset of the carrier of E_{T}^{2} is defined as follows:

(Def. 4) RSegment(P, p_{1}, p_{2}, q_{1}) = {q : LE q_{1}, q, P, p_{1}, p_{2}}.

Next we state several propositions:

(20) For every non empty subset P of the carrier of E_{T}^{2} and for all points p_{1}, p_{2}, q_{1} of E_{T}^{2} holds LSegment(P, p_{1}, p_{2}, q_{1}) ⊆ P.

(21) For every non empty subset P of the carrier of E_{T}^{2} and for all points p_{1}, p_{2}, q_{1} of E_{T}^{2} holds RSegment(P, p_{1}, p_{2}, q_{1}) ⊆ P.

(22) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then LSegment(P, p_{1}, p_{2}, p_{1}) = {p_{1}}.

(23) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and q ∈ P, then LE q, p_{2}, P, p_{1}, p_{2}.

(24) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and q ∈ P, then LE p_{1}, q, P, p_{1}, p_{2}.

(25) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then LSegment(P, p_{1}, p_{2}, p_{2}) = P.

(26) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then RSegment(P, p_{1}, p_{2}, p_{2}) = {p_{2}}.

(27) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2}, then RSegment(P, p_{1}, p_{2}, p_{1}) = P.

(28) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and q_{1} ∈ P, then RSegment(P, p_{1}, p_{2}, q_{1}) = LSegment(P, p_{2}, p_{1}, q_{1}).

Let P be a non empty subset of the carrier of E_{T}^{2} and let p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. The functor Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) yielding a subset of the carrier of E_{T}^{2} is defined by:

(Def. 5) Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) = RSegment(P, p_{1}, p_{2}, q_{1})∩LSegment(P, p_{1}, p_{2}, q_{2}).

Next we state four propositions:

(29) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. Then Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) = {q : LE q_{1}, q, P, p_{1}, p_{2} ∧ LE q, q_{2}, P, p_{1}, p_{2}}.

(30) Let P be a non empty 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}. Then LE q_{1}, q_{2}, P, p_{1}, p_{2} if and only if LE q_{2}, q_{1}, P, p_{2}, p_{1}.

(31) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and q ∈ P, then LSegment(P, p_{1}, p_{2}, q) = RSegment(P, p_{2}, p_{1}, q).

(32) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2} be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and q_{1} ∈ P and q_{2} ∈ P, then Segment(P, p_{1}, p_{2}, q_{1}, q_{2}) = Segment(P, p_{2}, p_{1}, q_{2}, q_{1}).

3. Decomposition of a Simple Closed Curve Into Two Arcs

Let s be a real number. The functor VerticalLines yields a subset of the carrier of E_{T}^{2} and is defined as follows:

(Def. 6) VerticalLines = {p; p ranges over points of E_{T}^{2}: p_{1} = s}.

The functor HorizontalLines yielding a subset of the carrier of E_{T}^{2} is defined as follows:

(Def. 7) HorizontalLines = {p : p_{2} = s}.

Next we state several propositions:

(33) For every real number r holds VerticalLiner is closed and HorizontalLiner is closed.

(34) For every real number r and for every point p of E_{T}^{2} such that p ∈ VerticalLiner holds p_{1} = r.

(35) For every real number r and for every point p of E_{T}^{2} such that p ∈ HorizontalLiner holds p_{2} = r.

(36) For every compact non empty subset P of E_{T}^{2} holds W-minP ∈ P and W-maxP ∈ P.

(37) For every compact non empty subset P of E_{T}^{2} holds N-minP ∈ P and N-maxP ∈ P.

(38) For every compact non empty subset P of E_{T}^{2} holds E-minP ∈ P and E-maxP ∈ P.

(39) For every compact non empty subset P of E_{T}^{2} holds S-minP ∈ P and S-maxP ∈ P.

