FORMALIZED MATHEMATICS Volume 9, Number 1, 2001 University of Białystok

Recursive Euclide Algorithm ^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. The earlier SCM computer did not contain recursive function, so Trybulec and Nakamura proved the correctness of the Euclid’s algorithm only by way of an iterative program. However, the recursive method is a very important programming method, furthermore, for some algorithms, for example Quicksort, only by employing a recursive method (note push-down stack is essentially also a recursive method) can they be implemented. The main goal of the article is to test the recursive function of the SCMPDS computer by proving the correctness of the Euclid’s algorithm by way of a recursive program. In this article, we observed that the memory required by the recursive Euclide algorithm is variable but it is still autonomic. Although the algorithm here is more complicated than the non-recursive algorithm, its focus is that the SCMPDS computer will be able to implement many algorithms like Quicksort which the SCM computer cannot do.

MML Identifier: SCMP GCD.

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

1. Preliminaries

For simplicity, we adopt the following rules: m, n denote natural numbers, i, j denote instructions of SCMPDS, s denotes a state of SCMPDS, and I, J denote Program-block.

One can prove the following three propositions:

(1) If m > 0, then gcd(n, m) = gcd(m, n mod m).

(2) For all integers i, j such that i 0 and j > 0 holds i gcdj = j gcdimodj.

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

(3) For every natural number m and for every integer j such that insposm = j holds insposm + 2 = 2 · (|j| ÷ 2) + 4.

Let k be a natural number. The functor intposk yields a Int position and is defined as follows:

(Def. 1) intposk = d_{k}.

Next we state three propositions:

(4) For all natural numbers n_{1}, n_{2} such that n_{1} ≠ n_{2} holds intposn_{1} ≠ intposn_{2}.

(5) For all natural numbers n_{1}, n_{2} holds DataLoc(n_{1}, n_{2}) = intposn_{1} + n_{2}.

(6) For every state s of SCMPDS and for all natural numbers m_{1}, m_{2} such that IC_{s} = insposm_{1} + m_{2} holds ICplusConst(s, − m_{2}) = insposm_{1}.

The Int position GBP is defined by: (Def. 2) GBP = intpos0.

The Int position SBP is defined as follows: (Def. 3) SBP = intpos1.

The following propositions are true:

(7) GBP ≠ SBP.

(8) card(I;i) = cardI + 1.

(9) card(i;j) = 2.

(10) (I;i)(insposcardI) = i and insposcardI ∈ dom(I;i).

(11) (I;i;J)(insposcardI) = i.

2. The Construction of Recursive Euclide Algorithm

The Program-block GCD − Algorithm is defined by: (Def. 4) GCD − Algorithm = (GBP:=0);(SBP:=7); saveIC(SBP, RetIC);goto 2;

halt_{SCMPDS} ;((SBP, 3) <= 0 goto9);((SBP, 6) := (SBP, 3));

Divide(SBP, 2, SBP, 3);((SBP, 7) := (SBP, 3));((SBP, 4 + RetSP) := (GBP, 1)); AddTo(GBP, 1, 4); saveIC(SBP, RetIC);goto (− 7);((SBP, 2) := (SBP, 6)); returnSBP.

3. The Computation of Recursive Euclide Algorithm

One can prove the following propositions:

(12) cardGCD − Algorithm = 15.

(13) n < 15 iff insposn ∈ domGCD − Algorithm .

(14) (GCD − Algorithm )(inspos0) = GBP:=0 and (GCD − Algorithm ) (inspos1) = SBP:=7 and (GCD − Algorithm )(inspos2) = saveIC(SBP, RetIC) and (GCD − Algorithm )(inspos3) = goto 2 and (GCD − Algorithm ) (inspos4) = halt_{SCMPDS} and (GCD − Algorithm )(inspos5) = (SBP, 3) <= 0 goto9 and (GCD − Algorithm )(inspos6) = (SBP, 6) := (SBP, 3) and (GCD − Algorithm )(inspos7) = Divide(SBP, 2, SBP, 3) and (GCD − Algorithm )(inspos8) = (SBP, 7) := (SBP, 3) and (GCD − Algorithm )(inspos9) = (SBP, 4 + RetSP) := (GBP, 1) and (GCD − Algorithm )(inspos10) = AddTo(GBP, 1, 4) and (GCD − Algorithm ) (inspos11) = saveIC(SBP, RetIC) and (GCD − Algorithm )(inspos12) = goto (− 7) and (GCD − Algorithm )(inspos13) = (SBP, 2) := (SBP, 6) and (GCD − Algorithm )(inspos14) = returnSBP.

(15) Let s be a state of SCMPDS. Suppose Initialized(GCD − Algorithm ) ⊆ s. Then IC_{(Computation (s))(4)} = inspos5 and (Computation(s))(4)(GBP) = 0 and (Computation(s))(4)(SBP) = 7 and (Computation(s))(4)(intpos7+ RetIC) = inspos2 and (Computation(s))(4)(intpos9) = s(intpos9) and (Computation(s))(4)(intpos10) = s(intpos10).

(16) Let s be a state of SCMPDS. Suppose GCD − Algorithm ⊆ s and IC_{s} = inspos5 and s(SBP) > 0 and s(GBP) = 0 and s(DataLoc(s(SBP), 3)) 0 and s(DataLoc(s(SBP), 2)) s(DataLoc(s(SBP), 3)). Then there exists n such that

(i) CurInstr((Computation(s))(n)) = returnSBP, (ii) s(SBP) = (Computation(s))(n)(SBP), (iii) (Computation(s))(n)(DataLoc(s(SBP), 2)) = s(DataLoc(s(SBP), 2))

gcds(DataLoc(s(SBP), 3)), and (iv) for every natural number j such that 1 < j and j s(SBP) + 1 holds

s(intposj) = (Computation(s))(n)(intposj).

(17) Let s be a state of SCMPDS. Suppose GCD − Algorithm ⊆ s and IC_{s} = inspos5 and s(SBP) > 0 and s(GBP) = 0 and s(DataLoc(s(SBP), 3)) 0 and s(DataLoc(s(SBP), 2)) 0. Then there exists n such that

(i) CurInstr((Computation(s))(n)) = returnSBP, (ii) s(SBP) = (Computation(s))(n)(SBP), (iii) (Computation(s))(n)(DataLoc(s(SBP), 2)) = s(DataLoc(s(SBP), 2))

gcds(DataLoc(s(SBP), 3)), and (iv) for every natural number j such that 1 < j and j s(SBP) + 1 holds

s(intposj) = (Computation(s))(n)(intposj).

4 jing-chao chen

4. The Correctness of Recursive Euclide Algorithm

The following proposition is true

(18) Let s be a state of SCMPDS. Suppose Initialized(GCD − Algorithm ) ⊆ s. Let x, y be integers. If s(intpos9) = x and s(intpos10) = y and x 0 and y 0, then (Result(s))(intpos9) = x gcdy.

5. The Autonomy of Recursive Euclide Algorithm

We now state the proposition

(19) Let p be a finite partial state of SCMPDS and x, y be integers. If y 0 and x y and p = [intpos9 ⟼→ x, intpos10 ⟼→ y], then Initialized(GCD − Algorithm )+·p is autonomic.

References

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

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

[3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

[5] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

[6] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Formalized Mathematics, 8(1):193–199, 1999.

[7] Jing-Chao Chen. The construction and shiftability of program blocks for SCMPDS. Formalized Mathematics, 8(1):201–210, 1999.

[8] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

[9] Jing-Chao Chen. A small computer model with push-down stack. Formalized Mathematics, 8(1):175–182, 1999.

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

[11] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[12] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Formalized Mathematics, 3(2):151–160, 1992.

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

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

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

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

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

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

Scott-Continuous Functions. Part II^{1}

Adam Grabowski University of Białystok

MML Identifier: WAYBEL24.

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

1. Preliminaries

One can prove the following proposition

(1) Let S, T be up-complete Scott top-lattices and M be a subset of SCMaps(S, T). Then ⨆_{SCMaps (S,T)} M is a continuous map from S into T.

Let S be a non empty relational structure and let T be a non empty reflexive relational structure. One can check that every map from S into T which is constant is also monotone.

Let S be a non empty relational structure, let T be a reflexive non empty relational structure, and let a be an element of the carrier of T. One can check that S ⟼→ a is monotone.

One can prove the following propositions:

(2) Let S be a non empty relational structure and T be a lower-bounded antisymmetric reflexive non empty relational structure. Then ⊥_{MonMaps (S,T)} = S ⟼→ ⊥_{T} .

(3) Let S be a non empty relational structure and T be an upperbounded antisymmetric reflexive non empty relational structure. Then ⊤_{MonMaps (S,T)} = S ⟼→ ⊤_{T} .

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

(4) Let S, T be complete lattices, f be a monotone map from S into T, and x be an element of S. Then f(x) = sup(f^{∘}↓x).

(5) Let S, T be complete lower-bounded lattices, f be a monotone map from S into T, and x be an element of S. Then f(x) = ⨆_{T} {f(w); w ranges over elements of S: w x}.

(6) Let S be a relational structure, T be a non empty relational structure, and F be a subset of T^{the carrier of S}. Then supF is a map from S into T.

2. On the Scott Continuity of Maps

Let X_{1}, X_{2}, Y be non empty relational structures, let f be a map from [: X_{1}, X_{2} :] into Y, and let x be an element of the carrier of X_{1}. The functor Proj(f, x) yields a map from X_{2} into Y and is defined as follows:

(Def. 1) Proj(f, x) = (curryf)(x).

For simplicity, we use the following convention: X_{1}, X_{2}, Y denote non empty relational structures, f denotes a map from [: X_{1}, X_{2} :] into Y, x denotes an element of the carrier of X_{1}, and y denotes an element of the carrier of X_{2}.

We now state the proposition

(7) For every element y of the carrier of X_{2} holds (Proj(f, x))(y) = f(⟨⟨x, y⟩⟩).

Let X_{1}, X_{2}, Y be non empty relational structures, let f be a map from [: X_{1}, X_{2} :] into Y, and let y be an element of the carrier of X_{2}. The functor Proj(f, y) yielding a map from X_{1} into Y is defined by:

(Def. 2) Proj(f, y) = (curry′f)(y). The following propositions are true:

(8) For every element x of the carrier of X_{1} holds (Proj(f, y))(x) = f(⟨⟨x, y⟩⟩).

(9) Let R, S, T be non empty relational structures, f be a map from [: R, S:] into T, a be an element of R, and b be an element of S. Then (Proj(f, a))(b) = (Proj(f, b))(a).

Let S be a non empty relational structure and let T be a non empty reflexive relational structure. Observe that there exists a map from S into T which is antitone.

The following two propositions are true:

(10) Let R, S, T be non empty reflexive relational structures, f be a map from [: R, S:] into T, a be an element of the carrier of R, and b be an element of the carrier of S. If f is monotone, then Proj(f, a) is monotone and Proj(f, b) is monotone.

(11) Let R, S, T be non empty reflexive relational structures, f be a map from [: R, S:] into T, a be an element of the carrier of R, and b be an element of the carrier of S. If f is antitone, then Proj(f, a) is antitone and Proj(f, b) is antitone.

Let R, S, T be non empty reflexive relational structures, let f be a monotone map from [: R, S:] into T, and let a be an element of the carrier of R. Note that Proj(f, a) is monotone.

Let R, S, T be non empty reflexive relational structures, let f be a monotone map from [: R, S:] into T, and let b be an element of the carrier of S. Note that Proj(f, b) is monotone.

Let R, S, T be non empty reflexive relational structures, let f be an antitone map from [: R, S:] into T, and let a be an element of the carrier of R. Observe that Proj(f, a) is antitone.

Let R, S, T be non empty reflexive relational structures, let f be an antitone map from [: R, S:] into T, and let b be an element of the carrier of S. Note that Proj(f, b) is antitone.

We now state several propositions:

(12) Let R, S, T be lattices and f be a map from [: R, S:] into T. Suppose that for every element a of R and for every element b of S holds Proj(f, a) is monotone and Proj(f, b) is monotone. Then f is monotone.

(13) Let R, S, T be lattices and f be a map from [: R, S:] into T. Suppose that for every element a of R and for every element b of S holds Proj(f, a) is antitone and Proj(f, b) is antitone. Then f is antitone.

(14) Let R, S, T be lattices, f be a map from [: R, S:] into T, b be an element of S, and X be a subset of R. Then (Proj(f, b))^{∘}X = f^{∘}[: X, {b}:].

(15) Let R, S, T be lattices, f be a map from [: R, S:] into T, b be an element of R, and X be a subset of S. Then (Proj(f, b))^{∘}X = f^{∘}[: {b}, X:].

(16) Let R, S, T be lattices, f be a map from [: R, S:] into T, a be an element of R, and b be an element of S. Suppose f is directed-sups-preserving. Then Proj(f, a) is directed-sups-preserving and Proj(f, b) is directed-sups-preserving.

(17) Let R, S, T be lattices, f be a monotone map from [: R, S:] into T, a be an element of R, b be an element of S, and X be a directed subset of [: R, S:]. If sup f^{∘}X exists in T and a ∈ π_{1}(X) and b ∈ π_{2}(X), then f(⟨⟨a, b⟩⟩) sup(f^{∘}X).

(18) Let R, S, T be complete lattices and f be a map from [: R, S:] into T. Suppose that for every element a of R and for every element b of S holds Proj(f, a) is directed-sups-preserving and Proj(f, b) is directed-sups-preserving. Then f is directed-sups-preserving.

(19) Let S be a non empty 1-sorted structure, T be a non empty relational structure, and f be a set. Then f is an element of T^{the carrier of S} if and only if f is a map from S into T.

3. The Poset of Continuous Maps

Let S be a topological structure and let T be a non empty FR-structure. The functor [S → T] yielding a strict relational structure is defined by the conditions (Def. 3).

(Def. 3)(i) [S → T] is a full relational substructure of T^{the carrier of S}, and

(ii) for every set x holds x ∈ the carrier of ([S → T]) iff there exists a map

f from S into T such that x = f and f is continuous. Let S be a non empty topological space and let T be a non empty topological

space-like FR-structure. Observe that [S → T] is non empty.

Let S be a non empty topological space and let T be a non empty topological space-like FR-structure. Note that [S → T] is constituted functions.

One can prove the following propositions:

(20) Let S be a non empty topological space, T be a non empty reflexive topological space-like FR-structure, and x, y be elements of [S → T]. Then x y if and only if for every element i of S holds ⟨⟨x(i), y(i)⟩⟩∈ the internal relation of T.

(21) Let S be a non empty topological space, T be a non empty reflexive topological space-like FR-structure, and x be a set. Then x is a continuous map from S into T if and only if x is an element of [S → T].

Let S be a non empty topological space and let T be a non empty reflexive topological space-like FR-structure. Note that [S → T] is reflexive.

Let S be a non empty topological space and let T be a non empty transitive topological space-like FR-structure. Note that [S → T] is transitive.

Let S be a non empty topological space and let T be a non empty antisymmetric topological space-like FR-structure. One can check that [S → T] is antisymmetric.

Let S be a non empty 1-sorted structure and let T be a non empty topological space-like FR-structure. One can verify that T^{the carrier of S} is constituted functions.

