FORMALIZED MATHEMATICS Volume 5, Number 3, 1996 Warsaw University - Białystok

Vertex Sequences Induced by Chains ^{1}

Yatsuka Nakamura Shinshu University Department of Information Engineering Nagano, Japan

Piotr Rudnicki University of Alberta Department of Computing Science Edmonton, Alberta, Canada

Summary. In the three preliminary sections to the article we define two operations on finite sequences which seem to be of general interest. The first is the cut operation that extracts a contiguous chunk of a finite sequence from a position to a position. The second operation is a glueing catenation that given two finite sequences catenates them with removal of the first element of the second sequence. The main topic of the article is to define an operation which for a given chain in a graph returns the sequence of vertices through which the chain passes. We define the exact conditions when such an operation is uniquely definable. This is done with the help of the so called two-valued alternating finite sequences. We also prove theorems about the existence of simple chains which are subchains of a given chain. In order to do this we define the notion of a finite subsequence of a typed finite sequence.

MML Identifier: GRAPH 2.

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

1This work was partially supported by Shinshu Endowment for Information Science, NSERC Grant OGP9207 and JSTF award 651-93-S009.

1. Preliminaries

We adopt the following convention: p, q are finite sequences, X, Y are sets, and i, k, l, m, n, r are natural numbers.

The scheme FinSegRng deals with natural numbers A, B, a unary functor F yielding a set, and a unary predicate P, and states that:

{F(i) : A ≤ i ∧ i ≤ B ∧ P[i]} is finite for all values of the parameters.

One can prove the following propositions:

(1) m+1 ≤ k and k ≤ n iff there exists a natural number i such that m ≤ i and i < n and k = i + 1.

(2) If q = p Seg n, then len q ≤ len p and for every i such that 1 ≤ i and i ≤ len q holds p(i) = q(i).

(3) If X ⊆ Seg k and Y ⊆ dom Sgm X, then Sgm X · Sgm Y = Sgm rng(Sgm X Y).

(4) For all natural numbers m, n holds \overline{\overline{\{k : m ≤ k ∧ k ≤ m + n\}}} = n + 1.

(5) For every l such that 1 ≤ l and l ≤ n holds (Sgm{k_{1} : k_{1} ranges over natural numbers, m + 1 ≤ k_{1} ∧ k_{1} ≤ m + n})(l) = m + l.

2. The cut operation for finite sequences

Let p be a finite sequence and let m, n be natural numbers. The functor ⟨p(m),…, p(n)⟩ yields a finite sequence and is defined by: (Def.1) (i) len⟨p(m),…, p(n)⟩ + m = n + 1 and for every natural number i such

that i < len⟨p(m),…, p(n)⟩ holds ⟨p(m),…, p(n)⟩(i + 1) = p(m + i) if 1 ≤ m and m ≤ n + 1 and n ≤ len p,

(ii) ⟨p(m),…, p(n)⟩ = ε, otherwise. We now state several propositions:

(6) If 1 ≤ m and m ≤ len p, then ⟨p(m),…, p(m)⟩ = ⟨p(m)⟩.

(7) ⟨p(1),…, p(len p)⟩ = p.

(8) If m ≤ n and n ≤ r and r ≤ len p, then ⟨p(m + 1),…, p(n)⟩^{}⟨p(n + 1),…, p(r)⟩ = ⟨p(m + 1),…, p(r)⟩.

(9) If 1 ≤ m and m ≤ len p, then ⟨p(1),…, p(m)⟩^{}⟨p(m+1),…, p(len p)⟩ = p.

(10) If 1 ≤ m and m ≤ n and n ≤ len p, then ⟨p(1),…, p(m)⟩ ^{}⟨p(m + 1),…, p(n)⟩^{}⟨p(n + 1),…, p(len p)⟩ = p.

(11) rng⟨p(m),…, p(n)⟩ ⊆ rng p.

Let D be a set, let p be a finite sequence of elements of D, and let m, n be natural numbers. Then ⟨p(m),…, p(n)⟩ is a finite sequence of elements of D.

Next we state the proposition (12) If p ≠ ε and 1 ≤ m and m ≤ n and n ≤ len p, then ⟨p(m),…, p(n)⟩(1) =

p(m) and ⟨p(m),…, p(n)⟩(len⟨p(m),…, p(n)⟩) = p(n).

3. The glueing catenation of finite sequences

Let p, q be finite sequences. The functor p q yielding a finite sequence is defined as follows: (Def.2) p q = p^{}⟨q(2),…, q(len q)⟩.

Next we state several propositions:

(13) If q ≠ ε, then len(p q) + 1 = len p + len q.

(14) If 1 ≤ k and k ≤ len p, then (p q)(k) = p(k).

(15) If 1 ≤ k and k < len q, then (p q)(len p + k) = q(k + 1).

(16) If 1 < len q, then (p q)(len(p q)) = q(len q).

(17) rng(p q) ⊆ rng p ∪ rng q.

Let D be a set and let p, q be finite sequences of elements of D. Then p q is a finite sequence of elements of D.

Next we state the proposition

(18) If p ≠ ε and q ≠ ε and p(len p) = q(1), then rng(p q) = rng p ∪ rng q.

4. Two valued alternating finite sequences

A finite sequence is two-valued if: (Def.3) card rng it = 2.

The following proposition is true

(19) p is two-valued iff len p > 1 and there exist arbitrary x, y such that

x ≠ y and rng p = {x, y}. A finite sequence is alternating if:

(Def.4) For every natural number i such that 1 ≤ i and i + 1 ≤ len it holds

it(i) ≠ it(i + 1). One can check that there exists a finite sequence which is two-valued and

alternating.

In the sequel a, a_{1}, a_{2} are two-valued alternating finite sequences.

One can prove the following propositions:

(20) If len a_{1} = len a_{2} and rng a_{1} = rng a_{2} and a_{1}(1) = a_{2}(1), then a_{1} = a_{2}.

(21) If a_{1} ≠ a_{2} and len a_{1} = len a_{2} and rng a_{1} = rng a_{2}, then for every i such that 1 ≤ i and i ≤ len a_{1} holds a_{1}(i) ≠ a_{2}(i).

(22) If a_{1} ≠ a_{2} and len a_{1} = len a_{2} and rng a_{1} = rng a_{2}, then for every a such that len a = len a_{1} and rng a = rng a_{1} holds a = a_{1} or a = a_{2}.

(23) If X ≠ Y and n > 1, then there exists a_{1} such that rng a_{1} = {X, Y } and len a_{1} = n and a_{1}(1) = X.

5. Finite subsequence of finite sequences

Let us consider X and let f_{1} be a finite sequence of elements of X. A finite subsequence is called a FinSubsequence of f_{1} if: (Def.5) It ⊆ f_{1}.

In the sequel s_{1} will denote a finite subsequence. The following propositions are true:

(24) If s_{1} is a finite sequence, then Seq s_{1} = s_{1}.

(25) If rng p ⊆ dom s_{1}, then s_{1} · p is a finite sequence.

(26) Let f be a finite subsequence and let g, h, f_{2}, f_{3}, f_{4} be finite sequences. If rng g ⊆ dom f and rng h ⊆ dom f and f_{2} = f · g and f_{3} = f · h and f_{4} = f · (g^{}h), then f_{4} = f_{2} ^{}f_{3}.

We follow the rules: f_{1}, f_{5}, f_{6} will be finite sequences of elements of X and f_{7}, f_{8} will be FinSubsequence of f_{1}.

We now state four propositions:

(27) dom f_{7} ⊆ dom f_{1} and rng f_{7} ⊆ rng f_{1}.

(28) f_{1} is a FinSubsequence of f_{1}.

(29) f_{7} Y is a FinSubsequence of f_{1}.

(30) For every FinSubsequence f_{9} of f_{5} such that Seq f_{7} = f_{5} and Seq f_{9} = f_{6} and f_{8} = f_{7} rng(Sgm dom f_{7} dom f_{9}) holds Seq f_{8} = f_{6}.

6. Vertex sequences induced by chains

In the sequel G is a graph. Let us consider G. One can verify that the vertices of G is non empty. We follow the rules: v, v_{1}, v_{2}, v_{3}, v_{4} will denote elements of the vertices of

G and e will be arbitrary.

We now state two propositions:

(31) If e joins v_{1} with v_{2}, then e joins v_{2} with v_{1}.

(32) If e joins v_{1} with v_{2} and e joins v_{3} with v_{4}, then v_{1} = v_{3} and v_{2} = v_{4} or

v_{1} = v_{4} and v_{2} = v_{3}. Let us consider G. We see that the chain of G is a finite sequence of elements

of the edges of G.

Let us consider G. A path of G is a path-like chain of G. We follow the rules: v_{5}, v_{6}, v_{7} will denote finite sequences of elements of the

vertices of G and c, c_{1}, c_{2} will denote chains of G. The following proposition is true (33) ε is a chain of G.

Let us consider G. One can check that there exists a chain of G which is empty.

Let us consider G, X. The functor (G) -VSet(X) yields a set and is defined as follows: (Def.6) (G) -VSet(X) = {v : ⋁_{e: element of the edges of G} e ∈ X ∧ (v = (the source

of G)(e) ∨ v = (the target of G)(e))}. Let us consider G, v_{5} and let c be a finite sequence. We say that v_{5} is vertex

sequence of c if and only if: (Def.7) len v_{5} = len c + 1 and for every n such that 1 ≤ n and n ≤ len c holds

c(n) joins π_{n}v_{5} with π_{n+1}v_{5}. One can prove the following four propositions:

(34) If c ≠ ε and v_{5} is vertex sequence of c, then (G) -VSet(rng c) = rng v_{5}.

(35) ⟨v⟩ is vertex sequence of ε.

(36) There exists v_{5} which is vertex sequence of c.

(37) Suppose c ≠ ε and v_{6} is vertex sequence of c and v_{7} is vertex sequence of c and v_{6} ≠ v_{7}. Then v_{6}(1) ≠ v_{7}(1) and for every v_{5} such that v_{5} is vertex sequence of c holds v_{5} = v_{6} or v_{5} = v_{7}.

Let us consider G and let c be a finite sequence. We say that c alternates vertices in G if and only if:

(Def.8) len c ≥ 1 and \overline{\overline{(G) -VSet(rng c)}} = 2 and for every n such that n ∈ dom c

holds (the source of G)(c(n)) ≠ (the target of G)(c(n)). One can prove the following propositions:

(38) If c alternates vertices in G and v_{5} is vertex sequence of c, then for every k such that k ∈ dom c holds v_{5}(k) ≠ v_{5}(k + 1).

(39) Suppose c alternates vertices in G and v_{5} is vertex sequence of c. Then rng v_{5} = {(the source of G)(c(1)), (the target of G)(c(1))}.

(40) Suppose c alternates vertices in G and v_{5} is vertex sequence of c. Then v_{5} is a two-valued alternating finite sequence.

(41) Suppose c alternates vertices in G. Then there exist v_{6}, v_{7} such that (i) v_{6} ≠ v_{7}, (ii) v_{6} is vertex sequence of c, (iii) v_{7} is vertex sequence of c, and (iv) for every v_{5} such that v_{5} is vertex sequence of c holds v_{5} = v_{6} or

v_{5} = v_{7}.

(42) Suppose v_{5} is vertex sequence of c. Then \overline{\overline{the vertices of G}} = 1 or c ≠ ε and c does not alternate vertices in G if and only if for every v_{6} such that v_{6} is vertex sequence of c holds v_{6} = v_{5}.

Let us consider G, c. Let us assume that \overline{\overline{the vertices of G}} = 1 or c ≠ ε and c does not alternate vertices in G. The functor vertex-seq(c) yielding a finite sequence of elements of the vertices of G is defined as follows:

(Def.9) vertex-seq(c) is vertex sequence of c.

We now state several propositions:

(43) If v_{5} is vertex sequence of c and c_{1} = c Seg n and v_{6} = v_{5} Seg(n + 1), then v_{6} is vertex sequence of c_{1}.

(44) If 1 ≤ m and m ≤ n and n ≤ len c and q = ⟨c(m),…, c(n)⟩, then q is a chain of G.

(45) If 1 ≤ m and m ≤ n and n ≤ len c and c_{1} = ⟨c(m),…, c(n)⟩ and v_{5} is vertex sequence of c and v_{6} = ⟨v_{5}(m),…, v_{5}(n + 1)⟩, then v_{6} is vertex sequence of c_{1}.

(46) If v_{6} is vertex sequence of c_{1} and v_{7} is vertex sequence of c_{2} and v_{6}(len v_{6}) = v_{7}(1), then c_{1} ^{}c_{2} is a chain of G.

(47) Suppose v_{6} is vertex sequence of c_{1} and v_{7} is vertex sequence of c_{2} and v_{6}(len v_{6}) = v_{7}(1) and c = c_{1} ^{}c_{2} and v_{5} = v_{6} v_{7}. Then v_{5} is vertex sequence of c.

7. Vertex sequences induced by simple chains, paths and ordered chains

Let us consider G. A chain of G is simple if it satisfies the condition (Def.10). (Def.10) There exists v_{5} such that v_{5} is vertex sequence of it and for all n, m

such that 1 ≤ n and n < m and m ≤ len v_{5} and v_{5}(n) = v_{5}(m) holds n = 1 and m = len v_{5}.

Let us consider G. One can check that there exists a chain of G which is simple.

In the sequel s_{2} denotes a simple chain of G. Next we state several propositions:

(49)^{2} s_{2} Seg n is a simple chain of G.

(50) If 2 < len s_{2} and v_{6} is vertex sequence of s_{2} and v_{7} is vertex sequence of s_{2}, then v_{6} = v_{7}.

(51) If v_{5} is vertex sequence of s_{2}, then for all n, m such that 1 ≤ n and n < m and m ≤ len v_{5} and v_{5}(n) = v_{5}(m) holds n = 1 and m = len v_{5}.

(52) Suppose c is not a simple chain of G and v_{5} is vertex sequence of c. Then there exists a FinSubsequence f_{10} of c and there exists a FinSubsequence f_{11} of v_{5} and there exist c_{1}, v_{6} such that len c_{1} < len c and v_{6} is vertex sequence of c_{1} and len v_{6} < len v_{5} and v_{5}(1) = v_{6}(1) and v_{5}(len v_{5}) = v_{6}(len v_{6}) and Seq f_{10} = c_{1} and Seq f_{11} = v_{6}.

(53) Suppose v_{5} is vertex sequence of c. Then there exists a FinSubsequence f_{10} of c and there exists a FinSubsequence f_{11} of v_{5} and there exist s_{2}, v_{6} such that Seq f_{10} = s_{2} and Seq f_{11} = v_{6} and v_{6} is vertex sequence of s_{2} and v_{5}(1) = v_{6}(1) and v_{5}(len v_{5}) = v_{6}(len v_{6}).

2The proposition (48) has been removed.

vertex sequences induced by chains 303

Let us consider G. One can check that every chain of G which is empty is also path-like.

We now state the proposition

(54) If p is a path of G, then p Seg n is a path of G.

Let us consider G. One can verify that there exists a path of G which is simple.

We now state two propositions:

(55) If 2 < len s_{2}, then s_{2} is a path of G.

(56) s_{2} is a path of G iff len s_{2} = 0 or len s_{2} = 1 or s_{2}(1) ≠ s_{2}(2). Let us consider G. Observe that every chain of G which is empty is also

oriented.

Let us consider G and let o_{1} be an oriented chain of G. Let us assume that o_{1} ≠ ε. The functor vertex-seq(o_{1}) yields a finite sequence of elements of the vertices of G and is defined as follows:

(Def.11) vertex-seq(o_{1}) is vertex sequence of o_{1} and (vertex-seq(o_{1}))(1) = (the source of G)(o_{1}(1)).

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On the Lattice of Subspaces of a Vector Space

Andrzej Iwaniuk Warsaw University Białystok

MML Identifier: VECTSP 8.

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

In this paper F denotes a field and V_{1} denotes a strict vector space over F. Let us consider F, V_{1}. The functor _{(V_{1})} yields a strict bounded lattice and

is defined as follows: (Def.1) _{(V_{1})} = ⟨Subspaces V_{1}, SubJoin V_{1}, SubMeet V_{1}⟩.

Let us consider F, V_{1}. Family of subspaces of V_{1} is defined as follows: (Def.2) For arbitrary x such that x ∈ it holds x is a subspace of V_{1}.

Let us consider F, V_{1}. Note that there exists a family of subspaces of V_{1} which is non empty.

Let us consider F, V_{1}. Then Subspaces V_{1} is a non empty family of subspaces of V_{1}. Let X be a non empty family of subspaces of V_{1}. We see that the element of X is a subspace of V_{1}.

Let us consider F, V_{1} and let x be an element of Subspaces V_{1}. The functor \overline{x} yielding a subset of the carrier of V_{1} is defined as follows: (Def.3) There exists a subspace X of V_{1} such that x = X and \overline{x} = the carrier

of X. Let us consider F, V_{1}. The functor \overline{V_{1}} yielding a function from Subspaces V_{1}

into 2^{^{the} ^{carrier} ^{of} V_{1}} is defined by: (Def.4) For every element h of Subspaces V_{1} and for every subspace H of V_{1}

such that h = H holds \overline{V_{1}}(h) = the carrier of H. We now state two propositions:

(1) For every strict vector space V_{1} over F and for every non empty subset H of Subspaces V_{1} holds \overline{V_{1}}^{∘}H is non empty.

(2) For every strict vector space V_{1} over F and for every strict subspace H of V_{1} holds 0_{(V_{1})} ∈ \overline{V_{1}}(H).

Let us consider F, V_{1} and let G be a non empty subset of Subspaces V_{1}. The functor ⋂ G yielding a strict subspace of V_{1} is defined by: (Def.5) The carrier of ⋂ G = ⋂(\overline{V_{1}}^{∘}G).

Next we state several propositions:

(3) Subspaces V_{1} = the carrier of _{(V_{1})}.

(4) The meet operation of _{(V_{1})} = SubMeet V_{1}.

(5) The join operation of _{(V_{1})} = SubJoin V_{1}.

(6) Let V_{1} be a strict vector space over F, and let p, q be elements of the carrier of _{(V_{1})}, and let H_{1}, H_{2} be strict subspaces of V_{1}. Suppose p = H_{1} and q = H_{2}. Then p ⊑ q if and only if the carrier of H_{1} ⊆ the carrier of H_{2}.

(7) Let V_{1} be a strict vector space over F, and let p, q be elements of the carrier of _{(V_{1})}, and let H_{1}, H_{2} be subspaces of V_{1}. If p = H_{1} and q = H_{2}, then p ⊔ q = H_{1} + H_{2}.

(8) Let V_{1} be a strict vector space over F, and let p, q be elements of the carrier of _{(V_{1})}, and let H_{1}, H_{2} be subspaces of V_{1}. If p = H_{1} and q = H_{2}, then p ⊓ q = H_{1} ∩ H_{2}.

Let us observe that a non empty lattice structure is complete if it satisfies the condition (Def.6). (Def.6) Let X be a subset of the carrier of it. Then there exists an element a

of the carrier of it such that a ⊑ X and for every element b of the carrier of it such that b ⊑ X holds b ⊑ a.

The following propositions are true:

(9) For every V_{1} holds _{(V_{1})} is complete.

(10) Let x be arbitrary, and let V_{1} be a strict vector space over F, and let S be a subset of the carrier of V_{1}. If S is non empty and linearly closed, then if x ∈ Lin(S), then x ∈ S.

Let F be a field, let A, B be strict vector spaces over F, and let f be a function from the carrier of A into the carrier of B. The functor FuncLatt(f) yields a function from the carrier of _{A} into the carrier of _{B} and is defined by the condition (Def.7). (Def.7) Let S be a strict subspace of A and let B_{0} be a subset of the carrier of

B. If B_{0} = f^{∘}(the carrier of S), then (FuncLatt(f))(S) = Lin(B_{0}). Let L_{1}, L_{2} be lattices. A function from the carrier of L_{1} into the carrier of

L_{2} is called a lower homomorphism between L_{1} and L_{2} if: (Def.8) For all elements a, b of the carrier of L_{1} holds it(a ⊓ b) = it(a) ⊓ it(b).

Let L_{1}, L_{2} be lattices. A function from the carrier of L_{1} into the carrier of L_{2} is called an upper homomorphism between L_{1} and L_{2} if: (Def.9) For all elements a, b of the carrier of L_{1} holds it(a ⊔ b) = it(a) ⊔ it(b).

on the lattice of subspaces of a vector space 307

One can prove the following propositions:

(11) Let L_{1}, L_{2} be lattices and let f be a function from the carrier of L_{1} into the carrier of L_{2}. Then f is a homomorphism from L_{1} to L_{2} if and only if f is an upper homomorphism between L_{1} and L_{2} and a lower homomorphism between L_{1} and L_{2}.

(12) Let F be a field, and let A, B be strict vector spaces over F, and let f be a function from the carrier of A into the carrier of B. If f is linear, then FuncLatt(f) is an upper homomorphism between _{A} and _{B}.

(13) Let F be a field, and let A, B be strict vector spaces over F, and let f be a function from the carrier of A into the carrier of B. Suppose f is one-to-one and linear. Then FuncLatt(f) is a homomorphism from _{A} to B.

(14) Let A, B be strict vector spaces over F and let f be a function from the carrier of A into the carrier of B. If f is linear and one-to-one, then FuncLatt(f) is one-to-one.

(15) Let A be a strict vector space over F and let f be a function from the carrier of A into the carrier of A. If f = id_{(the carrier of A)}, then FuncLatt(f) = id_{(the carrier of _{A})} .

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[13] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877–882, 1990.

[14] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics, 1(2):395–399, 1990.

[15] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871–876, 1990.

[16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865–870, 1990.

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

308 andrzej iwaniuk

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

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

[20] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

On the Lattice of Subgroups of a Group

Janusz Ganczarski Warsaw University Białystok

MML Identifier: LATSUBGR.

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

The following propositions are true:

(1) Let G be a group and let H_{1}, H_{2} be subgroups of G. Then the carrier of H_{1} ∩ H_{2} = (the carrier of H_{1}) ∩ (the carrier of H_{2}).

(2) For every group G and for arbitrary h holds h ∈ SubGr G iff there exists a strict subgroup H of G such that h = H.

(3) Let G be a group, and let A be a subset of the carrier of G, and let H be a strict subgroup of G. If A = the carrier of H, then gr(A) = H.

(4) Let G be a group, and let H_{1}, H_{2} be subgroups of G, and let A be a subset of the carrier of G. If A = (the carrier of H_{1}) ∪ (the carrier of H_{2}), then H_{1} ⊔ H_{2} = gr(A).

(5) Let G be a group, and let H_{1}, H_{2} be subgroups of G, and let g be an element of the carrier of G. If g ∈ H_{1} or g ∈ H_{2}, then g ∈ H_{1} ⊔ H_{2}.

(6) Let G_{1}, G_{2} be groups, and let f be a homomorphism from G_{1} to G_{2}, and let H_{1} be a subgroup of G_{1}. Then there exists a strict subgroup H_{2} of G_{2} such that the carrier of H_{2} = f^{∘}(the carrier of H_{1}).

(7) Let G_{1}, G_{2} be groups, and let f be a homomorphism from G_{1} to G_{2}, and let H_{2} be a subgroup of G_{2}. Then there exists a strict subgroup H_{1} of G_{1} such that the carrier of H_{1} = f ^{−1} (the carrier of H_{2}).

(8) Let G_{1}, G_{2} be groups, and let f be a homomorphism from G_{1} to G_{2}, and let H_{1}, H_{2} be subgroups of G_{1}. Suppose the carrier of H_{1} ⊆ the carrier of H_{2}. Then f^{∘}(the carrier of H_{1}) ⊆ f^{∘}(the carrier of H_{2}).

(9) Let G_{1}, G_{2} be groups, and let f be a homomorphism from G_{1} to G_{2}, and let H_{1}, H_{2} be subgroups of G_{2}. Suppose the carrier of H_{1} ⊆ the carrier of H_{2}. Then f ^{−1} (the carrier of H_{1}) ⊆ f ^{−1} (the carrier of H_{2}).

(10) Let G_{1}, G_{2} be groups, and let f be a homomorphism from G_{1} to G_{2}, and let H_{1}, H_{2} be subgroups of G_{1}, and let H_{3}, H_{4} be subgroups of G_{2}. Suppose the carrier of H_{3} = f^{∘}(the carrier of H_{1}) and the carrier of H_{4} = f^{∘}(the carrier of H_{2}). If H_{1} is a subgroup of H_{2}, then H_{3} is a subgroup of H_{4}.

(11) Let G_{1}, G_{2} be groups, and let f be a homomorphism from G_{1} to G_{2}, and let H_{1}, H_{2} be subgroups of G_{2}, and let H_{3}, H_{4} be subgroups of G_{1}. Suppose the carrier of H_{3} = f ^{−1} (the carrier of H_{1}) and the carrier of H_{4} = f ^{−1} (the carrier of H_{2}). If H_{1} is a subgroup of H_{2}, then H_{3} is a subgroup of H_{4}.

(12) Let G_{1}, G_{2} be groups, and let f be a function from the carrier of G_{1} into the carrier of G_{2}, and let A be a subset of the carrier of G_{1}. Then f^{∘}A ⊆ f^{∘}(the carrier of gr(A)).

(13) Let G_{1}, G_{2} be groups, and let H_{1}, H_{2} be subgroups of G_{1}, and let f be a function from the carrier of G_{1} into the carrier of G_{2}, and let A be a subset of the carrier of G_{1}. Suppose A = (the carrier of H_{1}) ∪ (the carrier of H_{2}). Then f^{∘}(the carrier of H_{1} ⊔ H_{2}) = f^{∘}(the carrier of gr(A)).

(14) For every group G and for every subset A of the carrier of G such that A = {1_{G}} holds gr(A) = {1}_{G}.

(15) For all non empty sets X, Y and for all subsets A_{1}, A_{2} of Y and for every function f from X into Y holds f ^{−1} (A_{1} ∪ A_{2}) = f ^{−1} A_{1} ∪ f ^{−1} A_{2}.

(16) For all non empty sets X, Y and for all subsets A_{1}, A_{2} of X and for

every function f from X into Y holds f^{∘}(A_{1} ∪ A_{2}) = f^{∘}A_{1} ∪ f^{∘}A_{2}. Let G be a group. The functor \overline{G} yields a function from SubGr G into

2^{the carrier of G} and is defined as follows: (Def.1) For every element h of SubGr G and for every subgroup H of G such

that h = H holds \overline{G}(h) = the carrier of H. Next we state several propositions:

(17) Let G be a group, and let h be an element of SubGr G, and let H be a subgroup of G. If h = H, then \overline{G}(h) = the carrier of H.

(18) Let G be a group, and let H be a strict subgroup of G, and let x be an element of the carrier of G. Then x ∈ \overline{G}(H) if and only if x ∈ H.

(19) For every group G and for every strict subgroup H of G holds 1_{G} ∈ \overline{G}(H).

(20) For every group G and for every strict subgroup H of G holds \overline{G}(H) ≠ ∅.

(21) Let G be a group, and let H be a strict subgroup of G, and let g_{1}, g_{2} be elements of the carrier of G. If g_{1} ∈ \overline{G}(H) and g_{2} ∈ \overline{G}(H), then g_{1} · g_{2} ∈ \overline{G}(H).

(22) Let G be a group, and let H be a strict subgroup of G, and let g be an element of the carrier of G. If g ∈ \overline{G}(H), then g^{−1} ∈ \overline{G}(H).

(23) For every group G and for all strict subgroups H_{1}, H_{2} of G holds the carrier of H_{1} ∩ H_{2} = \overline{G}(H_{1}) ∩ \overline{G}(H_{2}).

(24) For every group G and for all strict subgroups H_{1}, H_{2} of G holds \overline{G}(H_{1}∩

H_{2}) = \overline{G}(H_{1}) ∩ \overline{G}(H_{2}). Let G be a group and let F be a non empty subset of SubGr G. The functor

⋂ F yielding a strict subgroup of G is defined by: (Def.2) The carrier of ⋂ F = ⋂(\overline{G}^{∘}F).

Next we state several propositions:

(25) For every group G and for every non empty subset F of SubGr G such that {1}_{G} ∈ F holds ⋂ F = {1}_{G}.

(26) For every group G and for every element h of SubGr G and for every non empty subset F of SubGr G such that F = {h} holds ⋂ F = h.

(27) Let G be a group, and let H_{1}, H_{2} be subgroups of G, and let h_{1}, h_{2} be elements of the carrier of _{G}. If h_{1} = H_{1} and h_{2} = H_{2}, then h_{1} ⊔ h_{2} = H_{1} ⊔ H_{2}.

(28) Let G be a group, and let H_{1}, H_{2} be subgroups of G, and let h_{1}, h_{2} be elements of the carrier of _{G}. If h_{1} = H_{1} and h_{2} = H_{2}, then h_{1} ⊓ h_{2} = H_{1} ∩ H_{2}.

(29) Let G be a group, and let p be an element of the carrier of _{G}, and let H be a subgroup of G. If p = H, then H is a strict subgroup of G.

(30) Let G be a group, and let H_{1}, H_{2} be subgroups of G, and let p, q be elements of the carrier of _{G}. Suppose p = H_{1} and q = H_{2}. Then p ⊑ q if and only if the carrier of H_{1} ⊆ the carrier of H_{2}.

(31) Let G be a group, and let H_{1}, H_{2} be subgroups of G, and let p, q be elements of the carrier of _{G}. If p = H_{1} and q = H_{2}, then p ⊑ q iff H_{1} is a subgroup of H_{2}.

(32) For every group G holds _{G} is complete.

Let G_{1}, G_{2} be groups and let f be a function from the carrier of G_{1} into the carrier of G_{2}. The functor FuncLatt(f) yielding a function from the carrier of (G_{1}) into the carrier of _{(G_{2})} is defined by the condition (Def.3). (Def.3) Let H be a strict subgroup of G_{1} and let A be a subset of the carrier

of G_{2}. If A = f^{∘}(the carrier of H), then (FuncLatt(f))(H) = gr(A). One can prove the following propositions:

(33) Let G be a group and let f be a function from the carrier of G into the carrier of G. If f = id_{(the carrier of G)}, then FuncLatt(f) = id_{(the carrier of _{G})} .

(34) For all groups G_{1}, G_{2} and for every homomorphism f from G_{1} to G_{2} such that f is one-to-one holds FuncLatt(f) is one-to-one.

(35) For all groups G_{1}, G_{2} and for every homomorphism f from G_{1} to G_{2} holds (FuncLatt(f))({1}_{(G_{1})}) = {1}_{(G_{2})}.

(36) Let G_{1}, G_{2} be groups and let f be a homomorphism from G_{1} to G_{2}. Suppose f is one-to-one. Then FuncLatt(f) is a lower homomorphism between _{(G_{1})} and _{(G_{2})}.

312 janusz ganczarski

(37) Let G_{1}, G_{2} be groups and let f be a homomorphism from G_{1} to G_{2}. Then FuncLatt(f) is an upper homomorphism between _{(G_{1})} and _{(G_{2})}.

(38) Let G_{1}, G_{2} be groups and let f be a homomorphism from G_{1} to G_{2}. If f is one-to-one, then FuncLatt(f) is a homomorphism from _{(G_{1})} to (G_{2}).

References

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

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

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

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

[5] Andrzej Iwaniuk. On the lattice of subspaces of a vector space. Formalized Mathematics, 5(3):305–308, 1996.

[6] Jolanta Kamieńska and Jarosław Stanisław Walijewski. Homomorphisms of lattices, finite join and finite meet. Formalized Mathematics, 4(1):35–40, 1993.

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

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

[9] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955–962, 1990.

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

[11] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.

[12] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855–864, 1990.

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

[14] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.

[15] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

[17] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

On the Lattice of Subalgebras of a Universal Algebra

Mirosław Jan Paszek Warsaw University Białystok

MML Identifier: UNIALG 3.

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

In this paper U_{0} is a universal algebra, H is a non empty subset of the carrier of U_{0}, and o is an operation of U_{0}.

Let us consider U_{0}. Family of subalgebras of U_{0} is defined by: (Def.1) For arbitrary U_{1} such that U_{1} ∈ it holds U_{1} is a subalgebra of U_{0}.

Let us consider U_{0}. One can check that there exists a family of subalgebras of U_{0} which is non empty.

Let us consider U_{0}. Then Subalgebras(U_{0}) is a non empty family of subalgebras of U_{0}. Let U_{2} be a non empty family of subalgebras of U_{0}. We see that the element of U_{2} is a subalgebra of U_{0}.

Let us consider U_{0}. Then ⨆_{(U_{0})} is a binary operation on Subalgebras(U_{0}). Then ⌈−⌉_{(U_{0})} is a binary operation on Subalgebras(U_{0}).

Let us consider U_{0} and let u be an element of Subalgebras(U_{0}). The functor \overline{u} yielding a subset of the carrier of U_{0} is defined as follows: (Def.2) There exists a subalgebra U_{1} of U_{0} such that u = U_{1} and \overline{u} = the carrier

of U_{1}. Let us consider U_{0}. The functor Carr(U_{0}) yields a function from Subalgebras(U_{0})

into 2^{^{the} ^{carrier} ^{of} U_{0}} and is defined by: (Def.3) For every element u of Subalgebras(U_{0}) holds (Carr(U_{0}))(u) = \overline{u}.

We now state several propositions:

(1) For arbitrary u holds u ∈ Subalgebras(U_{0}) iff there exists a strict subalgebra U_{1} of U_{0} such that u = U_{1}.

(2) Let H be a non empty subset of the carrier of U_{0} and given o. If arity o = 0, then H is closed on o iff o(ε) ∈ H.

(3) For every subalgebra U_{1} of U_{0} holds the carrier of U_{1} ⊆ the carrier of U_{0}.

(4) For every non empty subset H of the carrier of U_{0} and for every o such that H is closed on o and arity o = 0 holds o_{H} = o.

(5) If U_{0} has constants, then Constants(U_{0}) = {o(ε) : o ranges over operation of U_{0}, arity o = 0}.

(6) For every universal algebra U_{0} with constants and for every subalgebra U_{1} of U_{0} holds Constants(U_{0}) = Constants(U_{1}).

Let U_{0} be a universal algebra with constants. Note that every subalgebra of U_{0} has constants.

The following proposition is true

(7) For every universal algebra U_{0} with constants and for all subalgebras U_{1}, U_{3} of U_{0} holds Constants(U_{1}) = Constants(U_{3}).

Let us consider U_{0}. Then Carr(U_{0}) is a function from Subalgebras(U_{0}) into 2^{^{the} ^{carrier} ^{of} U_{0}} and it can be characterized by the condition:

(Def.4) For every element u of Subalgebras(U_{0}) and for every subalgebra U_{1} of

U_{0} such that u = U_{1} holds (Carr(U_{0}))(u) = the carrier of U_{1}. One can prove the following propositions:

(8) For every strict subalgebra H of U_{0} and for every element u of U_{0} holds u ∈ (Carr(U_{0}))(H) iff u ∈ H.

(9) For every non empty subset H of Subalgebras(U_{0}) holds (Carr(U_{0}))^{∘}H is non empty.

(10) For every universal algebra U_{0} with constants and for every strict subalgebra U_{1} of U_{0} holds Constants(U_{0}) ⊆ (Carr(U_{0}))(U_{1}).

(11) Let U_{0} be a universal algebra with constants, and let U_{1} be a subalgebra of U_{0}, and let a be arbitrary. If a is an element of Constants(U_{0}), then a ∈ the carrier of U_{1}.

(12) Let U_{0} be a universal algebra with constants and let H be a non empty subset of Subalgebras(U_{0}). Then ⋂((Carr(U_{0}))^{∘}H) is a non empty subset of the carrier of U_{0}.

(13) For every universal algebra U_{0} with constants holds the carrier of the lattice of subalgebras of U_{0} = Subalgebras(U_{0}).

(14) Let U_{0} be a universal algebra with constants, and let H be a non empty subset of Subalgebras(U_{0}), and let S be a non empty subset of the carrier of U_{0}. If S = ⋂((Carr(U_{0}))^{∘}H), then S is operations closed.

Let U_{0} be a strict universal algebra with constants and let H be a non empty subset of Subalgebras(U_{0}). The functor ⋂ H yielding a strict subalgebra of U_{0} is defined as follows:

(Def.5) The carrier of ⋂ H = ⋂((Carr(U_{0}))^{∘}H).

One can prove the following propositions:

(15) Let U_{0} be a universal algebra with constants, and let l_{1}, l_{2} be elements of the carrier of the lattice of subalgebras of U_{0}, and let U_{1}, U_{3} be strict subalgebras of U_{0}. If l_{1} = U_{1} and l_{2} = U_{3}, then l_{1} ⊔ l_{2} = U_{1} ⨆ U_{3}.

(16) Let U_{0} be a universal algebra with constants, and let l_{1}, l_{2} be elements of the carrier of the lattice of subalgebras of U_{0}, and let U_{1}, U_{3} be strict subalgebras of U_{0}. If l_{1} = U_{1} and l_{2} = U_{3}, then l_{1} ⊓ l_{2} = U_{1} ∩ U_{3}.

(17) Let U_{0} be a universal algebra with constants, and let l_{1}, l_{2} be elements of the carrier of the lattice of subalgebras of U_{0}, and let U_{1}, U_{3} be strict subalgebras of U_{0}. Suppose l_{1} = U_{1} and l_{2} = U_{3}. Then l_{1} ⊑ l_{2} if and only if the carrier of U_{1} ⊆ the carrier of U_{3}.

(18) Let U_{0} be a universal algebra with constants, and let l_{1}, l_{2} be elements of the carrier of the lattice of subalgebras of U_{0}, and let U_{1}, U_{3} be strict subalgebras of U_{0}. If l_{1} = U_{1} and l_{2} = U_{3}, then l_{1} ⊑ l_{2} iff U_{1} is a subalgebra of U_{3}.

(19) For every strict universal algebra U_{0} with constants holds the lattice of subalgebras of U_{0} is bounded.

(20) For every strict universal algebra U_{0} with constants and for every strict subalgebra U_{1} of U_{0} holds Gen^{UA}(Constants(U_{0})) ∩ U_{1} = Gen^{UA}(Constants(U_{0})).

(21) For every strict universal algebra U_{0} with constants holds ⊥_{_{the} _{lattice} _{of} _{subalgebras} _{of} U_{0}} = Gen^{UA}(Constants(U_{0})).

(22) Let U_{0} be a strict universal algebra with constants, and let U_{1} be a subalgebra of U_{0}, and let H be a subset of the carrier of U_{0}. If H = the carrier of U_{0}, then Gen^{UA}(H) ⨆ U_{1} = Gen^{UA}(H).

(23) Let U_{0} be a strict universal algebra with constants and let H be a subset of the carrier of U_{0}. Suppose H = the carrier of U_{0}. Then ⊤_{_{the} _{lattice} _{of} _{subalgebras} _{of} U_{0}} = Gen^{UA}(H).

(24) For every strict universal algebra U_{0} with constants holds ⊤_{_{the} _{lattice} _{of} _{subalgebras} _{of} U_{0}} = U_{0}.

(25) For every strict universal algebra U_{0} with constants holds the lattice of

subalgebras of U_{0} is complete. Let U_{4}, U_{5} be universal algebras with constants and let F be a function from

the carrier of U_{4} into the carrier of U_{5}. The functor FuncLatt(F) yielding a function from the carrier of the lattice of subalgebras of U_{4} into the carrier of the lattice of subalgebras of U_{5} is defined by the condition (Def.6). (Def.6) Let U_{1} be a strict subalgebra of U_{4} and let H be a subset of the carrier of

U_{5}. If H = F^{∘}(the carrier of U_{1}), then (FuncLatt(F))(U_{1}) = Gen^{UA}(H). We now state the proposition

(26) Let U_{0} be a strict universal algebra with constants and let F be a function from the carrier of U_{0} into the carrier of U_{0}. Suppose F = id_{(the carrier of U_{0})}. Then FuncLatt(F) = id_{(the carrier of the lattice of subalgebras of U_{0})}.

316 mirosław jan paszek

References

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

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

[3] Ewa Burakowska. Subalgebras of the universal algebra. Lattices of subalgebras. Formalized Mathematics, 4(1):23–27, 1993.

[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.

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

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

[9] Jolanta Kamieńska and Jarosław Stanisław Walijewski. Homomorphisms of lattices, finite join and finite meet. Formalized Mathematics, 4(1):35–40, 1993.

[10] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra. Formalized Mathematics, 3(2):251–253, 1992.

[11] Agnieszka Julia Marasik. Boolean properties of lattices. Formalized Mathematics, 5(1):31–35, 1996.

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

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

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

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

[16] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

[18] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

On the Decomposition of Finite Sequences

Andrzej Trybulec Warsaw University Białystok

MML Identifier: FINSEQ 6.

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

1. Preliminaries

We introduce degenerated as a synonym of trivial. Let us observe that every set which is non trivial is also non empty. In the sequel x, y, z will be arbitrary. Let us consider x, y. Observe that ⟨x, y⟩ is non trivial. Let us consider x, y, z. Note that ⟨x, y, z⟩ is non trivial. Let f be a non empty finite sequence. One can check that Rev(f) is non

empty.

2. Decomposing a finite sequence

For simplicity we adopt the following rules: f_{1}, f_{2}, f_{3} will denote finite sequences, p, p_{1}, p_{2}, p_{3} will be arbitrary, f will denote a finite sequence, and i, k will denote natural numbers.

Next we state a number of propositions: (3)^{1} For every set X and for every i such that X ⊆ Seg i and 1 ∈ X holds

(Sgm X)(1) = 1.

(4) For every finite sequence f such that k ∈ dom f and for every i such that 1 ≤ i and i < k holds f(i) ≠ f(k) holds f(k) f = k.

1The propositions (1) and (2) have been removed.

(5) ⟨p_{1}, p_{2}⟩ Seg 1 = ⟨p_{1}⟩.

(6) ⟨p_{1}, p_{2}, p_{3}⟩ Seg 1 = ⟨p_{1}⟩.