(40) Let P be a compact non empty subset of E_{T}^{2}. Suppose P is a simple closed curve. Then there exist non empty subsets P_{1}, P_{2} of the carrier of E_{T}^{2} such that

(i) P_{1} is an arc from W-minP to E-maxP, (ii) P_{2} is an arc from E-maxP to W-minP, (iii) P_{1} ∩ P_{2} = {W-minP, E-maxP }, (iv) P_{1} ∪ P_{2} = P, and

(v) (FPoint(P_{1}, W-minP, E-maxP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2} > (LPoint(P_{2}, E-maxP, W-minP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2}.

4. Uniqueness of Decomposition of a Simple Closed Curve

One can prove the following propositions:

(41) For every subset P of the carrier of 𝕀 such that P = (the carrier of 𝕀) \ {0, 1} holds P is open.

(42) For all subsets B_{1}, B_{2} of ℝ such that B_{2} is lower bounded and B_{1} ⊆ B_{2} holds B_{1} is lower bounded.

(43) For all subsets B_{1}, B_{2} of ℝ such that B_{2} is upper bounded and B_{1} ⊆ B_{2} holds B_{1} is upper bounded.

(44) For all r, s holds ]r, s[ ∩ {r, s} = ∅.

(45) For all a, b, c holds c ∈ ]a, b[ iff a < c and c < b.

(46) For every subset P of the carrier of ℝ^{1} and for all r, s such that P = ]r, s[ holds P is open.

(47) Let S be a non empty topological space, P_{1}, P_{2} be subsets of the carrier of S, and P′_{1} be a subset of the carrier of S↾P_{2}. If P_{1} = P′_{1} and P_{1} ≠ ∅ and P_{1} ⊆ P_{2}, then S↾P_{1} = S↾P_{2}↾P′_{1}.

(48) For every subset P_{7} of the carrier of 𝕀 such that P_{7} = (the carrier of 𝕀) \ {0, 1} holds P_{7} ≠ ∅ and P_{7} is connected.

(49) Let P be a non empty subset of the carrier of E_{T}^{n} and p_{1}, p_{2} be points of E_{T}^{n}. If P is an arc from p_{1} to p_{2}, then p_{1} ≠ p_{2}.

(50) Let P be a non empty subset of the carrier of E_{T}^{n}, Q be a subset of the carrier of (E_{T}^{n})↾P, and p_{1}, p_{2} be points of E_{T}^{n}. If P is an arc from p_{1} to p_{2} and Q = P \ {p_{1}, p_{2}}, then Q is open.

(51) For all points p, q of E_{T}^{n} and for every non empty subset P of E_{T}^{n} such that P is an arc from p to q holds P is compact.

(52) Let P be a subset of the carrier of E_{T}^{n}, P_{1}, P_{2} be non empty subsets of the carrier of E_{T}^{n}, Q be a subset of the carrier of (E_{T}^{n})↾P, and p_{1}, p_{2} be points of E_{T}^{n}. Suppose p_{1} ∈ P and p_{2} ∈ P and P_{1} is an arc from p_{1} to p_{2} and P_{2} is an arc from p_{1} to p_{2} and P_{1} ∪ P_{2} = P and P_{1} ∩ P_{2} = {p_{1}, p_{2}} and Q = P_{1} \ {p_{1}, p_{2}}. Then Q is open.

(53) Let P be a non empty subset of the carrier of E_{T}^{n}, Q be a subset of the carrier of (E_{T}^{n})↾P, and p_{1}, p_{2} be points of E_{T}^{n}. If P is an arc from p_{1} to p_{2} and Q = P \ {p_{1}, p_{2}}, then Q is connected.

(54) Let G_{1} be a non empty topological space, P_{1}, P be non empty subsets of the carrier of G_{1}, Q′ be a subset of the carrier of G_{1}↾P_{1}, and Q be a non empty subset of the carrier of G_{1}↾P. If P_{1} ⊆ P and Q = Q′ and Q′ is connected, then Q is connected.

(55) Let P be a non empty subset of the carrier of E_{T}^{n} and p_{1}, p_{2} be points of E_{T}^{n}. Suppose P is an arc from p_{1} to p_{2}. Then there exists a point p_{3} of E_{T}^{n} such that p_{3} ∈ P and p_{3} ≠ p_{1} and p_{3} ≠ p_{2}.