One can prove the following three propositions:

(22) Let S be a non empty 1-sorted structure, T be a complete lattice, f, g, h be maps from S into T, and i be an element of S. If h = ⨆_{(T^{the carrier of S})}{f, g}, then h(i) = sup{f(i), g(i)}.

(23) Let I be a non empty set and J be a relational structure yielding nonempty reflexive-yielding many sorted set indexed by I. Suppose that for every element i of I holds J(i) is a complete lattice. Let X be a subset of ∏ J and i be an element of I. Then (inf X)(i) = inf π_{i}X.

(24) Let S be a non empty 1-sorted structure, T be a complete lattice, f, g, h be maps from S into T, and i be an element of S. If h = ⌈−⌉_{(T^{the carrier of S})}{f, g}, then h(i) = inf{f(i), g(i)}.

Let S be a non empty 1-sorted structure and let T be a lattice. Observe that every element of T^{the carrier of S} is function-like and relation-like.

Let S, T be top-lattices. One can check that every element of [S → T] is function-like and relation-like.

One can prove the following propositions:

(25) Let S be a non empty relational structure, T be a complete lattice, F be a non empty subset of T^{the carrier of S}, and i be an element of the carrier of S. Then (supF)(i) = ⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }.

(26) Let S, T be complete top-lattices, F be a non empty subset of [S → T], and i be an element of the carrier of S. Then (⨆_{(T^{the carrier of S})} F)(i) = ⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }.

In the sequel S denotes a non empty relational structure, T denotes a complete lattice, and i denotes an element of S.

Next we state two propositions:

(27) Let F be a non empty subset of T^{the carrier of S} and D be a non empty subset of S. Then (supF)^{∘}D = {⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }; i ranges over elements of S: i ∈ D}.

(28) Let S, T be complete Scott top-lattices, F be a non empty subset of [S → T], and D be a non empty subset of S. Then (⨆_{(T^{the carrier of S})} F)^{∘}D = {⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }; i ranges over elements of S: i ∈ D}.

The scheme FraenkelF’RSS deals with a non empty relational structure A, a unary functor F yielding a set, a unary functor G yielding a set, and and states that:

{F(v_{1}); v_{1} ranges over elements of A : P[v_{1}]} = {G(v_{2}); v_{2} ranges over elements of A : P[v_{2}]}

provided the following condition is met:

• For every element v of A such that P[v] holds F(v) = G(v). The following propositions are true:

(29) Let S, T be complete Scott top-lattices and F be a non empty subset of [S → T]. Then ⨆_{(T^{the carrier of S})} F is a monotone map from S into T.

(30) Let S, T be complete Scott top-lattices, F be a non empty subset of [S → T], and D be a directed non empty subset of S. Then ⨆_{T} {⨆_{T} {g(i); i ranges over elements of S: i ∈ D}; g ranges over elements of T^{the carrier of S}: g ∈ F } = ⨆_{T} {⨆_{T} {g′(i′); g′ ranges over elements of T^{the carrier of S}: g′ ∈ F }; i′ ranges over elements of S: i′∈ D}.

(31) Let S, T be complete Scott top-lattices, F be a non empty subset of [S → T], and D be a directed non empty subset of S. Then ⨆_{T} ((⨆_{(T^{the carrier of S})} F)^{∘}D) = (⨆_{(T^{the carrier of S})} F)(supD).

(32) Let S, T be complete Scott top-lattices and F be a non empty subset of [S → T]. Then ⨆_{(T^{the carrier of S})} F ∈ the carrier of ([S → T]).

(33) Let S be a non empty relational structure and T be a lower-bounded antisymmetric non empty relational structure. Then ⊥_{T^{the carrier of S}} = S ⟼→ ⊥_{T} .

(34) Let S be a non empty relational structure and T be an upper-bounded antisymmetric non empty relational structure. Then ⊤_{T^{the carrier of S}} = S ⟼→ ⊤_{T} .

Let S be a non empty reflexive relational structure, let T be a complete lattice, and let a be an element of T. Note that S ⟼→ a is directed-sups-preserving.

One can prove the following proposition

(35) Let S, T be complete Scott top-lattices. Then [S → T] is a sups-

inheriting relational substructure of T^{the carrier of S}. Let S, T be complete Scott top-lattices. Observe that [S → T] is complete. We now state three propositions:

(36) For all non empty Scott complete top-lattices S, T holds ⊥_{[S→T]} = S ⟼→ ⊥_{T} .

(37) For all non empty Scott complete top-lattices S, T holds ⊤_{[S→T]} = S ⟼→ ⊤_{T} .

(38) For all Scott complete top-lattices S, T holds SCMaps(S, T) = [S → T].

References

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

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

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

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

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

[6] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

[7] Adam Grabowski. Scott-continuous functions. Formalized Mathematics, 7(1):13–18, 1998.

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

[9] Artur Korniłowicz. Cartesian products of relations and relational structures. Formalized Mathematics, 6(1):145–152, 1997.

scott-continuous functions. part ii 11

[10] Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269–277, 1997.

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

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

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

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

[15] Andrzej Trybulec. Scott topology. Formalized Mathematics, 6(2):311–319, 1997.

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

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

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

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

12 adam grabowski

Some Properties of Isomorphism between Relational Structures.

On the Product of Topological Spaces^{1}

Jarosław Gryko University of Białystok

Artur Korniłowicz University of Białystok

MML Identifier: YELLOW14.

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

1. Preliminaries

The following propositions are true:

(1) 2^{1} = {0, 1}.

(2) For every set X and for every subset Y of X holds rng(id_{X}↾Y) = Y.

(3) For every function f and for all sets a, b holds (f+·(a⟼→ . b))(a) = b. Let us observe that there exists a relational structure which is strict and

empty.

Next we state four propositions:

(4) Let S be an empty 1-sorted structure, T be a 1-sorted structure, and f be a map from S into T. If rngf = Ω_{T} , then T is empty.

(5) Let S be a 1-sorted structure, T be an empty 1-sorted structure, and f be a map from S into T. If domf = Ω_{S}, then S is empty.

(6) Let S be a non empty 1-sorted structure, T be a 1-sorted structure, and f be a map from S into T. If domf = Ω_{S}, then T is non empty.

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

(7) Let S be a 1-sorted structure, T be a non empty 1-sorted structure, and f be a map from S into T. If rngf = Ω_{T} , then S is non empty.

Let S be a non empty reflexive relational structure, let T be a non empty relational structure, and let f be a map from S into T. Let us observe that f is directed-sups-preserving if and only if:

(Def. 1) For every non empty directed subset X of S holds f preserves sup of X.

Let R be a 1-sorted structure and let N be a net structure over R. We say that N is function yielding if and only if:

(Def. 2) The mapping of N is function yielding.

Let us note that there exists a 1-sorted structure which is strict, non empty, and constituted functions.

One can verify that there exists a relational structure which is strict, non empty, and constituted functions.

Let R be a constituted functions 1-sorted structure. One can verify that every net structure over R is function yielding.

Let R be a constituted functions 1-sorted structure. Note that there exists a net structure over R which is strict and function yielding.

Let R be a non empty constituted functions 1-sorted structure. Note that there exists a net structure over R which is strict, non empty, and function yielding.

Let R be a constituted functions 1-sorted structure and let N be a function yielding net structure over R. Observe that the mapping of N is function yielding.

Let R be a non empty constituted functions 1-sorted structure. Note that there exists a net in R which is strict and function yielding.

Let S be a non empty 1-sorted structure and let N be a non empty net structure over S. Note that rng(the mapping of N) is non empty.

Let S be a non empty 1-sorted structure and let N be a non empty net structure over S. Observe that rngnetmap(N, S) is non empty.

One can prove the following two propositions:

(8) Let A, B, C be non empty relational structures, f be a map from B into C, and g, h be maps from A into B. If g h and f is monotone, then f · g f · h.

(9) Let S be a non empty topological space, T be a non empty topological space-like FR-structure, f, g be maps from S into T, and x, y be elements of [S → T]. If x = f and y = g, then x y iff f g.

Let I be a set and let R be a non empty relational structure. Note that every element of the carrier of R^{I} is function-like and relation-like.

Let I be a non empty set, let R be a non empty relational structure, let f be an element of the carrier of R^{I}, and let i be an element of I. Then f(i) is an element of R.

2. Some Properties of Isomorphism between Relational Structures

One can prove the following proposition

(10) For all relational structures S, T and for every map f from S into T

such that f is isomorphic holds f is onto. Let S, T be relational structures. Note that every map from S into T which

is isomorphic is also onto.

We now state four propositions:

(11) Let S, T be non empty relational structures and f be a map from S into T. If f is isomorphic, then f^{−1} is isomorphic.

(12) For all non empty relational structures S, T such that S and T are isomorphic and S has g.l.b.’s holds T has g.l.b.’s.

(13) For all non empty relational structures S, T such that S and T are isomorphic and S has l.u.b.’s holds T has l.u.b.’s.

(14) For every relational structure L such that L is empty holds L is bounded. Let us note that every relational structure which is empty is also bounded. The following propositions are true:

(15) Let S, T be relational structures. Suppose S and T are isomorphic and S is lower-bounded. Then T is lower-bounded.

(16) Let S, T be relational structures. Suppose S and T are isomorphic and S is upper-bounded. Then T is upper-bounded.

(17) Let S, T be non empty relational structures, A be a subset of S, and f be a map from S into T. Suppose f is isomorphic and sup A exists in S. Then sup f^{∘}A exists in T.

(18) Let S, T be non empty relational structures, A be a subset of S, and f be a map from S into T. Suppose f is isomorphic and inf A exists in S. Then inf f^{∘}A exists in T.

3. On the Product of Topological Spaces

Next we state two propositions:

(19) Let S, T be topological structures. Suppose S and T are homeomorphic or there exists a map f from S into T such that domf = Ω_{S} and rngf = Ω_{T} . Then S is empty if and only if T is empty.

(20) For every non empty topological space T holds T and the topological structure of T are homeomorphic.

Let T be a Scott reflexive non empty FR-structure. One can verify that every subset of T which is open is also inaccessible and upper and every subset of T which is inaccessible and upper is also open.

Next we state several propositions:

(21) Let T be a topological structure, x, y be points of T, and X, Y be subsets of T. If X = \{x\} and \overline{X} ⊆ \overline{Y} , then x ∈ \overline{Y} .

(22) Let T be a topological structure, x, y be points of T, and Y, V be subsets of T. If Y = \{y\} and x ∈ \overline{Y} and V is open and x ∈ V, then y ∈ V.

(23) Let T be a topological structure, x, y be points of T, and X, Y be subsets of T. Suppose X = {x} and Y = {y}. Suppose that for every subset V of T such that V is open holds if x ∈ V, then y ∈ V. Then \overline{X} ⊆ \overline{Y} .

(24) Let S, T be non empty topological spaces, A be an irreducible subset of S, and B be a subset of T. Suppose A = B and the topological structure of S = the topological structure of T. Then B is irreducible.

(25) Let S, T be non empty topological spaces, a be a point of S, b be a point of T, A be a subset of the carrier of S, and B be a subset of the carrier of T. Suppose a = b and A = B and the topological structure of S = the topological structure of T and a is dense point of A. Then b is dense point of B.

(26) Let S, T be topological structures, A be a subset of S, and B be a subset of T. Suppose A = B and the topological structure of S = the topological structure of T and A is compact. Then B is compact.

(27) Let S, T be non empty topological spaces. Suppose the topological structure of S = the topological structure of T and S is sober. Then T is sober.

(28) Let S, T be non empty topological spaces. Suppose the topological structure of S = the topological structure of T and S is locally-compact. Then T is locally-compact.

(29) Let S, T be topological structures. Suppose the topological structure of

S = the topological structure of T and S is compact. Then T is compact. Let I be a non empty set, let T be a non empty topological space, let x be

a point of ∏(I ⟼→ T), and let i be an element of I. Then x(i) is an element of T.

The following propositions are true:

(30) Let M be a non empty set, J be a topological space yielding nonempty many sorted set indexed by M, and x, y be points of ∏ J. Then x ∈ \overline{\{y\}} if and only if for every element i of M holds x(i) ∈ \overline{\{y(i)\}}.

(31) Let M be a non empty set, T be a non empty topological space, and x, y be points of ∏(M ⟼→ T). Then x ∈ \overline{\{y\}} if and only if for every element i of M holds x(i) ∈ \overline{\{y(i)\}}.

(32) Let M be a non empty set, i be an element of M, J be a topological space yielding nonempty many sorted set indexed by M, and x be a point of ∏ J. Then π_{i}\overline{\{x\}} = \overline{\{x(i)\}}.

(33) Let M be a non empty set, i be an element of M, T be a non empty topological space, and x be a point of ∏(M ⟼→ T). Then π_{i}\overline{\{x\}} = \overline{\{x(i)\}}.

(34) Let X, Y be non empty topological structures, f be a map from X into Y, and g be a map from Y into X. Suppose f = id_{X} and g = id_{X} and f is continuous and g is continuous. Then the topological structure of X = the topological structure of Y.

(35) Let X, Y be non empty topological spaces and f be a map from X into

Y. If f^{∘} is continuous, then f is continuous. Let X, Y be non empty topological spaces. Observe that every continuous

map from X into Y is continuous.

Let X be a non empty topological space and let Y be a non empty subspace of X. Note that {Y \atop ↪} is continuous.

The following propositions are true:

(36) For every non empty topological space T and for every map f from T into T such that f · f = f holds f^{∘} · ({Im f \atop ↪}) = id_{Im f}.

(37) For every non empty topological space Y and for every non empty subspace W of Y holds ({W \atop ↪})^{∘} is a homeomorphism.

(38) Let M be a non empty set and J be a topological space yielding nonempty many sorted set indexed by M. Suppose that for every element i of M holds J(i) is a T_{0} topological space. Then ∏ J is T_{0}.

Let I be a non empty set and let T be a non empty T_{0} topological space. One can check that ∏(I ⟼→ T) is T_{0}.

The following proposition is true

(39) Let M be a non empty set and J be a topological space yielding nonempty many sorted set indexed by M. Suppose that for every element i of M holds J(i) is T_{1} and topological space-like. Then ∏ J is a T_{1} space.

Let I be a non empty set and let T be a non empty T_{1} topological space. Observe that ∏(I ⟼→ T) is T_{1}.

References

[1] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

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

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

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

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

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

18 jarosław gryko and artur korniłowicz

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

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

[10] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

[14] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

[15] Adam Grabowski. Scott-continuous functions. Part II. Formalized Mathematics, 9(1):5– 11, 2001.

[16] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57–62, 1998.

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

[18] Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269–277, 1997.

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

[20] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.

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

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

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

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

[25] Andrzej Trybulec. Baire spaces, Sober spaces. Formalized Mathematics, 6(2):289–294, 1997.

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

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

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

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

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

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

Cages - the External Approximation of Jordan’s Curve

Czesław Byliński^{1} University of Białystok

Mariusz Żynel^{2} University of Białystok

Summary. On the Euclidean plane Jordan’s curve may be approximated with a polygonal path of sides parallel to coordinate axes, either externally, or internally. The paper deals with the external approximation, and the existence of a Cage – an external polygonal path – is proved.

MML Identifier: JORDAN9.

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

1. Generalities

We adopt the following rules: k, n are natural numbers, D is a non empty set, and f, g are finite sequences of elements of D.