(7) ⟨p_{1}, p_{2}, p_{3}⟩ Seg 2 = ⟨p_{1}, p_{2}⟩.

(8) If p ∈ rng f_{1}, then p (f_{1} ^{}f_{2}) = p f_{1}.

(9) If p ∈ rng f_{2} \ rng f_{1}, then p (f_{1} ^{}f_{2}) = len f_{1} + p f_{2}.

(10) If p ∈ rng f_{1}, then f_{1} ^{}f_{2} → p = (f_{1} → p)^{}f_{2}.

(11) If p ∈ rng f_{2} \ rng f_{1}, then f_{1} ^{}f_{2} → p = f_{2} → p.

(12) f_{1} ⊆ f_{1} ^{}f_{2}.

(13) For every set A such that A ⊆ dom f_{1} holds (f_{1} ^{}f_{2}) A = f_{1} A.

(14) If p ∈ rng f_{1}, then f_{1} ^{}f_{2} ← p = f_{1} ← p.

Let us consider f_{1}, i. Observe that f_{1} Seg i is finite sequence-like.

The following propositions are true:

(15) If f_{1} ⊆ f_{2}, then f_{3} ^{}f_{1} ⊆ f_{3} ^{}f_{2}.

(16) (f_{1} ^{}f_{2}) Seg(len f_{1} + i) = f_{1} ^{}(f_{2} Seg i).

(17) If p ∈ rng f_{2} \ rng f_{1}, then f_{1} ^{}f_{2} ← p = f_{1} ^{}(f_{2} ← p).

(18) For every finite sequence f and for arbitrary p, q such that p ∈ rng f and q ∈ rng f and p f = q f holds p = q.

(19) If f_{1} ^{}f_{2} yields p just once, then p ∈ rng f_{1}− . rng f_{2}.

(20) If f_{1} ^{}f_{2} yields p just once and p ∈ rng f_{1}, then f_{1} yields p just once.

(21) If rng f is non trivial, then f is non trivial.

(22) p ⟨p⟩ = 1.

(23) p_{1} ⟨p_{1}, p_{2}⟩ = 1.

(24) If p_{1} ≠ p_{2}, then p_{2} ⟨p_{1}, p_{2}⟩ = 2.

(25) p_{1} ⟨p_{1}, p_{2}, p_{3}⟩ = 1.

(26) If p_{1} ≠ p_{2}, then p_{2} ⟨p_{1}, p_{2}, p_{3}⟩ = 2.

(27) If p_{1} ≠ p_{3} and p_{2} ≠ p_{3}, then p_{3} ⟨p_{1}, p_{2}, p_{3}⟩ = 3.

(28) For every finite sequence f holds Rev(⟨p⟩^{}f) = (Rev(f))^{}⟨p⟩.

(29) For every finite sequence f holds Rev(Rev(f)) = f.

(30) If x ≠ y, then ⟨x, y⟩ ← y = ⟨x⟩.

(31) If x ≠ y, then ⟨x, y, z⟩ ← y = ⟨x⟩.

(32) If x ≠ z and y ≠ z, then ⟨x, y, z⟩ ← z = ⟨x, y⟩.

(33) ⟨x, y⟩ → x = ⟨y⟩.

(34) If x ≠ y, then ⟨x, y, z⟩ → y = ⟨z⟩.

(35) ⟨x, y, z⟩ → x = ⟨y, z⟩.

(36) ⟨z⟩ → z = ε and ⟨z⟩ ← z = ε.

(37) If x ≠ y, then ⟨x, y⟩ → y = ε.

(38) If x ≠ z and y ≠ z, then ⟨x, y, z⟩ → z = ε.

(39) If x ∈ rng f and y ∈ rng(f ← x), then (f ← x) ← y = f ← y.

(40) If x ∉ rng f_{1}, then x (f_{1} ^{}⟨x⟩^{}f_{2}) = len f_{1} + 1.

(41) If f yields x just once, then x f + x Rev(f) = len f + 1.

(42) If f yields x just once, then Rev(f ← x) = Rev(f) → x.

(43) If f yields x just once, then Rev(f) yields x just once.

3. Finite sequences with elements from a non empty set

We adopt the following convention: D will denote a non empty set, p, p_{1}, p_{2}, p_{3} will denote elements of D, and f, f_{1}, f_{2} will denote finite sequences of elements of D.

One can prove the following propositions:

(44) If p ∈ rng f, then f −: p = (f ← p)^{}⟨p⟩.

(45) If p ∈ rng f, then f :− p = ⟨p⟩^{}(f → p).

(46) If f ≠ ε, then π_{1}f ∈ rng f.

(47) If f ≠ ε, then (π_{1}f) f = 1.

(48) If f ≠ ε and π_{1}f = p, then f −: p = ⟨p⟩ and f :− p = f.

(49) (⟨p_{1}⟩^{}f)_{1} = f.

(50) ⟨p_{1}, p_{2}⟩_{1} = ⟨p_{2}⟩.

(51) ⟨p_{1}, p_{2}, p_{3}⟩_{1} = ⟨p_{2}, p_{3}⟩.

(52) If k ∈ dom f and for every i such that 1 ≤ i and i < k holds π_{i}f ≠ π_{k}f, then (π_{k}f) f = k.

(53) If p_{1} ≠ p_{2}, then ⟨p_{1}, p_{2}⟩ −: p_{2} = ⟨p_{1}, p_{2}⟩.

(54) If p_{1} ≠ p_{2}, then ⟨p_{1}, p_{2}, p_{3}⟩ −: p_{2} = ⟨p_{1}, p_{2}⟩.

(55) If p_{1} ≠ p_{3} and p_{2} ≠ p_{3}, then ⟨p_{1}, p_{2}, p_{3}⟩ −: p_{3} = ⟨p_{1}, p_{2}, p_{3}⟩.

(56) ⟨p⟩:− p = ⟨p⟩ and ⟨p⟩ −: p = ⟨p⟩.

(57) If p_{1} ≠ p_{2}, then ⟨p_{1}, p_{2}⟩:− p_{2} = ⟨p_{2}⟩.

(58) If p_{1} ≠ p_{2}, then ⟨p_{1}, p_{2}, p_{3}⟩:− p_{2} = ⟨p_{2}, p_{3}⟩.

(59) If p_{1} ≠ p_{3} and p_{2} ≠ p_{3}, then ⟨p_{1}, p_{2}, p_{3}⟩:− p_{3} = ⟨p_{3}⟩.

(60) If x ∈ rng f and p ∈ rng f and x f ≤ p f, then x ∈ rng(f −: p).

(61) If p ∈ rng f and p f > k, then p f = k + p (f_{k}).

(62) If p ∈ rng f and p f > k, then p ∈ rng(f_{k}).

(63) If k < i and i ∈ dom f, then π_{i}f ∈ rng(f_{k}).

(64) If p ∈ rng f and p f > k, then f_{k} −: p = (f −: p)_{k}.

(65) If p ∈ rng f and p f ≠ 1, then f_{1} −: p = (f −: p)_{1}.

(66) p ∈ rng(f :− p).

(67) If x ∈ rng f and p ∈ rng f and x f ≥ p f, then x ∈ rng(f :− p).

(68) If p ∈ rng f and k ≤ len f and k ≥ p f, then π_{k}f ∈ rng(f :− p).

(69) If p ∈ rng f_{1}, then f_{1} ^{}f_{2} :− p = (f_{1} :− p)^{}f_{2}.

(70) If p ∈ rng f_{2} \ rng f_{1}, then f_{1} ^{}f_{2} :− p = f_{2} :− p.

(71) If p ∈ rng f_{1}, then f_{1} ^{}f_{2} −: p = f_{1} −: p.

(72) If p ∈ rng f_{2} \ rng f_{1}, then f_{1} ^{}f_{2} −: p = f_{1} ^{}(f_{2} −: p).

(73) f :− p:− p = f :− p.

(74) If p_{1} ∈ rng f and p_{2} ∈ rng f\rng(f−:p_{1}), then f → p_{2} = (f → p_{1}) → p_{2}.

(75) If p ∈ rng f, then rng f = rng(f −: p) ∪ rng(f :− p).

(76) If p_{1} ∈ rng f and p_{2} ∈ rng f \ rng(f −: p_{1}), then f :− p_{1} :− p_{2} = f :− p_{2}.

(77) If p ∈ rng f, then p (f −: p) = p f.

(78) f i i = f i.

(79) If p ∈ rng f, then f −: p −: p = f −: p.

(80) If p_{1} ∈ rng f and p_{2} ∈ rng(f −: p_{1}), then f −: p_{1} −: p_{2} = f −: p_{2}.

(81) If p ∈ rng f, then (f −: p)^{}((f :− p)_{1}) = f.

(82) If f_{1} ≠ ε, then (f_{1} ^{}f_{2})_{1} = ((f_{1})_{1})^{}f_{2}.

(83) If p_{2} ∈ rng f and p_{2} f ≠ 1, then p_{2} ∈ rng(f_{1}).

(84) If p ∈ rng f, then p (f :− p) = 1. (86)^{2} (ε_{D})_{k} = ε_{D}.

(87) f_{i+k} = (f_{i})_{k}.

(88) If p ∈ rng f and p f > k, then f_{k} :− p = f :− p.

(89) If p ∈ rng f and p f ≠ 1, then f_{1} :− p = f :− p.

(90) If i + k = len f, then Rev(f_{k}) = Rev(f) i.

(91) If i + k = len f, then Rev(f k) = (Rev(f))_{i}.

(92) If f yields p just once, then Rev(f → p) = Rev(f) ← p.

(93) If f yields p just once, then Rev(f :− p) = Rev(f) −: p.

(94) If f yields p just once, then Rev(f −: p) = Rev(f) :− p.

4. Rotating a finite sequence

Let D be a non empty set. A finite sequence of elements of D is circular if: (Def.1) π_{1}it = π_{len_{it}}it.

Let us consider D, f, p. The functor f p_{}yielding a finite sequence of elements of D is defined by: (Def.2) (i) f p_{}= (f :− p)^{}((f −: p)_{1}) if p ∈ rng f,

(ii) f p_{}= f, otherwise.

Let us consider D, let f be a non empty finite sequence of elements of D, and let p be an element of D. One can verify that f p_{}is non empty.

Let us consider D. Observe that there exists a finite sequence of elements of D which is circular non empty and trivial and there exists a finite sequence of elements of D which is circular non empty and non trivial.

The following proposition is true

(95) f π_{1}f_{}= f.

2The proposition (85) has been removed.

on the decomposition of finite sequences 321

Let us consider D, p and let f be a circular non empty finite sequence of elements of D. Observe that f p_{}is circular.

We now state a number of propositions:

(96) If f is circular and p ∈ rng f, then rng(f p_{}) = rng f.

(97) If p ∈ rng f, then p ∈ rng(f p_{}).

(98) If p ∈ rng f, then π_{1}f p_{}= p.

(99) (f p_{})p_{}= f p_{}.

(100) ⟨p⟩p_{}= ⟨p⟩.

(101) ⟨p_{1}, p_{2}⟩p_{1}_{}= ⟨p_{1}, p_{2}⟩.

(102) ⟨p_{1}, p_{2}⟩p_{2}_{}= ⟨p_{2}, p_{2}⟩.

(103) ⟨p_{1}, p_{2}, p_{3}⟩p_{1}_{}= ⟨p_{1}, p_{2}, p_{3}⟩.

(104) If p_{1} ≠ p_{2}, then ⟨p_{1}, p_{2}, p_{3}⟩p_{2}_{}= ⟨p_{2}, p_{3}, p_{2}⟩.

(105) If p_{2} ≠ p_{3}, then ⟨p_{1}, p_{2}, p_{3}⟩p_{3}_{}= ⟨p_{3}, p_{2}, p_{3}⟩.

(106) For every circular non trivial finite sequence f of elements of D holds rng(f_{1}) = rng f.

(107) rng(f_{1}) ⊆ rng(f p_{}).

(108) If p_{2} ∈ rng f \ rng(f −: p_{1}), then (f p_{1}_{})p_{2}_{}= f p_{2}_{}.

(109) If p_{2} f ≠ 1 and p_{2} ∈ rng f \ rng(f :− p_{1}), then (f p_{1}_{})p_{2}_{}= f p_{2}_{}.

(110) If p_{2} ∈ rng(f_{1}) and f yields p_{2} just once, then (f p_{1}_{})p_{2}_{}= f p_{2}_{}.

(111) If f is circular and f yields p_{2} just once, then (f p_{1}_{})p_{2}_{}= f p_{2}_{}.

(112) If f is circular and f yields p just once, then Rev(f p_{}) = (Rev(f))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] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

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

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

[6] 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.

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

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

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

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

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

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322 andrzej trybulec

[13] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received May 24, 1995

Decomposing a Go-Board into Cells

Yatsuka Nakamura Shinshu University Nagano Andrzej Trybulec Warsaw University Białystok

MML Identifier: GOBOARD5.

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

For simplicity we follow a convention: q will be a point of E_{T}^{2}, i, i_{1}, i_{2}, j, j_{1}, j_{2}, k will be natural numbers, r, s will be real numbers, and G will be a Go-board.

We now state the proposition

(1) Let M be a tabular finite sequence and given i, j. If ⟨⟨i, j⟩⟩ ∈ the indices of M, then 1 ≤ i and i ≤ len M and 1 ≤ j and j ≤ width M.

Let us consider G, i. The functor vstrip(G, i) yielding a subset of the carrier of E_{T}^{2} is defined as follows: (Def.1) (i) vstrip(G, i) = {[r, s] : (G_{i,1})_{1} ≤ r ∧ r ≤ (G_{i+1,1})_{1}} if 1 ≤ i and

i < len G, (ii) vstrip(G, i) = {[r, s] : (G_{i,1})_{1} ≤ r} if i ≥ len G, (iii) vstrip(G, i) = {[r, s] : r ≤ (G_{i+1,1})_{1}}, otherwise.

The functor hstrip(G, i) yields a subset of the carrier of E_{T}^{2} and is defined by: (Def.2) (i) hstrip(G, i) = {[r, s] : (G_{1,i})_{2} ≤ s ∧ s ≤ (G_{1,i+1})_{2}} if 1 ≤ i and

i < width G, (ii) hstrip(G, i) = {[r, s] : (G_{1,i})_{2} ≤ s} if i ≥ width G, (iii) hstrip(G, i) = {[r, s] : s ≤ (G_{1,i+1})_{2}}, otherwise. We now state a number of propositions:

(2) If 1 ≤ j and j ≤ width G and 1 ≤ i and i ≤ len G, then (G_{i,j})_{2} = (G_{1,j})_{2}.

(3) If 1 ≤ j and j ≤ width G and 1 ≤ i and i ≤ len G, then (G_{i,j})_{1} = (G_{i,1})_{1}.

(4) If 1 ≤ j and j ≤ width G and 1 ≤ i_{1} and i_{1} < i_{2} and i_{2} ≤ len G, then (G_{i_{1},j})_{1} < (G_{i_{2},j})_{1}.

(5) If 1 ≤ j_{1} and j_{1} < j_{2} and j_{2} ≤ width G and 1 ≤ i and i ≤ len G, then (G_{i,j_{1}})_{2} < (G_{i,j_{2}})_{2}.

(6) If 1 ≤ j and j < width G and 1 ≤ i and i ≤ len G, then hstrip(G, j) = {[r, s] : (G_{i,j})_{2} ≤ s ∧ s ≤ (G_{i,j+1})_{2}}.

(7) If 1 ≤ i and i ≤ len G, then hstrip(G, width G) = {[r, s] : (G_{i,width G})_{2} ≤ s}.

(8) If 1 ≤ i and i ≤ len G, then hstrip(G, 0) = {[r, s] : s ≤ (G_{i,1})_{2}}.

(9) If 1 ≤ i and i < len G and 1 ≤ j and j ≤ width G, then vstrip(G, i) = {[r, s] : (G_{i,j})_{1} ≤ r ∧ r ≤ (G_{i+1,j})_{1}}.

(10) If 1 ≤ j and j ≤ width G, then vstrip(G, len G) = {[r, s] : (G_{len G,j})_{1} ≤ r}.

(11) If 1 ≤ j and j ≤ width G, then vstrip(G, 0) = {[r, s] : r ≤ (G_{1,j})_{1}}. Let G be a Go-board and let us consider i, j. The functor cell(G, i, j) yields

a subset of the carrier of E_{T}^{2} and is defined as follows: (Def.3) cell(G, i, j) = vstrip(G, i) ∩ hstrip(G, j).

A finite sequence of elements of E_{T}^{2} is s.c.c. if: (Def.4) For all i, j such that i + 1 < j but i > 1 and j < len it or j + 1 < len it

holds L(it, i) ∩ L(it, j) = ∅. A non empty finite sequence of elements of E_{T}^{2} is standard if: (Def.5) It is a sequence which elements belong to the Go-board of it.

One can verify that there exists a non empty finite sequence of elements of E_{T}^{2} which is non constant special unfolded circular s.c.c. and standard.

We now state two propositions:

(12) Let f be a standard non empty finite sequence of elements of E_{T}^{2}. Suppose k ∈ dom f. Then there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of the Go-board of f and π_{k}f = (the Go-board of f)_{i,j}.

(13) Let f be a standard non empty finite sequence of elements of E_{T}^{2} and let n be a natural number. Suppose n ∈ dom f and n + 1 ∈ dom f. Let m, k, i, j be natural numbers. Suppose that

(i) ⟨⟨m, k⟩⟩ ∈ the indices of the Go-board of f, (ii) ⟨⟨i, j⟩⟩ ∈ the indices of the Go-board of f, (iii) π_{n}f = (the Go-board of f)_{m,k}, and (iv) π_{n+1}f = (the Go-board of f)_{i,j}.

Then |m − i| + |k − j| = 1. A special circular sequence is a special unfolded circular s.c.c. non empty

finite sequence of elements of E_{T}^{2}.

In the sequel f is a standard special circular sequence. Let us consider f, k. Let us assume that 1 ≤ k and k+1 ≤ len f. The functor

rightcell(f, k) yielding a subset of the carrier of E_{T}^{2} is defined by the condition (Def.6). (Def.6) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose that

(i) ⟨⟨i_{1}, j_{1}⟩⟩ ∈ the indices of the Go-board of f, (ii) ⟨⟨i_{2}, j_{2}⟩⟩ ∈ the indices of the Go-board of f, (iii) π_{k}f = (the Go-board of f)_{i_{1},j_{1}}, and (iv) π_{k+1}f = (the Go-board of f)_{i_{2},j_{2}}.

Then

(v) i_{1} = i_{2} and j_{1} + 1 = j_{2} and rightcell(f, k) = cell(the Go-board of f, i_{1}, j_{1}), or

(vi) i_{1} + 1 = i_{2} and j_{1} = j_{2} and rightcell(f, k) = cell(the Go-board of f,

i_{1}, j_{1} −′ 1), or (vii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and rightcell(f, k) = cell(the Go-board of f,

i_{2}, j_{2}), or (viii) i_{1} = i_{2} and j_{1} = j_{2} + 1 and rightcell(f, k) = cell(the Go-board of f,

i_{1} −′ 1, j_{2}). The functor leftcell(f, k) yielding a subset of the carrier of E_{T}^{2} is defined by the condition (Def.7). (Def.7) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose that

(i) ⟨⟨i_{1}, j_{1}⟩⟩ ∈ the indices of the Go-board of f, (ii) ⟨⟨i_{2}, j_{2}⟩⟩ ∈ the indices of the Go-board of f, (iii) π_{k}f = (the Go-board of f)_{i_{1},j_{1}}, and (iv) π_{k+1}f = (the Go-board of f)_{i_{2},j_{2}}.

Then

(v) i_{1} = i_{2} and j_{1} + 1 = j_{2} and leftcell(f, k) = cell(the Go-board of f, i_{1} −′ 1, j_{1}), or

(vi) i_{1} + 1 = i_{2} and j_{1} = j_{2} and leftcell(f, k) = cell(the Go-board of f,

i_{1}, j_{1}), or (vii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and leftcell(f, k) = cell(the Go-board of f,

i_{2}, j_{2} −′ 1), or (viii) i_{1} = i_{2} and j_{1} = j_{2} + 1 and leftcell(f, k) = cell(the Go-board of f,

i_{1}, j_{2}). Next we state a number of propositions:

(14) If i < len G and 1 ≤ j and j < width G, then L(G_{i+1,j}, G_{i+1,j+1}) ⊆ vstrip(G, i).

(15) If 1 ≤ i and i ≤ len G and 1 ≤ j and j < width G, then L(G_{i,j}, G_{i,j+1}) ⊆ vstrip(G, i).

(16) If j < width G and 1 ≤ i and i < len G, then L(G_{i,j+1}, G_{i+1,j+1}) ⊆ hstrip(G, j).

(17) If 1 ≤ j and j ≤ width G and 1 ≤ i and i < len G, then L(G_{i,j}, G_{i+1,j}) ⊆ hstrip(G, j).

(18) If 1 ≤ i and i ≤ len G and 1 ≤ j and j + 1 ≤ width G, then L(G_{i,j}, G_{i,j+1}) ⊆ hstrip(G, j).

(19) If i < len G and 1 ≤ j and j < width G, then L(G_{i+1,j}, G_{i+1,j+1}) ⊆ cell(G, i, j).

(20) If 1 ≤ i and i ≤ len G and 1 ≤ j and j < width G, then L(G_{i,j}, G_{i,j+1}) ⊆ cell(G, i, j).

(21) If 1 ≤ j and j ≤ width G and 1 ≤ i and i + 1 ≤ len G, then L(G_{i,j}, G_{i+1,j}) ⊆ vstrip(G, i).

(22) If j < width G and 1 ≤ i and i < len G, then L(G_{i,j+1}, G_{i+1,j+1}) ⊆ cell(G, i, j).

(23) If 1 ≤ i and i < len G and 1 ≤ j and j ≤ width G, then L(G_{i,j}, G_{i+1,j}) ⊆ cell(G, i, j).

(24) If i+1 ≤ len G, then vstrip(G, i)∩vstrip(G, i+1) = {q : q_{1} = (G_{i+1,1})_{1}}.

(25) If j + 1 ≤ width G, then hstrip(G, j) ∩ hstrip(G, j + 1) = {q : q_{2} = (G_{1,j+1})_{2}}.

(26) For every Go-board G such that i < len G and 1 ≤ j and j < width G holds cell(G, i, j) ∩ cell(G, i + 1, j) = L(G_{i+1,j}, G_{i+1,j+1}).

(27) For every Go-board G such that j < width G and 1 ≤ i and i < len G holds cell(G, i, j) ∩ cell(G, i, j + 1) = L(G_{i,j+1}, G_{i+1,j+1}).

(28) Suppose that (i) 1 ≤ k, (ii) k + 1 ≤ len f, (iii) ⟨⟨i + 1, j⟩⟩ ∈ the indices of the Go-board of f, (iv) ⟨⟨i + 1, j + 1⟩⟩ ∈ the indices of the Go-board of f,

(v) π_{k}f = (the Go-board of f)_{i+1,j}, and (vi) π_{k+1}f = (the Go-board of f)_{i+1,j+1}.

Then leftcell(f, k) = cell(the Go-board of f, i, j) and rightcell(f, k) = cell(the Go-board of f, i + 1, j).

(29) Suppose that (i) 1 ≤ k, (ii) k + 1 ≤ len f, (iii) ⟨⟨i, j + 1⟩⟩ ∈ the indices of the Go-board of f, (iv) ⟨⟨i + 1, j + 1⟩⟩ ∈ the indices of the Go-board of f,

(v) π_{k}f = (the Go-board of f)_{i,j+1}, and (vi) π_{k+1}f = (the Go-board of f)_{i+1,j+1}.

Then leftcell(f, k) = cell(the Go-board of f, i, j + 1) and rightcell(f, k) = cell(the Go-board of f, i, j).

(30) Suppose that (i) 1 ≤ k, (ii) k + 1 ≤ len f, (iii) ⟨⟨i, j + 1⟩⟩ ∈ the indices of the Go-board of f, (iv) ⟨⟨i + 1, j + 1⟩⟩ ∈ the indices of the Go-board of f,

(v) π_{k}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k+1}f = (the Go-board of f)_{i,j+1}.

Then leftcell(f, k) = cell(the Go-board of f, i, j) and rightcell(f, k) = cell(the Go-board of f, i, j + 1).

(31) Suppose that (i) 1 ≤ k, (ii) k + 1 ≤ len f,

decomposing a go-board into cells 327

(iii) ⟨⟨i + 1, j + 1⟩⟩ ∈ the indices of the Go-board of f, (iv) ⟨⟨i + 1, j⟩⟩ ∈ the indices of the Go-board of f,

(v) π_{k}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k+1}f = (the Go-board of f)_{i+1,j}.

Then leftcell(f, k) = cell(the Go-board of f, i + 1, j) and rightcell(f, k) = cell(the Go-board of f, i, j).

(32) If 1 ≤ k and k + 1 ≤ len f, then leftcell(f, k) ∩ rightcell(f, k) = L(f, k).

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

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

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

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

[8] 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.

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

[10] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

[11] Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711–717, 1991.

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

[13] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

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

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

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

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

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

[19] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5(3):317–322, 1996.

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

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

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

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

328 yatsuka nakamura and andrzej trybulec

[24] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received May 26, 1995

Indexed Category

Grzegorz Bancerek Institute of Mathematics Polish Academy of Sciences

Summary. The concept of indexing of a category (a part of indexed category, see [18]) is introduced as a pair formed by a many sorted category and a many sorted functor. The indexing of a category C against to [18] is not a functor but it can be treated as a functor from C into some categorial category (see [1]). The goal of the article is to work out the notation necessary to define institutions (see [13]).

MML Identifier: INDEX 1.

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

1. Category-yielding Functions

Let A be a non empty set. One can check that there exists a many sorted set indexed by A which is non empty yielding.

Let A be a non empty set. One can verify that every many sorted set indexed by A which is non-empty is also non empty yielding.

Let C be a categorial category and let f be a morphism of C. Then f_{2} is a functor from f_{1,1} to f_{1,2}.

We now state two propositions:

(1) For every categorial category C and for all morphisms f, g of C such that dom g = cod f holds g · f = ⟨⟨⟨⟨ dom f, cod g⟩⟩, g_{2} · f_{2}⟩⟩.

(2) Let C be a category, and let D, E be categorial categories, and let F be a functor from C to D, and let G be a functor from C to E. If F = G, then Obj F = Obj G.

A function is category-yielding if:

(Def.1) For arbitrary x such that x ∈ dom it holds it(x) is a category. Let us note that there exists a function which is category-yielding. Let X be a set. Observe that there exists a many sorted set indexed by X

which is category-yielding.

Let A be a set. A many sorted category indexed by A is a category-yielding many sorted set indexed by A.

Let C be a category. A many sorted set indexed by C is a many sorted set indexed by the objects of C. A many sorted category indexed by C is a many sorted category indexed by the objects of C.

Let X be a set and let x be a category. One can verify that X ⟼ x is category-yielding.

Let X be a set and let x be a function. One can check that X ⟼ x is function yielding.

Let X be a non empty set. One can check that every many sorted set indexed by X is non empty.

Let f be a non empty function. One can check that rng f is non empty. Let f be a category-yielding function. Observe that rng f is categorial. Let X be a non empty set, let f be a many sorted category indexed by X,

and let x be an element of X. Then f(x) is a category.

Let B be a set, let A be a non empty set, let f be a function from B into A, and let g be a many sorted category indexed by A. Observe that g · f is category-yielding.

Let F be a category-yielding function. The functor Objs(F) yields a nonempty function and is defined by the conditions (Def.2). (Def.2) (i) dom Objs(F) = dom F, and

(ii) for every set x such that x ∈ dom F and for every category C such

that C = F (x) holds (Objs(F))(x) = the objects of C. The functor Mphs(F) yields a non-empty function and is defined by the conditions (Def.3). (Def.3) (i) dom Mphs(F) = dom F, and

(ii) for every set x such that x ∈ dom F and for every category C such

that C = F (x) holds (Mphs(F))(x) = the morphisms of C. Let A be a non empty set and let F be a many sorted category indexed by A.

Then Objs(F) is a non-empty many sorted set indexed by A. Then Mphs(F) is a non-empty many sorted set indexed by A.

The following proposition is true

(3) For every set X and for every category C holds Objs(X ⟼ C) = X ⟼ the objects of C and Mphs(X ⟼ C) = X ⟼ the morphisms of C.

2. Pairs of Many Sorted Sets

Let A, B be sets. Pair of many sorted sets indexed by A and B is defined by: (Def.4) There exists a many sorted set f indexed by A and there exists a many

sorted set g indexed by B such that it = ⟨⟨f, g⟩⟩. Let A, B be sets, let f be a many sorted set indexed by A, and let g be a

many sorted set indexed by B. Then ⟨⟨f, g⟩⟩ is a pair of many sorted sets indexed by A and B.

Let A, B be sets and let X be a pair of many sorted sets indexed by A and B. Then X_{1} is a many sorted set indexed by A. Then X_{2} is a many sorted set indexed by B.

Let A, B be sets. A pair of many sorted sets indexed by A and B is categoryyielding on first if: (Def.5) it_{1} is category-yielding. A pair of many sorted sets indexed by A and B is function-yielding on second if: (Def.6) it_{2} is function yielding.

Let A, B be sets. One can check that there exists a pair of many sorted sets indexed by A and B which is category-yielding on first and function-yielding on second.

Let A, B be sets and let X be a category-yielding on first pair of many sorted sets indexed by A and B. Then X_{1} is a many sorted category indexed by A.

Let A, B be sets and let X be a function-yielding on second pair of many sorted sets indexed by A and B. Then X_{2} is a many sorted function of B.

Let f be a function yielding function. One can check that rng f is functional. Let A, B be sets, let f be a many sorted category indexed by A, and let

g be a many sorted function of B. Then ⟨⟨f, g⟩⟩ is a category-yielding on first function-yielding on second pair of many sorted sets indexed by A and B.

Let A be a non empty set and let F, G be many sorted categories indexed by A. A many sorted function of A is called a many sorted functor from F to G if: (Def.7) For every element a of A holds it(a) is a functor from F (a) to G(a).

The scheme LambdaMSFr deals with a non empty set A, many sorted categories B, C indexed by A, and a unary functor F yielding a set, and states that:

There exists a many sorted functor F from B to C such that for every element a of A holds F (a) = F(a)

provided the parameters meet the following requirement:

• For every element a of A holds F(a) is a functor from B(a) to C(a).

Let A be a non empty set, let F, G be many sorted categories indexed by A, let f be a many sorted functor from F to G, and let a be an element of A. Then f(a) is a functor from F (a) to G(a).

3. Indexing

Let A, B be non empty sets and let F, G be functions from B into A. A category-yielding on first pair of many sorted sets indexed by A and B is said to be an indexing of F and G if: (Def.8) it_{2} is a many sorted functor from it_{1} · F to it_{1} · G.

Next we state two propositions:

(4) Let A, B be non empty sets, and let F, G be functions from B into A, and let I be an indexing of F and G, and let m be an element of B. Then I_{2}(m) is a functor from I_{1}(F (m)) to I_{1}(G(m)).

(5) Let C be a category, and let I be an indexing of the dom-map of C and the cod-map of C, and let m be a morphism of C. Then I_{2}(m) is a functor from I_{1}(dom m) to I_{1}(cod m).

Let A, B be non empty sets, let F, G be functions from B into A, and let I be an indexing of F and G. Then I_{2} is a many sorted functor from I_{1} · F to I_{1} · G.

Let A, B be non empty sets, let F, G be functions from B into A, and let I be an indexing of F and G. A categorial category is called a target category of I if it satisfies the conditions (Def.9). (Def.9) (i) For every element a of A holds I_{1}(a) is an object of it, and

(ii) for every element b of B holds ⟨⟨⟨⟨I_{1}(F (b)), I_{1}(G(b))⟩⟩, I_{2}(b)⟩⟩ is a mor-

phism of it. Let A, B be non empty sets, let F, G be functions from B into A, and let I

be an indexing of F and G. One can verify that there exists a target category of I which is full and strict.

Let A, B be non empty sets, let F, G be functions from B into A, let c be a partial function from [: B, B:] to B, and let i be a function from A into B. Let us assume that there exists a category C such that C = ⟨A, B, F, G, c, i⟩. An indexing of F and G is called an indexing of F, G, c and i if it satisfies the conditions (Def.10).

(Def.10) (i) For every element a of A holds it_{2}(i(a)) = id_{it1(a)}, and

(ii) for all elements m_{1}, m_{2} of B such that F (m_{2}) = G(m_{1}) holds it_{2}(c(⟨⟨m_{2},

m_{1}⟩⟩)) = it_{2}(m_{2}) · it_{2}(m_{1}). Let C be a category. An indexing of C is an indexing of the dom-map of C,

the cod-map of C, the composition of C and the id-map of C. A coindexing of C is an indexing of the cod-map of C, the dom-map of C, (the composition of C) and the id-map of C.

One can prove the following propositions:

(6) Let C be a category and let I be an indexing of the dom-map of C and the cod-map of C. Then I is an indexing of C if and only if the following conditions are satisfied:

(i) for every object a of C holds I_{2}(id_{a}) = id_{I1(a)}, and (ii) for all morphisms m_{1}, m_{2} of C such that dom m_{2} = cod m_{1} holds

I_{2}(m_{2} · m_{1}) = I_{2}(m_{2}) · I_{2}(m_{1}).

(7) Let C be a category and let I be an indexing of the cod-map of C and the dom-map of C. Then I is a coindexing of C if and only if the following conditions are satisfied:

(i) for every object a of C holds I_{2}(id_{a}) = id_{I1(a)}, and (ii) for all morphisms m_{1}, m_{2} of C such that dom m_{2} = cod m_{1} holds

I_{2}(m_{2} · m_{1}) = I_{2}(m_{1}) · I_{2}(m_{2}).

(8) For every category C and for every set x holds x is a coindexing of C iff x is an indexing of C^{op}.

(9) Let C be a category, and let I be an indexing of C, and let c_{1}, c_{2} be objects of C. Suppose hom(c_{1}, c_{2}) is non empty. Let m be a morphism from c_{1} to c_{2}. Then I_{2}(m) is a functor from I_{1}(c_{1}) to I_{1}(c_{2}).

(10) Let C be a category, and let I be a coindexing of C, and let c_{1}, c_{2} be objects of C. Suppose hom(c_{1}, c_{2}) is non empty. Let m be a morphism from c_{1} to c_{2}. Then I_{2}(m) is a functor from I_{1}(c_{2}) to I_{1}(c_{1}).

Let C be a category, let I be an indexing of C, and let T be a target category of I. The functor I -functor(C, T) yielding a functor from C to T is defined as follows:

(Def.11) For every morphism f of C holds (I -functor(C, T))(f) = ⟨⟨⟨⟨I_{1}(dom f),

I_{1}(cod f)⟩⟩, I_{2}(f)⟩⟩. We now state three propositions:

(11) Let C be a category, and let I be an indexing of C, and let T_{1}, T_{2} be target categories of I. Then I -functor(C, T_{1}) = I -functor(C, T_{2}) and Obj(I -functor(C, T_{1})) = Obj(I -functor(C, T_{2})).

(12) For every category C and for every indexing I of C and for every target category T of I holds Obj(I -functor(C, T)) = I_{1}.

(13) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let x be an object of C. Then (I -functor(C, T))(x) = I_{1}(x).

Let C be a category and let I be an indexing of C. The functor rng I yielding a strict target category of I is defined by:

(Def.12) For every target category T of I holds rng I = Im(I -functor(C, T)).

Next we state the proposition

(14) Let C be a category, and let I be an indexing of C, and let D be a categorial category. Then rng I is a subcategory of D if and only if D is a target category of I.

Let C be a category, let I be an indexing of C, and let m be a morphism of C. The functor I(m) yielding a functor from I_{1}(dom m) to I_{1}(cod m) is defined by:

(Def.13) I(m) = I_{2}(m).

Let C be a category, let I be a coindexing of C, and let m be a morphism of C. The functor I(m) yielding a functor from I_{1}(cod m) to I_{1}(dom m) is defined as follows:

(Def.14) I(m) = I_{2}(m).

The following proposition is true

(15) Let C, D be categories. Then

(i) ⟨⟨(the objects of C) ⟼ (D), (the morphisms of C) ⟼ id_{D}⟩⟩ is an indexing of C, and

(ii) ⟨⟨(the objects of C) ⟼ (D), (the morphisms of C) ⟼ id_{D}⟩⟩ is a coindexing of C.

4. Indexing vs Functors

Let A be a set and let B be a non empty set. We see that the function from A into B is a many sorted set indexed by A.

Let C, D be categories and let F be a function from the morphisms of C into the morphisms of D. Then Obj F is a function from the objects of C into the objects of D.

Let C be a category, let D be a categorial category, and let F be a functor from C to D. Note that Obj F is category-yielding.

Let C be a category, let D be a categorial category, and let F be a functor from C to D. Then pr2(F) is a many sorted functor from Obj F · (the dom-map of C) to Obj F · (the cod-map of C).

Next we state the proposition

(16) Let C be a category, and let D be a categorial category, and let F be a

functor from C to D. Then ⟨⟨ Obj F, pr2(F)⟩⟩ is an indexing of C. Let C be a category, let D be a categorial category, and let F be a functor

from C to D. The functor F -indexing of C yields an indexing of C and is defined by:

(Def.15) F -indexing of C = ⟨⟨ Obj F, pr2(F)⟩⟩.

One can prove the following propositions:

(17) Let C be a category, and let D be a categorial category, and let F be a functor from C to D. Then D is a target category of F -indexing of C.

(18) Let C be a category, and let D be a categorial category, and let F be a functor from C to D, and let T be a target category of F -indexing of C. Then F = F -indexing of C -functor(C, T).

(19) Let C be a category, and let D, E be categorial categories, and let F be a functor from C to D, and let G be a functor from C to E. If F = G, then F -indexing of C = G-indexing of C.

(20) For every category C and for every indexing I of C and for every target category T of I holds pr2(I -functor(C, T)) = I_{2}.

(21) For every category C and for every indexing I of C and for every target category T of I holds (I -functor(C, T))-indexing of C = I.

5. Composing Indexings and Functors

Let C, D, E be categories, let F be a functor from C to D, and let I be an indexing of E. Let us assume that Im F is a subcategory of E. The functor I ·F yielding an indexing of C is defined by:

(Def.16) For every functor F′ from C to E such that F′ = F holds I · F =

((I -functor(E, rng I)) · F′)-indexing of C. Next we state several propositions:

(22) Let C, D_{1}, D_{2}, E be categories, and let I be an indexing of E, and let F be a functor from C to D_{1}, and let G be a functor from C to D_{2}. Suppose Im F is a subcategory of E and Im G is a subcategory of E and F = G. Then I · F = I · G.

(23) Let C, D be categories, and let F be a functor from C to D, and let I be an indexing of D, and let T be a target category of I. Then I · F = ((I -functor(D, T)) · F)-indexing of C.

(24) Let C, D be categories, and let F be a functor from C to D, and let I be an indexing of D. Then every target category of I is a target category of I · F.

(25) Let C, D be categories, and let F be a functor from C to D, and let I be an indexing of D, and let T be a target category of I. Then rng(I · F) is a subcategory of T.

(26) Let C, D, E be categories, and let F be a functor from C to D, and let G be a functor from D to E, and let I be an indexing of E. Then (I · G) · F = I · (G · F).

Let C be a category, let I be an indexing of C, and let D be a categorial category. Let us assume that D is a target category of I. Let E be a categorial category and let F be a functor from D to E. The functor F · I yielding an indexing of C is defined as follows:

(Def.17) For every target category T of I and for every functor G from T to E such that T = D and G = F holds F · I = (G · (I -functor(C, T)))-indexing of C.

One can prove the following propositions:

(27) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let D, E be categorial categories, and let F be a functor from T to D, and let G be a functor from T to E. If F = G, then F · I = G · I.

(28) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let D be a categorial category, and let F be a functor from T to D. Then Im F is a target category of F · I.

(29) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let D be a categorial category, and let F be a functor from T to D. Then D is a target category of F · I.

(30) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let D be a categorial category, and let F be a functor from T to D. Then rng(F · I) is a subcategory of Im F.

(31) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let D, E be categorial categories, and let F be a functor from T to D, and let G be a functor from D to E. Then (G · F) · I = G · (F · I).

Let C, D be categories, let I_{1} be an indexing of C, and let I_{2} be an indexing of D. The functor I_{2} · I_{1} yielding an indexing of C is defined as follows:

(Def.18) I_{2} · I_{1} = I_{2} · (I_{1} -functor(C, rng I_{1})).

We now state several propositions:

(32) Let C be a category, and let D be a categorial category, and let I_{1} be an indexing of C, and let I_{2} be an indexing of D, and let T be a target category of I_{1}. If D is a target category of I_{1}, then I_{2} · I_{1} = I_{2} · (I_{1} -functor(C, T)).

(33) Let C be a category, and let D be a categorial category, and let I_{1} be an indexing of C, and let I_{2} be an indexing of D, and let T be a target category of I_{2}. If D is a target category of I_{1}, then I_{2} · I_{1} = (I_{2} -functor(D, T)) · I_{1}.

(34) Let C, D be categories, and let F be a functor from C to D, and let I be an indexing of D, and let T be a target category of I, and let E be a categorial category, and let G be a functor from T to E. Then (G · I) · F = G · (I · F).

(35) Let C be a category, and let I be an indexing of C, and let T be a target category of I, and let D be a categorial category, and let F be a functor from T to D, and let J be an indexing of D. Then (J · F) · I = J · (F · I).

(36) Let C be a category, and let I be an indexing of C, and let T_{1} be a target category of I, and let J be an indexing of T_{1}, and let T_{2} be a target category of J, and let D be a categorial category, and let F be a functor from T_{2} to D. Then (F · J) · I = F · (J · I).

(37) Let C, D be categories, and let F be a functor from C to D, and let I be an indexing of D, and let T be a target category of I, and let J be an indexing of T. Then (J · I) · F = J · (I · F).