(56) Let P be a non empty subset of the carrier of E_{T}^{n} and p_{1}, p_{2} be points of E_{T}^{n}. If P is an arc from p_{1} to p_{2}, then P \ {p_{1}, p_{2}} ≠ ∅.

(57) Let P_{1} be a non empty subset of the carrier of E_{T}^{n}, P be a subset of the carrier of E_{T}^{n}, Q be a subset of the carrier of (E_{T}^{n})↾P, and p_{1}, p_{2} be points of E_{T}^{n}. If P_{1} is an arc from p_{1} to p_{2} and P_{1} ⊆ P and Q = P_{1} \ {p_{1}, p_{2}}, then Q is connected.

(58) Let T, S, V be non empty topological spaces, P_{1} be a non empty subset of the carrier of S, P_{2} be a subset of the carrier of S, f be a map from T into S↾P_{1}, and g be a map from S↾P_{2} into V. Suppose P_{1} ⊆ P_{2} and f is continuous and g is continuous. Then there exists a map h from T into V such that h = g · f and h is continuous.

(59) Let P_{1}, P_{2} be non empty subsets of the carrier of E_{T}^{n} and p_{1}, p_{2} be points of E_{T}^{n}. If P_{1} is an arc from p_{1} to p_{2} and P_{2} is an arc from p_{1} to p_{2} and P_{1} ⊆ P_{2}, then P_{1} = P_{2}.

(60) Let P be a non empty subset of the carrier of E_{T}^{2}, Q be a subset of the carrier of (E_{T}^{2})↾P, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose P is a simple closed curve and p_{1} ∈ P and p_{2} ∈ P and p_{1} ≠ p_{2} and Q = P \ {p_{1}, p_{2}}. Then Q is not connected.

(61) Let P, P_{1}, P_{2}, P′_{1}, P′_{2} be non empty subsets of the carrier of E_{T}^{2} and p_{1},

p_{2} be points of E_{T}^{2}. Suppose that

(i) P is a simple closed curve, (ii) P_{1} is an arc from p_{1} to p_{2}, (iii) P_{2} is an arc from p_{1} to p_{2}, (iv) P_{1} ∪ P_{2} = P,

(v) P_{1} ∩ P_{2} = {p_{1}, p_{2}}, (vi) P′_{1} is an arc from p_{1} to p_{2}, (vii) P′_{2} is an arc from p_{1} to p_{2}, (viii) P′_{1} ∪ P′_{2} = P, and (ix) P′_{1} ∩ P′_{2} = {p_{1}, p_{2}}.

Then P_{1} = P′_{1} and P_{2} = P′_{2} or P_{1} = P′_{2} and P_{2} = P′_{1}.

5. Lower Arcs and Upper Arcs

One can prove the following propositions:

(62) Let P_{1} be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. If P_{1} is an arc from p_{1} to p_{2}, then P_{1} is closed.

(63) Let G_{1}, G_{2} be non empty topological spaces, P be a non empty subset of the carrier of G_{2}, f be a map from G_{1} into G_{2}↾P, and f_{1} be a map from G_{1} into G_{2}. If f = f_{1} and f is continuous, then f_{1} is continuous.

(64) Let P_{1} be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2} be points of E_{T}^{2}. Suppose (p_{1})_{1} (p_{2})_{1} and P_{1} is an arc from p_{1} to p_{2}. Then P_{1} ∩ VerticalLine \frac{(p_{1})_{1} +(p_{2})_{1}}{2} ≠ ∅ and P_{1} ∩ VerticalLine \frac{(p_{1})_{1} +(p_{2})_{1}}{2} is closed.

Let P be a compact non empty subset of E_{T}^{2}. Let us assume that P is a simple closed curve. The functor UpperArcP yields a non empty subset of the carrier of E_{T}^{2} and is defined by the conditions (Def. 8).