One can prove the following propositions:

(1) For all sets A, B such that A meets B holds A ∩ B meets B.

(2) For every non empty set A and for all sets B_{1}, B_{2} such that A ⊆ B_{1} and A ⊆ B_{2} holds B_{1} meets B_{2}.

1The paper was started during the author’s visit in the Shinshu University, Nagano, Japan, summer 1998.

2The paper was finished during the author’s visit in the Shinshu University, Nagano, Japan, summer 1999.

(3) Let T be a non empty topological space and B, C_{1}, C_{2}, D be subsets of T. Suppose B is connected and C_{1} is a component of D and C_{2} is a component of D and B meets C_{1} and B meets C_{2} and B ⊆ D. Then C_{1} = C_{2}.

(4) If for every n holds f↾n = g↾n, then f = g.

(5) If n ∈ domf, then there exists k such that k ∈ domRev(f) and n + k = lenf + 1 and π_{n}f = π_{k} Rev(f).

(6) If n ∈ domRev(f), then there exists k such that k ∈ domf and n + k = lenf + 1 and π_{n} Rev(f) = π_{k}f.

2. Go-Board Preliminaries

For simplicity, we adopt the following convention: G denotes a Go-board, f, g denote finite sequences of elements of E_{T}^{2}, p denotes a point of E_{T}^{2}, r, s denote real numbers, i, j, k denote natural numbers, and x denotes a set.

Next we state a number of propositions:

(7) f is a sequence which elements belong to G iff Rev(f) is a sequence which elements belong to G.

(8) If f is a sequence which elements belong to G and 1 k and k lenf, then π_{k}f ∈ ValuesG.

(9) If n lenf and x ∈ L̃(f_{⇂n}), then there exists a natural number i such that n + 1 i and i + 1 lenf and x ∈ L(f, i).

(10) If f is a sequence which elements belong to G and 1 k and k+1 lenf, then π_{k}f ∈ left cell(f, k, G) and π_{k}f ∈ right cell(f, k, G).

(11) If f is a sequence which elements belong to G and 1 k and k+1 lenf, then Intleft cell(f, k, G) ≠ ∅ and Intright cell(f, k, G) ≠ ∅.

(12) Suppose f is a sequence which elements belong to G and 1 k and k + 1 lenf. Then Intleft cell(f, k, G) is connected and Intright cell(f, k, G) is connected.

(13) If f is a sequence which elements belong to G and 1 k and k+1 lenf, then \overline{Intleft cell(f, k, G)} = left cell(f, k, G) and \overline{Intright cell(f, k, G)} = right cell(f, k, G).

(14) Suppose f is a sequence which elements belong to G and L(f, k) is horizontal. Then there exists j such that 1 j and j widthG and for every p such that p ∈ L(f, k) holds p_{2} = (G_{1,j})_{2}.

(15) Suppose f is a sequence which elements belong to G and L(f, k) is vertical. Then there exists i such that 1 i and i lenG and for every p such that p ∈ L(f, k) holds p_{1} = (G_{i,1})_{1}.

(16) If f is a sequence which elements belong to G and special and i lenG and j widthG, then Intcell(G, i, j) misses L̃(f).

(17) Suppose f is a sequence which elements belong to G and special and 1 k and k + 1 lenf. Then Intleft cell(f, k, G) misses L̃(f) and Intright cell(f, k, G) misses L̃(f).

(18) Suppose 1 i and i + 1 lenG and 1 j and j + 1 widthG. Then (G_{i,j})_{1} = (G_{i,j+1})_{1} and (G_{i,j})_{2} = (G_{i+1,j})_{2} and (G_{i+1,j+1})_{1} = (G_{i+1,j})_{1} and (G_{i+1,j+1})_{2} = (G_{i,j+1})_{2}.

(19) Let i, j be natural numbers. Suppose 1 i and i + 1 lenG and 1 j and j + 1 widthG. Then p ∈ cell(G, i, j) if and only if the following conditions are satisfied:

(i) (G_{i,j})_{1} p_{1}, (ii) p_{1} (G_{i+1,j})_{1}, (iii) (G_{i,j})_{2} p_{2}, and (iv) p_{2} (G_{i,j+1})_{2}.

(20) If 1 i and i + 1 lenG and 1 j and j + 1 widthG, then cell(G, i, j) = {[r, s] : (G_{i,j})_{1} r ∧ r (G_{i+1,j})_{1} ∧ (G_{i,j})_{2} s ∧ s (G_{i,j+1})_{2}}.

(21) Suppose 1 i and i + 1 lenG and 1 j and j + 1 widthG and p ∈ ValuesG and p ∈ cell(G, i, j). Then p = G_{i,j} or p = G_{i,j+1} or p = G_{i+1,j+1} or p = G_{i+1,j}.

(22) If 1 i and i + 1 lenG and 1 j and j + 1 widthG, then G_{i,j} ∈ cell(G, i, j) and G_{i,j+1} ∈ cell(G, i, j) and G_{i+1,j+1} ∈ cell(G, i, j) and G_{i+1,j} ∈ cell(G, i, j).

(23) If 1 i and i+1 lenG and 1 j and j+1 widthG and p ∈ ValuesG and p ∈ cell(G, i, j), then p is extremal in cell(G, i, j).

(24) Suppose 2 lenG and 2 widthG and f is a sequence which elements belong to G and 1 k and k + 1 lenf. Then there exist i, j such that 1 i and i + 1 lenG and 1 j and j + 1 widthG and L(f, k) ⊆ cell(G, i, j).

(25) Suppose 2 lenG and 2 widthG and f is a sequence which elements belong to G and 1 k and k +1 lenf and p ∈ ValuesG and p ∈ L(f, k). Then p = π_{k}f or p = π_{k+1}f.

(26) If ⟨⟨i, j⟩⟩ ∈ the indices of G and 1 k and k widthG, then (G_{i,j})_{1} (G_{len G,k})_{1}.

(27) If ⟨⟨i, j⟩⟩ ∈ the indices of G and 1 k and k lenG, then (G_{i,j})_{2} (G_{k,width G})_{2}.

(28) Suppose f is a sequence which elements belong to G and special and L̃(g) ⊆ L̃(f) and 1 k and k + 1 lenf. Let A be a subset of E_{T}^{2}. If A = right cell(f, k, G) \ L̃(g) or A = left cell(f, k, G) \ L̃(g), then A is

connected.

(29) Let f be a non constant standard special circular sequence. Suppose f is a sequence which elements belong to G. Let given k. If 1 k and k + 1 lenf, then right cell(f, k, G)\L̃(f) ⊆ RightComp(f) and left cell(f, k, G)\ L̃(f) ⊆ LeftComp(f).

3. Cages

We follow the rules: C is a compact non vertical non horizontal non empty subset of E_{T}^{2} and i, k, n, i_{1}, i_{2} are natural numbers.

Next we state three propositions:

(30) There exists i such that 1 i and i + 1 lenGauge(C, n) and N-minC ∈ cell(Gauge(C, n), i, widthGauge(C, n) − ′ 1) and N-minC ≠ (Gauge(C, n))_{i,width Gauge(C,n)− ′1}.

(31) Suppose that 1 i_{1} and i_{1} + 1 lenGauge(C, n) and N-minC ∈ cell(Gauge(C, n), i_{1}, widthGauge(C, n) − ′1) and N-minC ≠ (Gauge(C, n))_{i_{1},width Gauge(C,n)− ′1} and 1 i_{2} and i_{2} + 1 lenGauge(C, n) and N-minC ∈ cell(Gauge(C, n), i_{2}, widthGauge(C, n) − ′ 1) and N-minC ≠ (Gauge(C, n))_{i_{2},width Gauge(C,n)− ′1}. Then i_{1} = i_{2}.

(32) Let f be a standard non constant special circular sequence. Suppose that (i) f is a sequence which elements belong to Gauge(C, n), (ii) for every k such that 1 k and k + 1 lenf holds

left cell(f, k, Gauge(C, n)) ∩ C = ∅ and right cell(f, k, Gauge(C, n)) ∩ C ≠ ∅, and

(iii) there exists i such that 1 i and i + 1 lenGauge(C, n) and π_{1}f = (Gauge(C, n))_{i,width Gauge(C,n)} and π_{2}f = (Gauge(C, n))_{i+1,width Gauge(C,n)} and N-minC ∈ cell(Gauge(C, n), i, widthGauge(C, n)− ′1) and N-minC ≠ (Gauge(C, n))_{i,width Gauge(C,n)− ′1}. Then N-min L̃(f) = π_{1}f.

Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2} and let n be a natural number. Let us assume that C is connected. The functor Cage(C, n) yields a clockwise oriented standard non constant special circular sequence and is defined by the conditions (Def. 1).

(Def. 1)(i) Cage(C, n) is a sequence which elements belong to Gauge(C, n),

(ii) there exists i such that 1 i and i + 1 lenGauge(C, n) and π_{1} Cage(C, n) = (Gauge(C, n))_{i,width Gauge(C,n)} and π_{2} Cage(C, n) = (Gauge(C, n))_{i+1,width Gauge(C,n)} and N-minC ∈ cell(Gauge(C, n), i, width Gauge(C, n) − ′1) and N-minC ≠ (Gauge(C, n))_{i,width Gauge(C,n)− ′1}, and

cages - the external approximation of … 23

(iii) for every k such that 1  k and k + 2  lenCage(C, n)

holds if front left cell(Cage(C, n), k, Gauge(C, n)) ∩ C = ∅ and front right cell(Cage(C, n), k, Gauge(C, n))∩C = ∅, then Cage(C, n) turns right k, Gauge(C, n) and if front left cell(Cage(C, n), k, Gauge(C, n)) ∩ C = ∅ and front right cell(Cage(C, n), k, Gauge(C, n)) ∩ C ≠ ∅, then Cage(C, n) goes straight k, Gauge(C, n) and if front left cell(Cage(C, n), k, Gauge(C, n)) ∩ C ≠ ∅, then Cage(C, n) turns left k, Gauge(C, n).

One can prove the following propositions:

(33) If C is connected and 1 k and k + 1 lenCage(C, n), then left cell(Cage(C, n), k, Gauge(C, n)) ∩ C = ∅ and right cell(Cage(C, n), k, Gauge(C, n)) ∩ C ≠ ∅.

(34) If C is connected, then N-min L̃(Cage(C, n)) = π_{1} Cage(C, n).

Acknowledgments

Thanks are due to Professor Yatsuka Nakamura and people in the Kiso Laboratory for their gracious support during the work on this paper. Authors would like also to express their gratitude to Andrzej Trybulec for his help in preparing the paper.

References

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

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

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

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

[5] Czesław Byliński. Gauges. Formalized Mathematics, 8(1):25–27, 1999.

[6] Czesław Byliński. Some properties of cells on go-board. Formalized Mathematics, 8(1):139–146, 1999.

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

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

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

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

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

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

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

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

[15] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[16] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

24 czesław byliński and mariusz żynel

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

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

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

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

[21] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

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

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

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

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

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

Components and Basis of Topological Spaces^{1}

Robert Milewski University in Białystok

Summary. This article contains many facts about components and basis of topological spaces.

MML Identifier: YELLOW15.

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

1. Preliminaries

The scheme SeqLambda1C deals with a natural number A, a non empty set B, a unary functor F yielding a set, a unary functor G yielding a set, and and states that:

There exists a finite sequence p of elements of B such that lenp = A and for every natural number i such that i ∈ SegA holds if P[i], then p(i) = F(i) and if not P[i], then p(i) = G(i)

provided the following requirement is met:

• For every natural number i such that i ∈ SegA holds if P[i], then F(i) ∈ B and if not P[i], then G(i) ∈ B.

Let X be a set and let p be a finite sequence of elements of 2^{X}. Then rngp is a family of subsets of X.

Let us observe that Boolean is finite.

We now state two propositions:

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

(2)^{2} For every natural number i and for every finite set D holds D^{i} is finite.

(3) For every finite set T holds every family of subsets of T is finite. Let T be a finite set. One can check that every family of subsets of T is

finite.

Let T be a finite 1-sorted structure. One can verify that every family of subsets of T is finite.

One can prove the following proposition

(4) For every infinite set X there exist sets x, y such that x ∈ X and y ∈ X and x ≠ y.

2. Components

Let X be a set, let p be a finite sequence of elements of 2^{X}, and let q be a finite sequence of elements of Boolean. The functor MergeSequence(p, q) yielding a finite sequence of elements of 2^{X} is defined as follows:

(Def. 1) lenMergeSequence(p, q) = lenp and for every natural number i such that

i ∈ domp holds (MergeSequence(p, q))(i) = (q(i) = true → p(i), X \ p(i)). One can prove the following propositions:

(5) Let X be a set, p be a finite sequence of elements of 2^{X}, and q be a finite sequence of elements of Boolean. Then domMergeSequence(p, q) = domp.

(6) Let X be a set, p be a finite sequence of elements of 2^{X}, q be a finite sequence of elements of Boolean, and i be a natural number. If q(i) = true, then (MergeSequence(p, q))(i) = p(i).

(7) Let X be a set, p be a finite sequence of elements of 2^{X}, q be a finite sequence of elements of Boolean, and i be a natural number. If i ∈ domp and q(i) = false, then (MergeSequence(p, q))(i) = X \ p(i).

(8) For every set X and for every finite sequence q of elements of Boolean holds lenMergeSequence(ε_{2^{X}} , q) = 0.

(9) For every set X and for every finite sequence q of elements of Boolean holds MergeSequence(ε_{2^{X}} , q) = ε_{2^{X}} .

(10) For every set X and for every element x of 2^{X} and for every finite sequence q of elements of Boolean holds lenMergeSequence(⟨x⟩, q) = 1.

(11) Let X be a set, x be an element of 2^{X}, and q be a finite sequence of

elements of Boolean. Then

(i) if q(1) = true, then (MergeSequence(⟨x⟩, q))(1) = x, and (ii) if q(1) = false, then (MergeSequence(⟨x⟩, q))(1) = X \ x.

2The proposition (1) has been removed.

(12) For every set X and for all elements x, y of 2^{X} and for every finite sequence q of elements of Boolean holds lenMergeSequence(⟨x, y⟩, q) = 2.

(13) Let X be a set, x, y be elements of 2^{X}, and q be a finite sequence of

elements of Boolean. Then

(i) if q(1) = true, then (MergeSequence(⟨x, y⟩, q))(1) = x, (ii) if q(1) = false, then (MergeSequence(⟨x, y⟩, q))(1) = X \ x, (iii) if q(2) = true, then (MergeSequence(⟨x, y⟩, q))(2) = y, and (iv) if q(2) = false, then (MergeSequence(⟨x, y⟩, q))(2) = X \ y.

(14) Let X be a set, x, y, z be elements of 2^{X}, and q be a finite sequence of elements of Boolean. Then lenMergeSequence(⟨x, y, z⟩, q) = 3.

(15) Let X be a set, x, y, z be elements of 2^{X}, and q be a finite sequence of

elements of Boolean. Then

(i) if q(1) = true, then (MergeSequence(⟨x, y, z⟩, q))(1) = x, (ii) if q(1) = false, then (MergeSequence(⟨x, y, z⟩, q))(1) = X \ x, (iii) if q(2) = true, then (MergeSequence(⟨x, y, z⟩, q))(2) = y, (iv) if q(2) = false, then (MergeSequence(⟨x, y, z⟩, q))(2) = X \ y,

(v) if q(3) = true, then (MergeSequence(⟨x, y, z⟩, q))(3) = z, and (vi) if q(3) = false, then (MergeSequence(⟨x, y, z⟩, q))(3) = X \ z.