(38) Let C be a category, and let I be an indexing of C, and let D be a target category of I, and let J be an indexing of D, and let E be a target category of J, and let K be an indexing of E. Then (K ·J)·I = K ·(J ·I). indexed category 337

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

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

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

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

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

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

[25] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

Received June 8, 1995 338

Associated Matrix of Linear Map

Robert Milewski Warsaw University Białystok

MML Identifier: MATRLIN.

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

1. Preliminaries

For simplicity we follow the rules: k, t, i, j, m, n are natural numbers, x is arbitrary, A is a set, and D is a non empty set.

We now state two propositions:

(1) For every finite sequence p of elements of D and for every i holds p_{i} is a finite sequence of elements of D.

(2) For every i and for every finite sequence p holds rng(p_{i}) ⊆ rng p.

Let D be a non empty set. A matrix over D is a tabular finite sequence of elements of D^{∗}.

Let K be a field. A matrix over K is a matrix over the carrier of K.

Let D be a non empty set, let us consider k, and let M be a matrix over D. Then M_{k} is a matrix over D.

Next we state four propositions:

(3) For every finite sequence M of elements of D such that len M = n + 1 holds len(M_{n+1}) = n.

(4) Let M be a matrix over D of dimension n + 1 × m and let M_{1} be a matrix over D. Then if n > 0, then width M = width(M_{n+1}) and if M_{1} = ⟨M(n + 1)⟩, then width M = width M_{1}.

(5) For every matrix M over D of dimension n + 1 × m holds M_{n+1} is a matrix over D of dimension n × m.

(6) For every finite sequence M of elements of D such that len M = n + 1 holds M = (M_{len M})^{}⟨M(len M)⟩.

Let us consider D and let P be a finite sequence of elements of D. Then ⟨P ⟩ is a matrix over D of dimension 1 × len P.

2. More on Finite Sequence

One can prove the following propositions:

(7) For every set A and for every finite sequence F holds (Sgm(F ^{−1} A))^{} Sgm(F ^{−1} (rng F \ A)) is a permutation of dom F.

(8) Let F be a finite sequence and let A be a subset of rng F. Suppose F is one-to-one. Then there exists a permutation p of dom F such that (F − A^{c})^{}(F − A) = F · p.

A function is finite sequence yielding if: (Def.1) For every x such that x ∈ dom it holds it(x) is a finite sequence.

Let us observe that there exists a function which is finite sequence yielding. Let F, G be finite sequence yielding functions. The functor F ^{⌢} G yields a

finite sequence yielding function and is defined by the conditions (Def.2). (Def.2) (i) dom(F ^{⌢} G) = dom F ∩ dom G, and

(ii) for arbitrary i such that i ∈ dom(F ^{⌢} G) and for all finite sequences f, g such that f = F (i) and g = G(i) holds (F ^{⌢} G)(i) = f ^{}g.

3. Matrices and Finite Sequences in Vector Space

For simplicity we adopt the following convention: K denotes a field, V denotes a vector space over K, a denotes an element of the carrier of K, W denotes an element of the carrier of V, K_{1}, K_{2}, K_{3} denote linear combinations of V, and X denotes a subset of the carrier of V.

Next we state four propositions:

(9) If X is linearly independent and support K_{1} ⊆ X and support K_{2} ⊆ X and ∑ K_{1} = ∑ K_{2}, then K_{1} = K_{2}.

(10) If X is linearly independent and support K_{1} ⊆ X and support K_{2} ⊆ X and support K_{3} ⊆ X and ∑ K_{1} = ∑ K_{2} + ∑ K_{3}, then K_{1} = K_{2} + K_{3}.

(11) If X is linearly independent and support K_{1} ⊆ X and support K_{2} ⊆ X and a ≠ 0_{K} and ∑ K_{1} = a · ∑ K_{2}, then K_{1} = a · K_{2}.

(12) For every basis b_{2} of V there exists a linear combination K_{4} of V such

that W = ∑ K_{4} and support K_{4} ⊆ b_{2}. Let K be a field and let V be a vector space over K. We say that V is finite

dimensional if and only if: (Def.3) There exists finite subset of the carrier of V which is a basis of V.

Let K be a field. Note that there exists a vector space over K which is strict and finite dimensional.

Let K be a field and let V be a finite dimensional vector space over K. A finite sequence of elements of the carrier of V is called an ordered basis of V if: (Def.4) It is one-to-one and rng it is a basis of V.

For simplicity we adopt the following convention: p will denote a finite sequence, M_{1} will denote a matrix over D of dimension n × m, M_{2} will denote a matrix over D of dimension k × m, V_{1}, V_{2}, V_{3} will denote finite dimensional vector spaces over K, f, f_{1}, f_{2} will denote maps from V_{1} into V_{2}, g will denote a map from V_{2} into V_{3}, b_{1} will denote an ordered basis of V_{1}, b_{2} will denote an ordered basis of V_{2}, b_{3} will denote an ordered basis of V_{3}, b will denote a basis of V_{1}, v_{1}, v_{2} will denote vectors of V_{2}, v will denote an element of the carrier of V_{1}, p_{2}, F will denote finite sequences of elements of the carrier of V_{1}, p_{1}, d will denote finite sequences of elements of the carrier of K, and K_{4} will denote a linear combination of V_{1}.

Let us consider K, let us consider V_{1}, V_{2}, and let us consider f_{1}, f_{2}. The functor f_{1} + f_{2} yielding a map from V_{1} into V_{2} is defined as follows: (Def.5) For every element v of the carrier of V_{1} holds (f_{1}+f_{2})(v) = f_{1}(v)+f_{2}(v).

Let us consider K, let us consider V_{1}, V_{2}, let us consider f, and let a be an element of the carrier of K. The functor a · f yielding a map from V_{1} into V_{2} is defined as follows: (Def.6) For every element v of the carrier of V_{1} holds (a · f)(v) = a · f(v).

The following propositions are true:

(13) Let a be an element of the carrier of V_{1}, and let F be a finite sequence of elements of the carrier of V_{1}, and let G be a finite sequence of elements of the carrier of K. Suppose len F = len G and for every k and for every element v of the carrier of K such that k ∈ dom F and v = G(k) holds F (k) = v · a. Then ∑ F = ∑ G · a.

(14) Let a be an element of the carrier of V_{1}, and let F be a finite sequence of elements of the carrier of K, and let G be a finite sequence of elements of the carrier of V_{1}. If len F = len G and for every k such that k ∈ dom F holds G(k) = π_{k}F · a, then ∑ G = ∑ F · a.

(15) If for every k such that k ∈ dom F holds π_{k}F = 0_{(V_{1})}, then ∑ F = 0_{(V_{1})}. Let us consider K, let us consider V_{1}, and let us consider p_{1}, p_{2}. The functor

lmlt(p_{1}, p_{2}) yielding a finite sequence of elements of the carrier of V_{1} is defined as follows: (Def.7) lmlt(p_{1}, p_{2}) = (the left multiplication of V_{1})^{∘}(p_{1}, p_{2}).

Next we state the proposition

(16) If dom p_{1} = dom p_{2}, then dom lmlt(p_{1}, p_{2}) = dom p_{1} and

dom lmlt(p_{1}, p_{2}) = dom p_{2}. Let us consider K, let us consider V_{1}, and let M be a matrix over the carrier

of V_{1}. The functor ∑ M yields a finite sequence of elements of the carrier of V_{1} and is defined as follows:

(Def.8) len ∑ M = len M and for every k such that k ∈ dom ∑ M holds

π_{k} ∑ M = ∑ Line(M, k). The following propositions are true:

(17) For every matrix M over the carrier of V_{1} such that len M = 0 holds ∑ ∑ M = 0_{(V_{1})}.

(18) For every matrix M over the carrier of V_{1} of dimension m + 1 × 0 holds ∑ ∑ M = 0_{(V_{1})}.

(19) For every element x of the carrier of V_{1} holds ⟨⟨x⟩⟩ = ⟨⟨x⟩⟩^{T}.

(20) For every finite sequence p of elements of the carrier of V_{1} such that f is linear holds f(∑ p) = ∑(f · p).

(21) Let a be a finite sequence of elements of the carrier of K and let p be a finite sequence of elements of the carrier of V_{1}. If len p = len a, then if f is linear, then f · lmlt(a, p) = lmlt(a, f · p).

(22) Let a be a finite sequence of elements of the carrier of K. If len a = len b_{2}, then if g is linear, then g(∑ lmlt(a, b_{2})) = ∑ lmlt(a, g · b_{2}).

(23) Let F, F_{1} be finite sequences of elements of the carrier of V_{1}, and let K_{4} be a linear combination of V_{1}, and let p be a permutation of dom F. If F_{1} = F · p, then K_{4} F_{1} = (K_{4} F) · p.

(24) If F is one-to-one and support K_{4} ⊆ rng F, then ∑(K_{4} F) = ∑ K_{4}.

(25) Let A be a set and let p be a finite sequence of elements of the carrier of V_{1}. Suppose rng p ⊆ A. Suppose f_{1} is linear and f_{2} is linear and for every v such that v ∈ A holds f_{1}(v) = f_{2}(v). Then f_{1}(∑ p) = f_{2}(∑ p).

(26) If f_{1} is linear and f_{2} is linear, then for every ordered basis b_{1} of V_{1} such that len b_{1} > 0 holds if f_{1} · b_{1} = f_{2} · b_{1}, then f_{1} = f_{2}.

Let D be a non empty set. Observe that every matrix over D is finite sequence yielding.

Let D be a non empty set and let F, G be matrices over D. Then F ^{⌢} G is a matrix over D.

Let D be a non empty set, let us consider n, m, k, let M_{1} be a matrix over D of dimension n × k, and let M_{2} be a matrix over D of dimension m × k. Then M_{1} ^{}M_{2} is a matrix over D of dimension n + m × k.

One can prove the following propositions:

(27) Given i, and let M_{1} be a matrix over D of dimension n × k, and let M_{2} be a matrix over D of dimension m × k. If i ∈ dom M_{1}, then Line(M_{1} ^{}M_{2}, i) = Line(M_{1}, i).

(28) Let M_{1} be a matrix over D of dimension n × k and let M_{2} be a matrix over D of dimension m × k. If width M_{1} = width M_{2}, then width(M_{1} ^{} M_{2}) = width M_{1} and width(M_{1} ^{}M_{2}) = width M_{2}.

(29) Given i, n, and let M_{1} be a matrix over D of dimension t × k, and let M_{2} be a matrix over D of dimension m × k. If n ∈ dom M_{2} and i = len M_{1} + n, then Line(M_{1} ^{}M_{2}, i) = Line(M_{2}, n).

(30) Let M_{1} be a matrix over D of dimension n × k and let M_{2} be a matrix over D of dimension m × k. If width M_{1} = width M_{2}, then for every i such that i ∈ Seg width M_{1} holds (M_{1} ^{}M_{2})_{,i} = ((M_{1})_{,i} )^{}((M_{2})_{,i} ).

(31) Let M_{1} be a matrix over the carrier of V_{1} of dimension n × k and let M_{2} be a matrix over the carrier of V_{1} of dimension m × k. Then ∑(M_{1} ^{}M_{2}) = (∑ M_{1})^{}∑ M_{2}.

(32) Let M_{1} be a matrix over D of dimension n × k and let M_{2} be a matrix over D of dimension m × k. If width M_{1} = width M_{2}, then (M_{1}^{}M_{2})^{T} = (M_{1}^{T})^{⌢} M_{2}^{T}.

(33) For all matrices M_{1}, M_{2} over the carrier of V_{1} holds (the addition of

V_{1})^{∘}(∑ M_{1}, ∑ M_{2}) = ∑(M_{1} ^{⌢} M_{2}). Let D be a non empty set, let F be a binary operation on D, and let P_{1},

P_{2} be finite sequences of elements of D. Then F^{∘}(P_{1}, P_{2}) is a finite sequence of elements of D.

Next we state several propositions:

(34) Let P_{1}, P_{2} be finite sequences of elements of the carrier of V_{1}. If len P_{1} = len P_{2}, then ∑((the addition of V_{1})^{∘}(P_{1}, P_{2})) = ∑ P_{1} + ∑ P_{2}.

(35) For all matrices M_{1}, M_{2} over the carrier of V_{1} such that len M_{1} = len M_{2} holds ∑ ∑ M_{1} + ∑ ∑ M_{2} = ∑ ∑(M_{1} ^{⌢} M_{2}).

(36) For every finite sequence P of elements of the carrier of V_{1} holds ∑ ∑⟨P ⟩ = ∑ ∑(⟨P ⟩^{T}).

(37) For every n and for every matrix M over the carrier of V_{1} such that len M = n holds ∑ ∑ M = ∑ ∑(M^{T}).

(38) Let M be a matrix over the carrier of K of dimension n × m. Suppose n > 0 and m > 0. Let p, d be finite sequences of elements of the carrier of K. Suppose len p = n and len d = m and for every j such that j ∈ dom d holds π_{j}d = ∑(p • M_{,j} ). Let b, c be finite sequences of elements of the carrier of V_{1}. Suppose len b = m and len c = n and for every i such that i ∈ dom c holds π_{i}c = ∑ lmlt(Line(M, i), b). Then ∑ lmlt(p, c) = ∑ lmlt(d, b).

4. Decomposition of a Vector in Basis

Let K be a field, let V be a finite dimensional vector space over K, let b_{1} be an ordered basis of V, and let W be an element of the carrier of V. The functor W → b_{1} yielding a finite sequence of elements of the carrier of K is defined by the conditions (Def.9). (Def.9) (i) len(W → b_{1}) = len b_{1}, and

(ii) there exists a linear combination K_{4} of V such that W = ∑ K_{4} and support K_{4} ⊆ rng b_{1} and for every k such that 1 ≤ k and k ≤ len(W → b_{1}) holds π_{k}(W → b_{1}) = K_{4}(π_{k}b_{1}).

The following four propositions are true:

(39) If v_{1} → b_{2} = v_{2} → b_{2}, then v_{1} = v_{2}.

(40) v = ∑ lmlt(v → b_{1}, b_{1}).

(41) If len d = len b_{1}, then d = ∑ lmlt(d, b_{1}) → b_{1}.

(42) Let a be a finite sequence of elements of the carrier of K. Suppose len a = len b_{2}. Let j be a natural number. Suppose j ∈ dom b_{3}. Let d be a finite sequence of elements of the carrier of K. Suppose len d = len b_{2} and for every k such that k ∈ dom b_{2} holds d(k) = π_{j}(g(π_{k}b_{2}) → b_{3}). If len b_{2} > 0 and len b_{3} > 0, then π_{j}(∑ lmlt(a, g · b_{2}) → b_{3}) = ∑(a • d).

5. Associated Matrix of Linear Map

Let K be a field, let V_{1}, V_{2} be finite dimensional vector spaces over K, let f be a function from the carrier of V_{1} into the carrier of V_{2}, let b_{1} be a finite sequence of elements of the carrier of V_{1}, and let b_{2} be an ordered basis of V_{2}. The functor AutMt(f, b_{1}, b_{2}) yielding a matrix over K is defined as follows:

(Def.10) len AutMt(f, b_{1}, b_{2}) = len b_{1} and for every k such that k ∈ dom b_{1} holds

π_{k} AutMt(f, b_{1}, b_{2}) = f(π_{k}b_{1}) → b_{2}. One can prove the following propositions:

(43) If len b_{1} = 0, then AutMt(f, b_{1}, b_{2}) = ε.

(44) If len b_{1} > 0, then width AutMt(f, b_{1}, b_{2}) = len b_{2}.

(45) If f_{1} is linear and f_{2} is linear, then if AutMt(f_{1}, b_{1}, b_{2}) = AutMt(f_{2}, b_{1}, b_{2}) and len b_{1} > 0, then f_{1} = f_{2}.

(46) If f is linear and g is linear and len b_{1} > 0 and len b_{2} > 0 and len b_{3} > 0, then AutMt(g · f, b_{1}, b_{3}) = AutMt(f, b_{1}, b_{2}) · AutMt(g, b_{2}, b_{3}).

(47) AutMt(f_{1} + f_{2}, b_{1}, b_{2}) = AutMt(f_{1}, b_{1}, b_{2}) + AutMt(f_{2}, b_{1}, b_{2}).

(48) If a ≠ 0_{K}, then AutMt(a · f, b_{1}, b_{2}) = a · AutMt(f, b_{1}, b_{2}).

References

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associated matrix of linear map 345

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Received June 30, 1995 346

On the Geometry of a Go-Board

Andrzej Trybulec Warsaw University Białystok

MML Identifier: GOBOARD6.

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

For simplicity we follow the rules: i, j, n will be natural numbers, r, s, r_{1}, s_{1}, r_{2}, s_{2} will be real numbers, p will be a point of E_{T}^{2}, G will be a Go-board, M will be a metric space, and u will be a point of E^{2}.

One can prove the following propositions: (4)^{1} For every metric space M and for every point u of M such that r > 0

holds u ∈ Ball(u, r). (6)^{2} For every subset B of the carrier of E_{T}^{n} and for every point u of E^{n} such

that B = Ball(u, r) holds B is open.

(7) Let M be a metric space, and let u be a point of M, and let P be a subset of the carrier of M_{top}. Then u ∈ Int P if and only if there exists r such that r > 0 and Ball(u, r) ⊆ P.

(8) Let u be a point of E^{n} and let P be a subset of the carrier of E_{T}^{n}. Then u ∈ Int P if and only if there exists r such that r > 0 and Ball(u, r) ⊆ P.

(9) For all points u, v of E^{2} such that u = [r_{1}, s_{1}] and v = [r_{2}, s_{2}] holds ρ(u, v) = \sqrt{(r_{1} − r_{2})^{2} + (s_{1} − s_{2})^{2}}.

(10) For every point u of E^{2} such that u = [r, s] holds if 0 ≤ r_{2} and r_{2} < r_{1}, then [r + r_{2}, s] ∈ Ball(u, r_{1}).

(11) For every point u of E^{2} such that u = [r, s] holds if 0 ≤ s_{2} and s_{2} < s_{1}, then [r, s + s_{2}] ∈ Ball(u, s_{1}).

(12) For every point u of E^{2} such that u = [r, s] holds if 0 ≤ r_{2} and r_{2} < r_{1}, then [r − r_{2}, s] ∈ Ball(u, r_{1}).

1The propositions (1)–(3) have been removed.

2The proposition (5) has been removed.

(13) For every point u of E^{2} such that u = [r, s] holds if 0 ≤ s_{2} and s_{2} < s_{1}, then [r, s − s_{2}] ∈ Ball(u, s_{1}).

(14) If 1 ≤ i and i < len G and 1 ≤ j and j < width G, then G_{i,j} +G_{i+1,j+1} = G_{i,j+1} + G_{i+1,j}.

(15) Int vstrip(G, 0) = {[r, s] : r < (G_{1,1})_{1}}.

(16) Int vstrip(G, len G) = {[r, s] : (G_{len G,1})_{1} < r}.

(17) If 1 ≤ i and i < len G, then Int vstrip(G, i) = {[r, s] : (G_{i,1})_{1} < r ∧ r < (G_{i+1,1})_{1}}.

(18) Int hstrip(G, 0) = {[r, s] : s < (G_{1,1})_{2}}.

(19) Int hstrip(G, width G) = {[r, s] : (G_{1,width G})_{2} < s}.

(20) If 1 ≤ j and j < width G, then Int hstrip(G, j) = {[r, s] : (G_{1,j})_{2} < s ∧ s < (G_{1,j+1})_{2}}.

(21) Int cell(G, 0, 0) = {[r, s] : r < (G_{1,1})_{1} ∧ s < (G_{1,1})_{2}}.

(22) Int cell(G, 0, width G) = {[r, s] : r < (G_{1,1})_{1} ∧ (G_{1,width G})_{2} < s}.

(23) If 1 ≤ j and j < width G, then Int cell(G, 0, j) = {[r, s] : r < (G_{1,1})_{1} ∧ (G_{1,j})_{2} < s ∧ s < (G_{1,j+1})_{2}}.

(24) Int cell(G, len G, 0) = {[r, s] : (G_{len G,1})_{1} < r ∧ s < (G_{1,1})_{2}}.

(25) Int cell(G, len G, width G) = {[r, s] : (G_{len G,1})_{1} < r ∧ (G_{1,width G})_{2} < s}.

(26) If 1 ≤ j and j < width G, then Int cell(G, len G, j) = {[r, s] : (G_{len G,1})_{1} < r ∧ (G_{1,j})_{2} < s ∧ s < (G_{1,j+1})_{2}}.

(27) If 1 ≤ i and i < len G, then Int cell(G, i, 0) = {[r, s] : (G_{i,1})_{1} < r ∧ r < (G_{i+1,1})_{1} ∧ s < (G_{1,1})_{2}}.

(28) If 1 ≤ i and i < len G, then Int cell(G, i, width G) = {[r, s] : (G_{i,1})_{1} < r ∧ r < (G_{i+1,1})_{1} ∧ (G_{1,width G})_{2} < s}.

(29) If 1 ≤ i and i < len G and 1 ≤ j and j < width G, then Int cell(G, i, j) = {[r, s] : (G_{i,1})_{1} < r ∧ r < (G_{i+1,1})_{1} ∧ (G_{1,j})_{2} < s ∧ s < (G_{1,j+1})_{2}}.

(30) If 1 ≤ j and j ≤ width G and p ∈ Int hstrip(G, j), then p_{2} > (G_{1,j})_{2}.

(31) If j < width G and p ∈ Int hstrip(G, j), then p_{2} < (G_{1,j+1})_{2}.

(32) If 1 ≤ i and i ≤ len G and p ∈ Int vstrip(G, i), then p_{1} > (G_{i,1})_{1}.

(33) If i < len G and p ∈ Int vstrip(G, i), then p_{1} < (G_{i+1,1})_{1}.

(34) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then \frac{1}{2} · (G_{i,j} + G_{i+1,j+1}) ∈ Int cell(G, i, j).

(35) If 1 ≤ i and i + 1 ≤ len G, then \frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1] ∈ Int cell(G, i, width G).

(36) If 1 ≤ i and i+1 ≤ len G, then \frac{1}{2} ·(G_{i,1}+G_{i+1,1})−[0, 1] ∈ Int cell(G, i, 0).

(37) If 1 ≤ j and j + 1 ≤ width G, then \frac{1}{2} · (G_{len G,j} + G_{len G,j+1}) + [1, 0] ∈ Int cell(G, len G, j).

(38) If 1 ≤ j and j + 1 ≤ width G, then \frac{1}{2} · (G_{1,j} + G_{1,j+1}) − [1, 0] ∈ Int cell(G, 0, j).

(39) G_{1,1} − [1, 1] ∈ Int cell(G, 0, 0).

(40) G_{len G,width G} + [1, 1] ∈ Int cell(G, len G, width G).

(41) G_{1,width G} + [−1, 1] ∈ Int cell(G, 0, width G).

(42) G_{len G,1} + [1, −1] ∈ Int cell(G, len G, 0).

(43) If 1 ≤ i and i < len G and 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{i,j} + G_{i+1,j+1}), \frac{1}{2} · (G_{i,j} + G_{i,j+1})) ⊆ Int cell(G, i, j) ∪ \{\frac{1}{2} · (G_{i,j} + G_{i,j+1})\}.

(44) Suppose 1 ≤ i and i < len G and 1 ≤ j and j < width G. Then L(\frac{1}{2} · (G_{i,j} + G_{i+1,j+1}), \frac{1}{2} · (G_{i,j+1} + G_{i+1,j+1})) ⊆ Int cell(G, i, j) ∪ \{\frac{1}{2} · (G_{i,j+1} + G_{i+1,j+1})}.

(45) Suppose 1 ≤ i and i < len G and 1 ≤ j and j < width G. Then L(\frac{1}{2} · (G_{i,j} + G_{i+1,j+1}), \frac{1}{2} · (G_{i+1,j} + G_{i+1,j+1})) ⊆ Int cell(G, i, j) ∪ \{\frac{1}{2} · (G_{i+1,j} + G_{i+1,j+1})}.

(46) If 1 ≤ i and i < len G and 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{i,j} + G_{i+1,j+1}), \frac{1}{2} · (G_{i,j} + G_{i+1,j})) ⊆ Int cell(G, i, j) ∪ \{\frac{1}{2} · (G_{i,j} + G_{i+1,j})\}.

(47) If 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{1,j} + G_{1,j+1}) − [1, 0], \frac{1}{2} · (G_{1,j} + G_{1,j+1})) ⊆ Int cell(G, 0, j) ∪ \{\frac{1}{2} · (G_{1,j} + G_{1,j+1})\}.

(48) If 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{len G,j} + G_{len G,j+1}) + [1, 0], \frac{1}{2} · (G_{len G,j} + G_{len G,j+1})) ⊆ Int cell(G, len G, j) ∪ \{\frac{1}{2} · (G_{len G,j} + G_{len G,j+1})\}.

(49) If 1 ≤ i and i < len G, then L(\frac{1}{2} · (G_{i,1} + G_{i+1,1}) − [0, 1], \frac{1}{2} · (G_{i,1} + G_{i+1,1})) ⊆ Int cell(G, i, 0) ∪ \{\frac{1}{2} · (G_{i,1} + G_{i+1,1})\}.

(50) If 1 ≤ i and i < len G, then L(\frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1], \frac{1}{2} ·(G_{i,width G} +G_{i+1,width G})) ⊆ Int cell(G, i, width G)∪\{\frac{1}{2} ·(G_{i,width G} + G_{i+1,width G})}.

(51) If 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{1,j} + G_{1,j+1}) − [1, 0], G_{1,j} − [1, 0]) ⊆ Int cell(G, 0, j) ∪ {G_{1,j} − [1, 0]}.

(52) If 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{1,j} + G_{1,j+1}) − [1, 0], G_{1,j+1} − [1, 0]) ⊆ Int cell(G, 0, j) ∪ {G_{1,j+1} − [1, 0]}.

(53) If 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{len G,j} + G_{len G,j+1}) + [1, 0], G_{len G,j} + [1, 0]) ⊆ Int cell(G, len G, j) ∪ {G_{len G,j} + [1, 0]}.

(54) If 1 ≤ j and j < width G, then L(\frac{1}{2} · (G_{len G,j} + G_{len G,j+1}) + [1, 0], G_{len G,j+1} + [1, 0]) ⊆ Int cell(G, len G, j) ∪ {G_{len G,j+1} + [1, 0]}.

(55) If 1 ≤ i and i < len G, then L(\frac{1}{2} · (G_{i,1} + G_{i+1,1}) − [0, 1], G_{i,1} − [0, 1]) ⊆ Int cell(G, i, 0) ∪ {G_{i,1} − [0, 1]}.

(56) If 1 ≤ i and i < len G, then L(\frac{1}{2} · (G_{i,1} + G_{i+1,1}) − [0, 1], G_{i+1,1} − [0, 1]) ⊆ Int cell(G, i, 0) ∪ {G_{i+1,1} − [0, 1]}.

(57) If 1 ≤ i and i < len G, then L(\frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1], G_{i,width G} + [0, 1]) ⊆ Int cell(G, i, width G) ∪ {G_{i,width G} + [0, 1]}.

(58) If 1 ≤ i and i < len G, then L(\frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1], G_{i+1,width G} + [0, 1]) ⊆ Int cell(G, i, width G) ∪ {G_{i+1,width G} + [0, 1]}.

(59) L(G_{1,1} − [1, 1], G_{1,1} − [1, 0]) ⊆ Int cell(G, 0, 0) ∪ {G_{1,1} − [1, 0]}.

(60) L(G_{len G,1} +[1, −1], G_{len G,1} +[1, 0]) ⊆ Int cell(G, len G, 0)∪{G_{len G,1} +[1, 0]}.

(61) L(G_{1,width G} + [−1, 1], G_{1,width G} − [1, 0]) ⊆ Int cell(G, 0, width G) ∪ {G_{1,width G} − [1, 0]}.

(62) L(G_{len G,width G}+[1, 1], G_{len G,width G}+[1, 0]) ⊆ Int cell(G, len G, width G)∪ {G_{len G,width G} + [1, 0]}.

(63) L(G_{1,1} − [1, 1], G_{1,1} − [0, 1]) ⊆ Int cell(G, 0, 0) ∪ {G_{1,1} − [0, 1]}.

(64) L(G_{len G,1} +[1, −1], G_{len G,1} −[0, 1]) ⊆ Int cell(G, len G, 0)∪{G_{len G,1} −[0, 1]}.

(65) L(G_{1,width G} + [−1, 1], G_{1,width G} + [0, 1]) ⊆ Int cell(G, 0, width G) ∪ {G_{1,width G} + [0, 1]}.

(66) L(G_{len G,width G}+[1, 1], G_{len G,width G}+[0, 1]) ⊆ Int cell(G, len G, width G)∪ {G_{len G,width G} + [0, 1]}.

(67) Suppose 1 ≤ i and i < len G and 1 ≤ j and j + 1 < width G. Then L(\frac{1}{2} · (G_{i,j} +G_{i+1,j+1}), \frac{1}{2} ·(G_{i,j+1}+G_{i+1,j+2})) ⊆ Int cell(G, i, j)∪Int cell(G, i, j + 1) ∪ \{\frac{1}{2} · (G_{i,j+1} + G_{i+1,j+1})\}.

(68) Suppose 1 ≤ j and j < width G and 1 ≤ i and i + 1 < len G. Then L(\frac{1}{2} · (G_{i,j} + G_{i+1,j+1}), \frac{1}{2} · (G_{i+1,j} + G_{i+2,j+1})) ⊆ Int cell(G, i, j) ∪ Int cell(G, i + 1, j) ∪ \{\frac{1}{2} · (G_{i+1,j} + G_{i+1,j+1})\}.

(69) If 1 ≤ i and i < len G and 1 < width G, then L(\frac{1}{2} · (G_{i,1} + G_{i+1,1}) − [0, 1], \frac{1}{2}·(G_{i,1}+G_{i+1,2})) ⊆ Int cell(G, i, 0)∪Int cell(G, i, 1)∪\{\frac{1}{2}·(G_{i,1}+G_{i+1,1})\}.

(70) Suppose 1 ≤ i and i < len G and 1 < width G. Then L(\frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1], \frac{1}{2} · (G_{i,width G} + G_{i+1,width G−′1})) ⊆ Int cell(G, i, width G −′ 1) ∪ Int cell(G, i, width G) ∪ \{\frac{1}{2} · (G_{i,width G} + G_{i+1,width G})}.

(71) If 1 ≤ j and j < width G and 1 < len G, then L(\frac{1}{2} · (G_{1,j} + G_{1,j+1}) − [1, 0], \frac{1}{2} · (G_{1,j} + G_{2,j+1})) ⊆ Int cell(G, 0, j) ∪ Int cell(G, 1, j) ∪ \{\frac{1}{2} · (G_{1,j} + G_{1,j+1})}.

(72) Suppose 1 ≤ j and j < width G and 1 < len G. Then L(\frac{1}{2} · (G_{len G,j} + G_{len G,j+1}) + [1, 0], \frac{1}{2} · (G_{len G,j} + G_{len G−′1,j+1})) ⊆ Int cell(G, len G −′ 1, j) ∪ Int cell(G, len G, j) ∪ \{\frac{1}{2} · (G_{len G,j} + G_{len G,j+1})\}.

(73) If 1 < len G and 1 ≤ j and j+1 < width G, then L(\frac{1}{2} ·(G_{1,j}+G_{1,j+1})−[1, 0], \frac{1}{2} · (G_{1,j+1} + G_{1,j+2}) − [1, 0]) ⊆ Int cell(G, 0, j) ∪ Int cell(G, 0, j + 1) ∪ {G_{1,j+1} − [1, 0]}.

(74) Suppose 1 < len G and 1 ≤ j and j +1 < width G. Then L(\frac{1}{2} ·(G_{len G,j} + G_{len G,j+1})+[1, 0], \frac{1}{2} ·(G_{len G,j+1}+G_{len G,j+2})+[1, 0]) ⊆ Int cell(G, len G, j)∪ Int cell(G, len G, j + 1) ∪ {G_{len G,j+1} + [1, 0]}.

(75) If 1 < width G and 1 ≤ i and i+1 < len G, then L(\frac{1}{2} ·(G_{i,1}+G_{i+1,1})−[0, 1], \frac{1}{2} · (G_{i+1,1} + G_{i+2,1}) − [0, 1]) ⊆ Int cell(G, i, 0) ∪ Int cell(G, i + 1, 0) ∪ {G_{i+1,1} − [0, 1]}.

(76) Suppose 1 < width G and 1 ≤ i and i + 1 < len G. Then L(\frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1], \frac{1}{2} · (G_{i+1,width G} + G_{i+2,width G}) + [0, 1]) ⊆ Int cell(G, i, width G) ∪ Int cell(G, i + 1, width G) ∪ {G_{i+1,width G} + [0, 1]}.

(77) If 1 < len G and 1 < width G, then L(G_{1,1} − [1, 1], \frac{1}{2} · (G_{1,1} + G_{1,2}) − [1, 0]) ⊆ Int cell(G, 0, 0) ∪ Int cell(G, 0, 1) ∪ {G_{1,1} − [1, 0]}.

(78) If 1 < len G and 1 < width G, then L(G_{len G,1} + [1, −1], \frac{1}{2} · (G_{len G,1} + G_{len G,2})+[1, 0]) ⊆ Int cell(G, len G, 0)∪Int cell(G, len G, 1)∪{G_{len G,1} +[1, 0]}.

(79) If 1 < len G and 1 < width G, then L(G_{1,width G}+[−1, 1], \frac{1}{2} ·(G_{1,width G}+ G_{1,width G−′1})−[1, 0]) ⊆ Int cell(G, 0, width G)∪Int cell(G, 0, width G−′1)∪ {G_{1,width G} − [1, 0]}.

(80) If 1 < len G and 1 < width G, then L(G_{len G,width G} + [1, 1], \frac{1}{2} · (G_{len G,width G} + G_{len G,width G−′1}) + [1, 0]) ⊆ Int cell(G, len G, width G) ∪ Int cell(G, len G, width G −′ 1) ∪ {G_{len G,width G} + [1, 0]}.

(81) If 1 < width G and 1 < len G, then L(G_{1,1} − [1, 1], \frac{1}{2} · (G_{1,1} + G_{2,1}) − [0, 1]) ⊆ Int cell(G, 0, 0) ∪ Int cell(G, 1, 0) ∪ {G_{1,1} − [0, 1]}.

(82) If 1 < width G and 1 < len G, then L(G_{1,width G}+[−1, 1], \frac{1}{2} ·(G_{1,width G}+ G_{2,width G}) + [0, 1]) ⊆ Int cell(G, 0, width G) ∪ Int cell(G, 1, width G) ∪ {G_{1,width G} + [0, 1]}.

(83) If 1 < width G and 1 < len G, then L(G_{len G,1} + [1, −1], \frac{1}{2} · (G_{len G,1} + G_{len G−′1,1}) − [0, 1]) ⊆ Int cell(G, len G, 0) ∪ Int cell(G, len G −′ 1, 0) ∪ {G_{len G,1} − [0, 1]}.

(84) If 1 < width G and 1 < len G, then L(G_{len G,width G} + [1, 1], \frac{1}{2} · (G_{len G,width G} + G_{len G−′1,width G}) + [0, 1]) ⊆ Int cell(G, len G, width G) ∪ Int cell(G, len G −′ 1, width G) ∪ {G_{len G,width G} + [0, 1]}.

(85) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then L(\frac{1}{2} · (G_{i,j} + G_{i+1,j+1}), p) meets Int cell(G, i, j).

(86) If 1 ≤ i and i + 1 ≤ len G, then L(p, \frac{1}{2} · (G_{i,width G} + G_{i+1,width G}) + [0, 1]) meets Int cell(G, i, width G).

(87) If 1 ≤ i and i + 1 ≤ len G, then L(\frac{1}{2} · (G_{i,1} + G_{i+1,1}) − [0, 1], p) meets Int cell(G, i, 0).

(88) If 1 ≤ j and j + 1 ≤ width G, then L(\frac{1}{2} · (G_{1,j} + G_{1,j+1}) − [1, 0], p) meets Int cell(G, 0, j).

(89) If 1 ≤ j and j + 1 ≤ width G, then L(p, \frac{1}{2} · (G_{len G,j} + G_{len G,j+1}) + [1, 0]) meets Int cell(G, len G, j).

(90) L(p, G_{1,1} − [1, 1]) meets Int cell(G, 0, 0).

(91) L(p, G_{len G,width G} + [1, 1]) meets Int cell(G, len G, width G).

(92) L(p, G_{1,width G} + [−1, 1]) meets Int cell(G, 0, width G).

(93) L(p, G_{len G,1} + [1, −1]) meets Int cell(G, len G, 0).

352 andrzej trybulec

References

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The Theorem of Weierstrass

Józef Białas Łódź University Łódź Yatsuka Nakamura Shinshu University Nagano

Summary. The basic purpose of this article is to prove the important Weierstrass’ theorem which states that a real valued continuous function f on a topological space T assumes a maximum and a minimum value on the compact subset S of T, i.e., there exist points x_{1}, x_{2} of T being elements of S, such that f(x_{1}) and f(x_{2}) are the supremum and the infimum, respectively, of f(S), which is the image of S under the function f. The paper is divided into three parts. In the first part, we prove some auxiliary theorems concerning properties of balls in metric spaces and define special families of subsets of topological spaces. These concepts are used in the next part of the paper which contains the essential part of the article, namely the formalization of the proof of Weierstrass’ theorem. Here, we also prove a theorem concerning the compactness of images of compact sets of T under a continuous function. The final part of this work is developed for the purpose of defining some measures of the distance between compact subsets of topological metric spaces. Some simple theorems about these measures are also proved.

MML Identifier: WEIERSTR.

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

1. Preliminaries

One can prove the following propositions:

(1) Let M be a metric space, and let x_{1}, x_{2} be points of M, and let r_{1}, r_{2} be real numbers. Then there exists a point x of M and there exists a real number r such that Ball(x_{1}, r_{1}) ∪ Ball(x_{2}, r_{2}) ⊆ Ball(x, r).

(2) Let M be a metric space, and let n be a natural number, and let F be a family of subsets of M, and let p be a finite sequence. Suppose F is finite and a family of balls and rng p = F and dom p = Seg(n + 1). Then there exists a family G of subsets of M such that

(i) G is finite and a family of balls, and (ii) there exists a finite sequence q such that rng q = G and dom q = Seg n

and there exists a point x of M and there exists a real number r such that ⋃ F ⊆ ⋃ G ∪ Ball(x, r).

(3) Let M be a metric space and let F be a family of subsets of M. Suppose F is finite and a family of balls. Then there exists a point x of M and there exists a real number r such that ⋃ F ⊆ Ball(x, r).

Let T, S be topological spaces, let f be a map from T into S, and let G be a family of subsets of S. The functor f ^{−1} G yields a family of subsets of T and is defined by the condition (Def.1). (Def.1) Let A be a subset of the carrier of T. Then A ∈ f ^{−1} G if and only if

there exists a subset B of the carrier of S such that B ∈ G and A = f^{−1}B. Next we state two propositions:

(4) Let T, S be topological spaces, and let f be a map from T into S, and let A, B be families of subsets of S. If A ⊆ B, then f ^{−1} A ⊆ f ^{−1} B.

(5) Let T, S be topological spaces, and let f be a map from T into S, and let B be a family of subsets of S. If f is continuous and B is open, then f ^{−1} B is open.

Let T, S be topological spaces, let f be a map from T into S, and let G be a family of subsets of T. The functor f^{∘}G yields a family of subsets of S and is defined by the condition (Def.2). (Def.2) Let A be a subset of the carrier of S. Then A ∈ f^{∘}G if and only if there

exists a subset B of the carrier of T such that B ∈ G and A = f^{∘}B. One can prove the following propositions:

(6) Let T, S be topological spaces, and let f be a map from T into S, and let A, B be families of subsets of T. If A ⊆ B, then f^{∘}A ⊆ f^{∘}B.

(7) Let T, S be topological spaces, and let f be a map from T into S, and let B be a family of subsets of S, and let P be a subset of the carrier of S. If f^{∘}f ^{−1} B is a cover of P, then B is a cover of P.

(8) Let T, S be topological spaces, and let f be a map from T into S, and let B be a family of subsets of T, and let P be a subset of the carrier of T. If B is a cover of P, then f ^{−1} f^{∘}B is a cover of P.

(9) Let T, S be topological spaces, and let f be a map from T into S, and let Q be a family of subsets of S. Then ⋃(f^{∘}f ^{−1} Q) ⊆ ⋃ Q.

(10) Let T, S be topological spaces, and let f be a map from T into S, and let P be a family of subsets of T. Then ⋃ P ⊆ ⋃(f ^{−1} f^{∘}P).

(11) Let T, S be topological spaces, and let f be a map from T into S, and let Q be a family of subsets of S. If Q is finite, then f ^{−1} Q is finite.

(12) Let T, S be topological spaces, and let f be a map from T into S, and let P be a family of subsets of T. If P is finite, then f^{∘}P is finite.

(13) Let T, S be topological spaces, and let f be a map from T into S, and let P be a subset of the carrier of T, and let F be a family of subsets of S. Given a family B of subsets of T such that B ⊆ f ^{−1} F and B is a cover of P and finite. Then there exists a family G of subsets of S such that G ⊆ F and G is a cover of f^{∘}P and finite.

2. The Weierstrass’ Theorem

One can prove the following three propositions:

(14) Let T, S be topological spaces, and let f be a map from T into S, and let P be a subset of the carrier of T. If P is compact and f is continuous, then f^{∘}P is compact.

(15) Let T be a topological space, and let f be a map from T into ^{1}, and let P be a subset of the carrier of T. If P is compact and f is continuous, then f^{∘}P is compact.

(16) Let f be a map from E_{T}^{2} into ^{1} and let P be a subset of the carrier of

E_{T}^{2}. If P is compact and f is continuous, then f^{∘}P is compact. Let P be a subset of the carrier of ^{1}. The functor Ω_{P} yields a subset of

and is defined as follows: (Def.3) Ω_{P} = P.