(Def. 8)(i) UpperArcP is an arc from W-minP to E-maxP, and

(ii) there exists a non empty subset P_{2} of the carrier of E_{T}^{2} such that P_{2} is an arc from E-maxP to W-minP and UpperArcP ∩ P_{2} = {W-minP, E-maxP } and UpperArcP ∪ P_{2} = P and (FPoint(UpperArcP, W-minP, E-maxP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2} > (LPoint(P_{2}, E-maxP, W-minP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2}.

Let P be a compact non empty subset of E_{T}^{2}. Let us assume that P is a simple closed curve. The functor LowerArcP yielding a non empty subset of the carrier of E_{T}^{2} is defined as follows:

(Def. 9) LowerArcP is an arc from E-maxP to W-minP and UpperArcP ∩ LowerArcP = {W-minP, E-maxP } and UpperArcP ∪ LowerArcP = P and (FPoint(UpperArcP, W-minP, E-maxP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2} > (LPoint(LowerArcP, E-maxP, W-minP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2}.

The following propositions are true:

(65) Let P be a compact non empty subset of E_{T}^{2}. Suppose P is a simple

closed curve. Then

(i) UpperArcP is an arc from W-minP to E-maxP, (ii) UpperArcP is an arc from E-maxP to W-minP, (iii) LowerArcP is an arc from E-maxP to W-minP, (iv) LowerArcP is an arc from W-minP to E-maxP,

(v) UpperArcP ∩ LowerArcP = {W-minP, E-maxP }, (vi) UpperArcP ∪ LowerArcP = P, and

(vii) (FPoint(UpperArcP, W-minP, E-maxP,

VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2} > (LPoint(LowerArcP, E-maxP, W-minP, VerticalLine \frac{W-bound P +E-bound P}{2} ))_{2}.

(66) Let P be a compact non empty subset of E_{T}^{2}. If P is a simple closed curve, then LowerArcP = (P \ UpperArcP) ∪ {W-minP, E-maxP } and UpperArcP = (P \ LowerArcP) ∪ {W-minP, E-maxP }.

(67) Let P be a compact non empty subset of E_{T}^{2} and P_{1} be a subset of the carrier of (E_{T}^{2})↾P. If P is a simple closed curve and UpperArcP ∩ P_{1} = {W-minP, E-maxP } and UpperArcP ∪ P_{1} = P, then P_{1} = LowerArcP.

(68) Let P be a compact non empty subset of E_{T}^{2} and P_{1} be a subset of the carrier of (E_{T}^{2})↾P. If P is a simple closed curve and P_{1} ∩ LowerArcP = {W-minP, E-maxP } and P_{1} ∪ LowerArcP = P, then P_{1} = UpperArcP.

6. An Order of Points in a Simple Closed Curve

One can prove the following propositions:

(69) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q be points of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and LE q, p_{1}, P, p_{1}, p_{2}, then q = p_{1}.

(70) Let P be a non empty subset of the carrier of E_{T}^{2} and p_{1}, p_{2}, q be points

of E_{T}^{2}. If P is an arc from p_{1} to p_{2} and LE p_{2}, q, P, p_{1}, p_{2}, then q = p_{2}. Let P be a compact non empty subset of E_{T}^{2} and let q_{1}, q_{2} be points of E_{T}^{2}.

The predicate LE(q_{1}, q_{2}, P) is defined by the conditions (Def. 10). (Def. 10)(i) q_{1} ∈ UpperArcP and q_{2} ∈ LowerArcP and q_{2} ≠ W-minP, or

(ii) q_{1} ∈ UpperArcP and q_{2} ∈ UpperArcP and LE q_{1}, q_{2}, UpperArcP, W-minP, E-maxP, or

a decomposition of a simple closed curves … 571

(iii) q_{1} ∈ LowerArcP and q_{2} ∈ LowerArcP and q_{2} ≠ W-minP and LE q_{1}, q_{2}, LowerArcP, E-maxP, W-minP.

Next we state three propositions:

(71) Let P be a compact non empty subset of E_{T}^{2} and q be a point of E_{T}^{2}. If P is a simple closed curve and q ∈ P, then LE(q, q, P).