(16) Let X be a set and p be a finite sequence of elements of 2^{X}. Then {Intersect(rngMergeSequence(p, q)); q ranges over finite sequences of elements of Boolean: lenq = lenp} is a family of subsets of X.

Let X be a set and let Y be a finite family of subsets of X. The functor ComponentsY yields a family of subsets of X and is defined by the condition (Def. 2).

(Def. 2) There exists a finite sequence p of elements of 2^{X} such that lenp = cardY and rngp = Y and ComponentsY = {Intersect(rngMergeSequence(p, q)); q ranges over finite sequences of elements of Boolean: lenq = lenp}.

Let X be a set and let Y be a finite family of subsets of X. Note that ComponentsY is finite.

One can prove the following four propositions:

(17) For every set X and for every empty family Y of subsets of X holds ComponentsY = {X}.

(18) For every set X and for all finite families Y, Z of subsets of X such that Z ⊆ Y holds ComponentsY is finer than ComponentsZ.

(19) For every set X and for every finite family Y of subsets of X holds ⋃ ComponentsY = X.

(20) Let X be a set, Y be a finite family of subsets of X, and A, B be sets. If

A ∈ ComponentsY and B ∈ ComponentsY and A ≠ B, then A ∩ B = ∅. Let X be a set and let Y be a finite family of subsets of X. We say that Y

is in general position if and only if:

(Def. 3) ∅ ∉ ComponentsY.

We now state three propositions:

(21) Let X be a set and Y, Z be finite families of subsets of X. If Z is in general position and Y ⊆ Z, then Y is in general position.

(22) For every non empty set X holds every empty family of subsets of X is in general position.

(23) Let X be a non empty set and Y be a finite family of subsets of X. If Y is in general position, then ComponentsY is a partition of X.

3. About Basis of Topological Spaces

We now state two propositions:

(24) For every non empty relational structure L holds Ω_{L} is infs-closed and sups-closed.

(25) For every non empty relational structure L holds Ω_{L} has bottom and top.

Let L be a non empty relational structure. Observe that Ω_{L} is infs-closed and sups-closed and has bottom and top.

The following propositions are true:

(26) For every continuous sup-semilattice L holds Ω_{L} is a CLbasis of L.

(27) For every up-complete non empty poset L such that L is finite holds the carrier of L = the carrier of CompactSublatt(L).

(28) For every lower-bounded sup-semilattice L and for every subset B of L such that B is infinite holds \overline{\overline{B}} = \overline{\overline{finsups(B)}}.

(29) For every T_{0} non empty topological space T holds \overline{\overline{the carrier of T}} ⊆ \overline{\overline{the topology of T}} .

(30) Let T be a topological structure and X be a subset of T. Suppose X is open. Let B be a finite family of subsets of T. Suppose B is a basis of T. Let Y be a set. If Y ∈ ComponentsB, then X ∩ Y = ∅ or Y ⊆ X.

(31) For every T_{0} topological space T such that T is infinite holds every basis of T is infinite.

(32) Let T be a non empty topological space. Suppose T is finite. Let B be a basis of T and x be an element of T. Then ⋂{A; A ranges over elements of the topology of T: x ∈ A} ∈ B. components and basis of topological spaces 29

References

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

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

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

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

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

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

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

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

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

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

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

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

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

[14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

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

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

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

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

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

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

30 robert milewski

Properties of the External Approximation of Jordan’s Curve

Artur Korniłowicz^{1} University of Białystok

MML Identifier: JORDAN10.

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

One can verify that there exists a subset of E_{T}^{2} which is connected, compact, non vertical, and non horizontal.

We adopt the following rules: i, j, k, n are natural numbers, P is a subset of E_{T}^{2}, and C is a connected compact non vertical non horizontal subset of E_{T}^{2}.

The following propositions are true:

(1) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i, j + 1⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i,j}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i,j+1}.

Then i < lenGauge(C, n).

(2) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i, j + 1⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i,j+1}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i,j}.

Then i > 1.

1This paper was written while the author visited Shinshu University, winter 1999.

(3) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i + 1, j⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i,j}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i+1,j}.

Then j > 1.

(4) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i + 1, j⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i+1,j}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i,j}.

Then j < widthGauge(C, n).

(5) C ∩ L̃(Cage(C, n)) = ∅.

(6) N-bound L̃(Cage(C, n)) = N-boundC + \frac{N-bound C−S-bound C}{2^{n}} .

(7) If i < j, then N-bound L̃(Cage(C, j)) < N-bound L̃(Cage(C, i)). Let C be a connected compact non vertical non horizontal subset of E_{T}^{2} and

let n be a natural number. Note that \overline{RightComp(Cage(C, n))} is compact. The following propositions are true:

(8) N-minC ∈ RightComp(Cage(C, n)).

(9) C ∩ RightComp(Cage(C, n)) ≠ ∅.

(10) C ∩ LeftComp(Cage(C, n)) = ∅.

(11) C ⊆ RightComp(Cage(C, n)).

(12) C ⊆ BDD L̃(Cage(C, n)).

(13) UBD L̃(Cage(C, n)) ⊆ UBDC. Let C be a compact non vertical non horizontal subset of E_{T}^{2}. The functor

UBD-FamilyC is defined as follows: (Def. 1) UBD-FamilyC = {UBD L̃(Cage(C, n)) : n ranges over natural

numbers}. The functor BDD-FamilyC is defined by:

(Def. 2) BDD-FamilyC = {BDD L̃(Cage(C, n)) : n ranges over natural

numbers}. Let C be a compact non vertical non horizontal subset of E_{T}^{2}. Then UBD-Family C is a family of subsets of E_{T}^{2}. Then BDD-FamilyC is a family of subsets of

E_{T}^{2}.

properties of the external approximation of … 33

Let C be a compact non vertical non horizontal subset of E_{T}^{2}. Note that UBD-FamilyC is non empty and BDD-FamilyC is non empty.

One can prove the following propositions:

(14) ⋃ UBD-FamilyC = UBDC.

(15) C ⊆ ⋂ BDD-FamilyC.

(16) BDDC ∩ LeftComp(Cage(C, n)) = ∅.

(17) BDDC ⊆ RightComp(Cage(C, n)).

(18) If P is inside component of C, then P ∩ L̃(Cage(C, n)) = ∅.

(19) BDDC ∩ L̃(Cage(C, n)) = ∅.

(20) ⋂ BDD-FamilyC = C ∪ BDDC.

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in the preparation of the article.

References

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

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

[3] Czesław Byliński. Gauges. Formalized Mathematics, 8(1):25–27, 1999.

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

[5] Czesław Byliński and Mariusz Żynel. Cages - the external approximation of Jordan’s curve. Formalized Mathematics, 9(1):19–24, 2001.

[6] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 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] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[12] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[13] Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1–13, 1999.

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

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

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

[17] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

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

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

34 artur korniłowicz

[20] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Ma-

thematics, 1(1):17–23, 1990.

Received June 24, 1999

Irrationality of e

Freek Wiedijk^{1} University of Nijmegen

Summary. We prove the irrationality of square roots of prime numbers and of the number e. In order to be able to prove the last, a proof is given that number e = exp(1) as defined in the Mizar library, that is that

lim_{n→∞} (1 + \frac{1}{n})^{n} = ∑_{k=0}^{∞} \frac{1}{k!}

MML Identifier: IRRAT 1.

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

1. Square Roots of Primes are Irrational

For simplicity, we follow the rules: k, n, p, K, N are natural numbers, x, y, e_{1} are real numbers, s_{1}, s_{2}, s_{3} are sequences of real numbers, and s_{4} is a finite sequence of elements of ℝ.

Let us consider x. We introduce x is irrational as an antonym of x is rational.

Let us consider x, y. We introduce x^{y} as a synonym of x^{y}.

One can prove the following two propositions:

(1) If p is prime, then \sqrt{p} is irrational.

(2) There exist x, y such that x is irrational and y is irrational and x^{y} is rational.

1Written while a guest of the Institute of Mathematics of the University of Białystok.

2. A proof that e = e

The scheme LambdaRealSeq deals with a unary functor F yielding a real number, and states that:

There exists s_{1} such that for every n holds s_{1}(n) = F(n) and for all s_{2}, s_{3} such that for every n holds s_{2}(n) = F(n) and for every n holds s_{3}(n) = F(n) holds s_{2} = s_{3}

for all values of the parameter.

Let us consider k. The functor a_{k} is a sequence of real numbers and is defined by:

(Def. 1) For every n holds a_{k}(n) = \frac{n−k}{n} .

Let us consider k. The functor b_{k} is a sequence of real numbers and is defined by:

(Def. 2) For every n holds b_{k}(n) = \binom{n}{k} · n^{− k}.

Let us consider n. The functor c_{n} is a sequence of real numbers and is defined as follows:

(Def. 3) For every k holds c_{n}(k) = \binom{n}{k} · n^{− k}. Next we state the proposition

(3) c_{n}(k) = b_{k}(n). The sequence d of real numbers is defined as follows:

(Def. 4) For every n holds d(n) = (1 + \frac{1}{n})^{n}.

The sequence e of real numbers is defined as follows: (Def. 5) For every k holds e(k) = \frac{1}{k!}.

We now state a number of propositions:

(4) If n > 0, then n^{− (k+1)} = \frac{n^{− k}}{n} .

(5) For all real numbers x, y, z, v, w such that y ≠ 0 and z ≠ 0 and v ≠ 0 and w ≠ 0 holds \frac{x}{y·z· \frac{v}{w}} = \frac{w}{z} · \frac{x}{y·v}.

(6) \binom{n}{k+1} = \frac{n−k}{k+1} · \binom{n}{k}.

(7) If n > 0, then b_{k+1}(n) = \frac{1}{k+1} · b_{k}(n) · a_{k}(n).

(8) If n > 0, then a_{k}(n) = 1 − \frac{k}{n}.

(9) a_{k} is convergent and lim(a_{k}) = 1.

(10) For every s_{1} such that for every n holds s_{1}(n) = x holds s_{1} is convergent and lims_{1} = x.

(11) For every n such that n > 0 holds b_{0}(n) = 1.

(12) \frac{1}{k+1} · \frac{1}{k!} = \frac{1}{(k+1)!}.

(13) b_{k} is convergent and lim(b_{k}) = \frac{1}{k!} and lim(b_{k}) = e(k).

(14) If k < n, then 0 < a_{k}(n) and a_{k}(n) 1.

irrationality of e 37

(15) If n > 0, then 0 b_{k}(n) and b_{k}(n) \frac{1}{k!} and b_{k}(n) e(k) and 0 c_{n}(k) and c_{n}(k) \frac{1}{k!} and c_{n}(k) e(k).

(16) For every s_{1} such that s_{1} ↑ 1 is summable holds s_{1} is summable and ∑ s_{1} = s_{1}(0) + ∑(s_{1} ↑ 1).

(17) For every s_{4} such that lens_{4} = n and 1 k and k < n holds (s_{4})_{⇂1}(k) = s_{4}(k + 1).

(18) For every s_{4} such that lens_{4} > 0 holds ∑ s_{4} = s_{4}(1) + ∑((s_{4})_{⇂1}).

(19) Let given n and given s_{1}, s_{4}. Suppose lens_{4} = n and for every k such that k < n holds s_{1}(k) = s_{4}(k + 1) and for every k such that k n holds s_{1}(k) = 0. Then s_{1} is summable and ∑ s_{1} = ∑ s_{4}.

(20) If x ≠ 0 and y ≠ 0 and k n, then ⟨\binom{n}{0}x^{0}y^{n}, …, \binom{n}{n}x^{n}y^{0}⟩(k + 1) = \binom{n}{k} · x^{n−k} · y^{k}.

(21) If n > 0 and k n, then c_{n}(k) = ⟨\binom{n}{0}1^{0}(\frac{1}{n})^{n}, …, \binom{n}{n}1^{n}(\frac{1}{n})^{0}⟩(k + 1).

(22) If n > 0, then c_{n} is summable and ∑(c_{n}) = (1+ \frac{1}{n})^{n} and ∑(c_{n}) = d(n).

(23) d is convergent and limd = e.

(24) e is summable and ∑ e = exp1.

(25) Let given K and d_{1} be a sequence of real numbers. If for every n holds d_{1}(n) = (∑_{α=0}^{κ}(c_{n})(α))_{κ∈ℕ}(K), then d_{1} is convergent and limd_{1} = (∑_{α=0}^{κ} e(α))_{κ∈ℕ}(K).

(26) If s_{1} is convergent and lims_{1} = x, then for every e_{1} such that e_{1} > 0 there exists N such that for every n such that n N holds s_{1}(n) > x−e_{1}.

(27) Suppose that

(i) for every e_{1} such that e_{1} > 0 there exists N such that for every n such that n N holds s_{1}(n) > x − e_{1}, and

(ii) there exists N such that for every n such that n N holds s_{1}(n) x.

Then s_{1} is convergent and lims_{1} = x.

(28) If s_{1} is summable, then for every e_{1} such that e_{1} > 0 there exists K such that (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(K) > ∑ s_{1} − e_{1}.

(29) If n 1, then d(n) ∑ e.

(30) If s_{1} is summable and for every k holds s_{1}(k) 0, then ∑ s_{1} (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(K).

(31) d is convergent and limd = ∑ e.

e can be characterized by the condition: (Def. 6) e = ∑ e.

e can be characterized by the condition: (Def. 7) e = exp1.

38 freek wiedijk

3. The Number e is Irrational

We now state a number of propositions:

(32) If x is rational, then there exists n such that n 2 and n! · x is integer.

(33) n! · e(k) = \frac{n!}{k!}.

(34) \frac{n!}{k!} > 0.

(35) If s_{1} is summable and for every n holds s_{1}(n) > 0, then ∑ s_{1} > 0.

(36) n! · ∑(e ↑ (n + 1)) > 0.

(37) If k n, then \frac{n!}{k!} is integer.

(38) n! · (∑_{α=0}^{κ} e(α))_{κ∈ℕ}(n) is integer.

(39) If x = \frac{1}{n+1}, then \frac{n!}{(n+k+1)!} x^{k+1}.

(40) If n > 0 and x = \frac{1}{n+1}, then n! · ∑(e ↑ (n + 1)) \frac{x}{1−x} .

(41) If n 2 and x = \frac{1}{n+1}, then \frac{x}{1−x} < 1.

(42) e is irrational.

References

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

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

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

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

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

[6] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

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

[8] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.

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

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

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

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

[13] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[14] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

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

[16] Konrad Raczkowski and Andrzej Nędzusiak. Serieses. Formalized Mathematics, 2(4):449– 452, 1991.

irrationality of e 39

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

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

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

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

[21] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

40 freek wiedijk

Injective Spaces. Part II^{1}

Artur Korniłowicz University of Białystok

Jarosław Gryko University of Białystok

MML Identifier: WAYBEL25.

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

1. Injective Spaces

The following propositions are true:

(1) For every point p of the Sierpiński space such that p = 0 holds {p} is closed.