Next we state three propositions:

(17) For every subset P of the carrier of ^{1} such that P is compact holds Ω_{P} is bounded.

(18) For every subset P of the carrier of ^{1} such that P is compact holds Ω_{P} is closed.

(19) For every subset P of the carrier of ^{1} such that P is compact holds

Ω_{P} is compact. Let P be a subset of the carrier of ^{1}. The functor sup P yields a real number

and is defined as follows: (Def.4) sup P = sup(Ω_{P}). The functor inf P yielding a real number is defined by: (Def.5) inf P = inf(Ω_{P}).

We now state two propositions:

(20) Let T be a topological space, and let f be a map from T into ^{1}, and let P be a subset of the carrier of T. Suppose P ≠ ∅ and P is compact and f is continuous. Then there exists a point x_{1} of T such that x_{1} ∈ P and f(x_{1}) = sup(f^{∘}P).

(21) Let T be a topological space, and let f be a map from T into ^{1}, and let P be a subset of the carrier of T. Suppose P ≠ ∅ and P is compact and f is continuous. Then there exists a point x_{2} of T such that x_{2} ∈ P and f(x_{2}) = inf(f^{∘}P).

3. The Measure of the Distance Between Compact Sets

Let M be a metric space and let x be a point of M. The functor dist(x) yielding a map from M_{top} into ^{1} is defined by: (Def.6) For every point y of M holds (dist(x))(y) = ρ(y, x).

The following three propositions are true:

(22) For every metric space M and for every point x of M holds dist(x) is continuous.

(23) Let M be a metric space, and let x be a point of M, and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Then there exists a point x_{1} of M_{top} such that x_{1} ∈ P and (dist(x))(x_{1}) = sup((dist(x))^{∘}P).

(24) Let M be a metric space, and let x be a point of M, and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Then there exists a point x_{2} of M_{top} such that x_{2} ∈ P and (dist(x))(x_{2}) = inf((dist(x))^{∘}P).

Let M be a metric space and let P be a subset of the carrier of M_{top}. Let us assume that P ≠ ∅ and P is compact. The functor dist_{max}(P) yielding a map from M_{top} into ^{1} is defined by: (Def.7) For every point x of M holds (dist_{max}(P))(x) = sup((dist(x))^{∘}P). The functor dist_{min}(P) yields a map from M_{top} into ^{1} and is defined by: (Def.8) For every point x of M holds (dist_{min}(P))(x) = inf((dist(x))^{∘}P).

One can prove the following propositions:

(25) Let M be a metric space and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Let p_{1}, p_{2} be points of M. If p_{1} ∈ P, then ρ(p_{1}, p_{2}) ≤ sup((dist(p_{2}))^{∘}P) and inf((dist(p_{2}))^{∘}P) ≤ ρ(p_{1}, p_{2}).

(26) Let M be a metric space and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Let p_{1}, p_{2} be points of M. Then | sup((dist(p_{1}))^{∘}P) − sup((dist(p_{2}))^{∘}P)| ≤ ρ(p_{1}, p_{2}).

(27) Let M be a metric space and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Let p_{1}, p_{2} be points of M and let x_{1}, x_{2} be real numbers. If x_{1} = (dist_{max}(P))(p_{1}) and x_{2} = (dist_{max}(P))(p_{2}), then |x_{1} − x_{2}| ≤ ρ(p_{1}, p_{2}).

(28) Let M be a metric space and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Let p_{1}, p_{2} be points of M. Then | inf((dist(p_{1}))^{∘}P) − inf((dist(p_{2}))^{∘}P)| ≤ ρ(p_{1}, p_{2}).

(29) Let M be a metric space and let P be a subset of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact. Let p_{1}, p_{2} be points of M and let x_{1}, x_{2} be real numbers. If x_{1} = (dist_{min}(P))(p_{1}) and x_{2} = (dist_{min}(P))(p_{2}), then |x_{1} − x_{2}| ≤ ρ(p_{1}, p_{2}).

(30) Let M be a metric space and let X be a subset of the carrier of M_{top}. If X ≠ ∅ and X is compact, then dist_{max}(X) is continuous.

(31) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exists a point x_{1} of M_{top} such that x_{1} ∈ Q and (dist_{max}(P))(x_{1}) = sup((dist_{max}(P))^{∘}Q).

(32) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exists a point x_{2} of M_{top} such that x_{2} ∈ Q and (dist_{max}(P))(x_{2}) = inf((dist_{max}(P))^{∘}Q).

(33) Let M be a metric space and let X be a subset of the carrier of M_{top}. If X ≠ ∅ and X is compact, then dist_{min}(X) is continuous.

(34) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exists a point x_{1} of M_{top} such that x_{1} ∈ Q and (dist_{min}(P))(x_{1}) = sup((dist_{min}(P))^{∘}Q).

(35) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exists a point x_{2} of M_{top} such that x_{2} ∈ Q and (dist_{min}(P))(x_{2}) = inf((dist_{min}(P))^{∘}Q).

Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Let us assume that P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. The functor dist_{min}^{min}(P, Q) yields a real number and is defined as follows: (Def.9) dist_{min}^{min}(P, Q) = inf((dist_{min}(P))^{∘}Q). The functor dist_{min}^{max}(P, Q) yielding a real number is defined as follows:

(Def.10) dist_{min}^{max}(P, Q) = sup((dist_{min}(P))^{∘}Q).

The functor dist_{max}^{min}(P, Q) yielding a real number is defined as follows: (Def.11) dist_{max}^{min}(P, Q) = inf((dist_{max}(P))^{∘}Q).

The functor dist_{max}^{max}(P, Q) yielding a real number is defined as follows: (Def.12) dist_{max}^{max}(P, Q) = sup((dist_{max}(P))^{∘}Q).

One can prove the following propositions:

(36) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exist points x_{1}, x_{2} of M such that x_{1} ∈ P and x_{2} ∈ Q and ρ(x_{1}, x_{2}) = dist_{min}^{min}(P, Q).

(37) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exist points x_{1}, x_{2} of M such that x_{1} ∈ P and x_{2} ∈ Q and ρ(x_{1}, x_{2}) = dist_{max}^{min}(P, Q).

358 józef białas and yatsuka nakamura

(38) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exist points x_{1}, x_{2} of M such that x_{1} ∈ P and x_{2} ∈ Q and ρ(x_{1}, x_{2}) = dist_{min}^{max}(P, Q).

(39) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then there exist points x_{1}, x_{2} of M such that x_{1} ∈ P and x_{2} ∈ Q and ρ(x_{1}, x_{2}) = dist_{max}^{max}(P, Q).

(40) Let M be a metric space and let P, Q be subsets of the carrier of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Let x_{1}, x_{2} be points of M. If x_{1} ∈ P and x_{2} ∈ Q, then dist_{min}^{min}(P, Q) ≤ ρ(x_{1}, x_{2}) and ρ(x_{1}, x_{2}) ≤ dist_{max}^{max}(P, Q).

References

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[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.

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the theorem of weierstrass 359

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360

Dyadic Numbers and T_{4} Topological Spaces

Józef Białas Łódź University Łódź Yatsuka Nakamura Shinshu University Nagano

Summary. This article is the first part of a paper proving the fundamental Urysohn’s Theorem concerning the existence of a real valued continuous function on a normal topological space. The paper is divided into four parts. In the first part, we prove some auxiliary theorems concerning properties of natural numbers and prove two useful schemes about recurrently defined functions; in the second part, we define a special set of rational numbers, which we call dyadic, and prove some of its properties. The next part of the paper contains the definitions of T_{1} space and normal space, and we prove related theorems used in later parts of the paper. The final part of this work is developed for proving the theorem about the existence of some special family of subsets of a topological space. This theorem is essential in proving Urysohn’s Lemma.

MML Identifier: URYSOHN1.

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

1. Preliminaries

The following propositions are true:

(1) 0 ≠ \frac{1}{2} and 1 ≠ \frac{1}{2}.

(2) 0 < \frac{1}{2} and \frac{1}{2} < 1.

(3) For every natural number n holds 1 ≤ 2^{n}.

(4) For every natural number n holds 0 < 2^{n}. In this article we present several logical schemes. The scheme

FuncEx2DChoice deals with a non empty set A, a non empty set B, a non empty set C, and a ternary predicate P, and states that:

There exists a function F from [: A, B:] into C such that for every element x of A and for every element y of B holds P[x, y, F (⟨⟨x, y⟩⟩)]

provided the parameters meet the following requirement:

• For every element x of A and for every element y of B there exists an element z of C such that P[x, y, z].

The scheme RecExDN RD concerns a non empty set A, an element B of A, and a ternary predicate P, and states that:

There exists a function F from into A such that F (0) = B and for every element n of holds P[n, F (n), F (n + 1)]

provided the parameters satisfy the following condition:

• For every natural number n and for every element x of A there exists an element y of A such that P[n, x, y].

2. Dyadic Numbers

The subset _{<0} of is defined by: (Def.1) For every real number x holds x ∈ _{<0} iff x < 0.

The subset _{>1} of is defined by: (Def.2) For every real number x holds x ∈ _{>1} iff 1 < x.

Let n be a natural number. The functor dyadic(n) yields a subset of and is defined by: (Def.3) For every real number x holds x ∈ dyadic(n) iff there exists a natural

number i such that 0 ≤ i and i ≤ 2^{n} and x = \frac{i}{2^{n}} . The subset DYADIC of is defined by:

(Def.4) For every real number a holds a ∈ DYADIC iff there exists a natural

number n such that a ∈ dyadic(n). The subset DOM of is defined by:

(Def.5) DOM = _{<0} ∪ DYADIC ∪ _{>1}.

Let T be a topological space, let A be a non empty subset of , let F be a function from A into 2^{the carrier of T}, and let r be an element of A. Then F (r) is a subset of the carrier of T.

One can prove the following three propositions:

(5) For every natural number n and for every real number x such that x ∈ dyadic(n) holds 0 ≤ x and x ≤ 1.

(6) dyadic(0) = {0, 1}.

(7) dyadic(1) = \{0, \frac{1}{2}, 1\}. Let n be a natural number. Note that dyadic(n) is non empty. Next we state the proposition

(8) For every natural number x and for every natural number n holds x^{n} is a natural number.

dyadic numbers and t_{4} topological … 363

Let x, n be natural numbers. Then x^{n} is a natural number.

The following proposition is true

(9) Let n be a natural number. Then there exists a finite sequence F_{1} such that dom F_{1} = Seg(2^{n} + 1) and for every natural number i such that i ∈ dom F_{1} holds F_{1}(i) = \frac{i−1}{2^{n}}.

Let n be a natural number. The functor dyad(n) yielding a finite sequence is defined by: (Def.6) dom dyad(n) = Seg(2^{n} + 1) and for every natural number i such that

i ∈ dom dyad(n) holds (dyad(n))(i) = \frac{i−1}{2^{n}}. We now state the proposition

(10) For every natural number n holds dom dyad(n) = Seg(2^{n} + 1) and

rng dyad(n) = dyadic(n). Let us note that DYADIC is non empty. Let us observe that DOM is non empty. One can prove the following propositions:

(11) For every natural number n holds dyadic(n) ⊆ dyadic(n + 1).

(12) For every natural number n holds 0 ∈ dyadic(n) and 1 ∈ dyadic(n).

(13) For every natural number n and for every natural number i such that 0 < i and i ≤ 2^{n} holds \frac{i·2−1}{2^{n+1}} ∈ dyadic(n + 1) \ dyadic(n).

(14) For every natural number n and for every natural number i such that 0 ≤ i and i < 2^{n} holds \frac{i·2+1}{2^{n+1}} ∈ dyadic(n + 1) \ dyadic(n).

(15) For every natural number n holds \frac{1}{2^{n+1}} ∈ dyadic(n + 1) \ dyadic(n). Let n be a natural number and let x be an element of dyadic(n). The functor

axis(x, n) yields a natural number and is defined by: (Def.7) x = \frac{axis(x,n)}{2^{n}}.

One can prove the following propositions:

(16) For every natural number n and for every element x of dyadic(n) holds x = \frac{axis(x,n)}{2^{n}} and 0 ≤ axis(x, n) and axis(x, n) ≤ 2^{n}.

(17) For every natural number n and for every element x of dyadic(n) holds \frac{axis(x,n)−1}{2^{n}} < x and x < \frac{axis(x,n)+1}{2^{n}}.

(18) For every natural number n and for every element x of dyadic(n) holds \frac{axis(x,n)−1}{2^{n}} < \frac{axis(x,n)+1}{2^{n}}.

(19) For every natural number n there exists a natural number k such that n = k · 2 or n = k · 2 + 1.

(20) Let n be a natural number and let x be an element of dyadic(n + 1). If x ∉ dyadic(n), then \frac{axis(x,n+1)−1}{2^{n+1}} ∈ dyadic(n) and \frac{axis(x,n+1)+1}{2^{n+1}} ∈ dyadic(n).

(21) For every natural number n and for all elements x_{1}, x_{2} of dyadic(n) such that x_{1} < x_{2} holds axis(x_{1}, n) < axis(x_{2}, n).

(22) For every natural number n and for all elements x_{1}, x_{2} of dyadic(n) such that x_{1} < x_{2} holds x_{1} ≤ \frac{axis(x_{2},n)−1}{2^{n}} and \frac{axis(x_{1},n)+1}{2^{n}} ≤ x_{2}.

(23) Let n be a natural number and let x_{1}, x_{2} be elements of dyadic(n + 1). If x_{1} < x_{2} and x_{1} ∉ dyadic(n) and x_{2} ∉ dyadic(n), then \frac{axis(x_{1},n+1)+1}{2^{n+1}} ≤ \frac{axis(x_{2},n+1)−1}{2^{n+1}}.

3. Normal Spaces

Let T be a topological space and let x be a point of T. A subset of the carrier of T is said to be a neighbourhood of x in T if: (Def.8) There exists a subset A of the carrier of T such that A is open and

x ∈ A and A ⊆ it. One can prove the following propositions:

(24) Let T be a topological space and let A be a subset of the carrier of T. Then A is open if and only if for every point x of T such that x ∈ A there exists a neighbourhood B of x in T such that B ⊆ A.

(25) Let T be a topological space, and let A be a subset of the carrier of T, and let x be a point of T. If A is open and x ∈ A, then A is a neighbourhood of x in T.

(26) Let T be a topological space and let A be a subset of the carrier of T. Suppose that for every point x of T such that x ∈ A holds A is a neighbourhood of x in T. Then A is open.

Let T be a topological space. We say that T is a T_{1} space if and only if the condition (Def.9) is satisfied. (Def.9) Let p, q be points of T. Suppose p ≠ q. Then there exist subsets W, V

of the carrier of T such that W is open and V is open and p ∈ W and q ∉ W and q ∈ V and p ∉ V.

Next we state the proposition

(27) For every topological space T holds T is a T_{1} space iff for every point

p of T holds {p} is closed. Let T be a topological space, let F be a map from T into ^{1}, and let x be a

point of T. Then F (x) is a real number.

The following four propositions are true:

(28) Let T be a topological space. Suppose T is a T_{4} space. Let A, B be subsets of the carrier of T. Suppose A ≠ ∅ and A is open and B is open and \overline{A} ⊆ B. Then there exists a subset C of the carrier of T such that C ≠ ∅ and C is open and \overline{A} ⊆ C and \overline{C} ⊆ B.

(29) Let T be a topological space. Then T is a T_{3} space if and only if for every subset A of the carrier of T and for every point p of T such that A is open and p ∈ A there exists a subset B of the carrier of T such that p ∈ B and B is open and \overline{B} ⊆ A.

(30) Let T be a topological space. Suppose T is a T_{4} space and a T_{1} space. Let A be a subset of the carrier of T. Suppose A is open and A ≠ ∅. Then there exists a subset B of the carrier of T such that B ≠ ∅ and \overline{B} ⊆ A.

(31) Let T be a topological space. Suppose T is a T_{4} space. Let A be a subset of the carrier of T. Suppose A is open and A ≠ ∅. Let B be a subset of the carrier of T. Suppose B is closed and B ≠ ∅ and B ⊆ A. Then there exists a subset C of the carrier of T such that C is open and B ⊆ C and \overline{C} ⊆ A.

4. Some Increasing Family of Sets in Normal Space

Let T be a topological space and let A, B, C be subsets of the carrier of T. We say that C is between A and B if and only if:

(Def.10) C ≠ ∅ and C is open and \overline{A} ⊆ C and \overline{C} ⊆ B.

One can prove the following proposition

(32) Let T be a topological space. Suppose T is a T_{4} space. Let A, B be subsets of the carrier of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let n be a natural number and let G be a function from dyadic(n) into 2^{the carrier of T}. Suppose that for all elements r_{1}, r_{2} of dyadic(n) such that r_{1} < r_{2} holds G(r_{1}) is open and G(r_{2}) is open and \overline{G(r_{1})} ⊆ G(r_{2}) and A ⊆ G(0) and B = Ω_{T} \ G(1). Then there exists a function F from dyadic(n+1) into 2^{the carrier of T} such that for all elements r_{1}, r_{2}, r of dyadic(n + 1) if r_{1} < r_{2}, then F (r_{1}) is open and F (r_{2}) is open and \overline{F (r_{1})} ⊆ F (r_{2}) and A ⊆ F (0) and B = Ω_{T} \F (1) and if r ∈ dyadic(n), then F (r) = G(r).

References

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

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

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

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

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

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366 józef białas and yatsuka nakamura

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Full Adder Circuit. Part I ^{1}

Grzegorz Bancerek Institute of Mathematics Polish Academy of Sciences Yatsuka Nakamura Shinshu University Nagano

Summary. We continue the formalisation of circuits started by Piotr Rudnicki, Andrzej Trybulec, Pauline Kawamoto, and the second author in [16,17,14,15]. The first step in proving properties of full n-bit adder circuit, i.e. 1-bit adder, is presented. We employ the notation of combining circuits introduced in [13].

MML Identifier: FACIRC 1.

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

1. Combining of Many Sorted Signatures

A set is pair if: (Def.1) There exist sets x, y such that it = ⟨⟨x, y⟩⟩.

Let us mention that every set which is pair is also non empty. Let x, y be sets. Observe that ⟨⟨x, y⟩⟩ is pair. Let us mention that there exists a set which is pair and there exists a set

which is non pair.

Let us observe that every natural number is non pair. A set has a pair if:

(Def.2) There exists a pair set x such that x ∈ it.

Note that every set which is empty has no pairs. Let x be a non pair set. Note that {x} has no pairs. Let y be a non pair set. Observe that {x, y} has no pairs. Let z be a non pair set. One can check that {x, y, z} has no pairs.

1This work was written while the first author visited Shinshu University, July–August 1994.

Let us note that there exists a non empty set which has no pairs. Let X, Y be sets with no pairs. One can verify that X ∪ Y has no pairs. Let X be a set with no pairs and let Y be a set. One can verify the following

observations:

∗ X \ Y has no pairs,

∗ X ∩ Y has no pairs, and

∗ Y ∩ X has no pairs. One can verify that every set which is empty is also relation-like. Let x be a

pair set. One can check that {x} is relation-like. Let y be a pair set. Observe that {x, y} is relation-like. Let z be a pair set. One can check that {x, y, z} is relation-like.

Let us note that every set which is relation-like and has no pairs is also empty. A function is nonpair yielding if:

(Def.3) For every set x such that x ∈ dom it holds it(x) is non pair.

Let x be a non pair set. Observe that ⟨x⟩ is nonpair yielding. Let y be a non pair set. One can check that ⟨x, y⟩ is nonpair yielding. Let z be a non pair set. Observe that ⟨x, y, z⟩ is nonpair yielding.

One can prove the following proposition

(1) For every function f such that f is nonpair yielding holds rng f has no pairs.

Let n be a natural number. Observe that there exists a finite sequence with length n which is one-to-one and nonpair yielding.

One can check that there exists a finite sequence which is one-to-one and nonpair yielding.

Let f be a nonpair yielding function. Note that rng f has no pairs. The following propositions are true:

(2) Let S_{1}, S_{2} be non empty many sorted signatures. Suppose S_{1} ≈ S_{2} and InnerVertices(S_{1}) is a binary relation and InnerVertices(S_{2}) is a binary relation. Then InnerVertices(S_{1}+·S_{2}) is a binary relation.

(3) Let S_{1}, S_{2} be unsplit non empty many sorted signatures with arity held in gates. Suppose InnerVertices(S_{1}) is a binary relation and InnerVertices(S_{2}) is a binary relation. Then InnerVertices(S_{1}+·S_{2}) is a binary relation.

(4) For all non empty many sorted signatures S_{1}, S_{2} such that S_{1} ≈ S_{2} and InnerVertices(S_{2}) misses InputVertices(S_{1}) holds InputVertices(S_{1}) ⊆ InputVertices(S_{1}+·S_{2}) and InputVertices(S_{1}+·S_{2}) = InputVertices(S_{1}) ∪ (InputVertices(S_{2}) \ InnerVertices(S_{1})).

(5) For all sets X, R such that X has no pairs and R is a binary relation holds X misses R.

(6) Let S_{1}, S_{2} be unsplit non empty many sorted signatures with arity held in gates. Suppose InputVertices(S_{1}) has no pairs and InnerVertices(S_{2}) is a binary relation. Then InputVertices(S_{1}) ⊆ InputVertices(S_{1}+·S_{2}) and InputVertices(S_{1}+·S_{2}) = InputVertices(S_{1}) ∪ (InputVertices(S_{2}) \ InnerVertices(S_{1})).

(7) Let S_{1}, S_{2} be unsplit non empty many sorted signatures with arity held in gates. Suppose InputVertices(S_{1}) has no pairs and InnerVertices(S_{1}) is a binary relation and InputVertices(S_{2}) has no pairs and InnerVertices(S_{2}) is a binary relation. Then InputVertices(S_{1}+·S_{2}) = InputVertices(S_{1}) ∪ InputVertices(S_{2}).

(8) For all non empty many sorted signatures S_{1}, S_{2} such that S_{1} ≈ S_{2} and InputVertices(S_{1}) has no pairs and InputVertices(S_{2}) has no pairs holds InputVertices(S_{1}+·S_{2}) has no pairs.

(9) Let S_{1}, S_{2} be unsplit non empty many sorted signatures with arity held in gates. If InputVertices(S_{1}) has no pairs and InputVertices(S_{2}) has no pairs, then InputVertices(S_{1}+·S_{2}) has no pairs.

2. Combinig of Circuits

In this article we present several logical schemes. The scheme 2AryBooleDef concerns a binary functor F yielding an element of Boolean, and states that:

(i) There exists a function f from Boolean^{2} into Boolean such that for all elements x, y of Boolean holds f(⟨x, y⟩) = F(x, y), and (ii) for all functions f_{1}, f_{2} from Boolean^{2} into Boolean such that for all elements x, y of Boolean holds f_{1}(⟨x, y⟩) = F(x, y) and for all elements x, y of Boolean holds f_{2}(⟨x, y⟩) = F(x, y) holds f_{1} = f_{2}

for all values of the parameter.

The scheme 3AryBooleDef deals with a ternary functor F yielding an element of Boolean, and states that:

(i) There exists a function f from Boolean^{3} into Boolean such that for all elements x, y, z of Boolean holds f(⟨x, y, z⟩) = F(x, y, z), and (ii) for all functions f_{1}, f_{2} from Boolean^{3} into Boolean such that for all elements x, y, z of Boolean holds f_{1}(⟨x, y, z⟩) = F(x, y, z) and for all elements x, y, z of Boolean holds f_{2}(⟨x, y, z⟩) = F(x, y, z) holds f_{1} = f_{2}

for all values of the parameter.

The function xor from Boolean^{2} into Boolean is defined by: (Def.4) For all elements x, y of Boolean holds xor(⟨x, y⟩) = x ⊕ y. The function or from Boolean^{2} into Boolean is defined by: (Def.5) For all elements x, y of Boolean holds or(⟨x, y⟩) = x ∨ y. The function & from Boolean^{2} into Boolean is defined as follows: (Def.6) For all elements x, y of Boolean holds &(⟨x, y⟩) = x ∧ y.

The function or_{3} from Boolean^{3} into Boolean is defined by: (Def.7) For all elements x, y, z of Boolean holds or_{3}(⟨x, y, z⟩) = x ∨ y ∨ z.

Let x be a set. Then ⟨x⟩ is a finite sequence with length 1. Let y be a set. Then ⟨x, y⟩ is a finite sequence with length 2. Let z be a set. Then ⟨x, y, z⟩ is a finite sequence with length 3.

Let n, m be natural numbers, let p be a finite sequence with length n, and let q be a finite sequence with length m. Then p^{}q is a finite sequence with length n + m.

3. Signatures with One Operation

The following proposition is true

(10) Let S be a circuit-like non void non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let g be a gate of S. Then (Following(s))(the result sort of g) = (Den(g, A))(s · Arity(g)).

Let S be a non void circuit-like non empty many sorted signature, let A be a non-empty circuit of S, let s be a state of A, and let n be a natural number. The functor Following(s, n) yielding a state of A is defined by the condition (Def.8).

(Def.8) There exists a function f from into ∏ (the sorts of A) such that Following(s, n) = f(n) and f(0) = s and for every natural number n and for every state x of A such that x = f(n) holds f(n + 1) = Following(x).

The following propositions are true:

(11) Let S be a circuit-like non void non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A. Then Following(s, 0) = s.

(12) Let S be a circuit-like non void non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let n be a natural number. Then Following(s, n + 1) = Following(Following(s, n)).

(13) Let S be a circuit-like non void non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let n, m be natural numbers. Then Following(s, n + m) = Following(Following(s, n), m).

(14) Let S be a non void circuit-like non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A. Then Following(s, 1) = Following(s).

(15) Let S be a non void circuit-like non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A. Then Following(s, 2) = Following(Following(s)).

(16) Let S be a circuit-like non void non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let n be a natural number. Then Following(s, n + 1) = Following(Following(s), n).

Let S be a non void circuit-like non empty many sorted signature, let A be a non-empty circuit of S, let s be a state of A, and let x be a set. We say that s is stable at x if and only if: (Def.9) For every natural number n holds (Following(s, n))(x) = s(x).

The following propositions are true:

(17) Let S be a non void circuit-like non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let x be a set. If s is stable at x, then for every natural number n holds Following(s, n) is stable at x.

(18) Let S be a non void circuit-like non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let x be a set. If x ∈ InputVertices(S), then s is stable at x.

(19) Let S be a non void circuit-like non empty many sorted signature, and let A be a non-empty circuit of S, and let s be a state of A, and let g be a gate of S. Suppose that for every set x such that x ∈ rng Arity(g) holds s is stable at x. Then Following(s) is stable at the result sort of g.

4. Unsplit Condition

The following propositions are true:

(20) Let S_{1}, S_{2} be non empty many sorted signatures and let v be a vertex of S_{1}. Then v ∈ the carrier of S_{1}+·S_{2} and v ∈ the carrier of S_{2}+·S_{1}.

(21) Let S_{1}, S_{2} be unsplit non empty many sorted signatures with arity held in gates and let x be a set. If x ∈ InnerVertices(S_{1}), then x ∈ InnerVertices(S_{1}+·S_{2}) and x ∈ InnerVertices(S_{2}+·S_{1}).

(22) For all non empty many sorted signatures S_{1}, S_{2} and for every set x such that x ∈ InnerVertices(S_{2}) holds x ∈ InnerVertices(S_{1}+·S_{2}).

(23) For all unsplit non empty many sorted signatures S_{1}, S_{2} with arity held in gates holds S_{1}+·S_{2} = S_{2}+·S_{1}.

(24) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates, and let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates. Then A_{1}+·A_{2} = A_{2}+·A_{1}.

(25) Let S_{1}, S_{2}, S_{3} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates, and let A_{1} be a Boolean circuit of S_{1}, and let A_{2} be a Boolean circuit of S_{2}, and let A_{3} be a Boolean circuit of S_{3}. Then (A_{1}+·A_{2})+·A_{3} = A_{1}+·(A_{2}+·A_{3}).

(26) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates, and let A_{1} be a Boolean non-empty circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean non-empty circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}. Then s (the carrier of S_{1}) is a state of A_{1} and s (the carrier of S_{2}) is a state of A_{2}.

(27) For all unsplit non empty many sorted signatures S_{1}, S_{2} with arity held in gates holds InnerVertices(S_{1}+·S_{2}) = InnerVertices(S_{1}) ∪ InnerVertices(S_{2}).

(28) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates. Suppose InnerVertices(S_{2}) misses InputVertices(S_{1}). Let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}, and let s_{1} be a state of A_{1}. If s_{1} = s (the carrier of S_{1}), then Following(s) (the carrier of S_{1}) = Following(s_{1}).

(29) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates. Suppose InnerVertices(S_{1}) misses InputVertices(S_{2}). Let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}, and let s_{2} be a state of A_{2}. If s_{2} = s (the carrier of S_{2}), then Following(s) (the carrier of S_{2}) = Following(s_{2}).

(30) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates. Suppose InnerVertices(S_{2}) misses InputVertices(S_{1}). Let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}, and let s_{1} be a state of A_{1}. Suppose s_{1} = s (the carrier of S_{1}). Let n be a natural number. Then Following(s, n) (the carrier of S_{1}) = Following(s_{1}, n).

(31) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates. Suppose InnerVertices(S_{1}) misses InputVertices(S_{2}). Let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}, and let s_{2} be a state of A_{2}. Suppose s_{2} = s (the carrier of S_{2}). Let n be a natural number. Then Following(s, n) (the carrier of S_{2}) = Following(s_{2}, n).

(32) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates. Suppose InnerVertices(S_{2}) misses InputVertices(S_{1}). Let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}, and let s_{1} be a state of A_{1}. Suppose s_{1} = s (the carrier of S_{1}). Let v be a set. Suppose v ∈ the carrier of S_{1}. Let n be a natural number. Then (Following(s, n))(v) = (Following(s_{1}, n))(v).

(33) Let S_{1}, S_{2} be unsplit non void non empty many sorted signatures with arity held in gates and Boolean denotation held in gates. Suppose InnerVertices(S_{1}) misses InputVertices(S_{2}). Let A_{1} be a Boolean circuit of S_{1} with denotation held in gates, and let A_{2} be a Boolean circuit of S_{2} with denotation held in gates, and let s be a state of A_{1}+·A_{2}, and let s_{2} be a state of A_{2}. Suppose s_{2} = s (the carrier of S_{2}). Let v be a set. Suppose v ∈ the carrier of S_{2}. Let n be a natural number. Then (Following(s, n))(v) = (Following(s_{2}, n))(v).

Let S be a non void non empty many sorted signature with denotation held in gates and let g be a gate of S. One can verify that g_{2} is function-like and relation-like.

Next we state four propositions:

(34) Let S be a circuit-like non void non empty many sorted signature with denotation held in gates and let A be a non-empty circuit of S. Suppose A has denotation held in gates. Let s be a state of A and let g be a gate of S. Then (Following(s))(the result sort of g) = g_{2}(s · Arity(g)).

(35) Let S be an unsplit non void non empty many sorted signature with arity held in gates and Boolean denotation held in gates, and let A be a Boolean non-empty circuit of S with denotation held in gates, and let s be a state of A, and let p be a finite sequence, and let f be a function. If ⟨⟨p, f⟩⟩ ∈ the operation symbols of S, then (Following(s))(⟨⟨p, f⟩⟩) = f(s · p).

(36) Let S be an unsplit non void non empty many sorted signature with arity held in gates and Boolean denotation held in gates, and let A be a Boolean non-empty circuit of S with denotation held in gates, and let s be a state of A, and let p be a finite sequence, and let f be a function. Suppose ⟨⟨p, f⟩⟩ ∈ the operation symbols of S and for every set x such that x ∈ rng p holds s is stable at x. Then Following(s) is stable at ⟨⟨p, f⟩⟩.

(37) For every unsplit non empty many sorted signature S holds InnerVertices(S) = the operation symbols of S.

5. One Gate Circuits

We now state a number of propositions:

(38) For every set f and for every finite sequence p holds InnerVertices(1GateCircStr(p, f)) is a binary relation.

(39) For every set f and for every nonpair yielding finite sequence p holds InputVertices(1GateCircStr(p, f)) has no pairs.

(40) For every set f and for all sets x, y holds InputVertices(1GateCircStr(⟨x, y⟩, f)) = {x, y}.

(41) For every set f and for all non pair sets x, y holds InputVertices(1GateCircStr(⟨x, y⟩, f)) has no pairs.

(42) For every set f and for all sets x, y, z holds InputVertices(1GateCircStr(⟨x, y, z⟩, f)) = {x, y, z}.

(43) Let x, y, f be sets. Then x ∈ the carrier of 1GateCircStr(⟨x, y⟩, f) and y ∈ the carrier of 1GateCircStr(⟨x, y⟩, f) and ⟨⟨⟨x, y⟩, f⟩⟩ ∈ the carrier of 1GateCircStr(⟨x, y⟩, f).

(44) Let x, y, z, f be sets. Then x ∈ the carrier of 1GateCircStr(⟨x, y, z⟩, f) and y ∈ the carrier of 1GateCircStr(⟨x, y, z⟩, f) and z ∈ the carrier of 1GateCircStr(⟨x, y, z⟩, f).

(45) Let f, x be sets and let p be a finite sequence. Then x ∈ the carrier of 1GateCircStr(p, f, x) and for every set y such that y ∈ rng p holds y ∈ the carrier of 1GateCircStr(p, f, x).

(46) For all sets f, x and for every finite sequence p holds 1GateCircStr(p, f, x) is circuit-like and has arity held in gates.

(47) For every finite sequence p and for every set f holds ⟨⟨p, f⟩⟩ ∈

InnerVertices(1GateCircStr(p, f)). Let x, y be sets and let f be a function from Boolean^{2} into Boolean. The func-

tor 1GateCircuit(x, y, f) yielding a Boolean strict circuit of 1GateCircStr(⟨x, y⟩, f) with denotation held in gates is defined by:

(Def.10) 1GateCircuit(x, y, f) = 1GateCircuit(⟨x, y⟩, f).

We adopt the following convention: x, y, z, c denote sets and f denotes a function from Boolean^{2} into Boolean.

We now state four propositions:

(48) Let X be a finite non empty set, and let f be a function from X^{2} into X, and let s be a state of 1GateCircuit(⟨x, y⟩, f). Then (Following(s))(⟨⟨⟨x, y⟩, f⟩⟩) = f(⟨s(x), s(y)⟩) and (Following(s))(x) = s(x) and (Following(s))(y) = s(y).

(49) Let X be a finite non empty set, and let f be a function from X^{2} into X, and let s be a state of 1GateCircuit(⟨x, y⟩, f). Then Following(s) is stable.

(50) For every state s of 1GateCircuit(x, y, f) holds (Following(s))(⟨⟨⟨x, y⟩, f⟩⟩) = f(⟨s(x), s(y)⟩) and (Following(s))(x) = s(x) and (Following(s))(y) = s(y).

(51) For every state s of 1GateCircuit(x, y, f) holds Following(s) is stable. Let x, y, z be sets and let f be a function from Boolean^{3} into

Boolean. The functor 1GateCircuit(x, y, z, f) yields a Boolean strict circuit of 1GateCircStr(⟨x, y, z⟩, f) with denotation held in gates and is defined by:

(Def.11) 1GateCircuit(x, y, z, f) = 1GateCircuit(⟨x, y, z⟩, f).

We now state four propositions:

(52) Let X be a finite non empty set, and let f be a function from X^{3} into X, and let s be a state of 1GateCircuit(⟨x, y, z⟩, f). Then (Following(s))(⟨⟨⟨x, y, z⟩, f⟩⟩) = f(⟨s(x), s(y), s(z)⟩) and (Following(s))(x) = s(x) and (Following(s))(y) = s(y) and (Following(s))(z) = s(z).

(53) Let X be a finite non empty set, and let f be a function from X^{3} into X, and let s be a state of 1GateCircuit(⟨x, y, z⟩, f). Then Following(s) is

stable.

(54) Let f be a function from Boolean^{3} into Boolean and let s be a state of 1GateCircuit(x, y, z, f). Then (Following(s))(⟨⟨⟨x, y, z⟩, f⟩⟩) = f(⟨s(x), s(y), s(z)⟩) and (Following(s))(x) = s(x) and (Following(s))(y) = s(y) and (Following(s))(z) = s(z).

(55) For every function f from Boolean^{3} into Boolean and for every state s of 1GateCircuit(x, y, z, f) holds Following(s) is stable.

6. Boolean Circuits

Let x, y, c be sets and let f be a function from Boolean^{2} into Boolean. The functor 2GatesCircStr(x, y, c, f) yielding an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined as follows:

(Def.12) 2GatesCircStr(x, y, c, f) = 1GateCircStr(⟨x, y⟩, f)+· 1GateCircStr(⟨⟨⟨⟨x,

y⟩, f⟩⟩, c⟩, f). Let x, y, c be sets and let f be a function from Boolean^{2} into

Boolean. The functor 2GatesCircOutput(x, y, c, f) yields an element of InnerVertices(2GatesCircStr(x, y, c, f)) and is defined as follows:

(Def.13) 2GatesCircOutput(x, y, c, f) = ⟨⟨⟨⟨⟨⟨x, y⟩, f⟩⟩, c⟩, f⟩⟩.

Let x, y, c be sets and let f be a function from Boolean^{2} into Boolean. One can verify that 2GatesCircOutput(x, y, c, f) is pair.

One can prove the following two propositions:

(56) InnerVertices(2GatesCircStr(x, y, c, f)) = {⟨⟨⟨x, y⟩, f⟩⟩, 2GatesCircOutput(x, y, c, f)}.

(57) If c ≠ ⟨⟨⟨x, y⟩, f⟩⟩, then InputVertices(2GatesCircStr(x, y, c, f)) =

{x, y, c}. Let x, y, c be sets and let f be a function from Boolean^{2} into

Boolean. The functor 2GatesCircuit(x, y, c, f) yields a strict Boolean circuit of 2GatesCircStr(x, y, c, f) with denotation held in gates and is defined by:

(Def.14) 2GatesCircuit(x, y, c, f) = 1GateCircuit(x, y, f)+· 1GateCircuit(⟨⟨⟨x,

y⟩, f⟩⟩, c, f). We now state four propositions:

(58) InnerVertices(2GatesCircStr(x, y, c, f)) is a binary relation.

(59) For all non pair sets x, y, c holds InputVertices(2GatesCircStr(x, y, c, f)) has no pairs.

(60) x ∈ the carrier of 2GatesCircStr(x, y, c, f) and y ∈ the carrier of 2GatesCircStr(x, y, c, f) and c ∈ the carrier of 2GatesCircStr(x, y, c, f).

(61) ⟨⟨⟨x, y⟩, f⟩⟩ ∈ the carrier of 2GatesCircStr(x, y, c, f) and ⟨⟨⟨⟨⟨⟨x, y⟩, f⟩⟩, c⟩, f⟩⟩ ∈ the carrier of 2GatesCircStr(x, y, c, f).

Let S be an unsplit non void non empty many sorted signature, let A be a Boolean circuit of S, let s be a state of A, and let v be a vertex of S. Then s(v) is an element of Boolean.

In the sequel s will be a state of 2GatesCircuit(x, y, c, f). One can prove the following propositions:

(62) Suppose c ≠ ⟨⟨⟨x, y⟩, f⟩⟩. Then (Following(s, 2))(2GatesCircOutput(x, y, c, f)) = f(⟨f(⟨s(x), s(y)⟩), s(c)⟩) and (Following(s, 2))(⟨⟨⟨x, y⟩, f⟩⟩) = f(⟨s(x), s(y)⟩) and (Following(s, 2))(x) = s(x) and (Following(s, 2))(y) = s(y) and (Following(s, 2))(c) = s(c).

(63) If c ≠ ⟨⟨⟨x, y⟩, f⟩⟩, then Following(s, 2) is stable.

(64) Suppose c ≠ ⟨⟨⟨x, y⟩, xor ⟩⟩. Let s be a state of 2GatesCircuit(x, y, c, xor) and let a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(2GatesCircOutput(x, y, c, xor)) = a_{1} ⊕ a_{2} ⊕ a_{3}.

(65) Suppose c ≠ ⟨⟨⟨x, y⟩, or ⟩⟩. Let s be a state of 2GatesCircuit(x, y, c, or) and let a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(2GatesCircOutput(x, y, c, or)) = a_{1} ∨ a_{2} ∨ a_{3}.

(66) Suppose c ≠ ⟨⟨⟨x, y⟩, &⟩⟩. Let s be a state of 2GatesCircuit(x, y, c, &) and let a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(2GatesCircOutput(x, y, c, &)) = a_{1} ∧ a_{2} ∧ a_{3}.

7. One Bit Adder

Let x, y, c be sets. The functor BitAdderOutput(x, y, c) yields an element of InnerVertices(2GatesCircStr(x, y, c, xor)) and is defined as follows:

(Def.15) BitAdderOutput(x, y, c) = 2GatesCircOutput(x, y, c, xor).

Let x, y, c be sets. The functor BitAdderCirc(x, y, c) yields a strict Boolean circuit of 2GatesCircStr(x, y, c, xor) with denotation held in gates and is defined as follows:

(Def.16) BitAdderCirc(x, y, c) = 2GatesCircuit(x, y, c, xor).

Let x, y, c be sets. The functor MajorityIStr(x, y, c) yielding an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined by:

(Def.17) MajorityIStr(x, y, c) = 1GateCircStr(⟨x, y⟩, &)+· 1GateCircStr(⟨y,

c⟩, &)+· 1GateCircStr(⟨c, x⟩, &). Let x, y, c be sets. The functor MajorityStr(x, y, c) yields an unsplit non void

strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined as follows:

(Def.18) MajorityStr(x, y, c) = MajorityIStr(x, y, c)+· 1GateCircStr(⟨⟨⟨⟨x, y⟩, &⟩⟩, ⟨⟨⟨y, c⟩, &⟩⟩, ⟨⟨⟨c, x⟩, &⟩⟩⟩, or_{3}).