(72) Let P be a compact non empty subset of E_{T}^{2} and q_{1}, q_{2} be points of E_{T}^{2}. If P is a simple closed curve and LE(q_{1}, q_{2}, P) and LE(q_{2}, q_{1}, P), then q_{1} = q_{2}.

(73) Let P be a compact non empty subset of E_{T}^{2} and q_{1}, q_{2}, q_{3} be points of E_{T}^{2}. If P is a simple closed curve and LE(q_{1}, q_{2}, P) and LE(q_{2}, q_{3}, P), then LE(q_{1}, q_{3}, P).

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The Chinese Remainder Theorem

Andrzej Kondracki AMS Management Systems Poland Warsaw

Summary. The article is a translation of the first chapters of a book Wstęp do teorii liczb (Eng. Introduction to Number Theory) by W. Sierpiński, WSiP, Biblioteczka Matematyczna, Warszawa, 1987. The first few pages of this book have already been formalized in MML. We prove the Chinese Remainder Theorem and Thue’s Theorem as well as several useful number theory propositions.

MML Identifier: WSIERP 1.

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

For simplicity, we follow the rules: x, y, z, w denote real numbers, a, b, c, d, e, f, g denote natural numbers, k, l, m, n, m_{1}, n_{1} denote integers, and q denotes a rational number.

The following propositions are true:

(1) If y ≠ 0, then (\frac{x}{y})^{a} = \frac{x^{a}}{y^{a}} .

(2) x^{2} = x · x and (− x)^{2} = x^{2}.

(3) (− x)^{2·a}= x^{2·a} and (− x)^{2·a+1}= − x^{2·a+1}.

(4) If x ≠ 0, then x_{ℤ}^{a} = x^{a}.

(5) If x 0 and y 0 and d > 0 and x^{d} = y^{d}, then x = y.

(6) x > max(y, z) iff x > y and x > z.

(7) If x 0 and y z, then y − x z and y z + x.

(8) If x 0 and y > z or x < 0 and y z, then y > z + x and y − x > z. Let us consider a, b. Then gcd(a, b) is a natural number. Let us observe that

the functor gcd(a, b) is commutative.

Let us consider m, n. Then m gcdn is an integer. Let us observe that the functor m gcdn is commutative.

Let us consider k, a. Then k^{a} is an integer.

Let us consider a, b. Then a^{b} is a natural number.

We now state a number of propositions:

(9) If k | m and k | n, then k | m + n.

(10) If k | m and k | n, then k | m · m_{1} + n · n_{1}.

(11) If m gcdn = 1 and k gcdn = 1, then m · k gcdn = 1.

(12) If gcd(a, b) = 1 and gcd(c, b) = 1, then gcd(a · c, b) = 1.

(13) 0 gcdm = |m| and 1 gcdm = 1.

(14) 1 and k are relative prime.

(15) If k and l are relative prime, then k^{a} and l are relative prime.

(16) If k and l are relative prime, then k^{a} and l^{b} are relative prime.

(17) If k gcdl = 1, then k gcdl^{b} = 1 and k^{a} gcdl^{b} = 1.

(18) |m| | k iff m | k.

(19) If a | b, then a^{c} | b^{c}.

(20) If a | 1, then a = 1.

(21) If d | a and gcd(a, b) = 1, then gcd(d, b) = 1.

(22) If k ≠ 0, then k | l iff \frac{l}{k} is an integer.

(23) If a b − c, then a b and c b.

In the sequel f_{1}, f_{2}, f_{3} are finite sequences.

Next we state two propositions:

(24) If a ∈ Seglenf_{2}, then a ∈ Seglen(f_{2} ^{⌢} f_{3}).

(25) If a ∈ Seglenf_{3}, then lenf_{2} + a ∈ Seglen(f_{2} ^{⌢} f_{3}).

Let f_{4} be a finite sequence of elements of ℝ and let us consider a. Then f_{4}(a) is a real number.

Let f_{5} be a finite sequence of elements of ℤ and let us consider a. Then f_{5}(a) is an integer.

Let f_{6} be a finite sequence of elements of ℕ and let us consider a. Then f_{6}(a) is a natural number.