(2) For every point p of the Sierpiński space such that p = 1 holds {p} is non closed.

Let us note that the Sierpiński space is non T_{1}. One can check that every top-lattice which is complete and Scott is also

discernible.

Let us observe that there exists a T_{0}-space which is injective and strict. Let us observe that there exists a top-lattice which is complete, Scott, and

strict.

Next we state several propositions:

(3) Let I be a non empty set and T be a Scott topological augmentation of ∏(I ⟼→ 2_{⊆}^{1} ). Then the carrier of T = the carrier of ∏(I ⟼→ the Sierpiński space).

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

(4) Let L_{1}, L_{2} be complete lattices, T_{1} be a Scott topological augmentation of L_{1}, T_{2} be a Scott topological augmentation of L_{2}, h be a map from L_{1} into L_{2}, and H be a map from T_{1} into T_{2}. If h = H and h is isomorphic, then H is a homeomorphism.

(5) Let L_{1}, L_{2} be complete lattices, T_{1} be a Scott topological augmentation of L_{1}, and T_{2} be a Scott topological augmentation of L_{2}. If L_{1} and L_{2} are isomorphic, then T_{1} and T_{2} are homeomorphic.

(6) Let S, T be non empty topological spaces. If S is injective and S and T are homeomorphic, then T is injective.

(7) Let L_{1}, L_{2} be complete lattices, T_{1} be a Scott topological augmentation of L_{1}, and T_{2} be a Scott topological augmentation of L_{2}. If L_{1} and L_{2} are isomorphic and T_{1} is injective, then T_{2} is injective.

Let X, Y be non empty topological spaces. Let us observe that X is a topological retract of Y if and only if:

(Def. 1) There exists a continuous map c from X into Y and there exists a con-

tinuous map r from Y into X such that r · c = id_{X}. One can prove the following propositions:

(8) Let S, T, X, Y be non empty topological spaces. Suppose that

(i) the topological structure of S = the topological structure of T, (ii) the topological structure of X = the topological structure of Y, and (iii) S is a topological retract of X.

Then T is a topological retract of Y.

(9) Let T, S_{1}, S_{2} be non empty topological structures. Suppose S_{1} and S_{2} are homeomorphic and S_{1} is a topological retract of T. Then S_{2} is a topological retract of T.

(10) Let S, T be non empty topological spaces. Suppose T is injective and S is a topological retract of T. Then S is injective.

(11) Let J be an injective non empty topological space and Y be a non empty topological space. If J is a subspace of Y, then J is a topological retract of Y.

(12) For every complete continuous lattice L holds every Scott topological

augmentation of L is injective. Let L be a complete continuous lattice. Observe that every topological au-

gmentation of L which is Scott is also injective.

Let T be an injective non empty topological space. Note that the topological structure of T is injective.

2. Specialization Order

Let T be a topological structure. The functor ΩT yielding a strict FRstructure is defined by the conditions (Def. 2).

(Def. 2)(i) The topological structure of ΩT = the topological structure of T, and

(ii) for all elements x, y of ΩT holds x y iff there exists a subset Y of T

such that Y = \{y\} and x ∈ \overline{Y} . Let T be an empty topological structure. Observe that ΩT is empty. Let T be a non empty topological structure. Note that ΩT is non empty. Let T be a topological space. Note that ΩT is topological space-like. Let T be a topological structure. One can verify that ΩT is reflexive. Let T be a topological structure. One can verify that ΩT is transitive. Let T be a T_{0}-space. One can verify that ΩT is antisymmetric. One can prove the following propositions:

(13) Let S, T be topological spaces. Suppose the topological structure of S = the topological structure of T. Then ΩS = ΩT.

(14) Let M be a non empty set and T be a non empty topological space. Then the relational structure of Ω ∏(M ⟼→ T) = the relational structure of ∏(M ⟼→ ΩT).

(15) For every Scott complete top-lattice S holds ΩS = the FR-structure of S.

(16) Let L be a complete lattice and S be a Scott topological augmentation

of L. Then the relational structure of ΩS = the relational structure of L. Let S be a Scott complete top-lattice. Note that ΩS is complete. We now state four propositions:

(17) Let T be a non empty topological space and S be a non empty subspace of T. Then ΩS is a full relational substructure of ΩT.

(18) Let S, T be topological spaces, h be a map from S into T, and g be a map from ΩS into ΩT. If h = g and h is a homeomorphism, then g is isomorphic.

(19) For all topological spaces S, T such that S and T are homeomorphic holds ΩS and ΩT are isomorphic.

(20) For every injective T_{0}-space T holds ΩT is a complete continuous lattice. Let T be an injective T_{0}-space. One can verify that ΩT is complete and

continuous.

We now state the proposition

(21) For all non empty topological spaces X, Y holds every continuous map from ΩX into ΩY is monotone.

Let X, Y be non empty topological spaces. Note that every map from ΩX into ΩY which is continuous is also monotone.

Next we state the proposition

(22) For every non empty topological space T and for every element x of the

carrier of ΩT holds \overline{\{x\}} = ↓x. Let T be a non empty topological space and let x be an element of the carrier

of ΩT. One can verify that \overline{\{x\}} is non empty lower and directed and ↓x is closed.

Next we state the proposition

(23) For every topological space X holds every open subset of ΩX is upper. Let T be a topological space. One can verify that every subset of ΩT which

is open is also upper.

Let I be a non empty set, let S, T be non empty relational structures, let N be a net in T, and let i be an element of I. Let us assume that the carrier of T ⊆ the carrier of S^{I}. The functor commute(N, i, S) yielding a strict net in S is defined by the conditions (Def. 3).

(Def. 3)(i) The relational structure of commute(N, i, S) = the relational struc-

ture of N, and (ii) the mapping of commute(N, i, S) = (commute(the mapping of N))(i). Next we state two propositions:

(24) Let X, Y be non empty topological spaces, N be a net in [X → ΩY], i be an element of the carrier of N, and x be a point of X. Then (the mapping of commute(N, x, ΩY))(i) = (the mapping of N)(i)(x).

(25) Let X, Y be non empty topological spaces, N be an eventually-directed net in [X → ΩY], and x be a point of X. Then commute(N, x, ΩY) is eventually-directed.

Let X, Y be non empty topological spaces, let N be an eventually-directed net in [X → ΩY], and let x be a point of X. One can verify that commute(N, x, ΩY) is eventually-directed.

Let X, Y be non empty topological spaces. Observe that every net in [X → ΩY] is function yielding.

Next we state the proposition

(26) Let X be a non empty topological space, Y be a T_{0}-space, and N be a net in [X → ΩY]. Suppose that for every point x of X holds sup commute(N, x, ΩY) exists. Then sup rng(the mapping of N) exists in (ΩY)^{the carrier of X}.

3. Monotone Convergence Topological Spaces

Let T be a non empty topological space. We say that T is monotone convergence if and only if the condition (Def. 4) is satisfied.

(Def. 4) Let D be a non empty directed subset of ΩT. Then sup D exists in ΩT

and for every open subset V of T such that supD ∈ V holds D meets V. One can prove the following proposition

(27) Let S, T be non empty topological spaces. Suppose the topological structure of S = the topological structure of T and S is monotone convergence. Then T is monotone convergence.

Let us observe that every T_{0}-space which is trivial is also monotone convergence.

Let us observe that there exists a topological space which is strict, trivial, and non empty.

One can prove the following two propositions:

(28) Let S be a monotone convergence T_{0}-space and T be a T_{0}-space. If S and T are homeomorphic, then T is monotone convergence.

(29) Every Scott complete top-lattice is monotone convergence.

Let L be a complete lattice. One can check that every Scott topological augmentation of L is monotone convergence.

Let L be a complete lattice and let S be a Scott topological augmentation of L. One can check that the topological structure of S is monotone convergence.

We now state the proposition

(30) For every monotone convergence T_{0}-space X holds ΩX is up-complete. Let X be a monotone convergence T_{0}-space. Observe that ΩX is up-complete. One can prove the following three propositions:

(31) Let X be a monotone convergence non empty topological space and N be an eventually-directed prenet over ΩX. Then sup N exists.

(32) Let X be a monotone convergence non empty topological space and N be an eventually-directed net in ΩX. Then supN ∈ LimN.

(33) For every monotone convergence non empty topological space X holds

every eventually-directed net in ΩX is convergent. Let X be a monotone convergence non empty topological space. Observe

that every eventually-directed net in ΩX is convergent.

We now state two propositions:

(34) Let X be a non empty topological space. Suppose that for every eventually-directed net N in ΩX holds sup N exists and supN ∈ LimN. Then X is monotone convergence.

(35) Let X be a monotone convergence non empty topological space and Y be a T_{0}-space. Then every continuous map from ΩX into ΩY is directedsups-preserving.

Let X be a monotone convergence non empty topological space and let Y be a T_{0}-space. One can check that every map from ΩX into ΩY which is continuous is also directed-sups-preserving.

Next we state four propositions:

(36) Let T be a monotone convergence T_{0}-space and R be a T_{0}-space. If R is a topological retract of T, then R is monotone convergence.

(37) Let T be an injective T_{0}-space and S be a Scott topological augmentation of ΩT. Then the topological structure of S = the topological structure of T.

(38) Every injective T_{0}-space is compact, locally-compact, and sober.

(39) Every injective T_{0}-space is monotone convergence.

One can verify that every T_{0}-space which is injective is also monotone convergence.

One can prove the following propositions:

(40) Let X be a non empty topological space, Y be a monotone convergence T_{0}-space, N be a net in [X → ΩY], and f, g be maps from X into ΩY. Suppose that

(i) f = ⨆_{((ΩY)^{the carrier of X})} rng(the mapping of N), (ii) sup rng(the mapping of N) exists in (ΩY)^{the carrier of X}, and (iii) g ∈ rng(the mapping of N).

Then g f.

(41) Let X be a non empty topological space, Y be a monotone convergence T_{0}-space, N be a net in [X → ΩY], x be a point of X, and f be a map from X into ΩY. Suppose for every point a of X holds commute(N, a, ΩY) is eventually-directed and f = ⨆_{((ΩY)^{the carrier of X})} rng(the mapping of N). Then f(x) = supcommute(N, x, ΩY).

(42) Let X be a non empty topological space, Y be a monotone convergence T_{0}-space, and N be a net in [X → ΩY]. Suppose that for every point x of X holds commute(N, x, ΩY) is eventually-directed. Then ⨆_{((ΩY)^{the carrier of X})} rng(the mapping of N) is a continuous map from X into Y.

(43) Let X be a non empty topological space and Y be a monotone convergence T_{0}-space. Then [X → ΩY] is a directed-sups-inheriting relational substructure of (ΩY)^{the carrier of X}.

References

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

injective spaces. part ii 47

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

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

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

[5] Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1):35–43, 1998.

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

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

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

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

[10] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

[11] Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55–59, 1999.

[12] Adam Grabowski. Scott-continuous functions. Part II. Formalized Mathematics, 9(1):5– 11, 2001.

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

[14] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57–62, 1998.

[15] Artur Korniłowicz. Meet–continuous lattices. Formalized Mathematics, 6(1):159–167, 1997.

[16] Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269–277, 1997.

[17] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.

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

[19] Andrzej Trybulec. Natural transformations. Discrete categories. Formalized Mathematics, 2(4):467–474, 1991.

[20] Andrzej Trybulec. Baire spaces, Sober spaces. Formalized Mathematics, 6(2):289–294, 1997.

[21] Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213–225, 1997.

[22] Andrzej Trybulec. Scott topology. Formalized Mathematics, 6(2):311–319, 1997.

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

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

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

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

48 artur korniłowicz and jarosław gryko

Propositional Calculus for Boolean Valued Functions. Part VI

Shunichi Kobayashi Shinshu University Nagano

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

MML Identifier: BVFUNC10.

The articles [1] and [2] provide the notation and terminology for this paper. In this paper Y is a non empty set. The following propositions are true:

(1) For all elements a, b, c of BVF(Y) holds a ∧ b ∨ b ∧ c ∨ c ∧ a = (a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a).

(2) For all elements a, b, c of BVF(Y) holds a ∧ ¬b ∨ b ∧ ¬c ∨ c ∧ ¬a = b ∧ ¬a ∨ c ∧ ¬b ∨ a ∧ ¬c.

(3) For all elements a, b, c of BVF(Y) holds (a ∨ ¬b) ∧ (b ∨ ¬c) ∧ (c ∨ ¬a) = (b ∨ ¬a) ∧ (c ∨ ¬b) ∧ (a ∨ ¬c).

(4) For all elements a, b, c of BVF(Y) such that c ⇒ a = true(Y) and c ⇒ b = true(Y) holds c ⇒ a ∨ b = true(Y).

(5) For all elements a, b, c of BVF(Y) such that a ⇒ c = true(Y) and b ⇒ c = true(Y) holds a ∧ b ⇒ c = true(Y).

(6) For all elements a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2} of BVF(Y) holds (a_{1} ⇒ a_{2}) ∧ (b_{1} ⇒ b_{2}) ∧ (c_{1} ⇒ c_{2}) ∧ (a_{1} ∨ b_{1} ∨ c_{1}) ⋐ a_{2} ∨ b_{2} ∨ c_{2}.

(7) For all elements a_{1}, a_{2}, b_{1}, b_{2} of BVF(Y) holds (a_{1} ⇒ b_{1}) ∧ (a_{2} ⇒ b_{2}) ∧ (a_{1} ∨ a_{2}) ∧ ¬(b_{1} ∧ b_{2}) = (b_{1} ⇒ a_{1}) ∧ (b_{2} ⇒ a_{2}) ∧ (b_{1} ∨ b_{2}) ∧ ¬(a_{1} ∧ a_{2}).

(8) For all elements a, b, c, d of BVF(Y) holds (a ∨ b) ∧ (c ∨ d) = a ∧ c ∨ a ∧ d ∨ b ∧ c ∨ b ∧ d.

(9) For all elements a_{1}, a_{2}, b_{1}, b_{2}, b_{3} of BVF(Y) holds a_{1} ∧ a_{2} ∨ b_{1} ∧ b_{2} ∧ b_{3} = (a_{1} ∨ b_{1}) ∧ (a_{1} ∨ b_{2}) ∧ (a_{1} ∨ b_{3}) ∧ (a_{2} ∨ b_{1}) ∧ (a_{2} ∨ b_{2}) ∧ (a_{2} ∨ b_{3}).

(10) For all elements a, b, c, d of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ d) = (a ⇒ b ∧ c ∧ d) ∧ (b ⇒ c ∧ d) ∧ (c ⇒ d).

(11) For all elements a, b, c, d of BVF(Y) holds (a ⇒ c) ∧ (b ⇒ d) ∧ (a ∨ b) ⋐ c ∨ d.

(12) For all elements a, b, c of BVF(Y) holds (a ∧ b ⇒ ¬c) ∧ a ∧ c ⋐ ¬b.

(13) For all elements a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} of BVF(Y) holds a_{1} ∧ a_{2} ∧ a_{3} ⇒ b_{1} ∨ b_{2} ∨ b_{3} = ¬b_{1} ∧ ¬b_{2} ∧ a_{3} ⇒ ¬a_{1} ∨ ¬a_{2} ∨ b_{3}.