Let x, y, c be sets. The functor MajorityICirc(x, y, c) yields a strict Boolean circuit of MajorityIStr(x, y, c) with denotation held in gates and is defined as follows:

(Def.19) MajorityICirc(x, y, c) = 1GateCircuit(x, y, &)+· 1GateCircuit(y, c, &)

+· 1GateCircuit(c, x, &). Next we state several propositions:

(67) InnerVertices(MajorityStr(x, y, c)) is a binary relation.

(68) For all non pair sets x, y, c holds InputVertices(MajorityStr(x, y, c)) has no pairs.

(69) For every state s of MajorityICirc(x, y, c) and for all elements a, b of Boolean such that a = s(x) and b = s(y) holds (Following(s))(⟨⟨⟨x, y⟩, &⟩⟩) = a ∧ b.

(70) For every state s of MajorityICirc(x, y, c) and for all elements a, b of Boolean such that a = s(y) and b = s(c) holds (Following(s))(⟨⟨⟨y, c⟩, &⟩⟩) = a ∧ b.

(71) For every state s of MajorityICirc(x, y, c) and for all elements a, b of Boolean such that a = s(c) and b = s(x) holds (Following(s))(⟨⟨⟨c, x⟩, &⟩⟩) = a ∧ b.

Let x, y, c be sets. The functor MajorityOutput(x, y, c) yields an element of InnerVertices(MajorityStr(x, y, c)) and is defined by:

(Def.20) MajorityOutput(x, y, c) = ⟨⟨⟨⟨⟨⟨x, y⟩, &⟩⟩, ⟨⟨⟨y, c⟩, &⟩⟩, ⟨⟨⟨c, x⟩, &⟩⟩⟩, or_{3} ⟩⟩.

Let x, y, c be sets. The functor MajorityCirc(x, y, c) yielding a strict Boolean circuit of MajorityStr(x, y, c) with denotation held in gates is defined by:

(Def.21) MajorityCirc(x, y, c) = MajorityICirc(x, y, c)+· 1GateCircuit(⟨⟨⟨x, y⟩,

&⟩⟩, ⟨⟨⟨y, c⟩, &⟩⟩, ⟨⟨⟨c, x⟩, &⟩⟩, or_{3}). Next we state a number of propositions:

(72) x ∈ the carrier of MajorityStr(x, y, c) and y ∈ the carrier of MajorityStr(x, y, c) and c ∈ the carrier of MajorityStr(x, y, c).

(73) ⟨⟨⟨x, y⟩, &⟩⟩ ∈ InnerVertices(MajorityStr(x, y, c)) and ⟨⟨⟨y, c⟩, &⟩⟩ ∈ InnerVertices(MajorityStr(x, y, c)) and ⟨⟨⟨c, x⟩, &⟩⟩ ∈ InnerVertices(MajorityStr(x, y, c)).

(74) For all non pair sets x, y, c holds x ∈ InputVertices(MajorityStr(x, y, c)) and y ∈ InputVertices(MajorityStr(x, y, c)) and c ∈ InputVertices(MajorityStr(x, y, c)).

(75) For all non pair sets x, y, c holds InputVertices(MajorityStr(x, y, c)) = {x, y, c} and InnerVertices(MajorityStr(x, y, c)) = {⟨⟨⟨x, y⟩, &⟩⟩, ⟨⟨⟨y, c⟩, &⟩⟩, ⟨⟨⟨c, x⟩, &⟩⟩} ∪ {MajorityOutput(x, y, c)}.

(76) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{1}, a_{2} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y), then (Following(s))(⟨⟨⟨x, y⟩, &⟩⟩) = a_{1} ∧ a_{2}.

(77) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{2}, a_{3} be elements of Boolean. If a_{2} = s(y) and a_{3} = s(c), then

(Following(s))(⟨⟨⟨y, c⟩, &⟩⟩) = a_{2} ∧ a_{3}.

(78) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{1}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{3} = s(c), then (Following(s))(⟨⟨⟨c, x⟩, &⟩⟩) = a_{3} ∧ a_{1}.

(79) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(⟨⟨⟨x, y⟩, &⟩⟩) and a_{2} = s(⟨⟨⟨y, c⟩, &⟩⟩) and a_{3} = s(⟨⟨⟨c, x⟩, &⟩⟩), then (Following(s))(MajorityOutput(x, y, c)) = a_{1} ∨ a_{2} ∨ a_{3}.

(80) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{1}, a_{2} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y), then (Following(s, 2))(⟨⟨⟨x, y⟩, &⟩⟩) = a_{1} ∧ a_{2}.

(81) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{2}, a_{3} be elements of Boolean. If a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(⟨⟨⟨y, c⟩, &⟩⟩) = a_{2} ∧ a_{3}.

(82) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{1}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{3} = s(c), then (Following(s, 2))(⟨⟨⟨c, x⟩, &⟩⟩) = a_{3} ∧ a_{1}.

(83) Let x, y, c be non pair sets, and let s be a state of MajorityCirc(x, y, c), and let a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(MajorityOutput(x, y, c)) = a_{1} ∧ a_{2} ∨ a_{2} ∧ a_{3} ∨ a_{3} ∧ a_{1}.

(84) For all non pair sets x, y, c and for every state s of MajorityCirc(x, y, c)

holds Following(s, 2) is stable. Let x, y, c be sets. The functor BitAdderWithOverflowStr(x, y, c) yields an

unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined as follows:

(Def.22) BitAdderWithOverflowStr(x, y, c) = 2GatesCircStr(x, y, c, xor)

+· MajorityStr(x, y, c). The following three propositions are true:

(85) For all non pair sets x, y, c holds InputVertices(BitAdderWithOverflowStr (x, y, c)) = {x, y, c}.

(86) For all non pair sets x, y, c holds InnerVertices(BitAdderWithOverflowStr (x, y, c)) = {⟨⟨⟨x, y⟩, xor ⟩⟩, 2GatesCircOutput(x, y, c, xor)} ∪ {⟨⟨⟨x, y⟩, &⟩⟩, ⟨⟨⟨y, c⟩, &⟩⟩, ⟨⟨⟨c, x⟩, &⟩⟩} ∪ {MajorityOutput(x, y, c)}.

(87) Let S be a non empty many sorted signature. Suppose S = BitAdderWithOverflowStr(x, y, c). Then x ∈ the carrier of S and y ∈ the carrier of S and c ∈ the carrier of S.

Let x, y, c be sets. The functor BitAdderWithOverflowCirc(x, y, c) yielding a strict Boolean circuit of BitAdderWithOverflowStr(x, y, c) with denotation held in gates is defined as follows:

(Def.23) BitAdderWithOverflowCirc(x, y, c) = BitAdderCirc(x, y, c) +· MajorityCirc(x, y, c).

full adder circuit. part i 379

We now state several propositions:

(88) InnerVertices(BitAdderWithOverflowStr(x, y, c)) is a binary relation.

(89) For all non pair sets x, y, c holds InputVertices(BitAdderWithOverflowStr (x, y, c)) has no pairs.

(90) BitAdderOutput(x, y, c) ∈ InnerVertices(BitAdderWithOverflowStr(x, y, c)) and MajorityOutput(x, y, c) ∈ InnerVertices(BitAdderWithOverflow Str(x, y, c)).

(91) Let x, y, c be non pair sets, and let s be a state of BitAdderWithOverflowCirc(x, y, c), and let a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(c). Then (Following(s, 2))(BitAdderOutput(x, y, c)) = a_{1} ⊕ a_{2} ⊕ a_{3} and (Following(s, 2))(MajorityOutput(x, y, c)) = a_{1} ∧ a_{2} ∨ a_{2} ∧ a_{3} ∨ a_{3} ∧ a_{1}.

(92) For all non pair sets x, y, c and for every state s of BitAdderWithOverflowCirc(x, y, c) holds Following(s, 2) is stable.

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Received August 10, 1995

Continuous, Stable, and Linear Maps of Coherence Spaces

Grzegorz Bancerek Institute of Mathematics Polish Academy of Sciences

MML Identifier: COHSP 1.

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

1. Directed Sets

One can check that there exists a coherent space which is finite. Let us observe that a set is binary complete if: (Def.1) For every set A such that for all sets a, b such that a ∈ A and b ∈ A

holds a ∪ b ∈ it holds ⋃ A ∈ it. Let X be a set. The functor FlatCoh(X) yielding a set is defined as follows:

(Def.2) FlatCoh(X) = CohSp(△_{X}). The functor SubFin(X) yielding a subset of X is defined by: (Def.3) For every set x holds x ∈ SubFin(X) iff x ∈ X and x is finite.

One can prove the following three propositions:

(1) For all sets X, x holds x ∈ FlatCoh(X) iff x = ∅ or there exists a set y such that x = {y} and y ∈ X.

(2) For every set X holds ⋃ FlatCoh(X) = X.

(3) For every finite down-closed set X holds SubFin(X) = X.

One can check that {∅} is down-closed and binary complete. Let X be a set. One can check that 2^{X} is down-closed and binary complete and FlatCoh(X) is non empty down-closed and binary complete.

Let C be a non empty down-closed set. Observe that SubFin(C) is non empty and down-closed.

We now state the proposition

(4) Web({∅}) = ∅. The scheme MinimalElement wrt Incl concerns sets A, B and a unary pred-

icate P, and states that:

There exists a set a such that a ∈ B and P[a] and for every set b such that b ∈ B and P[b] and b ⊆ a holds b = a

provided the following requirements are met: • A ∈ B,

• P[A],

• A is finite. Let X be a set. One can check that there exists a subset of X which is finite. Let C be a coherent space. Observe that there exists an element of C which

is finite.

Let X be a set. We say that X is ∪-directed if and only if: (Def.4) For every finite subset Y of X there exists a set a such that ⋃ Y ⊆ a

and a ∈ X. We say that X is ∩-directed if and only if: (Def.5) For every finite subset Y of X there exists a set a such that for every

set y such that y ∈ Y holds a ⊆ y and a ∈ X. Let us note that every set which is ∪-directed is also non empty and every

set which is ∩-directed is also non empty. We now state several propositions:

(5) Let X be a set. Suppose X is ∪-directed. Let a, b be sets. If a ∈ X and b ∈ X, then there exists a set c such that a ∪ b ⊆ c and c ∈ X.

(6) Let X be a non empty set. Suppose that for all sets a, b such that a ∈ X and b ∈ X there exists a set c such that a ∪ b ⊆ c and c ∈ X. Then X is ∪-directed.

(7) Let X be a set. Suppose X is ∩-directed. Let a, b be sets. If a ∈ X and b ∈ X, then there exists a set c such that c ⊆ a ∩ b and c ∈ X.

(8) Let X be a non empty set. Suppose that for all sets a, b such that a ∈ X and b ∈ X there exists a set c such that c ⊆ a ∩ b and c ∈ X. Then X is ∩-directed.

(9) For every set x holds {x} is ∪-directed and ∩-directed.

(10) For all sets x, y holds {x, y, x ∪ y} is ∪-directed.

(11) For all sets x, y holds {x, y, x ∩ y} is ∩-directed. Let us observe that there exists a set which is ∪-directed ∩-directed and

finite.

Let C be a non empty set. Observe that there exists a subset of C which is ∪-directed ∩-directed and finite.

We now state the proposition (12) For every set X holds Fin X is ∪-directed and ∩-directed.

Let X be a set. Observe that Fin X is ∪-directed and ∩-directed. Let C be a down-closed non empty set. Note that there exists a subset of C

which is preboolean and non empty.

Let C be a down-closed non empty set and let a be an element of C. Then Fin a is a preboolean non empty subset of C.

One can prove the following proposition

(13) Let X be a non empty set and let Y be a set. Suppose X is ∪-directed and Y ⊆ ⋃ X and Y is finite. Then there exists a set Z such that Z ∈ X and Y ⊆ Z.

Let X be a set. We say that X is ∩-closed if and only if: (Def.6) For all sets x, y such that x ∈ X and y ∈ X holds x ∩ y ∈ X. We say that X is closed under directed unions if and only if: (Def.7) For every subset A of X such that A is ∪-directed holds ⋃ A ∈ X.

One can check that every set which is down-closed is also ∩-closed. Next we state two propositions:

(14) For every coherent space C and for all elements x, y of C holds x∩y ∈ C.

(15) For every coherent space C and for every ∪-directed subset A of C holds

⋃ A ∈ C. Let us note that every coherent space is closed under directed unions. Let us note that there exists a coherent space which is ∩-closed and closed

under directed unions.

Let C be a closed under directed unions non empty set and let A be a ∪- directed subset of C. Then ⋃ A is an element of C.

Let X, Y be sets. We say that X includes lattice of Y if and only if: (Def.8) For all sets a, b such that a ∈ Y and b ∈ Y holds a∩b ∈ X and a∪b ∈ X.

The following proposition is true

(16) For every non empty set X such that X includes lattice of X holds X

is ∪-directed and ∩-directed. Let X, x, y be sets. We say that X includes lattice of x, y if and only if:

(Def.9) X includes lattice of {x, y}.

One can prove the following proposition

(17) For all sets X, x, y holds X includes lattice of x, y iff x ∈ X and y ∈ X and x ∩ y ∈ X and x ∪ y ∈ X.

2. Continuous, Stable, and Linear Functions

Let f be a function. We say that f is preserving arbitrary unions if and only if:

(Def.10) For every subset A of dom f such that ⋃ A ∈ dom f holds f(⋃ A) = ⋃(f^{∘}A).

We say that f is preserving directed unions if and only if: (Def.11) For every subset A of dom f such that A is ∪-directed and ⋃ A ∈ dom f

holds f(⋃ A) = ⋃(f^{∘}A). Let f be a function. We say that f is ⊆-monotone if and only if:

(Def.12) For all sets a, b such that a ∈ dom f and b ∈ dom f and a ⊆ b holds

f(a) ⊆ f(b). We say that f is preserving binary intersections if and only if:

(Def.13) For all sets a, b such that dom f includes lattice of a, b holds f(a ∩ b) =

f(a) ∩ f(b). Let us note that every function which is preserving directed unions is also

⊆-monotone and every function which is preserving arbitrary unions is also preserving directed unions.

Next we state two propositions:

(18) Let f be a function. Suppose f is preserving arbitrary unions. Let x, y be sets. If x ∈ dom f and y ∈ dom f and x ∪ y ∈ dom f, then f(x ∪ y) = f(x) ∪ f(y).

(19) For every function f such that f is preserving arbitrary unions holds

f(∅) = ∅. Let C_{1}, C_{2} be coherent spaces. Note that there exists a function from C_{1} into

C_{2} which is preserving arbitrary unions and preserving binary intersections.

Let C be a coherent space. One can verify that there exists a many sorted set indexed by C which is preserving arbitrary unions and preserving binary intersections.

Let f be a function. We say that f is continuous if and only if: (Def.14) dom f is closed under directed unions and f is preserving directed

unions. Let f be a function. We say that f is stable if and only if:

(Def.15) dom f is ∩-closed and f is continuous and preserving binary intersec-

tions. Let f be a function. We say that f is linear if and only if:

(Def.16) f is stable and preserving arbitrary unions.

One can check the following observations:

∗ every function which is continuous is also preserving directed unions,

∗ every function which is stable is also preserving binary intersections and continuous, and

∗ every function which is linear is also preserving arbitrary unions and stable.

Let X be a closed under directed unions set. Note that every many sorted set indexed by X which is preserving directed unions is also continuous.

Let X be a ∩-closed set. Observe that every many sorted set indexed by X which is continuous and preserving binary intersections is also stable.

Let us note that every function which is stable and preserving arbitrary unions is also linear.

Note that there exists a function which is linear. Let C be a coherent space. One can check that there exists a many sorted set indexed by C which is linear. Let B be a coherent space. One can check that there exists a function from B into C which is linear.

Let f be a continuous function. One can verify that dom f is closed under directed unions.

Let f be a stable function. One can verify that dom f is ∩-closed. We now state several propositions:

(20) For every set X holds ⋃ Fin X = X.

(21) For every continuous function f such that dom f is down-closed and for every set a such that a ∈ dom f holds f(a) = ⋃(f^{∘} Fin a).

(22) Let f be a function. Suppose dom f is down-closed. Then f is contin-

uous if and only if the following conditions are satisfied:

(i) dom f is closed under directed unions, (ii) f is ⊆-monotone, and (iii) for all sets a, y such that a ∈ dom f and y ∈ f(a) there exists a set b

such that b is finite and b ⊆ a and y ∈ f(b).

(23) Let f be a function. Suppose dom f is down-closed and closed under directed unions. Then f is stable if and only if the following conditions are satisfied:

(i) f is ⊆-monotone, and (ii) for all sets a, y such that a ∈ dom f and y ∈ f(a) there exists a set b

such that b is finite and b ⊆ a and y ∈ f(b) and for every set c such that c ⊆ a and y ∈ f(c) holds b ⊆ c.

(24) Let f be a function. Suppose dom f is down-closed and closed under directed unions. Then f is linear if and only if the following conditions are satisfied:

(i) f is ⊆-monotone, and (ii) for all sets a, y such that a ∈ dom f and y ∈ f(a) there exists a set x

such that x ∈ a and y ∈ f({x}) and for every set b such that b ⊆ a and y ∈ f(b) holds x ∈ b.

3. Graph of Continuous Function

Let f be a function. The functor graph(f) yielding a set is defined as follows: (Def.17) For every set x holds x ∈ graph(f) iff there exists a finite set y and

there exists a set z such that x = ⟨⟨y, z⟩⟩ and y ∈ dom f and z ∈ f(y). Let C_{1}, C_{2} be non empty sets and let f be a function from C_{1} into C_{2}. Then

graph(f) is a subset of [: C_{1}, ⋃ C_{2} :].

Let f be a function. Note that graph(f) is relation-like.

Next we state several propositions:

(25) For every function f and for all sets x, y holds ⟨⟨x, y⟩⟩ ∈ graph(f) iff x is finite and x ∈ dom f and y ∈ f(x).

(26) Let f be a ⊆-monotone function and let a, b be sets. Suppose b ∈ dom f and a ⊆ b and b is finite. Let y be a set. If ⟨⟨a, y⟩⟩ ∈ graph(f), then ⟨⟨b, y⟩⟩ ∈ graph(f).

(27) Let C_{1}, C_{2} be coherent spaces, and let f be a function from C_{1} into C_{2}, and let a be an element of C_{1}, and let y_{1}, y_{2} be sets. If ⟨⟨a, y_{1}⟩⟩ ∈ graph(f) and ⟨⟨a, y_{2}⟩⟩ ∈ graph(f), then {y_{1}, y_{2}} ∈ C_{2}.

(28) Let C_{1}, C_{2} be coherent spaces, and let f be a ⊆-monotone function from C_{1} into C_{2}, and let a, b be elements of C_{1}. Suppose a ∪ b ∈ C_{1}. Let y_{1}, y_{2} be sets. If ⟨⟨a, y_{1}⟩⟩ ∈ graph(f) and ⟨⟨b, y_{2}⟩⟩ ∈ graph(f), then {y_{1}, y_{2}} ∈ C_{2}.

(29) For all coherent spaces C_{1}, C_{2} and for all continuous functions f, g from C_{1} into C_{2} such that graph(f) = graph(g) holds f = g.

(30) Let C_{1}, C_{2} be coherent spaces and let X be a subset of [: C_{1}, ⋃ C_{2} :].

Suppose that

(i) for every set x such that x ∈ X holds x_{1} is finite, (ii) for all finite elements a, b of C_{1} such that a ⊆ b and for every set y

such that ⟨⟨a, y⟩⟩ ∈ X holds ⟨⟨b, y⟩⟩ ∈ X, and (iii) for every finite element a of C_{1} and for all sets y_{1}, y_{2} such that ⟨⟨a,

y_{1}⟩⟩ ∈ X and ⟨⟨a, y_{2}⟩⟩ ∈ X holds {y_{1}, y_{2}} ∈ C_{2}. Then there exists a continuous function f from C_{1} into C_{2} such that X = graph(f).

(31) Let C_{1}, C_{2} be coherent spaces, and let f be a continuous function from C_{1} into C_{2}, and let a be an element of C_{1}. Then f(a) = (graph(f))^{∘} Fin a.

4. Trace of Stable Function

Let f be a function. The functor Trace(f) yields a set and is defined by the condition (Def.18).

(Def.18) Let x be a set. Then x ∈ Trace(f) if and only if there exist sets a, y such that x = ⟨⟨a, y⟩⟩ and a ∈ dom f and y ∈ f(a) and for every set b such that b ∈ dom f and b ⊆ a and y ∈ f(b) holds a = b.

Next we state the proposition

(32) Let f be a function and let a, y be sets. Then ⟨⟨a, y⟩⟩ ∈ Trace(f) if and

only if the following conditions are satisfied:

(i) a ∈ dom f, (ii) y ∈ f(a), and (iii) for every set b such that b ∈ dom f and b ⊆ a and y ∈ f(b) holds a = b. Let C_{1}, C_{2} be non empty sets and let f be a function from C_{1} into C_{2}. Then

Trace(f) is a subset of [: C_{1}, ⋃ C_{2} :].

Let f be a function. One can check that Trace(f) is relation-like. Next we state a number of propositions:

(33) For every continuous function f such that dom f is down-closed holds Trace(f) ⊆ graph(f).

(34) Let f be a continuous function. Suppose dom f is down-closed. Let a, y be sets. If ⟨⟨a, y⟩⟩ ∈ Trace(f), then a is finite.

(35) Let C_{1}, C_{2} be coherent spaces, and let f be a ⊆-monotone function from C_{1} into C_{2}, and let a_{1}, a_{2} be sets. Suppose a_{1} ∪ a_{2} ∈ C_{1}. Let y_{1}, y_{2} be sets. If ⟨⟨a_{1}, y_{1}⟩⟩ ∈ Trace(f) and ⟨⟨a_{2}, y_{2}⟩⟩ ∈ Trace(f), then {y_{1}, y_{2}} ∈ C_{2}.

(36) Let C_{1}, C_{2} be coherent spaces, and let f be a preserving binary intersections function from C_{1} into C_{2}, and let a_{1}, a_{2} be sets. If a_{1} ∪ a_{2} ∈ C_{1}, then for every set y such that ⟨⟨a_{1}, y⟩⟩ ∈ Trace(f) and ⟨⟨a_{2}, y⟩⟩ ∈ Trace(f) holds a_{1} = a_{2}.

(37) Let C_{1}, C_{2} be coherent spaces and let f, g be stable functions from C_{1} into C_{2}. If Trace(f) ⊆ Trace(g), then for every element a of C_{1} holds f(a) ⊆ g(a).

(38) For all coherent spaces C_{1}, C_{2} and for all stable functions f, g from C_{1} into C_{2} such that Trace(f) = Trace(g) holds f = g.

(39) Let C_{1}, C_{2} be coherent spaces and let X be a subset of [: C_{1}, ⋃ C_{2} :].

Suppose that

(i) for every set x such that x ∈ X holds x_{1} is finite, (ii) for all elements a, b of C_{1} such that a ∪ b ∈ C_{1} and for all sets y_{1}, y_{2}

such that ⟨⟨a, y_{1}⟩⟩ ∈ X and ⟨⟨b, y_{2}⟩⟩ ∈ X holds {y_{1}, y_{2}} ∈ C_{2}, and (iii) for all elements a, b of C_{1} such that a ∪ b ∈ C_{1} and for every set y such

that ⟨⟨a, y⟩⟩ ∈ X and ⟨⟨b, y⟩⟩ ∈ X holds a = b. Then there exists a stable function f from C_{1} into C_{2} such that X = Trace(f).

(40) Let C_{1}, C_{2} be coherent spaces, and let f be a stable function from C_{1} into C_{2}, and let a be an element of C_{1}. Then f(a) = (Trace(f))^{∘} Fin a.

(41) Let C_{1}, C_{2} be coherent spaces, and let f be a stable function from C_{1} into C_{2}, and let a be an element of C_{1}, and let y be a set. Then y ∈ f(a) if and only if there exists an element b of C_{1} such that ⟨⟨b, y⟩⟩ ∈ Trace(f) and b ⊆ a.

(42) For all coherent spaces C_{1}, C_{2} there exists a stable function f from C_{1} into C_{2} such that Trace(f) = ∅.

(43) Let C_{1}, C_{2} be coherent spaces, and let a be a finite element of C_{1}, and let y be a set. If y ∈ ⋃ C_{2}, then there exists a stable function f from C_{1} into C_{2} such that Trace(f) = {⟨⟨a, y⟩⟩}.

(44) Let C_{1}, C_{2} be coherent spaces, and let a be an element of C_{1}, and let y be a set. Suppose y ∈ ⋃ C_{2}. Let f be a stable function from C_{1} into C_{2}. Suppose Trace(f) = {⟨⟨a, y⟩⟩}. Let b be an element of C_{1}. Then if a ⊆ b, then f(b) = {y} and if a ⊈ b, then f(b) = ∅.

(45) Let C_{1}, C_{2} be coherent spaces, and let f be a stable function from C_{1} into C_{2}, and let X be a subset of Trace(f). Then there exists a stable function g from C_{1} into C_{2} such that Trace(g) = X.

(46) Let C_{1}, C_{2} be coherent spaces and let A be a set. Suppose that for all sets x, y such that x ∈ A and y ∈ A there exists a stable function f from C_{1} into C_{2} such that x∪y = Trace(f). Then there exists a stable function f from C_{1} into C_{2} such that ⋃ A = Trace(f).

Let C_{1}, C_{2} be coherent spaces. The functor StabCoh(C_{1}, C_{2}) yielding a set is defined as follows:

(Def.19) For every set x holds x ∈ StabCoh(C_{1}, C_{2}) iff there exists a stable

function f from C_{1} into C_{2} such that x = Trace(f). Let C_{1}, C_{2} be coherent spaces. Note that StabCoh(C_{1}, C_{2}) is non empty

down-closed and binary complete.

We now state three propositions:

(47) For all coherent spaces C_{1}, C_{2} and for every stable function f from C_{1} into C_{2} holds Trace(f) ⊆ [: SubFin(C_{1}), ⋃ C_{2} :].

(48) For all coherent spaces C_{1}, C_{2} holds ⋃ StabCoh(C_{1}, C_{2}) = [: SubFin(C_{1}), ⋃ C_{2} :].

(49) Let C_{1}, C_{2} be coherent spaces, and let a, b be finite elements of C_{1}, and let y_{1}, y_{2} be sets. Then ⟨⟨⟨⟨a, y_{1}⟩⟩, ⟨⟨b, y_{2}⟩⟩⟩⟩ ∈ Web(StabCoh(C_{1}, C_{2})) if and only if one of the following conditions is satisfied:

(i) a ∪ b ∉ C_{1} and y_{1} ∈ ⋃ C_{2} and y_{2} ∈ ⋃ C_{2}, or (ii) ⟨⟨y_{1}, y_{2}⟩⟩ ∈ Web(C_{2}) and if y_{1} = y_{2}, then a = b.

5. Trace of Linear Function

The following proposition is true

(50) Let C_{1}, C_{2} be coherent spaces and let f be a stable function from C_{1} into C_{2}. Then f is linear if and only if for all sets a, y such that ⟨⟨a, y⟩⟩ ∈ Trace(f) there exists a set x such that a = {x}.

Let f be a function. The functor LinTrace(f) yielding a set is defined as follows:

(Def.20) For every set x holds x ∈ LinTrace(f) iff there exist sets y, z such that

x = ⟨⟨y, z⟩⟩ and ⟨⟨{y}, z⟩⟩ ∈ Trace(f). Next we state three propositions:

(51) For every function f and for all sets x, y holds ⟨⟨x, y⟩⟩ ∈ LinTrace(f) iff ⟨⟨{x}, y⟩⟩ ∈ Trace(f).

(52) For every function f such that f(∅) = ∅ and for all sets x, y such that {x} ∈ dom f and y ∈ f({x}) holds ⟨⟨x, y⟩⟩ ∈ LinTrace(f).

(53) For every function f and for all sets x, y such that ⟨⟨x, y⟩⟩ ∈ LinTrace(f) holds {x} ∈ dom f and y ∈ f({x}).

Let C_{1}, C_{2} be non empty sets and let f be a function from C_{1} into C_{2}. Then LinTrace(f) is a subset of [: ⋃ C_{1}, ⋃ C_{2} :].

Let f be a function. One can verify that LinTrace(f) is relation-like. Let C_{1}, C_{2} be coherent spaces. The functor LinCoh(C_{1}, C_{2}) yielding a set is

defined as follows: (Def.21) For every set x holds x ∈ LinCoh(C_{1}, C_{2}) iff there exists a linear function

f from C_{1} into C_{2} such that x = LinTrace(f). Next we state a number of propositions:

(54) Let C_{1}, C_{2} be coherent spaces, and let f be a ⊆-monotone function from C_{1} into C_{2}, and let x_{1}, x_{2} be sets. Suppose {x_{1}, x_{2}} ∈ C_{1}. Let y_{1}, y_{2} be sets. If ⟨⟨x_{1}, y_{1}⟩⟩ ∈ LinTrace(f) and ⟨⟨x_{2}, y_{2}⟩⟩ ∈ LinTrace(f), then {y_{1}, y_{2}} ∈ C_{2}.

(55) Let C_{1}, C_{2} be coherent spaces, and let f be a preserving binary intersections function from C_{1} into C_{2}, and let x_{1}, x_{2} be sets. If {x_{1}, x_{2}} ∈ C_{1}, then for every set y such that ⟨⟨x_{1}, y⟩⟩ ∈ LinTrace(f) and ⟨⟨x_{2}, y⟩⟩ ∈ LinTrace(f) holds x_{1} = x_{2}.

(56) For all coherent spaces C_{1}, C_{2} and for all linear functions f, g from C_{1} into C_{2} such that LinTrace(f) = LinTrace(g) holds f = g.

(57) Let C_{1}, C_{2} be coherent spaces and let X be a subset of [: ⋃ C_{1}, ⋃ C_{2} :].

Suppose that

(i) for all sets a, b such that {a, b} ∈ C_{1} and for all sets y_{1}, y_{2} such that ⟨⟨a, y_{1}⟩⟩ ∈ X and ⟨⟨b, y_{2}⟩⟩ ∈ X holds {y_{1}, y_{2}} ∈ C_{2}, and

(ii) for all sets a, b such that {a, b} ∈ C_{1} and for every set y such that ⟨⟨a, y⟩⟩ ∈ X and ⟨⟨b, y⟩⟩ ∈ X holds a = b. Then there exists a linear function f from C_{1} into C_{2} such that X = LinTrace(f).

(58) Let C_{1}, C_{2} be coherent spaces, and let f be a linear function from C_{1} into C_{2}, and let a be an element of C_{1}. Then f(a) = (LinTrace(f))^{∘}a.

(59) For all coherent spaces C_{1}, C_{2} there exists a linear function f from C_{1} into C_{2} such that LinTrace(f) = ∅.

(60) Let C_{1}, C_{2} be coherent spaces, and let x be a set, and let y be a set. Suppose x ∈ ⋃ C_{1} and y ∈ ⋃ C_{2}. Then there exists a linear function f from C_{1} into C_{2} such that LinTrace(f) = {⟨⟨x, y⟩⟩}.

(61) Let C_{1}, C_{2} be coherent spaces, and let x be a set, and let y be a set. Suppose x ∈ ⋃ C_{1} and y ∈ ⋃ C_{2}. Let f be a linear function from C_{1} into C_{2}. Suppose LinTrace(f) = {⟨⟨x, y⟩⟩}. Let a be an element of C_{1}. Then if x ∈ a, then f(a) = {y} and if x ∉ a, then f(a) = ∅.

(62) Let C_{1}, C_{2} be coherent spaces, and let f be a linear function from C_{1} into C_{2}, and let X be a subset of LinTrace(f). Then there exists a linear function g from C_{1} into C_{2} such that LinTrace(g) = X.

(63) Let C_{1}, C_{2} be coherent spaces and let A be a set. Suppose that for all sets x, y such that x ∈ A and y ∈ A there exists a linear function f from C_{1} into C_{2} such that x ∪ y = LinTrace(f). Then there exists a linear function f from C_{1} into C_{2} such that ⋃ A = LinTrace(f).

Let C_{1}, C_{2} be coherent spaces. One can check that LinCoh(C_{1}, C_{2}) is non empty down-closed and binary complete.

One can prove the following propositions:

(64) For all coherent spaces C_{1}, C_{2} holds ⋃ LinCoh(C_{1}, C_{2}) = [: ⋃ C_{1}, ⋃ C_{2} :].

(65) Let C_{1}, C_{2} be coherent spaces, and let x_{1}, x_{2} be sets, and let y_{1}, y_{2} be sets. Then ⟨⟨⟨⟨x_{1}, y_{1}⟩⟩, ⟨⟨x_{2}, y_{2}⟩⟩⟩⟩ ∈ Web(LinCoh(C_{1}, C_{2})) if and only if the following conditions are satisfied:

(i) x_{1} ∈ ⋃ C_{1}, (ii) x_{2} ∈ ⋃ C_{1}, and (iii) ⟨⟨x_{1}, x_{2}⟩⟩ ∉ Web(C_{1}) and y_{1} ∈ ⋃ C_{2} and y_{2} ∈ ⋃ C_{2} or ⟨⟨y_{1}, y_{2}⟩⟩ ∈

Web(C_{2}) and if y_{1} = y_{2}, then x_{1} = x_{2}.

6. Negation of Coherence Spaces

Let C be a coherent space. The functor ¬C yielding a set is defined by: (Def.22) ¬C = {a : a ranges over subsets of ⋃ C, ⋀_{b: element of C} ⋁_{x: set} a ∩ b ⊆

{x}}. One can prove the following proposition

(66) Let C be a coherent space and let x be a set. Then x ∈ ¬C if and only

if the following conditions are satisfied:

(i) x ⊆ ⋃ C, and (ii) for every element a of C there exists a set z such that x ∩ a ⊆ {z}. Let C be a coherent space. Observe that ¬C is non empty down-closed and

binary complete.

Next we state several propositions:

(67) For every coherent space C holds ⋃ ¬C = ⋃ C.

(68) For every coherent space C and for all sets x, y such that x ≠ y and {x, y} ∈ C holds {x, y} ∉ ¬C.

(69) For every coherent space C and for all sets x, y such that {x, y} ⊆ ⋃ C and {x, y} ∉ C holds {x, y} ∈ ¬C.

(70) For every coherent space C and for all sets x, y holds ⟨⟨x, y⟩⟩ ∈ Web(¬C) iff x ∈ ⋃ C but y ∈ ⋃ C but x = y or ⟨⟨x, y⟩⟩ ∉ Web(C).

(71) For every coherent space C holds ¬¬C = C.

(72) ¬{∅} = {∅}.

(73) For every set X holds ¬ FlatCoh(X) = 2^{X} and ¬(2^{X}) = FlatCoh(X).

7. Product and Coproduct on Coherence Spaces

Let x, y be sets. The functor x ⊎ y yielding a set is defined by: (Def.23) x ⊎ y = ⋃ disjoint⟨x, y⟩.

We now state a number of propositions:

(74) For all sets x, y holds x ⊎ y = [: x, {1}:] ∪ [: y, {2}:].

(75) For every set x holds x ⊎ ∅ = [: x, {1}:] and ∅ ⊎ x = [: x, {2}:].

(76) For all sets x, y, z such that z ∈ x ⊎ y holds z = ⟨⟨z_{1}, z_{2}⟩⟩ but z_{2} = 1 and z_{1} ∈ x or z_{2} = 2 and z_{1} ∈ y.

(77) For all sets x, y, z holds ⟨⟨z, 1⟩⟩ ∈ x ⊎ y iff z ∈ x.

(78) For all sets x, y, z holds ⟨⟨z, 2⟩⟩ ∈ x ⊎ y iff z ∈ y.

(79) For all sets x_{1}, y_{1}, x_{2}, y_{2} holds x_{1} ⊎y_{1} ⊆ x_{2} ⊎y_{2} iff x_{1} ⊆ x_{2} and y_{1} ⊆ y_{2}.

(80) For all sets x, y, z such that z ⊆ x ⊎ y there exist sets x_{1}, y_{1} such that z = x_{1} ⊎ y_{1} and x_{1} ⊆ x and y_{1} ⊆ y.

(81) For all sets x_{1}, y_{1}, x_{2}, y_{2} holds x_{1} ⊎y_{1} = x_{2} ⊎y_{2} iff x_{1} = x_{2} and y_{1} = y_{2}.

(82) For all sets x_{1}, y_{1}, x_{2}, y_{2} holds (x_{1} ⊎ y_{1}) ∪ (x_{2} ⊎ y_{2}) = x_{1} ∪ x_{2} ⊎ y_{1} ∪ y_{2}.

(83) For all sets x_{1}, y_{1}, x_{2}, y_{2} holds (x_{1} ⊎ y_{1}) ∩ (x_{2} ⊎ y_{2}) = x_{1} ∩ x_{2} ⊎ y_{1} ∩ y_{2}. Let C_{1}, C_{2} be coherent spaces. The functor C_{1}⊓C_{2} yields a set and is defined

by: (Def.24) C_{1} ⊓ C_{2} = {a ⊎ b : a ranges over elements of C_{1}, b ranges over elements

of C_{2}}. The functor C_{1} ⊔ C_{2} yielding a set is defined as follows:

(Def.25) C_{1} ⊔ C_{2} = {a ⊎ ∅ : a ranges over elements of C_{1}} ∪ {∅ ⊎ b : b ranges over

elements of C_{2}}. The following propositions are true:

(84) Let C_{1}, C_{2} be coherent spaces and let x be a set. Then x ∈ C_{1} ⊓ C_{2} if and only if there exists an element a of C_{1} and there exists an element b of C_{2} such that x = a ⊎ b.

(85) For all coherent spaces C_{1}, C_{2} and for all sets x, y holds x ⊎ y ∈ C_{1} ⊓ C_{2} iff x ∈ C_{1} and y ∈ C_{2}.

(86) For all coherent spaces C_{1}, C_{2} holds ⋃(C_{1} ⊓ C_{2}) = ⋃ C_{1} ⊎ ⋃ C_{2}.

(87) For all coherent spaces C_{1}, C_{2} and for all sets x, y holds x ⊎ y ∈ C_{1} ⊔ C_{2} iff x ∈ C_{1} and y = ∅ or x = ∅ and y ∈ C_{2}.

(88) Let C_{1}, C_{2} be coherent spaces and let x be a set. Suppose x ∈ C_{1} ⊔ C_{2}. Then there exists an element a of C_{1} and there exists an element b of C_{2} such that x = a ⊎ b but a = ∅ or b = ∅.

(89) For all coherent spaces C_{1}, C_{2} holds ⋃(C_{1} ⊔ C_{2}) = ⋃ C_{1} ⊎ ⋃ C_{2}. Let C_{1}, C_{2} be coherent spaces. Observe that C_{1} ⊓ C_{2} is non empty down-

closed and binary complete and C_{1} ⊔ C_{2} is non empty down-closed and binary complete.

In the sequel C_{1}, C_{2} will be coherent spaces. We now state several propositions:

(90) For all sets x, y holds ⟨⟨⟨⟨x, 1⟩⟩, ⟨⟨y, 1⟩⟩⟩⟩ ∈ Web(C_{1} ⊓ C_{2}) iff ⟨⟨x, y⟩⟩ ∈ Web(C_{1}).

(91) For all sets x, y holds ⟨⟨⟨⟨x, 2⟩⟩, ⟨⟨y, 2⟩⟩⟩⟩ ∈ Web(C_{1} ⊓ C_{2}) iff ⟨⟨x, y⟩⟩ ∈ Web(C_{2}).

(92) For all sets x, y such that x ∈ ⋃ C_{1} and y ∈ ⋃ C_{2} holds ⟨⟨⟨⟨x, 1⟩⟩, ⟨⟨y, 2⟩⟩⟩⟩ ∈ Web(C_{1} ⊓ C_{2}) and ⟨⟨⟨⟨y, 2⟩⟩, ⟨⟨x, 1⟩⟩⟩⟩ ∈ Web(C_{1} ⊓ C_{2}).

(93) For all sets x, y holds ⟨⟨⟨⟨x, 1⟩⟩, ⟨⟨y, 1⟩⟩⟩⟩ ∈ Web(C_{1} ⊔ C_{2}) iff ⟨⟨x, y⟩⟩ ∈ Web(C_{1}).

(94) For all sets x, y holds ⟨⟨⟨⟨x, 2⟩⟩, ⟨⟨y, 2⟩⟩⟩⟩ ∈ Web(C_{1} ⊔ C_{2}) iff ⟨⟨x, y⟩⟩ ∈ Web(C_{2}).

(95) For all sets x, y such that x ∈ ⋃ C_{1} and y ∈ ⋃ C_{2} holds ⟨⟨⟨⟨x, 1⟩⟩, ⟨⟨y, 2⟩⟩⟩⟩ ∉ Web(C_{1} ⊔ C_{2}) and ⟨⟨⟨⟨y, 2⟩⟩, ⟨⟨x, 1⟩⟩⟩⟩ ∉ Web(C_{1} ⊔ C_{2}).

(96) ¬(C_{1} ⊓ C_{2}) = ¬C_{1} ⊔ ¬C_{2}.

Let C_{1}, C_{2} be coherent spaces. The functor C_{1} ⊗ C_{2} yielding a set is defined as follows:

(Def.26) C_{1} ⊗ C_{2} = ⋃{2^{[: a, b:]} : a ranges over elements of C_{1}, b ranges over

elements of C_{2}}. We now state the proposition

(97) Let C_{1}, C_{2} be coherent spaces and let x be a set. Then x ∈ C_{1} ⊗ C_{2} if and only if there exists an element a of C_{1} and there exists an element b of C_{2} such that x ⊆ [: a, b:].