Let D be a non empty set and let D_{1} be a non empty subset of D. We see that the finite sequence of elements of D_{1} is a finite sequence of elements of D.

Let D be a non empty set, let D_{1} be a non empty subset of D, and let f_{7}, f_{8} be finite sequences of elements of D_{1}. Then f_{7} ^{⌢} f_{8} is a finite sequence of elements of D_{1}.

Let D be a non empty set and let D_{1} be a non empty subset of D. Then ε_{(D_{1})} is an empty finite sequence of elements of D_{1}.

ℤ is a non empty subset of ℝ.

For simplicity, we adopt the following convention: D, D_{1} are non empty sets, v_{1}, v_{2}, v_{3} are sets, f_{6} is a finite sequence of elements of ℕ, f_{5}, f_{9} are finite sequences of elements of ℤ, and f_{4} is a finite sequence of elements of ℝ.

Let us consider f_{5}. Then ∑ f_{5} is an integer. Then ∏ f_{5} is an integer.

Let us consider f_{6}. Then ∑ f_{6} is a natural number. Then ∏ f_{6} is a natural number.

Let us consider a, f_{1}. The functor f_{1}∼ a yielding a finite sequence is defined as follows:

(Def. 1)(i) f_{1}∼ a = f_{1} if a ∉ domf_{1},

(ii) len(f_{1}∼ a) + 1 = lenf_{1} and for every b holds if b < a, then (f_{1}∼ a)(b) =

f_{1}(b) and if b a, then (f_{1}∼ a)(b) = f_{1}(b + 1), otherwise. Let us consider D, let us consider a, and let f_{1} be a finite sequence of elements

of D. Then f_{1}∼ a is a finite sequence of elements of D.

Let us consider D, let D_{1} be a non empty subset of D, let us consider a, and let f_{1} be a finite sequence of elements of D_{1}. Then f_{1}∼ a is a finite sequence of elements of D_{1}.

One can prove the following propositions:

(26) ⟨v_{1}⟩∼ 1 = ε and ⟨v_{1}, v_{2}⟩∼ 1 = ⟨v_{2}⟩ and ⟨v_{1}, v_{2}⟩∼ 2 = ⟨v_{1}⟩ and ⟨v_{1}, v_{2}, v_{3}⟩∼ 1 = ⟨v_{2}, v_{3}⟩ and ⟨v_{1}, v_{2}, v_{3}⟩∼ 2 = ⟨v_{1}, v_{3}⟩ and ⟨v_{1}, v_{2}, v_{3}⟩∼ 3 = ⟨v_{1}, v_{2}⟩.

(27) If 1 a and a lenf_{4}, then ∑(f_{4}∼ a) + f_{4}(a) = ∑ f_{4}.

(28) If a ∈ Seglenf_{6} and f_{6}(a) ≠ 0, then \frac{∏ f_{6}}{f_{6}(a)} is a natural number.

(29) numq and denq are relative prime.

(30) If q ≠ 0 and q = \frac{k}{a} and a ≠ 0 and k and a are relative prime, then k = numq and a = denq.

(31) If there exists q such that a = q^{b}, then there exists k such that a = k^{b}.

(32) If there exists q such that a = q^{d}, then there exists b such that a = b^{d}.

(33) If e > 0 and a^{e} | b^{e}, then a | b.

(34) There exist m, n such that gcd(a, b) = a · m + b · n.

(35) There exist m_{1}, n_{1} such that m gcdn = m · m_{1} + n · n_{1}.

(36) If m | n · k and m gcdn = 1, then m | k.

(37) If gcd(a, b) = 1 and a | b · c, then a | c.

(38) If a ≠ 0 and b ≠ 0, then there exist c, d such that gcd(a, b) = a · c − b · d.

(39) If f > 0 and g > 0 and gcd(f, g) = 1 and a^{f} = b^{g}, then there exists e

such that a = e^{g} and b = e^{f}. In the sequel x, y, z, t denote integers. Next we state several propositions:

(40) There exist x, y such that m · x + n · y = k iff m gcdn | k.