(14) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ a) = a ∧ b ∧ c ∨ ¬a ∧ ¬b ∧ ¬c.

(15) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ a) ∧ (a ∨ b ∨ c) = a ∧ b ∧ c.

(16) For all elements a, b, c of BVF(Y) holds (a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a) ∧ ¬(a ∧ b ∧ c) = ¬a ∧ b ∧ c ∨ a ∧ ¬b ∧ c ∨ a ∧ b ∧ ¬c.

(17) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ⇒ b ∧ c.

(18) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ∨ b ⇒ c.

(19) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ⇒ b ∨ c.

(20) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ⇒ b ∨ ¬c.

(21) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ b ⇒ c ∨ a.

(22) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ b ⇒ c ∨ ¬a.

(23) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b) ∧ (b ⇒ c ∨ a).

(24) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ ¬c) ∧ (b ⇒ c).

(25) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ c) ∧ (b ⇒ c ∨ a).

(26) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ ¬c) ∧ (b ⇒ c ∨ a).

(27) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ ¬c) ∧ (b ⇒ c ∨ ¬a).

References

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

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

Predicate Calculus for Boolean Valued Functions. Part III

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC11.

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

1. Preliminaries

In this paper Y is a non empty set. We now state several propositions:

(1) For every element z of Y and for all partitions P_{1}, P_{2} of Y such that P_{1} ⋐ P_{2} holds EqClass(z, P_{1}) ⊆ EqClass(z, P_{2}).

(2) For every element z of Y and for all partitions P_{1}, P_{2} of Y holds EqClass(z, P_{1}) ⊆ EqClass(z, P_{1} ∨ P_{2}).

(3) For every element z of Y and for all partitions P_{1}, P_{2} of Y holds EqClass(z, P_{1} ∧ P_{2}) ⊆ EqClass(z, P_{1}).

(4) For every element z of Y and for every partition P_{1} of Y holds EqClass(z, P_{1}) ⊆ EqClass(z, O(Y)) and EqClass(z, I(Y)) ⊆ EqClass(z, P_{1}).

(5) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. Suppose G is an independent family of partitions and G = {A, B} and A ≠ B. Let a, b be sets. If a ∈ A and b ∈ B, then a ∩ b ≠ ∅.

(6) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ⋀ G = A ∧ B.

(7) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then CompF(A, G) = B and CompF(B, G) = A.

2. Predicate Calculus

One can prove the following propositions:

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B}, then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B}, then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(14) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

predicate calculus for boolean valued … 53

(17) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(19) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,A} G,B}G.

References

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

[2] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[3] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

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

[5] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

54 shunichi kobayashi and yatsuka nakamura

A Characterization of Concept Lattices. Dual Concept Lattices

Christoph Schwarzweller University of Tuebingen

Summary. In this article we continue the formalization of concept lattices following [4]. We give necessary and sufficient conditions for a complete lattice to be isomorphic to a given formal context. As a by-product we get that a lattice is complete if and only if it is isomorphic to a concept lattice. In addition we introduce dual formal concepts and dual concept lattices and prove that the dual of a concept lattice over a formal context is isomorphic to the concept lattice over the dual formal context.

MML Identifier: CONLAT 2.

The notation and terminology used in this paper have been introduced in the following articles: [8], [10], [2], [3], [11], [1], [5], [9], [15], [7], [14], [6], [13], [12], and [16].

1. Preliminaries

Let C be a FormalContext and let C_{1} be a strict FormalConcept of C. The functor ^{@}C_{1} yielding an element of ConceptLatticeC is defined as follows:

(Def. 1) ^{@}C_{1} = C_{1}. Next we state four propositions:

(1) For every FormalContext C holds ⊥_{ConceptLattice C} = Concept − with − all −Attributes C and ⊤_{ConceptLattice C} = Concept − with − all − Objects C.

(2) Let C be a FormalContext and D be a non empty subset of 2^{the objects of C}. Then (ObjectDerivationC)(⋃ D) = ⋂{(ObjectDerivationC) (O); O ranges over subsets of the objects of C: O ∈ D}.

(3) Let C be a FormalContext and D be a non empty subset of 2^{the Attributes of C}. Then (AttributeDerivationC)(⋃ D) = ⋂{(AttributeDerivationC)(A); A ranges over subsets of the Attributes of C: A ∈ D}.

(4) Let C be a FormalContext and D be a subset of the carrier of ConceptLatticeC. Then ⌈−⌉_{ConceptLattice C}D is a FormalConcept of C and ⨆_{ConceptLattice C} D is a FormalConcept of C.

Let C be a FormalContext and let D be a subset of the carrier of ConceptLatticeC. The functor ⌈−⌉_{C}D yields a FormalConcept of C and is

defined as follows: (Def. 2) ⌈−⌉_{C}D = ⌈−⌉_{ConceptLattice C}D.

The functor ⨆_{C} D yields a FormalConcept of C and is defined by: (Def. 3) ⨆_{C} D = ⨆_{ConceptLattice C} D.

Next we state several propositions:

(5) For every FormalContext C holds ⨆_{C}(∅_{ConceptLattice C}) = Concept − with −all − Attributes C and ⌈−⌉_{C}(∅_{ConceptLattice C}) = Concept − with − all − Objects C.

(6) For every FormalContext C holds ⨆_{C}(Ω_{the carrier of ConceptLattice C}) = Concept − with − all − Objects C and ⌈−⌉_{C}(Ω_{the carrier of ConceptLattice C}) = Concept − with − all − Attributes C.

(7) Let C be a FormalContext and D be a non empty subset of ConceptLatticeC. Then

(i) the Extent of ⨆_{C} D = (AttributeDerivationC)((ObjectDerivationC) (⋃{the Extent of ⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D})), and

(ii) the Intent of ⨆_{C} D = ⋂{the Intent of ⟨E, I⟩; E ranges over subsets of

the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D}.

(8) Let C be a FormalContext and D be a non empty subset of ConceptLatticeC. Then

(i) the Extent of ⌈−⌉_{C}D = ⋂{the Extent of ⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D}, and

(ii) the Intent of ⌈−⌉_{C}D = (ObjectDerivationC)((AttributeDerivationC) (⋃{the Intent of ⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D})).

(9) Let C be a FormalContext and C_{1} be a strict FormalConcept of C. Then ⨆_{ConceptLattice C}{⟨O, A⟩; O ranges over subsets of the objects of C, A ranges over subsets of the Attributes of C: ⋁_{o: object of C} (o ∈ the Extent of C_{1} ∧ O = (AttributeDerivationC)((ObjectDerivationC)({o})) ∧ A = (ObjectDerivationC)({o}))} = C_{1}.

(10) Let C be a FormalContext and C_{1} be a strict FormalConcept of C. Then ⌈−⌉_{ConceptLattice C}{⟨O, A⟩; O ranges over subsets of the objects of C, A ranges over subsets of the Attributes of C: ⋁_{a: Attribute of C} (a ∈ the Intent of C_{1} ∧ O = (AttributeDerivationC)({a}) ∧ A = (ObjectDerivationC)((AttributeDerivationC)({a})))} = C_{1}.

Let C be a FormalContext. The functor γ(C) yields a function from the objects of C into the carrier of ConceptLatticeC and is defined by the condition (Def. 4).

(Def. 4) Let o be an element of the objects of C. Then there exists a subset O of the objects of C and there exists a subset A of the Attributes of C such that (γ(C))(o) = ⟨O, A⟩ and O = (AttributeDerivationC)((ObjectDerivationC)({o})) and A = (ObjectDerivationC)({o}).

Let C be a FormalContext. The functor δ_{C} yielding a function from the Attributes of C into the carrier of ConceptLatticeC is defined by the condition (Def. 5).

(Def. 5) Let a be an element of the Attributes of C. Then there exists a subset O of the objects of C and there exists a subset A of the Attributes of C such that δ_{C}(a) = ⟨O, A⟩ and O = (AttributeDerivationC)({a}) and A = (ObjectDerivationC)((AttributeDerivationC)({a})).

The following propositions are true:

(11) Let C be a FormalContext, o be an object of C, and a be a Attribute of C. Then (γ(C))(o) is a FormalConcept of C and δ_{C}(a) is a FormalConcept of C.

(12) For every FormalContext C holds rngγ(C) is supremum-dense and rng(δ_{C}) is infimum-dense.

(13) Let C be a FormalContext, o be an object of C, and a be a Attribute of C. Then o is connected with a if and only if (γ(C))(o) ⊑ δ_{C}(a).

2. The Characterization

We now state the proposition

(14) Let L be a complete lattice and C be a FormalContext. Then ConceptLatticeC and L are isomorphic if and only if there exists a function g from the objects of C into the carrier of L and there exists a function d from the Attributes of C into the carrier of L such that rngg is supremum-dense and rngd is infimum-dense and for every object o of C and for every Attribute a of C holds o is connected with a iff g(o) ⊑ d(a).

Let L be a lattice. The functor ContextL yields a strict non quasi-empty ContextStr and is defined as follows:

(Def. 6) ContextL = ⟨the carrier of L, the carrier of L, LattRel(L)⟩.

One can prove the following proposition

(15) For every complete lattice L holds ConceptLatticeContextL and L are

isomorphic. Let L_{1}, L_{2} be lattices. Let us note that the predicate L_{1} and L_{2} are isomor-

phic is symmetric.

Next we state the proposition

(16) For every lattice L holds L is complete iff there exists a FormalContext C such that ConceptLatticeC and L are isomorphic.

3. Dual Concept Lattices

Let L be a complete lattice. Observe that L^{∘} is complete. Let C be a FormalContext. The functor C^{∘} yielding a strict non quasi-empty

ContextStr is defined as follows: (Def. 7) C^{∘} = ⟨the Attributes of C, the objects of C, (the Information of C)^{⌣}⟩.

We now state three propositions:

(17) For every strict FormalContext C holds (C^{∘})^{∘} = C.

(18) For every FormalContext C and for every subset O of the objects of C holds (ObjectDerivationC)(O) = (AttributeDerivationC^{∘})(O).

(19) For every FormalContext C and for every subset A of the Attributes of

C holds (AttributeDerivationC)(A) = (ObjectDerivationC^{∘})(A). Let C be a FormalContext and let C_{1} be a ConceptStr over C. The functor

C_{1}^{∘} yields a strict ConceptStr over C^{∘} and is defined as follows: (Def. 8) The Extent of C_{1}^{∘} = the Intent of C_{1} and the Intent of C_{1}^{∘} = the Extent

of C_{1}. Let C be a FormalContext and let C_{1} be a FormalConcept of C. Then C_{1}^{∘}

is a strict FormalConcept of C^{∘}.

We now state the proposition

(20) For every FormalContext C and for every strict FormalConcept C_{1} of C

holds (C_{1}^{∘})^{∘} = C_{1}. Let C be a FormalContext. The functor DualHomomorphismC yielding a

homomorphism from (ConceptLatticeC)^{∘} to ConceptLatticeC^{∘} is defined as follows:

(Def. 9) For every strict FormalConcept C_{1} of C holds (DualHomomorphismC)(C_{1}) = C_{1}^{∘}.

a characterization of concept lattices. … 59

We now state two propositions:

(21) For every FormalContext C holds DualHomomorphismC is isomorphism.

(22) For every FormalContext C holds ConceptLatticeC^{∘} and (ConceptLatticeC)^{∘} are isomorphic.

References

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

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

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

[4] Bernhard Ganter and Rudolf Wille. Formal Concept Analysis. Springer Verlag, Berlin, Heidelberg, Ney York, 1996. (written in German).

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

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

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

[8] Christoph Schwarzweller. Introduction to concept lattices. Formalized Mathematics, 7(2):233–241, 1998.

[9] Christoph Schwarzweller. Noetherian lattices. Formalized Mathematics, 8(1):169–174, 1999.

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

[11] Andrzej Trybulec. Finite join and finite meet and dual lattices. Formalized Mathematics, 1(5):983–988, 1990.

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

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

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

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

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

Received August 17, 1999 60 christoph schwarzweller

Predicate Calculus for Boolean Valued Functions. Part IV

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC12.

The terminology and notation used in this paper are introduced in the following papers: [1], [2], [3], [5], and [4].

In this paper Y is a non empty set. The following propositions are true:

(1) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∀_{∀_{a,A}G,B}G = ∃_{¬∀_{a,A}G,B} G.

(2) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∀_{a,A}G,B}G = ∀_{¬∀_{a,A}G,B} G.

(3) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{¬∀_{a,A}G,B} G = ∀_{∃_{¬a,A} G,B}G.

(4) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{¬∃_{a,A}G,B} G = ∀_{∀_{¬a,A} G,B}G.

(5) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∀_{∃_{a,A}G,B}G = ∃_{∀_{¬a,A} G,B}G.

(6) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∀_{a,A}G,B}G = ∀_{∃_{¬a,A} G,B}G.

(7) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∀_{∀_{a,A}G,B}G = ∃_{∃_{¬a,A} G,B}G.

(8) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∃_{¬∀_{a,A}G,B} G = ∃_{∃_{¬a,A} G,B}G.

(9) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∃_{¬∃_{a,A}G,B} G = ∃_{∀_{¬a,A} G,B}G.

(10) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∃_{a,A}G,B}G = ∀_{¬∃_{a,A}G,B} G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(12) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,A}G,B}G.

(13) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,A}G,B}G.

(14) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,A}G,B}G.

(15) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,A}G,B}G.

(16) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,A}G,B}G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

predicate calculus for boolean valued … 63

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

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

Received August 17, 1999 64 shunichi kobayashi and yatsuka nakamura

Predicate Calculus for Boolean Valued Functions. Part V

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC13.

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

In this paper Y denotes a non empty set. One can prove the following propositions:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(2) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(3) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(11) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(17) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(19) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(21) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(22) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(23) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(24) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(25) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(26) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(27) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(28) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(29) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

(30) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(31) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(32) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B,

then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∃_{a,B}G,A}G.

(33) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∃_{a,B}G,A}G.

(34) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(35) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(36) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(37) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(38) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(39) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(40) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∃_{a,B}G,A} G.

(41) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∃_{a,B}G,A} G.

(42) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(43) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(44) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(45) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(46) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∀_{¬a,B} G,A}G.

(47) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∀_{¬a,B} G,A}G.

(48) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(49) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(50) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(51) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(52) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(53) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(54) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(55) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(56) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(57) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(58) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(59) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and 70 shunichi kobayashi and yatsuka nakamura

A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(61)^{1} Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(62) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

Received August 17, 1999

1The proposition (60) has been removed.

Definitions of Radix-2^{k} Signed-Digit Number and its Adder Algorithm

Yoshinori Fujisawa Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Summary. In this article, a radix-2^{k} signed-digit number (Radix-2^{k} SD number) is defined and based on it a high-speed adder algorithm is discussed.

The processes of coding and encoding for public-key cryptograms require a great deal of addition operations of natural number of many figures. This results in a long time for the encoding and decoding processes. It is possible to reduce the processing time using the high-speed adder algorithm.

In the first section of this article, we prepared some useful theorems for natural numbers and integers. In the second section, we defined the concept of radix-2^{k}, a set named k-SD and proved some properties about them. In the third section, we provide some important functions for generating Radix-2^{k} SD numbers from natural numbers and natural numbers from Radix-2^{k} SD numbers. In the fourth section, we defined the carry and data components of addition with Radix-2^{k} SD numbers and some properties about them. In the fifth section, we defined a theorem for checking whether or not a natural number can be expressed as n digits Radix-2^{k} SD number.