Let C_{1}, C_{2} be coherent spaces. One can check that C_{1} ⊗ C_{2} is non empty. Next we state the proposition

(98) For all coherent spaces C_{1}, C_{2} and for every element a of C_{1} ⊗ C_{2} holds

π_{1}(a) ∈ C_{1} and π_{2}(a) ∈ C_{2} and a ⊆ [: π_{1}(a), π_{2}(a) :]. Let C_{1}, C_{2} be coherent spaces. One can check that C_{1} ⊗ C_{2} is down-closed

and binary complete.

Next we state two propositions:

(99) For all coherent spaces C_{1}, C_{2} holds ⋃(C_{1} ⊗ C_{2}) = [: ⋃ C_{1}, ⋃ C_{2} :].

(100) For all sets x_{1}, y_{1}, x_{2}, y_{2} holds ⟨⟨⟨⟨x_{1}, x_{2}⟩⟩, ⟨⟨y_{1}, y_{2}⟩⟩⟩⟩ ∈ Web(C_{1} ⊗ C_{2}) iff ⟨⟨x_{1}, y_{1}⟩⟩ ∈ Web(C_{1}) and ⟨⟨x_{2}, y_{2}⟩⟩ ∈ Web(C_{2}).

References

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

[2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 1990.

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

[4] Grzegorz Bancerek. Indexed category. Formalized Mathematics, 5(3):329–337, 1996.

[5] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

continuous, stable, and linear maps of … 393

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

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

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

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

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

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

[12] Krzysztof Hryniewiecki. Relations of tolerance. Formalized Mathematics, 2(1):105–109, 1991.

[13] Jarosław Kotowicz and Konrad Raczkowski. Coherent space. Formalized Mathematics, 3(2):255–261, 1992.

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

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

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

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

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

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

[20] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

[21] Zinaida Trybulec and Halina Świe‘czkowska. 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.

Received August 30, 1995 394

Some Basic Properties of Many Sorted Sets

Artur Korniłowicz Warsaw University Białystok

MML Identifier: PZFMISC1.

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

1. Preliminaries

For simplicity we follow a convention: i will be arbitrary, I will be a set, f will be a function, x, x_{1}, x_{2}, y, A, B, X, Y, Z will be many sorted sets indexed by I, J will be a non empty set, and N_{1} will be a many sorted set indexed by J.

We now state three propositions:

(1) For every set X and for every many sorted set M indexed by I such that i ∈ I holds dom(M+·(i⟼ . X)) = I.

(2) If f = ∅, then f is a many sorted set indexed by ∅.

(3) If I is non empty, then there exists no X which is empty yielding and non-empty.

2. Singelton and unordered pairs

Let us consider I, A. The functor {A} yielding a many sorted set indexed by I is defined as follows: (Def.1) For every i such that i ∈ I holds {A}(i) = {A(i)}.

Let us consider I, A. Observe that {A} is non-empty and locally-finite. Let us consider I, A, B. The functor {A, B} yields a many sorted set indexed

by I and is defined as follows:

(Def.2) For every i such that i ∈ I holds {A, B}(i) = {A(i), B(i)}.

Let us consider I, A, B. One can verify that {A, B} is non-empty and locally-finite.

We now state a number of propositions:

(4) X = {y} iff for every x holds x ∈ X iff x = y.

(5) If for every x holds x ∈ X iff x = x_{1} or x = x_{2}, then X = {x_{1}, x_{2}}.

(6) If X = {x_{1}, x_{2}}, then for every x such that x = x_{1} or x = x_{2} holds x ∈ X.

(7) {N_{1}} ≠ ∅_{I}.

(8) If x ∈ {A}, then x = A.

(9) x ∈ {x}.

(10) If x = A or x = B, then x ∈ {A, B}.

(11) {A} ∪ {B} = {A, B}.

(12) {x, x} = {x}.

(13) {A, B} = {B, A}.

(14) If {A} ⊆ {B}, then A = B.

(15) If {x} = {y}, then x = y.

(16) If {x} = {A, B}, then x = A and x = B.

(17) If {x} = {A, B}, then A = B.

(18) {x} ⊆ {x, y} and {y} ⊆ {x, y}.

(19) If {x} ∪ {y} = {x} or {x} ∪ {y} = {y}, then x = y.

(20) {x} ∪ {x, y} = {x, y}.

(21) {y} ∪ {x, y} = {x, y}.

(22) If I is non empty and {x} ∩ {y} = ∅_{I}, then x ≠ y.

(23) If {x} ∩ {y} = {x} or {x} ∩ {y} = {y}, then x = y.

(24) {x} ∩ {x, y} = {x} and {y} ∩ {x, y} = {y}.

(25) If I is non empty and {x} \ {y} = {x}, then x ≠ y.

(26) If {x} \ {y} = ∅_{I}, then x = y.

(27) {x} \ {x, y} = ∅_{I} and {y} \ {x, y} = ∅_{I}.

(28) If {x} ⊆ {y}, then {x} = {y}.

(29) If {x, y} ⊆ {A}, then x = A and y = A.

(30) If {x, y} ⊆ {A}, then {x, y} = {A}.

(31) 2^{{x}} = {∅_{I}, {x}}.

(32) {A} ⊆ 2^{A}.

(33) ⋃{x} = x.

(34) ⋃{{x}, {y}} = {x, y}.

(35) ⋃{A, B} = A ∪ B.

(36) {x} ⊆ X iff x ∈ X.

(37) {x_{1}, x_{2}} ⊆ X iff x_{1} ∈ X and x_{2} ∈ X.

(38) If A = ∅_{I} or A = {x_{1}} or A = {x_{2}} or A = {x_{1}, x_{2}}, then A ⊆ {x_{1}, x_{2}}.

3. Sum of unordered pairs (or a singelton) and a set

One can prove the following propositions:

(39) If x ∈ A or x = B, then x ∈ A ∪ {B}.

(40) A ∪ {x} ⊆ B iff x ∈ B and A ⊆ B.

(41) If {x} ∪ X = X, then x ∈ X.

(42) If x ∈ X, then {x} ∪ X = X.

(43) {x, y} ∪ A = A iff x ∈ A and y ∈ A.

(44) If I is non empty, then {x} ∪ X ≠ ∅_{I}.

(45) If I is non empty, then {x, y} ∪ X ≠ ∅_{I}.

4. Intersection of unordered pairs (or a singelton) and a set

We now state several propositions:

(46) If X ∩ {x} = {x}, then x ∈ X.

(47) If x ∈ X, then X ∩ {x} = {x}.

(48) x ∈ X and y ∈ X iff {x, y} ∩ X = {x, y}.

(49) If I is non empty and {x} ∩ X = ∅_{I}, then x ∉ X.

(50) If I is non empty and {x, y} ∩ X = ∅_{I}, then x ∉ X and y ∉ X.

5. Difference of unordered pairs (or a singelton) and a set

The following propositions are true:

(51) If y ∈ X \ {x}, then y ∈ X.

(52) If I is non empty and y ∈ X \ {x}, then y ≠ x.

(53) If I is non empty and X \ {x} = X, then x ∉ X.

(54) If I is non empty and {x} \ X = {x}, then x ∉ X.

(55) {x} \ X = ∅_{I} iff x ∈ X.

(56) If I is non empty and {x, y} \ X = {x}, then x ∉ X.

(57) If I is non empty and {x, y} \ X = {y}, then y ∉ X.

(58) If I is non empty and {x, y} \ X = {x, y}, then x ∉ X and y ∉ X.

(59) {x, y} \ X = ∅_{I} iff x ∈ X and y ∈ X.

(60) If X = ∅_{I} or X = {x} or X = {y} or X = {x, y}, then X \ {x, y} = ∅_{I}.

6. Cartesian product

One can prove the following propositions:

(61) If X = ∅_{I} or Y = ∅_{I}, then [[X, Y]] = ∅_{I}.

(62) If X is non-empty and Y is non-empty and [[X, Y]] = [[Y, X]], then X = Y.

(63) If [[X, X]] = [[Y, Y]], then X = Y.

(64) If Z is non-empty and if [[X, Z]] ⊆ [[Y, Z]] or [[Z, X]] ⊆ [[Z, Y]], then X ⊆ Y.

(65) If X ⊆ Y, then [[X, Z]] ⊆ [[Y, Z]] and [[Z, X]] ⊆ [[Z, Y]].

(66) If x_{1} ⊆ A and x_{2} ⊆ B, then [[x_{1}, x_{2}]] ⊆ [[A, B]].

(67) [[X ∪ Y, Z]] = [[X, Z]] ∪ [[Y, Z]] and [[Z, X ∪ Y]] = [[Z, X]] ∪ [[Z, Y]].

(68) [[x_{1} ∪ x_{2}, A ∪ B]] = [[x_{1}, A]] ∪ [[x_{1}, B]] ∪ [[x_{2}, A]] ∪ [[x_{2}, B]].

(69) [[X ∩ Y, Z]] = [[X, Z]] ∩ [[Y, Z]] and [[Z, X ∩ Y]] = [[Z, X]] ∩ [[Z, Y]].

(70) [[x_{1} ∩ x_{2}, A ∩ B]] = [[x_{1}, A]] ∩ [[x_{2}, B]].

(71) If A ⊆ X and B ⊆ Y, then [[A, Y]] ∩ [[X, B]] = [[A, B]].

(72) [[X \ Y, Z]] = [[X, Z]] \ [[Y, Z]] and [[Z, X \ Y]] = [[Z, X]] \ [[Z, Y]].

(73) [[x_{1}, x_{2}]] \ [[A, B]] = [[x_{1} \ A, x_{2}]] ∪ [[x_{1}, x_{2} \ B]].

(74) If x_{1} ∩ x_{2} = ∅_{I} or A ∩ B = ∅_{I}, then [[x_{1}, A]] ∩ [[x_{2}, B]] = ∅_{I}.

(75) If X is non-empty, then [[{x}, X]] is non-empty and [[X, {x}]] is nonempty.

(76) [[{x, y}, X]] = [[{x}, X]] ∪ [[{y}, X]] and [[X, {x, y}]] = [[X, {x}]] ∪ [[X, {y}]].

(77) If x_{1} is non-empty and A is non-empty and [[x_{1}, A]] = [[x_{2}, B]], then x_{1} = x_{2} and A = B.

(78) If X ⊆ [[X, Y]] or X ⊆ [[Y, X]], then X = ∅_{I}.

(79) If A ∈ [[x, y]] and A ∈ [[X, Y]], then A ∈ [[x ∩ X, y ∩ Y]].

(80) If [[x, X]] ⊆ [[y, Y]] and [[x, X]] is non-empty, then x ⊆ y and X ⊆ Y.

(81) If A ⊆ X, then [[A, A]] ⊆ [[X, X]].

(82) If X ∩ Y = ∅_{I}, then [[X, Y]] ∩ [[Y, X]] = ∅_{I}.

(83) If A is non-empty and if [[A, B]] ⊆ [[X, Y]] or [[B, A]] ⊆ [[Y, X]], then B ⊆ Y.

(84) If x ⊆ [[A, B]] and y ⊆ [[X, Y]], then x ∪ y ⊆ [[A ∪ X, B ∪ Y]].

References

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

some basic properties of many sorted sets 399

[4] 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.

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

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

[7] Artur Korniłowicz. Definitions and basic properties of boolean & union of many sorted sets. Formalized Mathematics, 5(2):279–281, 1996.

[8] Beata Madras. Products of many sorted algebras. Formalized Mathematics, 5(1):55–60, 1996.

[9] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.

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

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

[12] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

Received September 29, 1995 400

Replacement of Subtrees in a Tree

Oleg Okhotnikov Ural University Ekaterinburg

Summary. This paper is based on previous works [1], [3] in which the operation replacement of subtree in a tree has been defined. We extend this notion for arbitrary non empty antichain.

MML Identifier: TREES A.

The notation and terminology used in this paper are introduced in the following papers: [8], [9], [6], [10], [5], [7], [4], [1], [3], and [2].

We follow the rules: T, T_{1} will denote trees, P will denote an antichain of prefixes of T, and p, q, r will denote finite sequences of elements of .

We now state the proposition

(1) For all finite sequences p, q, r, s such that p^{}q = s^{}r holds p and s are comparable.

Let us consider T, T_{1} and let us consider P. Let us assume that P ≠ ∅. The functor T (P/T_{1}) yields a tree and is defined as follows: (Def.1) q ∈ T (P/T_{1}) iff q ∈ T and for every p such that p ∈ P holds p q or

there exist p, r such that p ∈ P and r ∈ T_{1} and q = p^{}r. One can prove the following propositions:

(2) Suppose P ≠ ∅. Then T (P/T_{1}) = {t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}} ∪ {p^{}s : p ranges over elements of T, s ranges over elements of T_{1}, p ∈ P }.

(3) {t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}} ⊆ {t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}}.

(4) P ⊆ {t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}}.

(5) {t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}}\{t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}} = P.

(6) For all T, T_{1}, P holds P ⊆ {p^{}s : p ranges over elements of T, s ranges over elements of T_{1}, p ∈ P }.

(7) Suppose P ≠ ∅. Then T (P/T_{1}) = {t_{1} : t_{1} ranges over elements of T, ⋀_{p} p ∈ P ⇒ p t_{1}} ∪ {p^{}s : p ranges over elements of T, s ranges over elements of T_{1}, p ∈ P }.

(8) If p ∈ P and q ∈ T_{1}, then p^{}q ∈ T (P/T_{1}).

(9) If p ∈ P, then T_{1} = T (P/T_{1}) p. Let us consider T. Observe that there exists an antichain of prefixes of T

which is non empty.

Let us consider T and let t be an element of T. Then {t} is a non empty antichain of prefixes of T.

In the sequel t will be an element of T. We now state the proposition

(10) T ({t}/T_{1}) = T (t/T_{1}).

In the sequel T, T_{1} denote decorated trees, P denotes an antichain of prefixes of dom T, and t denotes an element of dom T.

Let us consider T, P, T_{1}. Let us assume that P ≠ ∅. The functor T (P/T_{1}) yields a decorated tree and is defined by the conditions (Def.2). (Def.2) (i) dom(T (P/T_{1})) = (dom T)(P/ dom T_{1}), and

(ii) for every q such that q ∈ (dom T)(P/ dom T_{1}) holds for every p such that p ∈ P holds p q and T (P/T_{1})(q) = T (q) or there exist p, r such that p ∈ P and r ∈ dom T_{1} and q = p^{}r and T (P/T_{1})(q) = T_{1}(r).

We now state several propositions:

(11) If P ≠ ∅, then dom(T (P/T_{1})) = (dom T)(P/ dom T_{1}).

(12) If p ∈ dom T, then dom(T (p/T_{1})) = (dom T)(p/ dom T_{1}).

(13) Suppose P ≠ ∅. Given q. Suppose q ∈ dom(T (P/T_{1})). Then for every p such that p ∈ P holds p q and T (P/T_{1})(q) = T (q) or there exist p, r such that p ∈ P and r ∈ dom T_{1} and q = p^{}r and T (P/T_{1})(q) = T_{1}(r).

(14) Suppose p ∈ dom T. Given q. Suppose q ∈ dom(T (p/T_{1})). Then p q and T (p/T_{1})(q) = T (q) or there exists r such that r ∈ dom T_{1} and q = p^{}r and T (p/T_{1})(q) = T_{1}(r).

(15) Suppose P ≠ ∅. Given q. Suppose q ∈ dom(T (P/T_{1})) and q ∈ {t_{1} : t_{1} ranges over elements of dom T, ⋀_{p} p ∈ P ⇒ p t_{1}}. Then T (P/T_{1})(q) = T (q).

(16) If p ∈ dom T, then for every q such that q ∈ dom(T (p/T_{1})) and q ∈ {t_{1} : t_{1} ranges over elements of dom T, p t_{1}} holds T (p/T_{1})(q) = T (q).

(17) Suppose P ≠ ∅. Given q. Suppose q ∈ dom(T (P/T_{1})) and q ∈ {p^{}s : p ranges over elements of dom T, s ranges over elements of dom T_{1}, p ∈ P }. Then there exists an element p′ of dom T and there exists an element r of dom T_{1} such that q = p′ ^{}r and p′ ∈ P and T (P/T_{1})(q) = T_{1}(r).

(18) Suppose p ∈ dom T. Given q. Suppose q ∈ dom(T (p/T_{1})) and q ∈ {p ^{}s : s ranges over elements of dom T_{1}, s = s}. Then there exists an element r of dom T_{1} such that q = p^{}r and T (p/T_{1})(q) = T_{1}(r).

(19) T ({t}/T_{1}) = T (t/T_{1}).

replacement of subtrees in a tree 403

In the sequel D will denote a non empty set, T, T_{1} will denote trees decorated with elements of D, and P will denote an antichain of prefixes of dom T.

Let us consider D, T, P, T_{1}. Let us assume that P ≠ ∅. The functor T (P/T_{1}) yields a tree decorated with elements of D and is defined by: (Def.3) T (P/T_{1}) = T (P/T_{1}).

Acknowledgments

The author wishes to thank to G. Bancerek for his assistance during the preparation of this paper.

References

[1] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[2] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77–82, 1993.

[3] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

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

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 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] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[9] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

404

Minimal Signature for Partial Algebra

Grzegorz Bancerek Institute of Mathematics Polish Academy of Sciences

Summary. The concept of characterizing of partial algebras by many sorted signature is introduced, i.e. we say that a signature S characterizes a partial algebra A if there is an S-algebra whose sorts form a partition of the carrier of algebra A and operations are formed from operations of A by the partition. The main result is that for any partial algebra there is the minimal many sorted signature which characterizes the algebra. The minimality means that there are signature endomorphisms from any signature which characterizes the algebra A onto the minimal one.

MML Identifier: PUA2MSS1.

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

1. Preliminary

Let f be a non empty binary relation. Observe that dom f is non empty and rng f is non empty.

Let f be a non-empty function. One can verify that rng f has non empty elements.

Let X, Y be non empty sets. One can verify that there exists a partial function from X to Y which is non empty.

Let X be a set with non empty elements. Note that every finite sequence of elements of X is non-empty.

Let A be a non empty set. One can verify that there exists a finite sequence of operational functions of A which is homogeneous quasi total non-empty and non empty.

Let us observe that every universal algebra structure which is non-empty is also non empty.

Let X be a non empty set with non empty elements. One can verify that every element of X is non empty.

Next we state two propositions:

(1) For all non-empty functions f, g such that ∏ f ⊆ ∏ g holds dom f = dom g and for every set x such that x ∈ dom f holds f(x) ⊆ g(x).

(2) For all non-empty functions f, g such that ∏ f = ∏ g holds f = g. Let A be a non empty set and let f be a finite sequence of operational

functions of A. Then rng f is a subset of A^{∗} ˙→A.

Let A, B be non empty sets and let S be a non empty subset of A ˙→B. We see that the element of S is a partial function from A to B.

Let A be a non-empty universal algebra structure. An operation symbol of A is an element of dom (the characteristic of A). An operation of A is an element of rng (the characteristic of A).

Let A be a non-empty universal algebra structure and let o be an operation symbol of A. The functor Den(o, A) yielding an operation of A is defined by: (Def.1) Den(o, A) = (the characteristic of A)(o).

2. Partitions

Let X be a set. Note that every partition of X has non empty elements. Let X be a non empty set. One can verify that every partition of X is non

empty.

Let X be a set and let R be an equivalence relation of X. Then Classes R is a partition of X.

Next we state a number of propositions:

(3) Let X be a set, and let P be a partition of X, and let x, a, b be sets. If x ∈ a and a ∈ P and x ∈ b and b ∈ P, then a = b.

(4) Let X, Y be sets. Suppose X is finer than Y. Let p be a finite sequence of elements of X. Then there exists a finite sequence q of elements of Y such that ∏ p ⊆ ∏ q.

(5) Let X be a set, and let P, Q be partitions of X, and let f be a function from P into Q. Suppose that for every set a such that a ∈ P holds a ⊆ f(a). Let p be a finite sequence of elements of P and let q be a finite sequence of elements of Q. Then ∏ p ⊆ ∏ q if and only if f · p = q.

(6) For every set P and for every function f such that rng f ⊆ ⋃ P there exists a function p such that dom p = dom f and rng p ⊆ P and f ∈ ∏ p.

(7) Let X be a set, and let P be a partition of X, and let f be a finite sequence of elements of X. Then there exists a finite sequence p of elements of P such that f ∈ ∏ p.

(8) Let X, Y be non empty sets, and let P be a partition of X, and let Q be a partition of Y. Then {[: p, q:] : p ranges over elements of P, q ranges over elements of Q} is a partition of [: X, Y :].

(9) For every non empty set X and for every partition P of X holds {∏ p : p ranges over elements of P^{∗}} is a partition of X^{∗}.

(10) Let X be a non empty set, and let n be a natural number, and let P be a partition of X. Then {∏ p : p ranges over elements of P^{n}} is a partition of X^{n}.

(11) Let X be a non empty set and let Y be a set. Suppose Y ⊆ X. Let P be a partition of X. Then {a ∩ Y : a ranges over elements of P, a meets Y } is a partition of Y.

(12) Let f be a non empty function and let P be a partition of dom f. Then

{f a : a ranges over elements of P } is a partition of f. Let X be a set. The functor SmallestPartition(X) yielding a partition of X

is defined as follows: (Def.2) SmallestPartition(X) = Classes(△_{X}).

One can prove the following propositions:

(13) For every non empty set X holds SmallestPartition(X) = {{x} : x ranges over elements of X}.

(14) Let X be a set and let p be a finite sequence of elements of SmallestPartition(X). Then there exists a finite sequence q of elements of X such that ∏ p = {q}.

Let X be a set. A function is said to be an indexed partition of X if: (Def.3) rng it is a partition of X and it is one-to-one.

Let X be a set. Note that every indexed partition of X is one-to-one and non-empty. Let P be an indexed partition of X. Then rng P is a partition of X.

Let X be a non empty set. Observe that every indexed partition of X is non empty.

Let X be a set and let P be a partition of X. Then △_{P} is an indexed partition of X.

Let X be a set, let P be an indexed partition of X, and let x be a set. Let us assume that x ∈ X. The P -index of x is a set and is defined by: (Def.4) The P -index of x ∈ dom P and x ∈ P (the P -index of x).

Next we state two propositions:

(15) Let X be a set and let P be a non-empty function. Suppose ⋃ P = X and for all sets x, y such that x ∈ dom P and y ∈ dom P and x ≠ y holds P (x) misses P (y). Then P is an indexed partition of X.

(16) Let X, Y be non empty sets, and let P be a partition of Y, and let f be a function from X into P. If P ⊆ rng f and f is one-to-one, then f is an indexed partition of Y.

3. Relations Generated by Operations of Partial Algebra

In this article we present several logical schemes. The scheme RelationEx concerns non empty sets A, B and a binary predicate P, and states that:

There exists a relation R between A and B such that for every element x of A and for every element y of B holds ⟨⟨x, y⟩⟩ ∈ R if and only if P[x, y]

for all values of the parameters.

The scheme IndRelationEx concerns non empty sets A, B, a natural number C, a relation D between A and B, and a binary functor F yielding a relation between A and B, and states that:

There exists a relation R between A and B and there exists a many sorted set F indexed by such that

(i) R = F (C), (ii) F (0) = D, and (iii) for every natural number i and for every relation R between A and B such that R = F (i) holds F (i + 1) = F(R, i)

for all values of the parameters.

The scheme RelationUniq concerns non empty sets A, B and a binary predicate P, and states that:

Let R_{1}, R_{2} be relations between A and B. Suppose that

(i) for every element x of A and for every element y of B holds ⟨⟨x, y⟩⟩ ∈ R_{1} iff P[x, y], and (ii) for every element x of A and for every element y of B holds ⟨⟨x, y⟩⟩ ∈ R_{2} iff P[x, y].

Then R_{1} = R_{2}

for all values of the parameters.

The scheme IndRelationUniq concerns non empty sets A, B, a natural number C, a relation D between A and B, and a binary functor F yielding a relation between A and B, and states that:

Let R_{1}, R_{2} be relations between A and B. Suppose that

(i) there exists a many sorted set F indexed by such that R_{1} = F (C) and F (0) = D and for every natural number i and for every relation R between A and B such that R = F (i) holds F (i + 1) = F(R, i), and (ii) there exists a many sorted set F indexed by such that R_{2} = F (C) and F (0) = D and for every natural number i and for every relation R between A and B such that R = F (i) holds F (i + 1) = F(R, i).

Then R_{1} = R_{2}

for all values of the parameters.

Let A be a partial non-empty universal algebra structure. The functor DomRel(A) yields a binary relation on the carrier of A and is defined by the condition (Def.5).

(Def.5) Let x, y be elements of the carrier of A. Then ⟨⟨x, y⟩⟩ ∈ DomRel(A) if and only if for every operation f of A and for all finite sequences p, q holds p^{}⟨x⟩^{}q ∈ dom f iff p^{}⟨y⟩^{}q ∈ dom f.

Let A be a partial non-empty universal algebra structure. Note that DomRel(A) is equivalence relation-like.

Let A be a non-empty partial universal algebra structure and let R be a binary relation on the carrier of A. The functor R^{A} yielding a binary relation on the carrier of A is defined by the condition (Def.6). (Def.6) Let x, y be elements of the carrier of A. Then ⟨⟨x, y⟩⟩ ∈ R^{A} if and only

if the following conditions are satisfied:

(i) ⟨⟨x, y⟩⟩ ∈ R, and (ii) for every operation f of A and for all finite sequences p, q such that

p^{}⟨x⟩^{}q ∈ dom f and p^{}⟨y⟩^{}q ∈ dom f holds ⟨⟨f(p^{}⟨x⟩^{}q), f(p^{}⟨y⟩^{}q)⟩⟩ ∈ R.

Let A be a non-empty partial universal algebra structure, let R be a binary relation on the carrier of A, and let i be a natural number. The functor R^{A,i} yielding a binary relation on the carrier of A is defined by the condition (Def.7). (Def.7) There exists a many sorted set F indexed by such that

(i) R^{A,i} = F (i), (ii) F (0) = R, and (iii) for every natural number i and for every binary relation R on the

carrier of A such that R = F (i) holds F (i + 1) = R^{A}. Next we state several propositions:

(17) Let A be a non-empty partial universal algebra structure and let R be a binary relation on the carrier of A. Then R^{A,0} = R and R^{A,1} = R^{A}.

(18) Let A be a non-empty partial universal algebra structure, and let i be a natural number, and let R be a binary relation on the carrier of A. Then R^{A,i+1} = (R^{A,i})^{A}.

(19) Let A be a non-empty partial universal algebra structure, and let i, j be natural numbers, and let R be a binary relation on the carrier of A. Then R^{A,i+j} = (R^{A,i})^{A,j}.

(20) Let A be a non-empty partial universal algebra structure and let R be an equivalence relation of the carrier of A. If R ⊆ DomRel(A), then R^{A} is equivalence relation-like.

(21) Let A be a non-empty partial universal algebra structure and let R be a binary relation on the carrier of A. Then R^{A} ⊆ R.

(22) Let A be a non-empty partial universal algebra structure and let R be an equivalence relation of the carrier of A. Suppose R ⊆ DomRel(A). Let i be a natural number. Then R^{A,i} is equivalence relation-like.

Let A be a non-empty partial universal algebra structure. The functor LimDomRel(A) yields a binary relation on the carrier of A and is defined by the condition (Def.8).

(Def.8) Let x, y be elements of the carrier of A. Then ⟨⟨x, y⟩⟩ ∈ LimDomRel(A)

if and only if for every natural number i holds ⟨⟨x, y⟩⟩ ∈ (DomRel(A))^{A,i}. The following proposition is true

(23) For every non-empty partial universal algebra structure A holds

LimDomRel(A) ⊆ DomRel(A). Let A be a non-empty partial universal algebra structure. Note that

LimDomRel(A) is equivalence relation-like.

4. Partitability

Let X be a non empty set, let f be a partial function from X^{∗} to X, and let P be a partition of X. We say that f is partitable w.r.t. P if and only if: (Def.9) For every finite sequence p of elements of P there exists an element a

of P such that f^{∘} ∏ p ⊆ a. Let X be a non empty set, let f be a partial function from X^{∗} to X, and let

P be a partition of X. We say that f is exactly partitable w.r.t. P if and only if:

(Def.10) f is partitable w.r.t. P and for every finite sequence p of elements of P

such that ∏ p meets dom f holds ∏ p ⊆ dom f. We now state the proposition

(24) Let A be a non-empty partial universal algebra structure. Then every operation of A is exactly partitable w.r.t. SmallestPartition(the carrier of A).

The scheme FiniteTransitivity concerns finite sequences A, B, a unary predicate P, and a binary predicate Q, and states that:

P[B] provided the following conditions are met:

• P[A],

• len A = len B,

• For all finite sequences p, q and for all sets z_{1}, z_{2} such that P[p^{} ⟨z_{1}⟩^{}q] and Q[z_{1}, z_{2}] holds P[p^{}⟨z_{2}⟩^{}q],

• For every natural number i such that i ∈ dom A holds Q[A(i), B(i)]. One can prove the following proposition

(25) For every non-empty partial universal algebra structure A holds every

operation of A is exactly partitable w.r.t. Classes LimDomRel(A). Let A be a partial non-empty universal algebra structure. A partition of the

carrier of A is said to be a partition of A if: (Def.11) Every operation of A is exactly partitable w.r.t. it.

Let A be a partial non-empty universal algebra structure. An indexed partition of the carrier of A is called an indexed partition of A if:

(Def.12) rng it is a partition of A.

Let A be a partial non-empty universal algebra structure and let P be an indexed partition of A. Then rng P is a partition of A.

One can prove the following propositions:

(26) For every non-empty partial universal algebra structure A holds Classes LimDomRel(A) is a partition of A.

(27) Let X be a non empty set, and let P be a partition of X, and let p be a finite sequence of elements of P, and let q_{1}, q_{2} be finite sequences, and let x, y be sets. Suppose q_{1} ^{}⟨x⟩^{}q_{2} ∈ ∏ p and there exists an element a of P such that x ∈ a and y ∈ a. Then q_{1} ^{}⟨y⟩^{}q_{2} ∈ ∏ p.

(28) For every partial non-empty universal algebra structure A holds every partition of A is finer than Classes LimDomRel(A).

5. Signature Morphisms

Let S_{1}, S_{2} be many sorted signatures and let f, g be functions. We say that f and g form morphism between S_{1} and S_{2} if and only if the conditions (Def.13) are satisfied.

(Def.13) (i) dom f = the carrier of S_{1}, (ii) dom g = the operation symbols of S_{1}, (iii) rng f ⊆ the carrier of S_{2}, (iv) rng g ⊆ the operation symbols of S_{2},

(v) f · (the result sort of S_{1}) = (the result sort of S_{2}) · (g), and (vi) for every set o and for every function p such that o ∈ the operation

symbols of S_{1} and p = (the arity of S_{1})(o) holds f · p = (the arity of S_{2})(g(o)).

Next we state two propositions:

(29) Let S be a non void non empty many sorted signature. Then id_{(the carrier of S)} and id_{(the operation symbols of S)} form morphism between S and S.

(30) Let S_{1}, S_{2}, S_{3} be many sorted signatures and let f_{1}, f_{2}, g_{1}, g_{2} be functions. Suppose f_{1} and g_{1} form morphism between S_{1} and S_{2} and f_{2} and g_{2} form morphism between S_{2} and S_{3}. Then f_{2} · f_{1} and g_{2} · g_{1} form morphism between S_{1} and S_{3}.

Let S_{1}, S_{2} be many sorted signatures. We say that S_{1} is rougher than S_{2} if and only if the condition (Def.14) is satisfied.

(Def.14) There exist functions f, g such that f and g form morphism between S_{2} and S_{1} and rng f = the carrier of S_{1} and rng g = the operation symbols of S_{1}.

Let S_{1}, S_{2} be non void non empty many sorted signatures. Let us observe that the predicate defined above is reflexive.

One can prove the following proposition (31) For all many sorted signatures S_{1}, S_{2}, S_{3} such that S_{1} is rougher than

S_{2} and S_{2} is rougher than S_{3} holds S_{1} is rougher than S_{3}.

6. Many Sorted Signature of Partial Algebra

Let A be a partial non-empty universal algebra structure and let P be a partition of A. The functor MSSign(A, P) yields a strict many sorted signature and is defined by the conditions (Def.15).

(Def.15) (i) The carrier of MSSign(A, P) = P,

(ii) the operation symbols of MSSign(A, P) = {⟨⟨o, p⟩⟩ : o ranges over operation symbols of A, p ranges over elements of P^{∗}, ∏ p meets dom Den(o, A)}, and

(iii) for every operation symbol o of A and for every element p of P^{∗} such that ∏ p meets dom Den(o, A) holds (the arity of MSSign(A, P))(⟨⟨o, p⟩⟩) = p and (Den(o, A))^{∘} ∏ p ⊆ (the result sort of MSSign(A, P))(⟨⟨o, p⟩⟩).

Let A be a partial non-empty universal algebra structure and let P be a partition of A. One can verify that MSSign(A, P) is non empty and non void.

Let A be a partial non-empty universal algebra structure, let P be a partition of A, and let o be an operation symbol of MSSign(A, P). Then o_{1} is an operation symbol of A. Then o_{2} is an element of P^{∗}.

Let A be a partial non-empty universal algebra structure, let S be a non void non empty many sorted signature, let G be an algebra over S, and let P be an indexed partition of the operation symbols of S. We say that A can be characterized by S, G, and P if and only if the conditions (Def.16) are satisfied.

(Def.16) (i) The sorts of G is an indexed partition of A, (ii) dom P = dom (the characteristic of A), and (iii) for every operation symbol o of A holds (the characteristics of G) P (o)

is an indexed partition of Den(o, A). Let A be a partial non-empty universal algebra structure and let S be a non

void non empty many sorted signature. We say that A can be characterized by S if and only if the condition (Def.17) is satisfied.

(Def.17) There exists an algebra G over S and there exists an indexed partition P of the operation symbols of S such that A can be characterized by S, G, and P.

One can prove the following propositions:

(32) Let A be a partial non-empty universal algebra structure and let P be a partition of A. Then A can be characterized by MSSign(A, P).

(33) Let A be a partial non-empty universal algebra structure, and let S be a non void non empty many sorted signature, and let G be an algebra over S, and let Q be an indexed partition of the operation symbols of S. Suppose A can be characterized by S, G, and Q. Let o be an operation symbol of A and let r be a finite sequence of elements of rng (the sorts of

minimal signature for partial algebra 413

G). Suppose ∏ r ⊆ dom Den(o, A). Then there exists an operation symbol s of S such that (the sorts of G) · Arity(s) = r and s ∈ Q(o).

(34) Let A be a partial non-empty universal algebra structure and let P be a partition of A. Suppose P = Classes LimDomRel(A). Let S be a non void non empty many sorted signature. If A can be characterized by S, then MSSign(A, P) is rougher than S.

Acknowledgments

I would like to thank Professor Andrzej Trybulec for suggesting me the problem solved here.

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Received October 1, 1995

The Subformula Tree of a Formula of the First Order Language

Oleg Okhotnikov Ural University Ekaterinburg

Summary. A continuation of [12]. The notions of list of immediate constituents of a formula and subformula tree of a formula are introduced. The some propositions related to these notions are proved.

MML Identifier: QC LANG4.

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

1. Preliminaries

The following propositions are true:

(1) For all real numbers x, y, z such that x ≤ y and y < z holds x < z.

(2) For all natural numbers m, k holds m + 1 ≤ k iff m < k.

(3) For every finite sequence r holds r = r Seg len r.

(4) For every natural number n and for every finite sequence r there exists a finite sequence q such that q = r Seg n and q ⪯ r.

(5) For all finite sequences p, q, r such that q ⪯ r holds p^{}q ⪯ p^{}r.

(6) Let D be a non empty set, and let r be a finite sequence of elements of D, and let r_{1}, r_{2} be finite sequences, and let k be a natural number. Suppose k + 1 ≤ len r and r_{1} = r Seg(k + 1) and r_{2} = r Seg k. Then there exists an element x of D such that r_{1} = r_{2} ^{}⟨x⟩.

(7) Let D be a non empty set, and let r be a finite sequence of elements of D, and let r_{1} be a finite sequence. If 1 ≤ len r and r_{1} = r Seg 1, then there exists an element x of D such that r_{1} = ⟨x⟩.

Let D be a non empty set and let T be a tree decorated with elements of D. Observe that every element of dom T is function-like and relation-like.

Let D be a non empty set and let T be a tree decorated with elements of D. One can verify that every element of dom T is finite sequence-like.

Let D be a non empty set. One can check that there exists a tree decorated with elements of D which is finite.

In the sequel T will be a decorated tree and p will be a finite sequence of elements of .

Next we state the proposition

(8) If p ∈ dom T, then T (p) = (T p)(ε). In the sequel T is a finite-branching decorated tree, t is an element of dom T,

x is a finite sequence, and n is a natural number.

The following propositions are true:

(9) succ(T, t) = T · Succ t.

(10) dom(T · Succ t) = dom Succ t.

(11) dom succ(T, t) = dom Succ t.

(12) t^{}⟨n⟩ ∈ dom T iff n + 1 ∈ dom Succ t.

(13) For all T, x, n such that x ^{}⟨n⟩ ∈ dom T holds T (x ^{}⟨n⟩) =

(succ(T, x))(n + 1). In the sequel x, x′ will be elements of dom T and y′ will be arbitrary. One can prove the following two propositions:

(14) If x′ ∈ succ x, then T (x′) ∈ rng succ(T, x).

(15) If y′ ∈ rng succ(T, x), then there exists x′ such that y′ = T (x′) and

x′ ∈ succ x. In the sequel n, k, m will denote natural numbers. The scheme ExDecTrees deals with a non empty set A, an element B of A,

and a unary functor F yielding a finite sequence of elements of A, and states that:

There exists a finite-branching tree T decorated with elements of A such that T (ε) = B and for every element t of dom T and for every element w of A such that w = T (t) holds succ(T, t) = F(w)

for all values of the parameters.

The following propositions are true:

(16) For every tree T and for every element t of T holds Seg_{⪯}(t) is a finite chain of T.

(17) For every tree T holds T -level(0) = {ε}.

(18) For every tree T holds T -level(n + 1) = ⋃{succ w : w ranges over elements of T, len w = n}.

(19) For every finite-branching tree T and for every natural number n holds T -level(n) is finite.

(20) For every finite-branching tree T holds T is finite iff there exists a natural number n such that T -level(n) = ∅.

(21) For every finite-branching tree T such that T is not finite holds there exists chain of T which is not finite.

(22) For every finite-branching tree T such that T is not finite holds there exists branch of T which is not finite.

(23) Let T be a tree, and let C be a chain of T, and let t be an element of T. If t ∈ C and C is not finite, then there exists an element t′ of T such that t′ ∈ C and t ≺ t′.

(24) Let T be a tree, and let B be a branch of T, and let t be an element of T. Suppose t ∈ B and B is not finite. Then there exists an element t′ of T such that t′ ∈ B and t′ ∈ succ t.

(25) Let f be a function from into . Suppose that for every n holds f(n + 1) qua natural number ≤ f(n) qua natural number. Then there exists m such that for every n such that m ≤ n holds f(n) = f(m).

The scheme FinDecTree concerns a non empty set A, a finite-branching tree B decorated with elements of A, and a unary functor F yielding a natural number, and states that:

B is finite provided the parameters meet the following requirement:

• For all elements t, t′ of dom B and for every element d of A such that t′ ∈ succ t and d = B(t′) holds F(d) < F(B(t)).

In the sequel D will denote a non empty set and T will denote a tree decorated with elements of D.

Next we state two propositions:

(26) For arbitrary y such that y ∈ rng T holds y is an element of D.

(27) For arbitrary x such that x ∈ dom T holds T (x) is an element of D.

2. Subformula tree

In the sequel F, G, H will denote elements of WFF. One can prove the following propositions:

(28) If F is a subformula of G, then len(^{@}F) ≤ len(^{@}G).

(29) If F is a subformula of G and len(^{@}F) = len(^{@}G), then F = G. Let p be an element of WFF. The list of immediate constituents of p yields

a finite sequence of elements of WFF and is defined by: (Def.1) (i) The list of immediate constituents of p = ε_{WFF} if p = VERUM or p

is atomic, (ii) the list of immediate constituents of p = ⟨Arg(p)⟩ if p is negative, (iii) the list of immediate constituents of p = ⟨LeftArg(p), RightArg(p)⟩ if

p is conjunctive, (iv) the list of immediate constituents of p = ⟨Scope(p)⟩, otherwise. Next we state two propositions:

(30) Suppose k ∈ dom (the list of immediate constituents of F) and G = (the list of immediate constituents of F)(k). Then G is an immediate constituent of F.

(31) rng (the list of immediate constituents of F) = {G : G ranges over

elements of WFF, G is an immediate constituent of F }. Let p be an element of WFF. The tree of subformulae of p yields a finite tree

decorated with elements of WFF and is defined by the conditions (Def.2). (Def.2) (i) (The tree of subformulae of p)(ε) = p, and

(ii) for every element x of dom (the tree of subformulae of p) holds succ(the tree of subformulae of p, x) = the list of immediate constituents of (the tree of subformulae of p)(x).

In the sequel t, t′ will be elements of dom (the tree of subformulae of F). One can prove the following propositions:

(32) (The tree of subformulae of F)(ε) = F.

(33) succ(the tree of subformulae of F, t) = the list of immediate constituents of (the tree of subformulae of F)(t).

(34) F ∈ rng (the tree of subformulae of F).

(35) Suppose t^{}⟨n⟩ ∈ dom (the tree of subformulae of F). Then there exists

G such that

(i) G = (the tree of subformulae of F)(t^{}⟨n⟩), and

(ii) G is an immediate constituent of (the tree of subformulae of F)(t).

(36) The following statements are equivalent (i) H is an immediate constituent of (the tree of subformulae of F)(t), (ii) there exists n such that t^{}⟨n⟩ ∈ dom (the tree of subformulae of F)

and H = (the tree of subformulae of F)(t^{}⟨n⟩).

(37) Suppose G ∈ rng (the tree of subformulae of F) and H is an immediate constituent of G. Then H ∈ rng (the tree of subformulae of F).