(41) Suppose m ≠ 0 and n ≠ 0 and m · m_{1} + n · n_{1} = k. Let given x, y. If m · x + n · y = k, then there exists t such that x = m_{1} + t · \frac{n}{m gcd n} and y = n_{1} − t · \frac{m}{m gcd n}.

(42) If gcd(a, b) = 1 and a · b = c^{d}, then there exist e, f such that a = e^{d} and b = f^{d}.

576 andrzej kondracki

(43) For every d such that for every a such that a ∈ Seglenf_{6} holds gcd(f_{6}(a), d) = 1 holds gcd(∏ f_{6}, d) = 1.

(44) Suppose lenf_{6} 2 and for all b, c such that b ∈ Seglenf_{6} and c ∈ Seglenf_{6} and b ≠ c holds gcd(f_{6}(b), f_{6}(c)) = 1. Let given f_{5}. Suppose lenf_{5} = lenf_{6}. Then there exists f_{9} such that lenf_{9} = lenf_{6} and for every b such that b ∈ Seglenf_{6} holds f_{6}(b) · f_{9}(b) + f_{5}(b) = f_{6}(1) · f_{9}(1) + f_{5}(1).

(45) If x < y and z w or x y and z > w or x < y and z > w, then x − z < y − w.

(46) If a ≠ 0 and a gcdk = 1, then there exist b, e such that 0 ≠ b and 0 ≠ e and b \sqrt{a} and e \sqrt{a} and a | k · b + e or a | k · b − e.

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[12] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[13] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[14] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[15] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

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

[17] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

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

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

[20] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

the chinese remainder theorem 577

[21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received December 30, 1997

Index of MML Identifiers

ALTCAT 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .475

BORSUK 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .449

EULER 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

JORDAN2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

JORDAN5A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

JORDAN5B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

JORDAN5C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

JORDAN6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

LATTICE5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

SCMFSA8C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

SCMFSA 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .553

SPRECT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531

SPRECT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .541

TOPREAL5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

UNIFORM1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

WAYBEL15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

WSIERP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573

Contents Formaliz. Math. 6 (4)

Introduction to the Homotopy Theory

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Some Properties of Real Maps

By Adam Grabowski and Yatsuka Nakamura  . . . . . . . . . . . . 455

The Ordering of Points on a Curve. Part I

By Adam Grabowski and Yatsuka Nakamura  . . . . . . . . . . . . 461

The Ordering of Points on a Curve. Part II

By Adam Grabowski and Yatsuka Nakamura  . . . . . . . . . . . . 467

On the Categories Without Uniqueness of cod and dom . Some

Properties of the Morphisms and the Functors

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

The loop and Times Macroinstruction for SCM _{FSA}

By Noriko Asamoto  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Algebraic and Arithmetic Lattices. Part II

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Projections in n-Dimensional Euclidean Space to Each Coordina-

tes

By Roman Matuszewski and Yatsuka Nakamura  . . . . . . . . . 505

Intermediate Value Theorem and Thickness of Simple Closed Cu-

rves

By Yatsuka Nakamura and Andrzej Trybulec  . . . . . . . . . . . 511

The Jónson’s Theorem

By Jarosław Gryko  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Continued on inside back cover

Lebesgue’s Covering Lemma, Uniform Continuity and Segmenta-

tion of Arcs

By Yatsuka Nakamura and Andrzej Trybulec  . . . . . . . . . . . 525

On the Rectangular Finite Sequences of the Points of the Plane

By Andrzej Trybulec and Yatsuka Nakamura  . . . . . . . . . . . 531

On the Order on a Special Polygon

By Andrzej Trybulec and Yatsuka Nakamura  . . . . . . . . . . . 541

The Euler’s Function

By Yoshinori Fujisawa and Yasushi Fuwa  . . . . . . . . . . . . . . . . . 549

While Macro Instructions of SCM _{FSA}

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

A Decomposition of a Simple Closed Curves and the Order of

Their Points

By Yatsuka Nakamura and Andrzej Trybulec  . . . . . . . . . . . 563

The Chinese Remainder Theorem

By Andrzej Kondracki  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578