In the last section, a high-speed adder algorithm on Radix-2^{k} SD numbers is proposed and we provided some properties. In this algorithm, the carry of each digit has an effect on only the next digit. Properties of the relationships of the results of this algorithm to the operations of natural numbers are also given.

MML Identifier: RADIX 1.

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

1. Some Useful Theorems

We adopt the following convention: i, k, m, n, x, y are natural numbers, i_{1}, i_{2}, i_{3} are integers, and e is a set.

The following propositions are true:

(1) If n ≠ 0, then m ÷ n = (m qua integer) ÷n qua integer and m modn = (m qua integer) mod n qua integer.

(2) If k ≠ 0 and n mod k = k − 1, then (n + 1) mod k = 0.

(3) If k ≠ 0 and n mod k < k − 1, then (n + 1) mod k = (n mod k) + 1.

(4) If m ≠ 0 and n ≠ 0, then k mod m · n mod n = k mod n.

(5) If k ≠ 0, then (n + 1) mod k = 0 or (n + 1) mod k = (n mod k) + 1.

(6) If i ≠ 0 and k ≠ 0, then (n mod i_{ℕ}^{k}) ÷ i_{ℕ}^{k− ′1} < i.

(7) If k n, then m_{ℕ}^{k} | m_{ℕ}^{n}.

(8) If i_{3} > 0, then i_{1} mod i_{2} · i_{3} mod i_{3} = i_{1} mod i_{3}.

2. Definition for Radix-2^{k}, k-SD

Let us consider n. The functor Radixn yields a natural number and is defined by:

(Def. 1) Radixn = 2^{n}.

Let us consider k. The functor k −SD yields a set and is defined by: (Def. 2) k −SD = {e; e ranges over integers: e Radixk−1 ∧ e − Radixk+1}.

The following propositions are true:

(9) Radixn ≠ 0.

(10) For every e holds e ∈ 0 −SD iff e = 0.

(11) 0 −SD = {0}.

(12) k −SD ⊆ k + 1 −SD .

(13) If e ∈ k −SD , then e is an integer.

(14) k −SD ⊆ ℤ.

(15) If i_{1} ∈ k −SD , then i_{1} Radixk − 1 and i_{1} − Radixk + 1.

(16) 0 ∈ k −SD .

Let us consider k. Note that k −SD is non empty. Let us consider k. Then k −SD is a non empty subset of ℤ.

definitions of radix-2^{k} signed-digit … 73

3. Functions for Generating Radix-2^{k} SD Numbers from Natural Numbers and Natural Numbers from Radix-2^{k} SD Numbers

In the sequel a denotes a tuple of n and k −SD . We now state the proposition

(18)^{1} If i ∈ Segn, then a(i) is an element of k −SD .

Let i, k, n be natural numbers and let x be a tuple of n and k −SD . The functor DigA(x, i) yields an integer and is defined by:

(Def. 3)(i) DigA(x, i) = x(i) if i ∈ Segn, (ii) DigA(x, i) = 0 if i = 0. Let i, k, n be natural numbers and let x be a tuple of n and k −SD . The

functor DigB(x, i) yielding an element of ℤ is defined as follows:

(Def. 4) DigB(x, i) = DigA(x, i).

One can prove the following propositions:

(19) If i ∈ Segn, then DigA(a, i) is an element of k −SD .

(20) For every tuple x of 1 and ℤ such that π_{1}x = m holds x = ⟨m⟩.

Let i, k, n be natural numbers and let x be a tuple of n and k −SD . The functor SubDigit(x, i, k) yielding an element of ℤ is defined by:

(Def. 5) SubDigit(x, i, k) = ((Radixk)_{ℕ}^{i− ′1}) · DigB(x, i).

Let n, k be natural numbers and let x be a tuple of n and k −SD . The functor DigitSDx yielding a tuple of n and ℤ is defined as follows:

(Def. 6) For every natural number i such that i ∈ Segn holds π_{i} DigitSDx =

SubDigit(x, i, k). Let n, k be natural numbers and let x be a tuple of n and k −SD . The

functor SDDecx yields an integer and is defined as follows:

(Def. 7) SDDecx = ∑ DigitSDx.

Let i, k, x be natural numbers. The functor DigitDC(x, i, k) yielding an element of k −SD is defined as follows:

(Def. 8) DigitDC(x, i, k) = (x mod (Radixk)_{ℕ}^{i}) ÷ (Radixk)_{ℕ}^{i− ′1}.

Let k, n, x be natural numbers. The functor DecSD(x, n, k) yields a tuple of n and k −SD and is defined as follows:

(Def. 9) For every natural number i such that i ∈ Segn holds DigA(DecSD(x, n, k), i) = DigitDC(x, i, k).

1The proposition (17) has been removed.

4. Definition for Carry and Data Components of Addition

Let x be an integer. The functor SD Add Carryx yielding an integer is defined as follows:

(Def. 10) SD Add Carryx = \{ \begin{matrix}1, if x > 2, \\ − 1, if x < − 2, \\ 0, otherwise. \end{matrix}

One can prove the following proposition

(21) SD Add Carry0 = 0.

Let x be an integer and let k be a natural number. The functor SD Add Data(x, k) yields an integer and is defined by:

(Def. 11) SD Add Data(x, k) = x − SD Add Carryx · Radixk.

Next we state two propositions:

(22) SD Add Data(0, k) = 0.

(23) If k 2 and i_{1} ∈ k −SD and i_{2} ∈ k −SD , then − Radixk + 2 SD Add Data(i_{1} + i_{2}, k) and SD Add Data(i_{1} + i_{2}, k) Radixk − 2.

5. Definition for Checking whether or not a Natural Number can be Expressed as n Digits Radix-2^{k} SD Number

Let n, x, k be natural numbers. We say that x is represented by n, k if and only if:

(Def. 12) x < (Radixk)_{ℕ}^{n}.

Next we state four propositions:

(24) If m is represented by 1, k, then DigA(DecSD(m, 1, k), 1) = m.

(25) For every n such that n 1 and for every m such that m is represented by n, k holds m = SDDecDecSD(m, n, k).

(26) If k 2 and m is represented by 1, k and n is represented by 1, k, then SD Add CarryDigA(DecSD(m, 1, k), 1) + DigA(DecSD(n, 1, k), 1) = SD Add Carrym + n.

(27) If m is represented by n + 1, k, then DigA(DecSD(m, n + 1, k), n + 1) = m ÷ (Radixk)_{ℕ}^{n}.

definitions of radix-2^{k} signed-digit … 75

6. Definition for Addition Operation for a High-Speed Adder Algorithm on Radix-2^{k} SD Number

Let k, i, n be natural numbers and let x, y be tuples of n and k −SD . Let us assume that i ∈ Segn and k 2. The functor Add(x, y, i, k) yields an element of k −SD and is defined as follows:

(Def. 13) Add(x, y, i, k) = SD Add Data(DigA(x, i)+DigA(y, i), k)+SD Add Carry DigA(x, i − ′1) + DigA(y, i − ′1).

Let n, k be natural numbers and let x, y be tuples of n and k −SD . The functor x′+′y yielding a tuple of n and k −SD is defined by:

(Def. 14) For every i such that i ∈ Segn holds DigA(x′+′y, i) = Add(x, y, i, k). One can prove the following two propositions:

(28) If k 2 and m is represented by 1, k and n is represented by 1, k, then SDDecDecSD(m, 1, k)′+′DecSD(n, 1, k) = SD Add Data(m + n, k).

(29) Let given n. Suppose n 1. Let given k, x, y. Suppose k 2 and x is represented by n, k and y is represented by n, k. Then x + y = SDDecDecSD(x, n, k)′+′DecSD(y, n, k) + ((Radixk)_{ℕ}^{n}) · SD Add CarryDigA(DecSD(x, n, k), n) + DigA(DecSD(y, n, k), n).

References

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

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

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

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

[5] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

[6] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Euler’s Theorem and small Fermat’s Theorem. Formalized Mathematics, 7(1):123–126, 1998.

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

[8] Andrzej Kondracki. The chinese remainder theorem. Formalized Mathematics, 6(4):573– 577, 1997.

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

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

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

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

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

76 yoshinori fujisawa and yasushi fuwa

Retracts and Inheritance^{1}

Grzegorz Bancerek University of Białystok

MML Identifier: YELLOW16.

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

1. Poset Retracts

The following three propositions are true:

(1) For all binary relations a, b holds a · b = a b.

(2) Let X be a set, L be a non empty relational structure, S be a non empty relational substructure of L, f, g be functions from X into the carrier of S, and f′, g′be functions from X into the carrier of L. If f′= f and g′= g and f g, then f′ g′.

(3) Let X be a set, L be a non empty relational structure, S be a full non empty relational substructure of L, f, g be functions from X into the carrier of S, and f′, g′be functions from X into the carrier of L. If f′= f and g′= g and f′ g′, then f g.

Let S be a non empty relational structure and let T be a non empty reflexive antisymmetric relational structure. Note that there exists a map from S into T which is directed-sups-preserving and monotone.

The following proposition is true

(4) For all functions f, g such that f is idempotent and rngg ⊆ rngf and rngg ⊆ domf holds f · g = g.

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

Let S be a 1-sorted structure. Note that there exists a map from S into S which is idempotent.

One can prove the following propositions:

(5) For every up-complete non empty poset L holds every directed-sups-inheriting full non empty relational substructure of L is up-complete.

(6) Let L be an up-complete non empty poset and f be a map from L into L. Suppose f is idempotent and directed-sups-preserving. Then Imf is directed-sups-inheriting.

(7) Let T be an up-complete non empty poset and S be a directed-sups-inheriting full non empty relational substructure of T. Then incl(S, T) is directed-sups-preserving.

(8) Let S, T be non empty relational structures, f be a map from T into S, and g be a map from S into T. If f · g = id_{S}, then rngf = the carrier of S.

(9) Let T be a non empty relational structure, S be a non empty relational substructure of T, and f be a map from T into S. If f · incl(S, T) = id_{S}, then f is an idempotent map from T into T.

Let S, T be non empty posets and let f be a function. We say that f is a retraction of T into S if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) f is a directed-sups-preserving map from T into S, (ii) f↾the carrier of S = id_{S}, and (iii) S is a directed-sups-inheriting full relational substructure of T.

We say that f is a UPS retraction of T into S if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) f is a directed-sups-preserving map from T into S, and

(ii) there exists a directed-sups-preserving map g from S into T such that

f · g = id_{S}. Let S, T be non empty posets. We say that S is a retract of T if and only if:

(Def. 3) There exists a map f from T into S such that f is a retraction of T into

S. We say that S is a UPS retract of T if and only if:

(Def. 4) There exists a map f from T into S such that f is a UPS retraction of

T into S. The following propositions are true:

(10) For all non empty posets S, T and for every function f such that f is a retraction of T into S holds f · incl(S, T) = id_{S}.

(11) Let S be a non empty poset, T be an up-complete non empty poset, and f be a function. Suppose f is a retraction of T into S. Then f is a UPS retraction of T into S.

(12) Let S, T be non empty posets and f be a function. If f is a retraction of T into S, then rngf = the carrier of S.

(13) Let S, T be non empty posets and f be a function. If f is a UPS retraction of T into S, then rngf = the carrier of S.

(14) Let S, T be non empty posets and f be a function. Suppose f is a retraction of T into S. Then f is an idempotent map from T into T.

(15) Let T, S be non empty posets and f be a map from T into T. Suppose f is a retraction of T into S. Then Imf = the relational structure of S.

(16) Let T be an up-complete non empty poset, S be a non empty poset, and f be a map from T into T. Suppose f is a retraction of T into S. Then f is directed-sups-preserving and projection.

(17) Let S, T be non empty reflexive transitive relational structures and f be a map from S into T. Then f is isomorphic if and only if the following conditions are satisfied:

(i) f is monotone, and (ii) there exists a monotone map g from T into S such that f · g = id_{T} and

g · f = id_{S}.

(18) Let S, T be non empty posets. Then S and T are isomorphic if and only if there exists a monotone map f from S into T and there exists a monotone map g from T into S such that f · g = id_{T} and g · f = id_{S}.

(19) Let S, T be up-complete non empty posets. Suppose S and T are isomorphic. Then S is a UPS retract of T and T is a UPS retract of S.

(20) Let T, S be non empty posets, f be a monotone map from T into S, and g be a monotone map from S into T. Suppose f · g = id_{S}. Then there exists a projection map h from T into T such that h = g · f and h↾the carrier of Imh = id_{Im h} and S and Imh are isomorphic.

(21) Let T be an up-complete non empty poset, S be a non empty poset, and f be a function. Suppose f is a UPS retraction of T into S. Then there exists a directed-sups-preserving projection map h from T into T such that h is a retraction of T into Imh and S and Imh are isomorphic.

(22) For every up-complete non empty poset L and for every non empty poset S such that S is a retract of L holds S is up-complete.

(23) For every complete non empty poset L and for every non empty poset S such that S is a retract of L holds S is complete.

(24) Let L be a continuous complete lattice and S be a non empty poset. If S is a retract of L, then S is continuous.

(25) Let L be an up-complete non empty poset and S be a non empty poset. If S is a UPS retract of L, then S is up-complete.

(26) Let L be a complete non empty poset and S be a non empty poset. If S is a UPS retract of L, then S is complete.

(27) Let L be a continuous complete lattice and S be a non empty poset. If S is a UPS retract of L, then S is continuous.

(28) Let L be a relational structure, S be a full relational substructure of L, and R be a relational substructure of S. Then R is full if and only if R is a full relational substructure of L.

(29) Let L be a non empty transitive relational structure and S be a directed-sups-inheriting non empty full relational substructure of L. Then every directed-sups-inheriting non empty relational substructure of S is a directed-sups-inheriting relational substructure of L.

(30) Let L be an up-complete non empty poset and S_{1}, S_{2} be directed-sups-inheriting full non empty relational substructures of L. Suppose S_{1} is a relational substructure of S_{2}. Then S_{1} is a directed-sups-inheriting full relational substructure of S_{2}.

Let X, Y be non empty topological spaces. One can check that every continuous map from X into Y is continuous.

2. Products

Let R be a binary relation. We say that R is poset-yielding if and only if: (Def. 5) For every set x such that x ∈ rngR holds x is a poset.

Let us observe that every binary relation which is poset-yielding is also relational structure yielding and reflexive-yielding.

Let X be a non empty set, let L be a non empty relational structure, let i be an element of X, and let Y be a subset of L^{X}. Then π_{i}Y is a subset of L.

Let X be a set and let S be a poset. Note that X ⟼→ S is poset-yielding. Let I be a set. Observe that there exists a many sorted set indexed by I

which is poset-yielding and nonempty.

Let I be a non empty set and let J be a poset-yielding nonempty many sorted set indexed by I. Note that ∏ J is transitive and antisymmetric.

Let I be a non empty set, let J be a poset-yielding nonempty many sorted set indexed by I, and let i be an element of I. Then J(i) is a non empty poset.

Next we state a number of propositions:

(31) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, f be an element of ∏ J, and X be a subset of ∏ J. Then f X if and only if for every element i of I holds f(i) π_{i}X.