(38) If G ∈ rng (the tree of subformulae of F) and H is a subformula of G, then H ∈ rng (the tree of subformulae of F).

(39) G ∈ rng (the tree of subformulae of F) iff G is a subformula of F.

(40) rng (the tree of subformulae of F) = Subformulae F.

(41) Suppose t′ ∈ succ t. Then (the tree of subformulae of F)(t′) is an immediate constituent of (the tree of subformulae of F)(t).

(42) If t ⪯ t′, then (the tree of subformulae of F)(t′) is a subformula of (the tree of subformulae of F)(t).

(43) If t ≺ t′, then len(^{@}(the tree of subformulae of F)(t′)) < len(^{@}(the tree of subformulae of F)(t)).

(44) If t ≺ t′, then (the tree of subformulae of F)(t′) ≠ (the tree of subformulae of F)(t).

(45) If t ≺ t′, then (the tree of subformulae of F)(t′) is a proper subformula of (the tree of subformulae of F)(t).

(46) (The tree of subformulae of F)(t) = F iff t = ε.

(47) Suppose t ≠ t′ and (the tree of subformulae of F)(t) = (the tree of

subformulae of F)(t′). Then t and t′ are not comparable. Let F, G be elements of WFF. The F -entry points in subformula tree of

G yields an antichain of prefixes of dom (the tree of subformulae of F) and is defined by the condition (Def.3). (Def.3) Let t be an element of dom (the tree of subformulae of F). Then t ∈

the F -entry points in subformula tree of G if and only if (the tree of subformulae of F)(t) = G.

We now state several propositions:

(48) t ∈ the F -entry points in subformula tree of G iff (the tree of subformulae of F)(t) = G.

(49) The F -entry points in subformula tree of G = {t : t ranges over elements of dom (the tree of subformulae of F), (the tree of subformulae of F)(t) = G}.

(50) G is a subformula of F iff the F -entry points in subformula tree of G ≠ ∅.

(51) Suppose t′ = t^{}⟨m⟩ and (the tree of subformulae of F)(t) is negative. Then (the tree of subformulae of F)(t′) = Arg((the tree of subformulae of F)(t)) and m = 0.

(52) Suppose t′ = t^{}⟨m⟩ and (the tree of subformulae of F)(t) is conjunctive.

Then

(i) (the tree of subformulae of F)(t′) = LeftArg((the tree of subformulae of F)(t)) and m = 0, or

(ii) (the tree of subformulae of F)(t′) = RightArg((the tree of subformulae

of F)(t)) and m = 1.

(53) Suppose t′ = t^{}⟨m⟩ and (the tree of subformulae of F)(t) is universal. Then (the tree of subformulae of F)(t′) = Scope((the tree of subformulae of F)(t)) and m = 0.

(54) Suppose (the tree of subformulae of F)(t) is negative. Then (i) t^{}⟨0⟩ ∈ dom (the tree of subformulae of F), and (ii) (the tree of subformulae of F)(t^{}⟨0⟩) = Arg((the tree of subformulae

of F)(t)).

(55) Suppose (the tree of subformulae of F)(t) is conjunctive. Then (i) t^{}⟨0⟩ ∈ dom (the tree of subformulae of F), (ii) (the tree of subformulae of F)(t^{}⟨0⟩) = LeftArg((the tree of subfor-

mulae of F)(t)), (iii) t^{}⟨1⟩ ∈ dom (the tree of subformulae of F), and (iv) (the tree of subformulae of F)(t^{}⟨1⟩) = RightArg((the tree of subfor-

mulae of F)(t)).

(56) Suppose (the tree of subformulae of F)(t) is universal. Then (i) t^{}⟨0⟩ ∈ dom (the tree of subformulae of F), and (ii) (the tree of subformulae of F)(t^{}⟨0⟩) = Scope((the tree of subformulae

of F)(t)).

In the sequel t will be an element of dom (the tree of subformulae of F) and s will be an element of dom (the tree of subformulae of G).

Next we state the proposition

(57) Suppose t ∈ the F -entry points in subformula tree of G and s ∈ the G-entry points in subformula tree of H. Then t^{}s ∈ the F -entry points in subformula tree of H.

In the sequel t will be an element of dom (the tree of subformulae of F) and s will be a finite sequence.

Next we state several propositions:

(58) Suppose t ∈ the F -entry points in subformula tree of G and t^{}s ∈ the F -entry points in subformula tree of H. Then s ∈ the G-entry points in subformula tree of H.

(59) Given F, G, H. Then {t^{}s : t ranges over elements of dom (the tree of subformulae of F), s ranges over elements of dom (the tree of subformulae of G), t ∈ the F -entry points in subformula tree of G ∧ s ∈ the G-entry points in subformula tree of H} ⊆ the F -entry points in subformula tree of H.

(60) (The tree of subformulae of F) t = the tree of subformulae of (the tree of subformulae of F)(t).

(61) t ∈ the F -entry points in subformula tree of G if and only if (the tree of subformulae of F) t = the tree of subformulae of G.

(62) The F -entry points in subformula tree of G = {t : t ranges over elements of dom (the tree of subformulae of F), (the tree of subformulae of F) t = the tree of subformulae of G}.

In the sequel C is a chain of dom (the tree of subformulae of F). Next we state the proposition

(63) Given F, G, H, C. Suppose that

(i) G ∈ {(the tree of subformulae of F)(t) : t ranges over elements of dom (the tree of subformulae of F), t ∈ C}, and

(ii) H ∈ {(the tree of subformulae of F)(t) : t ranges over elements of dom (the tree of subformulae of F), t ∈ C}. Then G is a subformula of H or H is a subformula of G.

Let F be an element of WFF. An element of WFF is said to be a subformula of F if: (Def.4) It is a subformula of F.

Let F be an element of WFF and let G be a subformula of F. An element of dom (the tree of subformulae of F) is said to be an entry point in subformula tree of G if: (Def.5) (The tree of subformulae of F)(it) = G.

In the sequel G will denote a subformula of F. Next we state the proposition (64) t is an entry point in subformula tree of G iff (the tree of subformulae

of F)(t) = G. In the sequel t, t′ are entry points in subformula tree of G. The following proposition is true

(65) If t ≠ t′, then t and t′ are not comparable.

Let F be an element of WFF and let G be a subformula of F. The entry points in subformula tree of G yields a non empty antichain of prefixes of dom (the tree of subformulae of F) and is defined as follows: (Def.6) The entry points in subformula tree of G = the F -entry points in sub-

formula tree of G. We now state three propositions:

(66) The entry points in subformula tree of G = the F -entry points in subformula tree of G.

(67) t ∈ the entry points in subformula tree of G.

(68) The entry points in subformula tree of G = {t : t ranges over entry

points in subformula tree of G, t = t}. In the sequel G_{1}, G_{2} will denote subformulae of F, t_{1} will denote an entry

point in subformula tree of G_{1}, and s will denote an element of dom (the tree of subformulae of G_{1}).

We now state the proposition

(69) If s ∈ the G_{1}-entry points in subformula tree of G_{2}, then t_{1} ^{}s is an

entry point in subformula tree of G_{2}. In the sequel s will be a finite sequence. Next we state three propositions:

(70) If t_{1}^{}s is an entry point in subformula tree of G_{2}, then s ∈ the G_{1}-entry points in subformula tree of G_{2}.

(71) Given F, G_{1}, G_{2}. Then {t^{}s : t ranges over entry points in subformula tree of G_{1}, s ranges over elements of dom (the tree of subformulae of G_{1}), s ∈ the G_{1}-entry points in subformula tree of G_{2}} = {t^{}s : t ranges over elements of dom (the tree of subformulae of F), s ranges over elements of dom (the tree of subformulae of G_{1}), t ∈ the F -entry points in subformula tree of G_{1} ∧ s ∈ the G_{1}-entry points in subformula tree of G_{2}}.

(72) Given F, G_{1}, G_{2}. Then {t^{}s : t ranges over entry points in subformula tree of G_{1}, s ranges over elements of dom (the tree of subformulae of G_{1}), s ∈ the G_{1}-entry points in subformula tree of G_{2}} ⊆ the entry points in subformula tree of G_{2}.

In the sequel G_{1}, G_{2} will denote subformulae of F, t_{1} will denote an entry point in subformula tree of G_{1}, and t_{2} will denote an entry point in subformula tree of G_{2}.

The following two propositions are true:

(73) If there exist t_{1}, t_{2} such that t_{1} ⪯ t_{2}, then G_{2} is a subformula of G_{1}.

422 oleg okhotnikov

(74) If G_{2} is a subformula of G_{1}, then for every t_{1} there exists t_{2} such that t_{1} ⪯ t_{2}.

Acknowledgments

The author wishes to thank to G. Bancerek for his assistance during the preparation of this paper.

References

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

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

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

[4] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[5] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77–82, 1993.

[6] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

[7] Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185–190, 1996.

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

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

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

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

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

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

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

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

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

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

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

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

The Steinitz Theorem and the Dimension of a Vector Space

Mariusz Żynel Warsaw University Białystok

Summary. The main purpose of the paper is to define the dimension of an abstract vector space. The dimension of a finite-dimensional vector space is, by the most common definition, the number of vectors in a basis. Obviously, each basis contains the same number of vectors. We prove the Steinitz Theorem together with Exchange Lemma in the second section. The Steinitz Theorem says that each linearly-independent subset of a vector space has cardinality less than any subset that generates the space, moreover it can be extended to a basis. Further we review some of the standard facts involving the dimension of a vector space. Additionally, in the last section, we introduce two notions: the family of subspaces of a fixed dimension and the pencil of subspaces. Both of them can be applied in the algebraic representation of several geometries.

MML Identifier: VECTSP 9.

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

1. Preliminaries

For simplicity we follow the rules: G_{1} is a field, V is a vector space over G_{1}, W is a subspace of V, x is arbitrary, and n is a natural number.

Let S be a non empty 1-sorted structure. Observe that there exists a subset of S which is non empty.

One can prove the following proposition

(1) For every finite set X such that n ≤ \overline{\overline{X}} there exists a finite subset A of X such that \overline{\overline{A}} = n.

In the sequel f, g will be functions. We now state a number of propositions:

(2) For every f such that f is one-to-one holds if x ∈ rng f, then \overline{\overline{f ^{−1} \{x\}}} = 1.

(3) For every f such that x ∉ rng f holds \overline{\overline{f ^{−1} \{x\}}} = 0.

(4) For all f, g such that rng f = rng g and f is one-to-one and g is one-to-one holds f and g are fiberwise equipotent.

(5) Let L be a linear combination of V, and let F, G be finite sequences of elements of the carrier of V, and let P be a permutation of dom F. If G = F · P, then ∑(L F) = ∑(L G).

(6) Let L be a linear combination of V and let F be a finite sequence of elements of the carrier of V. If support L misses rng F, then ∑(L F) = 0_{V} .

(7) Let F be a finite sequence of elements of the carrier of V. Suppose F is one-to-one. Let L be a linear combination of V. If support L ⊆ rng F, then ∑(L F) = ∑ L.

(8) Let L be a linear combination of V and let F be a finite sequence of elements of the carrier of V. Then there exists a linear combination K of V such that support K = rng F ∩ support L and L F = K F.

(9) Let L be a linear combination of V, and let A be a subset of V, and let F be a finite sequence of elements of the carrier of V. Suppose rng F ⊆ the carrier of Lin(A). Then there exists a linear combination K of A such that ∑(L F) = ∑ K.

(10) Let L be a linear combination of V and let A be a subset of V. Suppose support L ⊆ the carrier of Lin(A). Then there exists a linear combination K of A such that ∑ L = ∑ K.

(11) Let L be a linear combination of V. Suppose support L ⊆ the carrier of W. Let K be a linear combination of W. If K = L (the carrier of W), then support L = support K and ∑ L = ∑ K.

(12) For every linear combination K of W there exists a linear combination L of V such that support K = support L and ∑ K = ∑ L.

(13) Let L be a linear combination of V. Suppose support L ⊆ the carrier of W. Then there exists a linear combination K of W such that support K = support L and ∑ K = ∑ L.

(14) For every basis I of V and for every vector v of V holds v ∈ Lin(I).

(15) Let A be a subset of W. Suppose A is linearly independent. Then there exists a subset B of V such that B is linearly independent and B = A.

(16) Let A be a subset of V. Suppose A is linearly independent and A ⊆ the carrier of W. Then there exists a subset B of W such that B is linearly independent and B = A.

(17) For every basis A of W there exists a basis B of V such that A ⊆ B.

(18) Let A be a subset of V. Suppose A is linearly independent. Let v be a vector of V. If v ∈ A, then for every subset B of V such that B = A \ {v}

holds v ∉ Lin(B).

(19) Let I be a basis of V and let A be a non empty subset of V. Suppose A misses I. Let B be a subset of V. If B = I ∪ A, then B is linearlydependent.

(20) For every subset A of V such that A ⊆ the carrier of W holds Lin(A) is a subspace of W.

(21) For every subset A of V and for every subset B of W such that A = B holds Lin(A) = Lin(B).

2. The Steinitz Theorem

The following two propositions are true:

(22) Let A, B be finite subsets of V and let v be a vector of V. Suppose v ∈ Lin(A ∪ B) and v ∉ Lin(B). Then there exists a vector w of V such that w ∈ A and w ∈ Lin(((A ∪ B) \ {w}) ∪ {v}).

(23) Let A, B be finite subsets of V. Suppose the vector space structure of V = Lin(A) and B is linearly independent. Then \overline{\overline{B}} ≤ \overline{\overline{A}} and there exists a finite subset C of V such that C ⊆ A and \overline{\overline{C}} = \overline{\overline{A}} − \overline{\overline{B}} and the vector space structure of V = Lin(B ∪ C).

3. Finite-Dimensional Vector Spaces

Let G_{1} be a field and let V be a vector space over G_{1}. Let us observe that V is finite dimensional if and only if: (Def.1) There exists finite subset of V which is a basis of V.

Next we state several propositions:

(24) If V is finite dimensional, then every basis of V is finite.

(25) If V is finite dimensional, then for every subset A of V such that A is linearly independent holds A is finite.

(26) If V is finite dimensional, then for all bases A, B of V holds \overline{\overline{A}} = \overline{\overline{B}} .

(27) 0_{V} is finite dimensional.

(28) If V is finite dimensional, then W is finite dimensional.

Let G_{1} be a field and let V be a vector space over G_{1}. Observe that there exists a subspace of V which is strict and finite dimensional.

Let G_{1} be a field and let V be a finite dimensional vector space over G_{1}. Note that every subspace of V is finite dimensional.

Let G_{1} be a field and let V be a finite dimensional vector space over G_{1}. One can check that there exists a subspace of V which is strict.

4. The Dimension of a Vector Space

Let G_{1} be a field and let V be a vector space over G_{1}. Let us assume that V is finite dimensional. The functor dim(V) yields a natural number and is defined by: (Def.2) For every basis I of V holds dim(V) = \overline{\overline{I}} .

We adopt the following rules: V denotes a finite dimensional vector space over G_{1}, W, W_{1}, W_{2} denote subspaces of V, and u, v denote vectors of V.

The following propositions are true:

(29) dim(W) ≤ dim(V).

(30) For every subset A of V such that A is linearly independent holds \overline{\overline{A}} = dim(Lin(A)).

(31) dim(V) = dim(Ω_{V}).

(32) dim(V) = dim(W) iff Ω_{V} = Ω_{W} .

(33) dim(V) = 0 iff Ω_{V} = 0_{V} .

(34) dim(V) = 1 iff there exists v such that v ≠ 0_{V} and Ω_{V} = Lin({v}).

(35) dim(V) = 2 iff there exist u, v such that u ≠ v and {u, v} is linearly independent and Ω_{V} = Lin({u, v}).

(36) dim(W_{1} + W_{2}) + dim(W_{1} ∩ W_{2}) = dim(W_{1}) + dim(W_{2}).

(37) dim(W_{1} ∩ W_{2}) ≥ (dim(W_{1}) + dim(W_{2})) − dim(V).

(38) If V is the direct sum of W_{1} and W_{2}, then dim(V) = dim(W_{1}) + dim(W_{2}).

5. The Fixed-Dimensional Subspace Family and the Pencil of Subspaces

One can prove the following proposition

(39) n ≤ dim(V) iff there exists a strict subspace W of V such that

dim(W) = n. Let G_{1} be a field, let V be a finite dimensional vector space over G_{1}, and

let n be a natural number. The functor Sub_{n}(V) yields a set and is defined as follows: (Def.3) x ∈ Sub_{n}(V) iff there exists a strict subspace W of V such that W = x

and dim(W) = n. We now state three propositions:

(40) If n ≤ dim(V), then Sub_{n}(V) is non empty.

(41) If dim(V) < n, then Sub_{n}(V) = ∅.

(42) Sub_{n}(W) ⊆ Sub_{n}(V).

the steinitz theorem and the dimension of … 427

Let G_{1} be a field, let V be a finite dimensional vector space over G_{1}, let W_{2} be a subspace of V, and let W_{1} be a strict subspace of W_{2}. Let us assume that dim(W_{2}) = dim(W_{1}) + 2. The functor p(W_{1}, W_{2}) yields a non empty set and is defined by: (Def.4) x ∈ p(W_{1}, W_{2}) iff there exists a strict subspace W of W_{2} such that

W = x and dim(W) = dim(W_{1}) + 1 and W_{1} is a subspace of W. We now state two propositions:

(43) Let W_{1} be a strict subspace of W_{2}. Suppose dim(W_{2}) = dim(W_{1}) + 2. Then x ∈ p(W_{1}, W_{2}) if and only if there exists a strict subspace W of V such that W = x and dim(W) = dim(W_{1}) + 1 and W_{1} is a subspace of W and W is a subspace of W_{2}.

(44) For every strict subspace W_{1} of W_{2} such that dim(W_{2}) = dim(W_{1}) + 2 holds p(W_{1}, W_{2}) ⊆ Sub_{dim(W_{1})+1}(V).

References

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[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.

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428 mariusz żynel

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

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

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

On the Go-Board of a Standard Special Circular Sequence

Andrzej Trybulec Warsaw University Białystok

MML Identifier: GOBOARD7.

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

1. Preliminaries

For simplicity we adopt the following convention: f will denote a standard special circular sequence, i, j, k, n, i_{1}, i_{2}, j_{1}, j_{2} will denote natural numbers, r, s, r_{1}, r_{2} will denote real numbers, p, q, p_{1} will denote points of E_{T}^{2}, and G will denote a Go-board.

The following propositions are true:

(1) If |r_{1} − r_{2}| > s, then r_{1} + s < r_{2} or r_{2} + s < r_{1}.

(2) |r − s| = 0 iff r = s.

(3) For all points p, p_{1}, q of E_{T}^{n} such that p + p_{1} = q + p_{1} holds p = q.

(4) For all points p, p_{1}, q of E_{T}^{n} such that p_{1} + p = p_{1} + q holds p = q.

(5) If p_{1} ∈ L(p, q) and p_{1} = q_{1}, then (p_{1})_{1} = q_{1}.

(6) If p_{1} ∈ L(p, q) and p_{2} = q_{2}, then (p_{1})_{2} = q_{2}.

(7) \frac{1}{2} · (p + q) ∈ L(p, q).

(8) If p_{1} = q_{1} and q_{1} = (p_{1})_{1} and p_{2} ≤ q_{2} and q_{2} ≤ (p_{1})_{2}, then q ∈ L(p, p_{1}).

(9) If p_{1} ≤ q_{1} and q_{1} ≤ (p_{1})_{1} and p_{2} = q_{2} and q_{2} = (p_{1})_{2}, then q ∈ L(p, p_{1}).

(10) Let D be a non empty set, and given i, j, and let M be a matrix over D. If 1 ≤ i and i ≤ len M and 1 ≤ j and j ≤ width M, then ⟨⟨i, j⟩⟩ ∈ the indices of M.

(11) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then \frac{1}{2} · (G_{i,j} + G_{i+1,j+1}) = \frac{1}{2} · (G_{i,j+1} + G_{i+1,j}).

(12) Suppose L(f, k) is horizontal. Then there exists j such that 1 ≤ j and j ≤ width the Go-board of f and for every p such that p ∈ L(f, k) holds p_{2} = ((the Go-board of f)_{1,j})_{2}.

(13) Suppose L(f, k) is vertical. Then there exists i such that 1 ≤ i and i ≤ len the Go-board of f and for every p such that p ∈ L(f, k) holds p_{1} = ((the Go-board of f)_{i,1})_{1}.

(14) If i ≤ len the Go-board of f and j ≤ width the Go-board of f, then Int cell(the Go-board of f, i, j) misses L̃(f).

2. Segments on a Go-Board

Next we state a number of propositions:

(15) If 1 ≤ i and i ≤ len G and 1 ≤ j and j + 2 ≤ width G, then L(G_{i,j}, G_{i,j+1}) ∩ L(G_{i,j+1}, G_{i,j+2}) = {G_{i,j+1}}.

(16) If 1 ≤ i and i + 2 ≤ len G and 1 ≤ j and j ≤ width G, then L(G_{i,j}, G_{i+1,j}) ∩ L(G_{i+1,j}, G_{i+2,j}) = {G_{i+1,j}}.

(17) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then L(G_{i,j}, G_{i,j+1}) ∩ L(G_{i,j+1}, G_{i+1,j+1}) = {G_{i,j+1}}.

(18) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then L(G_{i,j+1}, G_{i+1,j+1}) ∩ L(G_{i+1,j}, G_{i+1,j+1}) = {G_{i+1,j+1}}.

(19) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then L(G_{i,j}, G_{i+1,j}) ∩ L(G_{i,j}, G_{i,j+1}) = {G_{i,j}}.

(20) If 1 ≤ i and i + 1 ≤ len G and 1 ≤ j and j + 1 ≤ width G, then L(G_{i,j}, G_{i+1,j}) ∩ L(G_{i+1,j}, G_{i+1,j+1}) = {G_{i+1,j}}.

(21) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose 1 ≤ i_{1} and i_{1} ≤ len G and 1 ≤ j_{1} and j_{1} +1 ≤ width G and 1 ≤ i_{2} and i_{2} ≤ len G and 1 ≤ j_{2} and j_{2} + 1 ≤ width G and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) meets L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}). Then i_{1} = i_{2} and |j_{1} − j_{2}| ≤ 1.

(22) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose 1 ≤ i_{1} and i_{1} + 1 ≤ len G and 1 ≤ j_{1} and j_{1} ≤ width G and 1 ≤ i_{2} and i_{2} + 1 ≤ len G and 1 ≤ j_{2} and j_{2} ≤ width G and L(G_{i_{1},j_{1}}, G_{i_{1}+1,j_{1}}) meets L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}). Then j_{1} = j_{2} and |i_{1} − i_{2}| ≤ 1.

(23) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose 1 ≤ i_{1} and i_{1} ≤ len G and 1 ≤ j_{1} and j_{1} + 1 ≤ width G and 1 ≤ i_{2} and i_{2} + 1 ≤ len G and 1 ≤ j_{2} and j_{2} ≤ width G and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) meets L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}). Then i_{1} = i_{2} or i_{1} = i_{2} + 1 but j_{1} = j_{2} or j_{1} + 1 = j_{2}.

(24) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose 1 ≤ i_{1} and i_{1} ≤ len G and 1 ≤ j_{1} and j_{1} +1 ≤ width G and 1 ≤ i_{2} and i_{2} ≤ len G and 1 ≤ j_{2} and j_{2} + 1 ≤ width G and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) meets L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}). Then (i) j_{1} = j_{2} and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) = L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}), or (ii) j_{1} = j_{2} + 1 and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) ∩ L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}) = {G_{i_{1},j_{1}}}, or (iii) j_{1} + 1 = j_{2} and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) ∩ L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}) = {G_{i_{2},j_{2}}}.

(25) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose 1 ≤ i_{1} and i_{1} + 1 ≤ len G and 1 ≤ j_{1} and j_{1} ≤ width G and 1 ≤ i_{2} and i_{2} + 1 ≤ len G and 1 ≤ j_{2} and j_{2} ≤ width G and L(G_{i_{1},j_{1}}, G_{i_{1}+1,j_{1}}) meets L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}). Then

(i) i_{1} = i_{2} and L(G_{i_{1},j_{1}}, G_{i_{1}+1,j_{1}}) = L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}), or (ii) i_{1} = i_{2} + 1 and L(G_{i_{1},j_{1}}, G_{i_{1}+1,j_{1}}) ∩ L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}) = {G_{i_{1},j_{1}}}, or (iii) i_{1} + 1 = i_{2} and L(G_{i_{1},j_{1}}, G_{i_{1}+1,j_{1}}) ∩ L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}) = {G_{i_{2},j_{2}}}.

(26) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose 1 ≤ i_{1} and i_{1} ≤ len G and 1 ≤ j_{1} and j_{1} + 1 ≤ width G and 1 ≤ i_{2} and i_{2} + 1 ≤ len G and 1 ≤ j_{2} and j_{2} ≤ width G and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) meets L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}). Then j_{1} = j_{2} and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1})∩L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}) = {G_{i_{1},j_{1}}} or j_{1}+1 = j_{2} and L(G_{i_{1},j_{1}}, G_{i_{1},j_{1}+1}) ∩ L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}) = {G_{i_{1},j_{1}+1}}.

(27) Suppose 1 ≤ i_{1} and i_{1} ≤ len G and 1 ≤ j_{1} and j_{1} + 1 ≤ width G and 1 ≤ i_{2} and i_{2} ≤ len G and 1 ≤ j_{2} and j_{2} + 1 ≤ width G and \frac{1}{2} · (G_{i_{1},j_{1}} + G_{i_{1},j_{1}+1}) ∈ L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}). Then i_{1} = i_{2} and j_{1} = j_{2}.

(28) Suppose 1 ≤ i_{1} and i_{1} + 1 ≤ len G and 1 ≤ j_{1} and j_{1} ≤ width G and 1 ≤ i_{2} and i_{2} + 1 ≤ len G and 1 ≤ j_{2} and j_{2} ≤ width G and \frac{1}{2} · (G_{i_{1},j_{1}} + G_{i_{1}+1,j_{1}}) ∈ L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}). Then i_{1} = i_{2} and j_{1} = j_{2}.

(29) Suppose 1 ≤ i_{1} and i_{1} + 1 ≤ len G and 1 ≤ j_{1} and j_{1} ≤ width G. Then it is not true that there exist i_{2}, j_{2} such that 1 ≤ i_{2} and i_{2} ≤ len G and 1 ≤ j_{2} and j_{2} +1 ≤ width G and \frac{1}{2} ·(G_{i_{1},j_{1}} +G_{i_{1}+1,j_{1}}) ∈ L(G_{i_{2},j_{2}}, G_{i_{2},j_{2}+1}).

(30) Suppose 1 ≤ i_{1} and i_{1} ≤ len G and 1 ≤ j_{1} and j_{1} +1 ≤ width G. Then it is not true that there exist i_{2}, j_{2} such that 1 ≤ i_{2} and i_{2} + 1 ≤ len G and 1 ≤ j_{2} and j_{2} ≤ width G and \frac{1}{2} · (G_{i_{1},j_{1}} + G_{i_{1},j_{1}+1}) ∈ L(G_{i_{2},j_{2}}, G_{i_{2}+1,j_{2}}).

3. Standard Special Circular Sequences

In the sequel f will be a non constant standard special circular sequence. The following propositions are true:

(31) For every standard non empty finite sequence f of elements of E_{T}^{2} such that i ∈ dom f and i + 1 ∈ dom f holds π_{i}f ≠ π_{i+1}f.

(32) There exists i such that i ∈ dom f and (π_{i}f)_{1} ≠ (π_{1}f)_{1}.

(33) There exists i such that i ∈ dom f and (π_{i}f)_{2} ≠ (π_{1}f)_{2}.

(34) len the Go-board of f > 1.

(35) width the Go-board of f > 1.

(36) len f > 4.

(37) Let f be a circular s.c.c. finite sequence of elements of E_{T}^{2}. Suppose len f > 4. Let i, j be natural numbers. If 1 ≤ i and i < j and j < len f, then π_{i}f ≠ π_{j}f.

(38) For all natural numbers i, j such that 1 ≤ i and i < j and j < len f holds π_{i}f ≠ π_{j}f.

(39) For all natural numbers i, j such that 1 < i and i < j and j ≤ len f holds π_{i}f ≠ π_{j}f.

(40) For every natural number i such that 1 < i and i ≤ len f and π_{i}f = π_{1}f holds i = len f.

(41) Suppose that (i) 1 ≤ i, (ii) i ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f, and

(v) \frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i,j+1}) ∈ L̃(f). Then there exists k such that 1 ≤ k and k+1 ≤ len f and L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) = L(f, k).

(42) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f (iii) 1 ≤ j, (iv) j ≤ width the Go-board of f and

(v) \frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j}) ∈ L̃(f). Then there exists k such that 1 ≤ k and k+1 ≤ len f and L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) = L(f, k).

(43) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+1}f = (the Go-board of f)_{i+1,j+1} and π_{k+2}f = (the Go-board of f)_{i+1,j}.

(44) Suppose that (i) 1 ≤ i, (ii) i ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 < width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i,j+2}) = L(f, k), and (viii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j+2} and π_{k+1}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i,j}.

(45) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i+1,j+1} and π_{k+1}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i,j}.

(46) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+1}f = (the Go-board of f)_{i+1,j+1} and π_{k+2}f = (the Go-board of f)_{i,j+1}.

(47) Suppose that (i) 1 ≤ i, (ii) i + 1 < len the Go-board of f, (iii) 1 ≤ j, (iv) j ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+2,j}) = L(f, k), and (viii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i+2,j} and π_{k+1}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i,j}.

(48) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i+1,j+1} and π_{k+1}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i,j}.

(49) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+1}f = (the Go-board of f)_{i+1,j+1} and π_{k+2}f = (the Go-board of f)_{i,j+1}.

(50) Suppose that (i) 1 ≤ i, (ii) i ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 < width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i,j+2}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j} and π_{k+1}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i,j+2}.

(51) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j} and π_{k+1}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i+1,j+1}.

(52) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) = L(f, k), and (viii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+1}f = (the Go-board of f)_{i+1,j+1} and π_{k+2}f = (the Go-board of f)_{i+1,j}.

(53) Suppose that (i) 1 ≤ i, (ii) i + 1 < len the Go-board of f, (iii) 1 ≤ j, (iv) j ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) = L(f, k), and (viii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+2,j}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j} and π_{k+1}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i+2,j}.

(54) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) 1 ≤ k, (vi) k + 1 < len f, (vii) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) = L(f, k), and (viii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) = L(f, k + 1).

Then π_{k}f = (the Go-board of f)_{i,j} and π_{k+1}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i+1,j+1}.

(55) Suppose that (i) 1 ≤ i, (ii) i ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 < width the Go-board of f,

(v) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) ⊆ L̃(f), and (vi) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i,j+2}) ⊆ L̃(f).

Then (vii) π_{1}f = (the Go-board of f)_{i,j+1} but π_{2}f = (the Go-board of f)_{i,j} and

π_{len f−′1}f = (the Go-board of f)_{i,j+2} or π_{2}f = (the Go-board of f)_{i,j+2} and π_{len f−′1}f = (the Go-board of f)_{i,j}, or

(viii) there exists k such that 1 ≤ k and k + 1 < len f and π_{k+1}f = (the Go-board of f)_{i,j+1} and π_{k}f = (the Go-board of f)_{i,j} and π_{k+2}f = (the Go-board of f)_{i,j+2} or π_{k}f = (the Go-board of f)_{i,j+2} and π_{k+2}f = (the Go-board of f)_{i,j}.

(56) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) ⊆ L̃(f), and (vi) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) ⊆ L̃(f).

Then (vii) π_{1}f = (the Go-board of f)_{i,j+1} but π_{2}f = (the Go-board of f)_{i,j}

and π_{len f−′1}f = (the Go-board of f)_{i+1,j+1} or π_{2}f = (the Go-board of f)_{i+1,j+1} and π_{len f−′1}f = (the Go-board of f)_{i,j}, or

(viii) there exists k such that 1 ≤ k and k + 1 < len f and π_{k+1}f = (the Goboard of f)_{i,j+1} and π_{k}f = (the Go-board of f)_{i,j} and π_{k+2}f = (the Goboard of f)_{i+1,j+1} or π_{k}f = (the Go-board of f)_{i+1,j+1} and π_{k+2}f = (the Go-board of f)_{i,j}.

(57) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) ⊆ L̃(f), and (vi) L((the Go-board of f)_{i+1,j+1}, (the Go-board of f)_{i+1,j}) ⊆ L̃(f).

Then (vii) π_{1}f = (the Go-board of f)_{i+1,j+1} but π_{2}f = (the Go-board of f)_{i,j+1}

and π_{len f−′1}f = (the Go-board of f)_{i+1,j} or π_{2}f = (the Go-board of f)_{i+1,j} and π_{len f−′1}f = (the Go-board of f)_{i,j+1}, or

(viii) there exists k such that 1 ≤ k and k + 1 < len f and π_{k+1}f = (the Goboard of f)_{i+1,j+1} and π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i+1,j} or π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i,j+1}.

(58) Suppose that (i) 1 ≤ i, (ii) i + 1 < len the Go-board of f, (iii) 1 ≤ j, (iv) j ≤ width the Go-board of f,

(v) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) ⊆ L̃(f), and (vi) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+2,j}) ⊆ L̃(f).

Then (vii) π_{1}f = (the Go-board of f)_{i+1,j} but π_{2}f = (the Go-board of f)_{i,j} and

π_{len f−′1}f = (the Go-board of f)_{i+2,j} or π_{2}f = (the Go-board of f)_{i+2,j} and π_{len f−′1}f = (the Go-board of f)_{i,j}, or

(viii) there exists k such that 1 ≤ k and k + 1 < len f and π_{k+1}f = (the Go-board of f)_{i+1,j} and π_{k}f = (the Go-board of f)_{i,j} and π_{k+2}f = (the Go-board of f)_{i+2,j} or π_{k}f = (the Go-board of f)_{i+2,j} and π_{k+2}f = (the Go-board of f)_{i,j}.

(59) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) ⊆ L̃(f), and (vi) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) ⊆ L̃(f).

Then (vii) π_{1}f = (the Go-board of f)_{i+1,j} but π_{2}f = (the Go-board of f)_{i,j}

and π_{len f−′1}f = (the Go-board of f)_{i+1,j+1} or π_{2}f = (the Go-board of f)_{i+1,j+1} and π_{len f−′1}f = (the Go-board of f)_{i,j}, or

(viii) there exists k such that 1 ≤ k and k + 1 < len f and π_{k+1}f = (the Goboard of f)_{i+1,j} and π_{k}f = (the Go-board of f)_{i,j} and π_{k+2}f = (the Goboard of f)_{i+1,j+1} or π_{k}f = (the Go-board of f)_{i+1,j+1} and π_{k+2}f = (the Go-board of f)_{i,j}.

(60) Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 1 ≤ width the Go-board of f,

(v) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) ⊆ L̃(f), and (vi) L((the Go-board of f)_{i+1,j+1}, (the Go-board of f)_{i,j+1}) ⊆ L̃(f).

Then (vii) π_{1}f = (the Go-board of f)_{i+1,j+1} but π_{2}f = (the Go-board of f)_{i+1,j}

and π_{len f−′1}f = (the Go-board of f)_{i,j+1} or π_{2}f = (the Go-board of f)_{i,j+1} and π_{len f−′1}f = (the Go-board of f)_{i+1,j}, or

(viii) there exists k such that 1 ≤ k and k + 1 < len f and π_{k+1}f = (the Goboard of f)_{i+1,j+1} and π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i,j+1} or π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i+1,j}.

(61) Suppose 1 ≤ i and i < len the Go-board of f and 1 ≤ j and j + 1 <

width the Go-board of f. Then

(i) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i,j+1}) ⊈ L̃(f), or (ii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i,j+2}) ⊈ L̃(f), or (iii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) ⊈ L̃(f).

(62) Suppose 1 ≤ i and i < len the Go-board of f and 1 ≤ j and j + 1 <

width the Go-board of f. Then

(i) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) ⊈ L̃(f), or (ii) L((the Go-board of f)_{i+1,j+1}, (the Go-board of f)_{i+1,j+2}) ⊈ L̃(f), or (iii) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) ⊈ L̃(f).

(63) Suppose 1 ≤ j and j < width the Go-board of f and 1 ≤ i and i + 1 <

len the Go-board of f. Then

(i) L((the Go-board of f)_{i,j}, (the Go-board of f)_{i+1,j}) ⊈ L̃(f), or (ii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+2,j}) ⊈ L̃(f), or (iii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) ⊈ L̃(f).

(64) Suppose 1 ≤ j and j < width the Go-board of f and 1 ≤ i and i + 1 <

len the Go-board of f. Then

(i) L((the Go-board of f)_{i,j+1}, (the Go-board of f)_{i+1,j+1}) ⊈ L̃(f), or (ii) L((the Go-board of f)_{i+1,j+1}, (the Go-board of f)_{i+2,j+1}) ⊈ L̃(f), or (iii) L((the Go-board of f)_{i+1,j}, (the Go-board of f)_{i+1,j+1}) ⊈ L̃(f).

References

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On the Monoid of Endomorphisms of Universal Algebra and Many Sorted Algebra

Jarosław Gryko Warsaw University Białystok

MML Identifier: ENDALG.

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

In this paper U_{1} is a universal algebra and f is a function from U_{1} into U_{1}. Let us consider U_{1}. The functor end(U_{1}) yields a non empty set of functions

from the carrier of U_{1} to the carrier of U_{1} and is defined as follows: (Def.1) For every function h from U_{1} into U_{1} holds h ∈ end(U_{1}) iff h is a

homomorphism of U_{1} into U_{1}. Next we state four propositions:

(1) end(U_{1}) ⊆ (the carrier of U_{1})^{^{the} ^{carrier} ^{of} U_{1}}.

(2) For every f holds f ∈ end(U_{1}) iff f is a homomorphism of U_{1} into U_{1}.

(3) id_{(the carrier of U_{1})} ∈ end(U_{1}).

(4) For all elements f_{1}, f_{2} of end(U_{1}) holds f_{1} · f_{2} ∈ end(U_{1}). Let us consider U_{1}. The functor Comp(U_{1}) yielding a binary operation on

end(U_{1}) is defined as follows: (Def.2) For all elements x, y of end(U_{1}) holds (Comp(U_{1}))(x, y) = y · x.

Let us consider U_{1}. The functor End(U_{1}) yields a strict multiplicative loop structure and is defined by: (Def.3) The carrier of End(U_{1}) = end(U_{1}) and the multiplication of End(U_{1}) =

Comp(U_{1}) and the unity of End(U_{1}) = id_{(the carrier of U_{1})}. Let us consider U_{1}. One can check that End(U_{1}) is non empty. Let us consider U_{1}. One can verify that End(U_{1}) is left unital well unital and

associative.

Next we state two propositions:

(5) Let x, y be elements of the carrier of End(U_{1}) and let f, g be elements of end(U_{1}). If x = f and y = g, then x · y = g · f.

(6) id_{(the carrier of U_{1})} = 1_{End(U_{1})}. In the sequel S will be a non void non empty many sorted signature and U_{2}

will be a non-empty algebra over S.

Let us consider S, U_{2}. The functor end(U_{2}) yields a set of many sorted functions from the sorts of U_{2} into the sorts of U_{2} and is defined by the conditions (Def.4). (Def.4) (i) Every element of end(U_{2}) is a many sorted function from U_{2} into U_{2},

and (ii) for every many sorted function h from U_{2} into U_{2} holds h ∈ end(U_{2})

iff h is a homomorphism of U_{2} into U_{2}. One can prove the following propositions:

(7) For every many sorted function F from U_{2} into U_{2} holds F ∈ end(U_{2}) iff F is a homomorphism of U_{2} into U_{2}.

(8) For every element f of end(U_{2}) holds f ∈ ∏ MSFuncs(the sorts of U_{2}, the sorts of U_{2}).

(9) end(U_{2}) ⊆ ∏ MSFuncs(the sorts of U_{2}, the sorts of U_{2}).

(10) id_{(the sorts of U_{2})} ∈ end(U_{2}).

(11) For all elements f_{1}, f_{2} of end(U_{2}) holds f_{1} ∘ f_{2} ∈ end(U_{2}).

(12) For every many sorted function F from MSAlg(U_{1}) into MSAlg(U_{1}) and for every element f of end(U_{1}) such that F = {0} ⟼ f holds F ∈ end(MSAlg(U_{1})).

Let us consider S, U_{2}. The functor Comp(U_{2}) yielding a binary operation on end(U_{2}) is defined as follows: (Def.5) For all elements x, y of end(U_{2}) holds (Comp(U_{2}))(x, y) = y ∘ x.

Let us consider S, U_{2}. The functor End(U_{2}) yields a strict multiplicative loop structure and is defined by: (Def.6) The carrier of End(U_{2}) = end(U_{2}) and the multiplication of End(U_{2}) =

Comp(U_{2}) and the unity of End(U_{2}) = id_{(the sorts of U_{2})}. Let us consider S, U_{2}. Note that End(U_{2}) is non empty. Let us consider S, U_{2}. Note that End(U_{2}) is left unital well unital and

associative.

The following four propositions are true:

(13) Let x, y be elements of the carrier of End(U_{2}) and let f, g be elements of end(U_{2}). If x = f and y = g, then x · y = g ∘ f.

(14) id_{(the sorts of U_{2})} = 1_{End(U_{2})}.

(15) Let U_{3}, U_{4} be universal algebras. Suppose U_{3} and U_{4} are similar. Let F be a many sorted function from MSAlg(U_{3}) into (MSAlg(U_{4}) over MSSign(U_{3})). Then F (0) is a function from U_{3} into U_{4}.

(16) For every element f of end(U_{1}) holds {0} ⟼ f is a many sorted function from MSAlg(U_{1}) into MSAlg(U_{1}).

Let G, H be multiplicative loop structures.

(Def.7) A function from the carrier of G into the carrier of H is called a map

from G into H. Let G, H be non empty multiplicative loop structures. A map from G into

H is multiplicative if:

(Def.8) For all elements x, y of the carrier of G holds it(x · y) = it(x) · it(y). A map from G into H is unity-preserving if:

(Def.9) It(1_{G}) = 1_{H}.

Let us mention that there exists a non empty multiplicative loop structure which is left unital.

Let G, H be left unital non empty multiplicative loop structures. Observe that there exists a map from G into H which is multiplicative and unitypreserving.

Let G, H be left unital non empty multiplicative loop structures. A homomorphism from G to H is a multiplicative unity-preserving map from G into H.

Let G, H be left unital non empty multiplicative loop structures and let h be a map from G into H. We say that h is a monomorphism if and only if:

(Def.10) h is one-to-one.

We say that h is an epimorphism if and only if:

(Def.11) rng h = the carrier of H.

Let G, H be left unital non empty multiplicative loop structures and let h be a map from G into H. We say that h is an isomorphism if and only if:

(Def.12) h is an epimorphism and a monomorphism.

We now state the proposition

(17) Let G be a left unital non empty multiplicative loop structure. Then id_{(the carrier of G)} is a homomorphism from G to G.

Let G, H be left unital non empty multiplicative loop structures. We say that G and H are isomorphic if and only if:

(Def.13) There exists homomorphism from G to H which is an isomorphism.

Let us observe that this predicate is reflexive.

One can prove the following propositions:

(18) Let h be a function. Suppose dom h = end(U_{1}) and for arbitrary x such that x ∈ end(U_{1}) holds h(x) = {0} ⟼ x. Then h is a homomorphism from End(U_{1}) to End(MSAlg(U_{1})).

(19) Let h be a homomorphism from End(U_{1}) to End(MSAlg(U_{1})). Suppose that for arbitrary x such that x ∈ end(U_{1}) holds h(x) = {0} ⟼ x. Then h is an isomorphism.

(20) End(U_{1}) and End(MSAlg(U_{1})) are isomorphic.

442 jarosław gryko

References

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[2] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.

[3] Ewa Burakowska. Subalgebras of the universal algebra. Lattices of subalgebras. Formalized Mathematics, 4(1):23–27, 1993.

[4] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Adam Grabowski. The correspondence between homomorphisms of universal algebra & many sorted algebra. Formalized Mathematics, 5(2):211–214, 1996.

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[12] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra. Formalized Mathematics, 3(2):251–253, 1992.

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

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

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More on Segments on a Go-Board

Andrzej Trybulec Warsaw University Białystok

Summary. We continue the preparatory work for the Jordan Curve Theorem.

MML Identifier: GOBOARD8.

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

We adopt the following rules: i, j, k will be natural numbers, p will be a point of E_{T}^{2}, and f will be a non constant standard special circular sequence.

One can prove the following propositions:

(1) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 2 ≤ width the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of

f)_{i+1,j+2} or π_{k+2}f = (the Go-board of f)_{i+1,j} and π_{k}f = (the Go-board of f)_{i+1,j+2}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i,j+1} + (the Go-board of f)_{i+1,j+2})) misses L̃(f).

(2) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 2 ≤ width the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+2,j+1} and π_{k+2}f = (the Go-board of

f)_{i+1,j+2} or π_{k+2}f = (the Go-board of f)_{i+2,j+1} and π_{k}f = (the Goboard of f)_{i+1,j+2}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i,j+1} + (the Go-board of f)_{i+1,j+2})) misses L̃(f).

(3) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 2 ≤ width the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+2,j+1} and π_{k+2}f = (the Go-board of

f)_{i+1,j} or π_{k+2}f = (the Go-board of f)_{i+2,j+1} and π_{k}f = (the Go-board of f)_{i+1,j}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i,j+1} + (the Go-board of f)_{i+1,j+2})) misses L̃(f).

(4) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ i, (ii) i + 1 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 2 ≤ width the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i,j} and π_{k+2}f = (the Go-board of f)_{i,j+2} or

π_{k+2}f = (the Go-board of f)_{i,j} and π_{k}f = (the Go-board of f)_{i,j+2}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i,j+1} + (the Go-board of f)_{i+1,j+2})) misses L̃(f).

(5) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 2 ≤ width the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of

f)_{i+1,j+2} or π_{k+2}f = (the Go-board of f)_{i,j+1} and π_{k}f = (the Go-board of f)_{i+1,j+2}. Then L(\frac{1}{2} ·((the Go-board of f)_{i+1,j} +(the Go-board of f)_{i+2,j+1}), \frac{1}{2} ·((the Go-board of f)_{i+1,j+1} + (the Go-board of f)_{i+2,j+2})) misses L̃(f).

(6) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) 1 ≤ j, (iv) j + 2 ≤ width the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of f)_{i+1,j}

or π_{k+2}f = (the Go-board of f)_{i,j+1} and π_{k}f = (the Go-board of f)_{i+1,j}. Then L(\frac{1}{2} ·((the Go-board of f)_{i+1,j} +(the Go-board of f)_{i+2,j+1}), \frac{1}{2} ·((the Go-board of f)_{i+1,j+1} + (the Go-board of f)_{i+2,j+2})) misses L̃(f).

(7) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i. Suppose that

(i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{i+1,1}, and (iv) π_{k}f = (the Go-board of f)_{i+2,1} and π_{k+2}f = (the Go-board of f)_{i+1,2}

or π_{k+2}f = (the Go-board of f)_{i+2,1} and π_{k}f = (the Go-board of f)_{i+1,2}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,1} + (the Go-board of f)_{i+1,1}) − [0, 1], \frac{1}{2} · ((the Go-board of f)_{i,1} + (the Go-board of f)_{i+1,2})) misses L̃(f).

(8) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i. Suppose that

(i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{i+1,1}, and (iv) π_{k}f = (the Go-board of f)_{i,1} and π_{k+2}f = (the Go-board of f)_{i+1,2} or

π_{k+2}f = (the Go-board of f)_{i,1} and π_{k}f = (the Go-board of f)_{i+1,2}. Then L(\frac{1}{2} · ((the Go-board of f)_{i+1,1} + (the Go-board of f)_{i+2,1}) − [0, 1], \frac{1}{2} · ((the Go-board of f)_{i+1,1} + (the Go-board of f)_{i+2,2})) misses L̃(f).

(9) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i. Suppose that

(i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{i+1,width the Go-board of f}, and (iv) π_{k}f = (the Go-board of f)_{i+2,width the Go-board of f} and π_{k+2}f =

(the Go-board of f)_{i+1,width the Go-board of f−′1} or π_{k+2}f = (the Goboard of f)_{i+2,width the Go-board of f} and π_{k}f = (the Go-board of f)_{i+1,width the Go-board of f−′1}. Then L(\frac{1}{2} ·((the Go-board of f)_{i,width the Go-board of f−′1} +(the Go-board of f)_{i+1,width the Go-board of f}), \frac{1}{2} · ((the Go-board of f)_{i,width the Go-board of f} + (the Go-board of f)_{i+1,width the Go-board of f}) + [0, 1]) misses L̃(f).

(10) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i. Suppose that (i) 1 ≤ i, (ii) i + 2 ≤ len the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{i+1,width the Go-board of f}, and (iv) π_{k}f = (the Go-board of f)_{i,width the Go-board of f} and π_{k+2}f =

(the Go-board of f)_{i+1,width the Go-board of f−′1} or π_{k+2}f = (the Goboard of f)_{i,width the Go-board of f} and π_{k}f = (the Go-board of f)_{i+1,width the Go-board of f−′1}. Then L(\frac{1}{2} ·((the Go-board of f)_{i+1,width the Go-board of f−′1} +(the Go-board of f)_{i+2,width the Go-board of f}), \frac{1}{2}·((the Go-board of f)_{i+1,width the Go-board of f}+ (the Go-board of f)_{i+2,width the Go-board of f}) + [0, 1]) misses L̃(f).

(11) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given i, j. Suppose that (i) 1 ≤ j, (ii) j + 1 ≤ width the Go-board of f, (iii) 1 ≤ i, (iv) i + 2 ≤ len the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i,j+1} and π_{k+2}f = (the Go-board of

f)_{i+2,j+1} or π_{k+2}f = (the Go-board of f)_{i,j+1} and π_{k}f = (the Go-board of f)_{i+2,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i+1,j} + (the Go-board of f)_{i+2,j+1})) misses L̃(f).

(12) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j, i. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) 1 ≤ i, (iv) i + 2 ≤ len the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+1,j+2} and π_{k+2}f = (the Go-board of

f)_{i+2,j+1} or π_{k+2}f = (the Go-board of f)_{i+1,j+2} and π_{k}f = (the Goboard of f)_{i+2,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i+1,j} + (the Go-board of f)_{i+2,j+1})) misses L̃(f).

(13) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j, i. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) 1 ≤ i, (iv) i + 2 ≤ len the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+1,j+2} and π_{k+2}f = (the Go-board of

f)_{i,j+1} or π_{k+2}f = (the Go-board of f)_{i+1,j+2} and π_{k}f = (the Go-board of f)_{i,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i+1,j} + (the Go-board of f)_{i+2,j+1})) misses L̃(f).

(14) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j, i. Suppose that (i) 1 ≤ j, (ii) j + 1 ≤ width the Go-board of f, (iii) 1 ≤ i, (iv) i + 2 ≤ len the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j}, and (vi) π_{k}f = (the Go-board of f)_{i,j} and π_{k+2}f = (the Go-board of f)_{i+2,j} or

π_{k+2}f = (the Go-board of f)_{i,j} and π_{k}f = (the Go-board of f)_{i+2,j}. Then L(\frac{1}{2} · ((the Go-board of f)_{i,j} + (the Go-board of f)_{i+1,j+1}), \frac{1}{2} · ((the Go-board of f)_{i+1,j} + (the Go-board of f)_{i+2,j+1})) misses L̃(f).

(15) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j, i. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) 1 ≤ i, (iv) i + 2 ≤ len the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of

f)_{i+2,j+1} or π_{k+2}f = (the Go-board of f)_{i+1,j} and π_{k}f = (the Go-board of f)_{i+2,j+1}. Then L(\frac{1}{2} ·((the Go-board of f)_{i,j+1}+(the Go-board of f)_{i+1,j+2}), \frac{1}{2} ·((the Go-board of f)_{i+1,j+1} + (the Go-board of f)_{i+2,j+2})) misses L̃(f).

(16) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j, i. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) 1 ≤ i, (iv) i + 2 ≤ len the Go-board of f,

(v) π_{k+1}f = (the Go-board of f)_{i+1,j+1}, and (vi) π_{k}f = (the Go-board of f)_{i+1,j} and π_{k+2}f = (the Go-board of f)_{i,j+1}

or π_{k+2}f = (the Go-board of f)_{i+1,j} and π_{k}f = (the Go-board of f)_{i,j+1}. Then L(\frac{1}{2} ·((the Go-board of f)_{i,j+1}+(the Go-board of f)_{i+1,j+2}), \frac{1}{2} ·((the Go-board of f)_{i+1,j+1} + (the Go-board of f)_{i+2,j+2})) misses L̃(f).

(17) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{1,j+1}, and (iv) π_{k}f = (the Go-board of f)_{1,j+2} and π_{k+2}f = (the Go-board of f)_{2,j+1}

or π_{k+2}f = (the Go-board of f)_{1,j+2} and π_{k}f = (the Go-board of f)_{2,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{1,j} + (the Go-board of f)_{1,j+1}) − [1, 0], \frac{1}{2} · ((the Go-board of f)_{1,j} + (the Go-board of f)_{2,j+1})) misses L̃(f).

(18) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{1,j+1}, and (iv) π_{k}f = (the Go-board of f)_{1,j} and π_{k+2}f = (the Go-board of f)_{2,j+1}

or π_{k+2}f = (the Go-board of f)_{1,j} and π_{k}f = (the Go-board of f)_{2,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{1,j+1} + (the Go-board of f)_{1,j+2}) − [1, 0], \frac{1}{2} · ((the Go-board of f)_{1,j+1} + (the Go-board of f)_{2,j+2})) misses L̃(f).

(19) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{len the Go-board of f, j+1}, and (iv) π_{k}f = (the Go-board of f)_{len the Go-board of f, j+2} and π_{k+2}f =

(the Go-board of f)_{len the Go-board of f−′1,j+1} or π_{k+2}f = (the Goboard of f)_{len the Go-board of f, j+2} and π_{k}f = (the Go-board of f)_{len the Go-board of f−′1,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{len the Go-board of f−′1,j} + (the Go-board of f)_{len the Go-board of f, j+1}), \frac{1}{2} ·((the Go-board of f)_{len the Go-board of f, j} +(the Go-board of f)_{len the Go-board of f, j+1}) + [1, 0]) misses L̃(f).

(20) Given k. Suppose 1 ≤ k and k + 2 ≤ len f. Given j. Suppose that (i) 1 ≤ j, (ii) j + 2 ≤ width the Go-board of f, (iii) π_{k+1}f = (the Go-board of f)_{len the Go-board of f, j+1}, and (iv) π_{k}f = (the Go-board of f)_{len the Go-board of f, j} and π_{k+2}f =

(the Go-board of f)_{len the Go-board of f−′1,j+1} or π_{k+2}f = (the Goboard of f)_{len the Go-board of f, j} and π_{k}f = (the Go-board of f)_{len the Go-board of f−′1,j+1}. Then L(\frac{1}{2} · ((the Go-board of f)_{len the Go-board of f−′1,j+1} + (the Go-board of f)_{len the Go-board of f, j+2}), \frac{1}{2} ·((the Go-board of f)_{len the Go-board of f, j+1} + (the Go-board of f)_{len the Go-board of f, j+2}) + [1, 0]) misses L̃(f).

In the sequel P will be a subset of the carrier of E_{T}^{2}. We now state a number of propositions:

(21) If for every p such that p ∈ P holds p_{1} < ((the Go-board of f)_{1,1})_{1}, then P misses L̃(f).

(22) If for every p such that p ∈ P holds p_{1} > ((the Go-board of f)_{len the Go-board of f, 1})_{1}, then P misses L̃(f).

(23) If for every p such that p ∈ P holds p_{2} < ((the Go-board of f)_{1,1})_{2}, then P misses L̃(f).

(24) If for every p such that p ∈ P holds p_{2} > ((the Go-board of f)_{1,width the Go-board of f})_{2}, then P misses L̃(f).

(25) Given i. Suppose 1 ≤ i and i + 2 ≤ len the Go-board of f. Then L(\frac{1}{2} · ((the Go-board of f)_{i,1} + (the Go-board of f)_{i+1,1}) − [0, 1], \frac{1}{2} · ((the Go-board of f)_{i+1,1} + (the Go-board of f)_{i+2,1}) − [0, 1]) misses L̃(f).

(26) L((the Go-board of f)_{1,1} − [1, 1], \frac{1}{2} · ((the Go-board of f)_{1,1} + (the Goboard of f)_{2,1}) − [0, 1]) misses L̃(f).

(27) L(\frac{1}{2} · ((the Go-board of f)_{len the Go-board of f−′1,1} + (the Go-board of f)_{len the Go-board of f, 1}) − [0, 1], (the Go-board of f)_{len the Go-board of f, 1} + [1, −1]) misses L̃(f).

(28) Given i. Suppose 1 ≤ i and i + 2 ≤ len the Go-board of f. Then L(\frac{1}{2} · ((the Go-board of f)_{i,width the Go-board of f} + (the Go-board of f)_{i+1,width the Go-board of f}) + [0, 1], \frac{1}{2} · ((the Go-board of f)_{i+1,width the Go-board of f} +(the Go-board of f)_{i+2,width the Go-board of f})+[0, 1]) misses L̃(f).

(29) L((the Go-board of f)_{1,width the Go-board of f} + [−1, 1], \frac{1}{2} · ((the Go-board of f)_{1,width the Go-board of f} + (the Go-board of f)_{2,width the Go-board of f}) + [0,

1]) misses L̃(f).

(30) L(\frac{1}{2} · ((the Go-board of f)_{len the Go-board of f−′1,width the Go-board of f} + (the Go-board of f)_{len the Go-board of f, width the Go-board of f}) + [0, 1], (the Goboard of f)_{len the Go-board of f, width the Go-board of f} + [1, 1]) misses L̃(f).

(31) Given j. Suppose 1 ≤ j and j + 2 ≤ width the Go-board of f. Then L(\frac{1}{2} · ((the Go-board of f)_{1,j} + (the Go-board of f)_{1,j+1}) − [1, 0], \frac{1}{2} · ((the Go-board of f)_{1,j+1} + (the Go-board of f)_{1,j+2}) − [1, 0]) misses L̃(f).

(32) L((the Go-board of f)_{1,1} − [1, 1], \frac{1}{2} · ((the Go-board of f)_{1,1} + (the Goboard of f)_{1,2}) − [1, 0]) misses L̃(f).

(33) L(\frac{1}{2} · ((the Go-board of f)_{1,width the Go-board of f−′1} + (the Go-board of f)_{1,width the Go-board of f})−[1, 0], (the Go-board of f)_{1,width the Go-board of f} + [−1, 1]) misses L̃(f).

(34) Given j. Suppose 1 ≤ j and j + 2 ≤ width the Go-board of f. Then L(\frac{1}{2} · ((the Go-board of f)_{len the Go-board of f, j} + (the Go-board of f)_{len the Go-board of f, j+1}) + [1, 0], \frac{1}{2} · ((the Go-board of f)_{len the Go-board of f, j+1} + (the Go-board of f)_{len the Go-board of f, j+2}) + [1, 0]) misses L̃(f).

(35) L((the Go-board of f)_{len the Go-board of f, 1} + [1, −1], \frac{1}{2} · ((the Go-board of f)_{len the Go-board of f, 1} + (the Go-board of f)_{len the Go-board of f, 2}) + [1, 0]) misses L̃(f).

(36) L(\frac{1}{2} · ((the Go-board of f)_{len the Go-board of f, width the Go-board of f−′1} + (the Go-board of f)_{len the Go-board of f, width the Go-board of f})+[1, 0], (the Goboard of f)_{len the Go-board of f, width the Go-board of f} + [1, 1]) misses L̃(f).

(37) If 1 ≤ k and k + 1 ≤ len f, then Int leftcell(f, k) misses L̃(f).

(38) If 1 ≤ k and k + 1 ≤ len f, then Int rightcell(f, k) misses L̃(f).

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Certain Facts about Families of Subsets of Many Sorted Sets

Artur Korniłowicz Warsaw University Białystok

MML Identifier: MSSUBFAM.

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

1. Preliminaries

For simplicity we follow the rules: I, G, H will denote sets, i will be arbitrary, A, B, M will denote many sorted sets indexed by I, s_{1}, s_{2}, s_{3} will denote families of subsets of I, v, w will denote subsets of I, and F will denote a many sorted function of I.

The scheme MSFExFunc deals with a set A, a many sorted set B indexed by A, a many sorted set C indexed by A, and a ternary predicate P, and states that:

There exists a many sorted function F from B into C such that for arbitrary i if i ∈ A, then there exists a function f from B(i) into C(i) such that f = F (i) and for arbitrary x such that x ∈ B(i) holds P[f(x), x, i]

provided the following condition is satisfied:

• Let i be arbitrary. Suppose i ∈ A. Let x be arbitrary. If x ∈ B(i), then there exists arbitrary y such that y ∈ C(i) and P[y, x, i].

We now state a number of propositions:

(1) If s_{1} ≠ ∅, then Intersect(s_{1}) ⊆ ⋃ s_{1}.

(2) If G ∈ s_{1}, then Intersect(s_{1}) ⊆ G.

(3) If ∅ ∈ s_{1}, then Intersect(s_{1}) = ∅.

(4) For every subset Z of I such that for arbitrary Z_{1} such that Z_{1} ∈ s_{1} holds Z ⊆ Z_{1} holds Z ⊆ Intersect(s_{1}).

(5) If s_{1} ≠ ∅ and for every set Z_{1} such that Z_{1} ∈ s_{1} holds G ⊆ Z_{1}, then G ⊆ Intersect(s_{1}).

(6) If G ∈ s_{1} and G ⊆ H, then Intersect(s_{1}) ⊆ H.

(7) If G ∈ s_{1} and G ∩ H = ∅, then Intersect(s_{1}) ∩ H = ∅.

(8) If s_{3} = s_{1} ∪ s_{2}, then Intersect(s_{3}) = Intersect(s_{1}) ∩ Intersect(s_{2}).

(9) If s_{1} = {v}, then Intersect(s_{1}) = v.

(10) If s_{1} = {v, w}, then Intersect(s_{1}) = v ∩ w.

(11) If A ∈ B, then A is an element of B.

(12) For every non-empty many sorted set B indexed by I such that A is an element of B holds A ∈ B.

(13) For every function f such that i ∈ I and f = F (i) holds (rng_{κ} F (κ))(i) = rng f.

(14) For every function f such that i ∈ I and f = F (i) holds (dom_{κ} F (κ))(i) = dom f.

(15) For all many sorted functions F, G of I holds G ∘ F is a many sorted function of I.

(16) Let A be a non-empty many sorted set indexed by I and let F be a many sorted function from A into ∅_{I}. Then F = ∅_{I}.

(17) If A is transformable to B and F is a many sorted function from A into B, then dom_{κ} F (κ) = A and rng_{κ} F (κ) ⊆ B.

2. Finite Many Sorted Sets

Let us consider I. Note that every many sorted set indexed by I which is empty yielding is also locally-finite.

Let us consider I. Note that ∅_{I} is empty yielding and locally-finite.

Let us consider I, A. Note that there exists a many sorted subset of A which is empty yielding and locally-finite.

Next we state the proposition

(18) If A ⊆ B and B is locally-finite, then A is locally-finite.

Let us consider I and let A be a locally-finite many sorted set indexed by I. One can check that every many sorted subset of A is locally-finite.

Let us consider I and let A, B be locally-finite many sorted sets indexed by I. Note that A ∪ B is locally-finite.

Let us consider I, A and let B be a locally-finite many sorted set indexed by I. Note that A ∩ B is locally-finite.

Let us consider I, B and let A be a locally-finite many sorted set indexed by I. Observe that A ∩ B is locally-finite.

Let us consider I, B and let A be a locally-finite many sorted set indexed by I. Note that A \ B is locally-finite.

Let us consider I, F and let A be a locally-finite many sorted set indexed by I. Observe that F ^{∘} A is locally-finite.

Let us consider I and let A, B be locally-finite many sorted sets indexed by I. Observe that [[A, B]] is locally-finite.

The following propositions are true:

(19) If B is non-empty and [[A, B]] is locally-finite, then A is locally-finite.

(20) If A is non-empty and [[A, B]] is locally-finite, then B is locally-finite.

(21) A is locally-finite iff 2^{A} is locally-finite.

Let us consider I and let M be a locally-finite many sorted set indexed by I. Observe that 2^{M} is locally-finite.

The following propositions are true:

(22) Let A be a non-empty many sorted set indexed by I. Suppose A is locally-finite and for every many sorted set M indexed by I such that M ∈ A holds M is locally-finite. Then ⋃ A is locally-finite.

(23) If ⋃ A is locally-finite, then A is locally-finite and for every M such that M ∈ A holds M is locally-finite.

(24) If dom_{κ} F (κ) is locally-finite, then rng_{κ} F (κ) is locally-finite.

(25) Suppose A ⊆ rng_{κ} F (κ) and for arbitrary i and for every function f such

that i ∈ I and f = F (i) holds f ^{−1} A(i) is finite. Then A is locally-finite. Let us consider I and let A, B be locally-finite many sorted sets indexed by

I. Observe that MSFuncs(A, B) is locally-finite.

Let us consider I and let A, B be locally-finite many sorted sets indexed by I. Note that A− . B is locally-finite.

In the sequel X, Y, Z denote many sorted sets indexed by I. One can prove the following propositions:

(26) Suppose X is locally-finite and X ⊆ [[Y, Z]]. Then there exist A, B such that A is locally-finite and A ⊆ Y and B is locally-finite and B ⊆ Z and X ⊆ [[A, B]].

(27) Suppose X is locally-finite and Z is locally-finite and X ⊆ [[Y, Z]]. Then there exists A such that A is locally-finite and A ⊆ Y and X ⊆ [[A, Z]].

(28) Let M be a non-empty locally-finite many sorted set indexed by I. Suppose that for all many sorted sets A, B indexed by I such that A ∈ M and B ∈ M holds A ⊆ B or B ⊆ A. Then there exists a many sorted set m indexed by I such that m ∈ M and for every many sorted set K indexed by I such that K ∈ M holds m ⊆ K.

(29) Let M be a non-empty locally-finite many sorted set indexed by I. Suppose that for all many sorted sets A, B indexed by I such that A ∈ M and B ∈ M holds A ⊆ B or B ⊆ A. Then there exists a many sorted set m indexed by I such that m ∈ M and for every many sorted set K indexed by I such that K ∈ M holds K ⊆ m.

(30) If Z is locally-finite and Z ⊆ rng_{κ} F (κ), then there exists Y such that Y ⊆ dom_{κ} F (κ) and Y is locally-finite and F ^{∘} Y = Z.

3. A Family of Subsets of Many Sorted Sets

Let us consider I, M. (Def.1) A many sorted subset of 2^{M} is said to be a subset family of M.

Let us consider I, M. Note that there exists a subset family of M which is non-empty.

Let us consider I, M. Then 2^{M} is a subset family of M.

Let us consider I, M. One can check that there exists a subset family of M which is empty yielding and locally-finite.

One can prove the following proposition

(31) ∅_{I} is an empty yielding locally-finite subset family of M.

Let us consider I and let M be a locally-finite many sorted set indexed by I. Note that there exists a subset family of M which is non-empty and locallyfinite.

We follow the rules: S_{1}, S_{2}, S_{3} will be subset families of M, S_{4} will be a non-empty subset family of M, and V, W will be many sorted subsets of M.

Let I be a non empty set, let M be a many sorted set indexed by I, let S_{1} be a subset family of M, and let i be an element of I. Then S_{1}(i) is a family of subsets of M(i).

The following propositions are true:

(32) If i ∈ I, then S_{1}(i) is a family of subsets of M(i).

(33) If A ∈ S_{1}, then A is a many sorted subset of M.

(34) S_{1} ∪ S_{2} is a subset family of M.

(35) S_{1} ∩ S_{2} is a subset family of M.

(36) S_{1} \ A is a subset family of M.

(37) S_{1}− . S_{2} is a subset family of M.

(38) If A ⊆ M, then {A} is a subset family of M.

(39) If A ⊆ M and B ⊆ M, then {A, B} is a subset family of M.

(40) ⋃ S_{1} ⊆ M.

4. Intersection of a Family of Many Sorted Sets

Let us consider I, M, S_{1}. The functor ⋂ S_{1} yields a many sorted set indexed by I and is defined by: (Def.2) For arbitrary i such that i ∈ I there exists a family Q of subsets of

M(i) such that Q = S_{1}(i) and (⋂ S_{1})(i) = Intersect(Q).

Let us consider I, M, S_{1}. Then ⋂ S_{1} is a many sorted subset of M.

We now state a number of propositions:

(41) If S_{1} = ∅_{I}, then ⋂ S_{1} = M.

(42) ⋂ S_{4} ⊆ ⋃ S_{4}.

(43) If A ∈ S_{1}, then ⋂ S_{1} ⊆ A.

(44) If ∅_{I} ∈ S_{1}, then ⋂ S_{1} = ∅_{I}.

(45) Let Z, M be many sorted sets indexed by I and let S_{1} be a non-empty subset family of M. Suppose that for every many sorted set Z_{1} indexed by I such that Z_{1} ∈ S_{1} holds Z ⊆ Z_{1}. Then Z ⊆ ⋂ S_{1}.

(46) If S_{1} ⊆ S_{2}, then ⋂ S_{2} ⊆ ⋂ S_{1}.

(47) If A ∈ S_{1} and A ⊆ B, then ⋂ S_{1} ⊆ B.

(48) If A ∈ S_{1} and A ∩ B = ∅_{I}, then ⋂ S_{1} ∩ B = ∅_{I}.

(49) If S_{3} = S_{1} ∪ S_{2}, then ⋂ S_{3} = ⋂ S_{1} ∩ ⋂ S_{2}.

(50) If S_{1} = {V }, then ⋂ S_{1} = V.

(51) If S_{1} = {V, W }, then ⋂ S_{1} = V ∩ W.

(52) If A ∈ ⋂ S_{1}, then for every B such that B ∈ S_{1} holds A ∈ B.

(53) Let A, M be many sorted sets indexed by I and let S_{1} be a non-empty subset family of M. Suppose A ∈ M and for every many sorted set B indexed by I such that B ∈ S_{1} holds A ∈ B. Then A ∈ ⋂ S_{1}.

Let us consider I, M. A subset family of M is additive if: (Def.3) For all A, B such that A ∈ it and B ∈ it holds A ∪ B ∈ it. A subset family of M is absolutely-additive if: (Def.4) For every subset family F of M such that F ⊆ it holds ⋃ F ∈ it. A subset family of M is multiplicative if: (Def.5) For all A, B such that A ∈ it and B ∈ it holds A ∩ B ∈ it. A subset family of M is absolutely-multiplicative if: (Def.6) For every subset family F of M such that F ⊆ it holds ⋂ F ∈ it. A subset family of M is properly-upper-bound if: (Def.7) M ∈ it. A subset family of M is properly-lower-bound if: (Def.8) ∅_{I} ∈ it.

Let us consider I, M. Observe that there exists a subset family of M which is non-empty additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound and properly-lower-bound.

Let us consider I, M. Then 2^{M} is an additive absolutely-additive multiplicative absolutely-multiplicative properly-upper-bound properly-lower-bound subset family of M.

Let us consider I, M. Note that every subset family of M which is absolutelyadditive is also additive.

Let us consider I, M. Note that every subset family of M which is absolutelymultiplicative is also multiplicative.

456 artur korniłowicz

Let us consider I, M. One can check that every subset family of M which is absolutely-multiplicative is also properly-upper-bound.

Let us consider I, M. Observe that every subset family of M which is properly-upper-bound is also non-empty.

Let us consider I, M. Note that every subset family of M which is absolutelyadditive is also properly-lower-bound.

Let us consider I, M. Note that every subset family of M which is properlylower-bound is also non-empty.

References

[1] Ewa Burakowska. Subalgebras of many sorted algebra. Lattice of subalgebras. Formalized Mathematics, 5(1):47–54, 1996.

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

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

[8] Artur Korniłowicz. Definitions and basic properties of boolean & union of many sorted sets. Formalized Mathematics, 5(2):279–281, 1996.

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[11] Artur Korniłowicz. Some basic properties of many sorted sets. Formalized Mathematics, 5(3):395–399, 1996.

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

[13] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103–108, 1993.

[14] Beata Madras. Products of many sorted algebras. Formalized Mathematics, 5(1):55–60, 1996.

[15] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.

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

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

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

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

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

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

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

On the Concept of the Triangulation

Beata Madras Warsaw University Białystok

MML Identifier: TRIANG 1.

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

1. Introduction

In this paper A will be a set and k, m, n will be natural numbers.

The scheme Regr1 concerns a natural number A and a unary predicate P, and states that:

For every k such that k ≤ A holds P[k] provided the parameters meet the following conditions:

• P[A],

• For every k such that k < A and P[k + 1] holds P[k].

Let n be a natural number. Observe that Seg(n + 1) is non empty.

Let X be a non empty set and let R be an order in X. Note that ⟨X, R⟩ is non empty.

One can prove the following proposition

(1) ∅ |^{2} A = ∅.

Let X be a set. Note that there exists a subset of Fin X which is non empty. Let X be a non empty set. Note that there exists a subset of Fin X which is

non empty and has non empty elements.

Let X be a non empty set and let F be a non empty subset of Fin X with non empty elements. Observe that there exists an element of F which is non empty.

A set has a non-empty element if:

(Def.1) There exists a non empty set X such that X ∈ it. Let us mention that there exists a set which has a non-empty element. Let X be a set with a non-empty element. Note that there exists an element

of X which is non empty.

One can check that every set which has a non-empty element is non empty. Let X be a non empty set. Note that there exists a subset of Fin X which

has a non-empty element.

Let X be a non empty set, let F be a subset of Fin X with a non-empty element, let R be an order in X, and let A be an element of F. Then R |^{2} A is an order in A.

The scheme SubFinite concerns a set A, a subset B of A, and a unary predicate P, and states that:

P[B] provided the following conditions are satisfied:

• B is finite,

• P[∅_{A}],

• For every element x of A and for every subset B of A such that x ∈ B and B ⊆ B and P[B] holds P[B ∪ {x}].

We now state the proposition

(2) Let F be a non empty poset and let A be a subset of F. Suppose A is finite and A ≠ ∅ and for all elements B, C of F such that B ∈ A and C ∈ A holds B ≤ C or C ≤ B. Then there exists an element m of F such that m ∈ A and for every element C of F such that C ∈ A holds m ≤ C.

Let X be a non empty set and let F be a subset of Fin X with a non-empty element. Observe that there exists an element of F which is finite and non empty.

Let A be a non empty poset and let a_{1}, a_{2} be elements of A. We introduce a_{2} ≥ a_{1} as a synonym of a_{1} ≤ a_{2} We introduce a_{2} > a_{1} as a synonym of a_{1} < a_{2}.

Let P be a non empty poset. Note that there exists a subset of P which is non empty and finite.

Let P be a non empty poset, let A be a non empty finite subset of P, and let x be an element of P. One can check that InitSegm(A, x) is finite.

The following proposition is true

(3) For all finite sets A, B such that A ⊆ B and card A = card B holds A = B.

Let A, B be non empty sets, let f be a function from A into B, and let x be an element of A. Then f(x) is an element of B.

Let F be a non empty poset and let A be a non empty subset of F. We see that the element of A is an element of F.

Let X be a non empty set, let F be a subset of Fin X with a non-empty element, let A be a non empty element of F, and let R be an order in X. Let us assume that R linearly orders A. The functor SgmX(R, A) yields a finite sequence of elements of the carrier of ⟨A, R|^{2} A⟩ and is defined by the conditions (Def.2).

(Def.2) (i) rng SgmX(R, A) = A, and

(ii) for all natural numbers n, m and for all elements p, q of ⟨A, R |^{2} A⟩ such that n ∈ dom SgmX(R, A) and m ∈ dom SgmX(R, A) and n < m and p = π_{n} SgmX(R, A) and q = π_{m} SgmX(R, A) holds p > q.

Next we state the proposition

(4) Let X be a non empty set, and let F be a subset of Fin X with a nonempty element, and let A be a non empty element of F, and let R be an order in X, and let f be a finite sequence of elements of the carrier of ⟨X, R⟩. Suppose that

(i) rng f = A, and (ii) for all natural numbers n, m and for all elements p, q of ⟨X, R⟩ such

that n ∈ dom f and m ∈ dom f and n < m and p = π_{n}f and q = π_{m}f holds p > q. Then f = SgmX(R, A).

2. Abstract Complexes

Let C be a non empty poset. The functor symplexes(C) yields a subset of Fin (the carrier of C) and is defined by: (Def.3) symplexes(C) = {A : A ranges over elements of Fin (the carrier of C),

the internal relation of C linearly orders A}. Let C be a non empty poset. Note that symplexes(C) has a non-empty

element.

In the sequel C denotes a non empty poset. Next we state three propositions:

(5) For every element x of C holds {x} ∈ symplexes(C).

(6) ∅ ∈ symplexes(C).

(7) For arbitrary x, s such that x ⊆ s and s ∈ symplexes(C) holds x ∈ symplexes(C).

Let us consider C. Observe that every element of symplexes(C) is finite. One can prove the following propositions:

(8) For every non empty poset C and for every non empty element A of symplexes(C) holds SgmX(the internal relation of C, A) is one-to-one.

(9) Let C be a non empty poset and let A be a non empty element of symplexes(C). If \overline{\overline{A}} = n, then len SgmX(the internal relation of C, A) = n.

(10) Let C be a non empty poset and let A be a non empty element of symplexes(C). If \overline{\overline{A}} = n, then dom SgmX(the internal relation of C, A) = Seg n.

Let C be a non empty poset. One can verify that there exists an element of symplexes(C) which is non empty.

3. Triangulations

A set sequence is a many sorted set indexed by . A set sequence is lower non-empty if:

(Def.4) For every n such that it(n) is non empty and for every m such that

m < n holds it(m) is non empty. Let us observe that there exists a set sequence which is lower non-empty. Let X be a set sequence. The functor FuncsSeq(X) yields a set sequence and

is defined by:

(Def.5) For every natural number n holds (FuncsSeq(X))(n) = X(n)^{X(n+1)}.

Let X be a lower non-empty set sequence and let n be a natural number. Observe that (FuncsSeq(X))(n) is non empty.

Let us consider n and let f be an element of (Seg(n + 1))^{Seg n}. The functor @f yields a finite sequence of elements of and is defined as follows:

(Def.6) ^{@}f = f.

The set sequence NatEmbSeq is defined by:

(Def.7) For every natural number n holds (NatEmbSeq)(n) = {f : f ranges

over elements of (Seg(n + 1))^{Seg n}, ^{@}f is increasing}. Let us consider n. Observe that (NatEmbSeq)(n) is non empty. Let n be a natural number. Note that every element of NatEmbSeq(n) is

function-like and relation-like.

Let X be a set sequence.

(Def.8) A many sorted function from NatEmbSeq into FuncsSeq(X) is called a

triangulation of X. We consider triangulation structures as systems ⟨ a skeleton sequence, a faces assignment ⟩,

where the skeleton sequence is a set sequence and the faces assignment is a many sorted function from NatEmbSeq into FuncsSeq(the skeleton sequence).

Let T be a triangulation structure. We say that T is lower non-empty if and only if:

(Def.9) The skeleton sequence of T is lower non-empty.

Let us note that there exists a triangulation structure which is lower nonempty and strict.

Let T be a lower non-empty triangulation structure. Note that the skeleton sequence of T is lower non-empty.

Let S be a lower non-empty set sequence and let F be a many sorted function from NatEmbSeq into FuncsSeq(S). Note that ⟨S, F ⟩ is lower non-empty.

on the concept of the triangulation 461

4. Relationship Between Abstract Complexes and Triangulations

Let T be a triangulation structure and let n be a natural number. A symplex of T and n is an element of (the skeleton sequence of T)(n).

Let n be a natural number. A face of n is an element of (NatEmbSeq)(n). Let T be a lower non-empty triangulation structure, let n be a natural num-

ber, let x be a symplex of T and n + 1, and let f be a face of n. Let us assume that (the skeleton sequence of T)(n + 1) ≠ ∅. The functor face(x, f) yields a symplex of T and n and is defined by:

(Def.10) For all functions F, G such that F = (the faces assignment of T)(n) and G = F (f) holds face(x, f) = G(x).

Let C be a non empty poset. The functor Triang(C) yielding a lower nonempty strict triangulation structure is defined by the conditions (Def.11).

(Def.11) (i) (The skeleton sequence of Triang(C))(0) = {∅},

(ii) for every natural number n such that n > 0 holds (the skeleton sequence of Triang(C))(n) = {SgmX(the internal relation of C, A) : A ranges over non empty elements of symplexes(C), \overline{\overline{A}} = n\}, and

(iii) for every natural number n and for every face f of n and for every element s of (the skeleton sequence of Triang(C))(n+1) such that s ∈ (the skeleton sequence of Triang(C))(n + 1) and for every non empty element A of symplexes(C) such that SgmX(the internal relation of C, A) = s holds face(s, f) = SgmX(the internal relation of C, A) · f.

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

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

462 beata madras

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on the concept of the triangulation 463

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Index of MML Identifiers

COHSP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 ENDALG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 FACIRC 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 FINSEQ 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 GOBOARD5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 GOBOARD6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 GOBOARD7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 GOBOARD8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 GRAPH 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 INDEX 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 LATSUBGR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 MATRLIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 MSSUBFAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 PUA2MSS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 PZFMISC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 QC LANG4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 TREES A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 TRIANG 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 UNIALG 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 URYSOHN1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 VECTSP 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 VECTSP 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 WEIERSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Contents Formaliz. Math. 5 (3)

Vertex Sequences Induced by Chains

By Yatsuka Nakamura and Piotr Rudnicki . . . . . . . . . . . . . . 297

On the Lattice of Subspaces of a Vector Space

By Andrzej Iwaniuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

On the Lattice of Subgroups of a Group

By Janusz Ganczarski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

On the Lattice of Subalgebras of a Universal Algebra

By Mirosław Jan Paszek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

On the Decomposition of Finite Sequences

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Decomposing a Go-Board into Cells

By Yatsuka Nakamura and Andrzej Trybulec . . . . . . . . . . 323

Indexed Category

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Associated Matrix of Linear Map

By Robert Milewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

On the Geometry of a Go-Board

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

The Theorem of Weierstrass

By Józef Białas and Yatsuka Nakamura . . . . . . . . . . . . . . . . . 353

Dyadic Numbers and T_{4} Topological Spaces

By Józef Białas and Yatsuka Nakamura . . . . . . . . . . . . . . . . . 361

Full Adder Circuit. Part I

By Grzegorz Bancerek and Yatsuka Nakamura . . . . . . . 367

Continued on inside back cover

Continuous, Stable, and Linear Maps of Coherence Spaces

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Some Basic Properties of Many Sorted Sets

By Artur Korniłowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Replacement of Subtrees in a Tree

By Oleg Okhotnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Minimal Signature for Partial Algebra

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

The Subformula Tree of a Formula of the First Order Language

By Oleg Okhotnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

The Steinitz Theorem and the Dimension of a Vector Space

By Mariusz Żynel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

On the Go-Board of a Standard Special Circular Sequence

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

On the Monoid of Endomorphisms of Universal Algebra and Many Sorted Algebra

By Jarosław Gryko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

More on Segments on a Go-Board

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Certain Facts about Families of Subsets of Many Sorted Sets

By Artur Korniłowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

On the Concept of the Triangulation

By Beata Madras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464