(32) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, f be an element of ∏ J, and X be a subset of ∏ J. Then f X if and only if for every element i of I holds f(i) π_{i}X.

(33) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. Then sup X exists in ∏ J if and only if for every element i of I holds sup π_{i}X exists in J(i).

(34) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. Then inf X exists in ∏ J if and only if for every element i of I holds inf π_{i}X exists in J(i).

(35) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. If sup X exists in ∏ J, then for every element i of I holds (supX)(i) = supπ_{i}X.

(36) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. If inf X exists in ∏ J, then for every element i of I holds (inf X)(i) = inf π_{i}X.

(37) Let I be a non empty set, J be a relational structure yielding nonempty reflexive-yielding many sorted set indexed by I, X be a directed subset of ∏ J, and i be an element of I. Then π_{i}X is directed.

(38) Let I be a non empty set and J, K be relational structure yielding nonempty many sorted sets indexed by I. Suppose that for every element i of I holds K(i) is a relational substructure of J(i). Then ∏ K is a relational substructure of ∏ J.

(39) Let I be a non empty set and J, K be relational structure yielding nonempty many sorted sets indexed by I. Suppose that for every element i of I holds K(i) is a full relational substructure of J(i). Then ∏ K is a full relational substructure of ∏ J.

(40) Let L be a non empty relational structure, S be a non empty relational substructure of L, and X be a set. Then S^{X} is a relational substructure of L^{X}.

(41) Let L be a non empty relational structure, S be a full non empty relational substructure of L, and X be a set. Then S^{X} is a full relational substructure of L^{X}.

3. Inheritance

Let S, T be non empty relational structures and let X be a set. We say that S inherits sup of X from T if and only if:

(Def. 6) If sup X exists in T, then ⨆_{T} X ∈ the carrier of S.

We say that S inherits inf of X from T if and only if: (Def. 7) If inf X exists in T, then ⌈−⌉_{T} X ∈ the carrier of S.

Next we state two propositions:

(42) Let T be a non empty transitive relational structure, S be a full non empty relational substructure of T, and X be a subset of S. Then S inherits sup of X from T if and only if if sup X exists in T, then sup X exists in S and supX = ⨆_{T} X.

(43) Let T be a non empty transitive relational structure, S be a full non empty relational substructure of T, and X be a subset of S. Then S inherits inf of X from T if and only if if inf X exists in T, then inf X exists in S and inf X = ⌈−⌉_{T} X.

In this article we present several logical schemes. The scheme ProductSupsInher deals with a non empty set A, poset-yielding nonempty many sorted sets B, C indexed by A, and and states that:

For every subset X of ∏ C such that P[X, ∏ C] holds ∏ C inherits sup of X from ∏ B

provided the following conditions are satisfied:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L],

• For every subset X of ∏ C such that P[X, ∏ C] and for every element i of A holds P[π_{i}X, C(i)],

• For every element i of A holds C(i) is a full relational substructure of B(i), and

• For every element i of A and for every subset X of C(i) such that P[X, C(i)] holds C(i) inherits sup of X from B(i).

The scheme PowerSupsInherit deals with a non empty set A, a non empty poset B, a non empty full relational substructure C of B, and and states that:

For every subset X of C^{A} such that P[X, C^{A}] holds C^{A} inherits sup of X from B^{A}

provided the following requirements are met:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L],

• For every subset X of C^{A} such that P[X, C^{A}] and for every element i of A holds P[π_{i}X, C], and

• For every subset X of C such that P[X, C] holds C inherits sup of X from B.

The scheme ProductInfsInher deals with a non empty set A, poset-yielding nonempty many sorted sets B, C indexed by A, and and states that:

For every subset X of ∏ C such that P[X, ∏ C] holds ∏ C inherits inf of X from ∏ B

provided the parameters meet the following conditions:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L], • For every subset X of ∏ C such that P[X, ∏ C] and for every

element i of A holds P[π_{i}X, C(i)],

• For every element i of A holds C(i) is a full relational substructure of B(i), and

• For every element i of A and for every subset X of C(i) such that P[X, C(i)] holds C(i) inherits inf of X from B(i).

The scheme PowerInfsInherit deals with a non empty set A, a non empty poset B, a non empty full relational substructure C of B, and and states that:

For every subset X of C^{A} such that P[X, C^{A}] holds C^{A} inherits inf of X from B^{A}

provided the following conditions are satisfied:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L],

• For every subset X of C^{A} such that P[X, C^{A}] and for every element i of A holds P[π_{i}X, C], and

• For every subset X of C such that P[X, C] holds C inherits inf of X from B.

Let I be a set, let L be a non empty relational structure, let X be a non empty subset of L^{I}, and let i be a set. Observe that π_{i}X is non empty.

The following proposition is true

(44) Let L be a non empty poset, S be a directed-sups-inheriting non empty full relational substructure of L, and X be a non empty set. Then S^{X} is a directed-sups-inheriting relational substructure of L^{X}.

Let I be a non empty set, let J be a relational structure yielding nonempty many sorted set indexed by I, let X be a non empty subset of ∏ J, and let i be a set. Observe that π_{i}X is non empty.

One can prove the following proposition

(45) For every non empty set X and for every up-complete non empty poset

L holds L^{X} is up-complete. Let L be an up-complete non empty poset and let X be a non empty set.

Note that L^{X} is up-complete.

4. Topological Retracts

Let T be a topological space. Note that the topology of T is non empty. We now state a number of propositions:

(46) Let T be a non empty topological space, S be a non empty subspace of T, and f be a continuous map from T into S. If f is a retraction, then rngf = the carrier of S.

(47) Let T be a non empty topological space, S be a non empty subspace of T, and f be a continuous map from T into S. If f is a retraction, then f is idempotent.

(48) Let T be a non empty topological space and V be an open subset of T. Then χ_{V,the carrier of T} is a continuous map from T into the Sierpiński space.

(49) Let T be a non empty topological space and x, y be elements of T. Suppose that for every open subset W of T such that y ∈ W holds x ∈ W. Then [0 ⟼→ y, 1 ⟼→ x] is a continuous map from the Sierpiński space into T.

(50) Let T be a non empty topological space, x, y be elements of T, and V be an open subset of T. Suppose x ∈ V and y ∉ V. Then χ_{V,the carrier of T} · [0 ⟼→ y, 1 ⟼→ x] = id_{the Sierpiński space}.

(51) Let T be a non empty 1-sorted structure, V, W be subsets of T, S be a topological augmentation of 2_{⊆}^{1} , and f, g be maps from T into S. Suppose f = χ_{V,the carrier of T} and g = χ_{W,the carrier of T} . Then V ⊆ W if and only if f g.

(52) Let L be a non empty relational structure, X be a non empty set, and R be a full non empty relational substructure of L^{X}. Suppose that for every set a holds a is an element of R iff there exists an element x of L such that a = X ⟼→ x. Then L and R are isomorphic.

(53) Let S, T be non empty topological spaces. Then S and T are homeomorphic if and only if there exists a continuous map f from S into T and there exists a continuous map g from T into S such that f · g = id_{T} and g · f = id_{S}.

(54) Let T, S, R be non empty topological spaces, f be a map from T into S, g be a map from S into T, and h be a map from S into R. If f · g = id_{S} and h is a homeomorphism, then h · f · (g · h^{−1} ) = id_{R}.

(55) Let T, S, R be non empty topological spaces. Suppose S is a topological retract of T and S and R are homeomorphic. Then R is a topological retract of T.

(56) For every non empty topological space T and for every non empty subspace S of T holds incl(S, T) is continuous.

(57) Let T be a non empty topological space, S be a non empty subspace of T, and f be a continuous map from T into S. If f is a retraction, then f · incl(S, T) = id_{S}.

(58) Let T be a non empty topological space and S be a non empty subspace of T. If S is a retract of T, then S is a topological retract of T.

(59) Let R, T be non empty topological spaces. Then R is a topological retract of T if and only if there exists a non empty subspace S of T such that S

retracts and inheritance 85

is a retract of T and S and R are homeomorphic.

References

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

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

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

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

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

[6] Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1):35–43, 1998.

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

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

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

[10] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

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

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

[13] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

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

[15] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57–62, 1998.

[16] Andrzej Kondracki. Mostowski’s fundamental operations - Part I. Formalized Mathematics, 2(3):371–375, 1991.

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

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

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

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

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

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

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

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

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

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

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

86 grzegorz bancerek

Technical Preliminaries to Algebraic Specifications

Grzegorz Bancerek University of Białystok

MML Identifier: ALGSPEC1.

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

1. Preliminaries

One can prove the following propositions:

(1) For all functions f, g, h such that domf ∩ domg ⊆ domh holds f+·g+·h = g+·f+·h.

(2) For all functions f, g, h such that f ⊆ g and rngh ∩ domg ⊆ domf holds g · h = f · h.

(3) For all functions f, g, h such that domf ⊆ rngg and domf misses rngh and g^{∘} domh misses domf holds f · (g+·h) = f · g.

(4) For all functions f_{1}, f_{2}, g_{1}, g_{2} such that f_{1} ≈ f_{2} and g_{1} ≈ g_{2} holds f_{1} · g_{1} ≈ f_{2} · g_{2}.

(5) Let X_{1}, Y_{1}, X_{2}, Y_{2} be non empty sets, f be a function from X_{1} into X_{2}, and g be a function from Y_{1} into Y_{2}. If f ⊆ g, then f^{∗} ⊆ g^{∗}.

(6) Let X_{1}, Y_{1}, X_{2}, Y_{2} be non empty sets, f be a function from X_{1} into X_{2}, and g be a function from Y_{1} into Y_{2}. If f ≈ g, then f^{∗} ≈ g^{∗}.

Let X be a set and let f be a function. The functor X -indexingf yielding a many sorted set indexed by X is defined as follows:

(Def. 1) X -indexingf = id_{X}+·f↾X. We now state a number of propositions:

(7) For every set X and for every function f holds rng(X -indexingf) = (X \ domf) ∪ f^{∘}X.

(8) For every non empty set X and for every function f and for every element x of X holds (X -indexingf)(x) = (id_{X}+·f)(x).

(9) For all sets X, x and for every function f such that x ∈ X holds if x ∈ domf, then (X -indexingf)(x) = f(x) and if x ∉ domf, then (X -indexingf)(x) = x.

(10) For every set X and for every function f such that domf = X holds X -indexingf = f.

(11) For every set X and for every function f holds X -indexing(X -indexingf) = X -indexingf.

(12) For every set X and for every function f holds X -indexing(id_{X}+·f) = X -indexingf.

(13) For every set X and for every function f such that f ⊆ id_{X} holds X -indexingf = id_{X}.

(14) For every set X holds X -indexing∅ = id_{X}.

(15) For every set X and for every function f holds X -indexingf↾X = X -indexingf.

(16) For every set X and for every function f such that X ⊆ domf holds X -indexingf = f↾X.

(17) For every function f holds ∅ -indexingf = ∅.

(18) For all sets X, Y and for every function f such that X ⊆ Y holds (Y -indexingf)↾X = X -indexingf.

(19) For all sets X, Y and for every function f holds (X ∪ Y) -indexingf = (X -indexingf)+·(Y -indexingf).

(20) For all sets X, Y and for every function f holds X -indexingf ≈ Y -indexingf.

(21) For all sets X, Y and for every function f holds (X ∪ Y) -indexingf = (X -indexingf) ∪ (Y -indexingf).

(22) For every non empty set X and for all functions f, g such that rngg ⊆ X holds (X -indexingf) · g = (id_{X}+·f) · g.

(23) For all functions f, g such that domf misses domg and rngg misses

domf and for every set X holds f · (X -indexingg) = f↾X. Let f be a function. A function is called a rng-retraction of f if:

(Def. 2) domit = rngf and f · it = id_{rng f}.

We now state several propositions:

(24) For every function f and for every rng-retraction g of f holds rngg ⊆ domf.

(25) Let f be a function, g be a rng-retraction of f, and x be a set. If x ∈ rngf, then g(x) ∈ domf and f(g(x)) = x.

(26) For every function f such that f is one-to-one holds f^{−1} is a rngretraction of f.

(27) For every function f such that f is one-to-one and for every rngretraction g of f holds g = f^{−1} .

(28) Let f_{1}, f_{2} be functions. Suppose f_{1} ≈ f_{2}. Let g_{1} be a rng-retraction of f_{1} and g_{2} be a rng-retraction of f_{2}. Then g_{1}+·g_{2} is a rng-retraction of f_{1}+·f_{2}.

(29) Let f_{1}, f_{2} be functions. Suppose f_{1} ⊆ f_{2}. Let g_{1} be a rng-retraction of f_{1}. Then there exists a rng-retraction g_{2} of f_{2} such that g_{1} ⊆ g_{2}.

2. Replacement in Signature

Let S be a non empty non void many sorted signature and let f, g be functions. We say that f and g form a replacement in S if and only if the condition (Def. 3) is satisfied.

(Def. 3) Let o_{1}, o_{2} be operation symbols of S. Suppose (id_{the operation symbols of S}+·g)

(o_{1}) = (id_{the operation symbols of S}+·g)(o_{2}). Then

(i) (id_{the carrier of S}+·f) · Arity(o_{1}) = (id_{the carrier of S}+·f) · Arity(o_{2}), and (ii) (id_{the carrier of S}+·f)(the result sort of o_{1}) = (id_{the carrier of S}+·f)(the

result sort of o_{2}). One can prove the following propositions:

(30) Let S be a non empty non void many sorted signature and f, g be functions. Then f and g form a replacement in S if and only if for all operation symbols o_{1}, o_{2} of S such that ((the operation symbols of S) -indexingg)(o_{1}) = ((the operation symbols of S) -indexingg)(o_{2}) holds ((the carrier of S) -indexingf)·Arity(o_{1}) = ((the carrier of S) -indexingf)· Arity(o_{2}) and ((the carrier of S) -indexingf)(the result sort of o_{1}) = ((the carrier of S) -indexingf)(the result sort of o_{2}).

(31) Let S be a non empty non void many sorted signature and f, g be functions. Then f and g form a replacement in S if and only if (the carrier of S) -indexingf and (the operation symbols of S) -indexingg form a replacement in S.

In the sequel S, S′ denote non void signatures and f, g denote functions. One can prove the following four propositions:

(32) If f and g form morphism between S and S′, then f and g form a replacement in S.

(33) f and ∅ form a replacement in S.

(34) If g is one-to-one and (the operation symbols of S)∩rngg ⊆ domg, then f and g form a replacement in S.

(35) If g is one-to-one and rngg misses the operation symbols of S, then f

and g form a replacement in S. Let X be a set, let Y be a non empty set, let a be a function from Y into

X^{∗}, and let r be a function from Y into X. Observe that ⟨X, Y, a, r⟩ is non void.

Let S be a non empty non void many sorted signature and let f, g be functions. Let us assume that f and g form a replacement in S. The functor S with-replacement(f, g) yields a strict non empty non void many sorted signature and is defined by the conditions (Def. 4).

(Def. 4)(i) (The carrier of S) -indexingf and (the operation symbols of

S) -indexingg form morphism between S and S with-replacement(f, g), (ii) the carrier of S with-replacement(f, g) = rng((the carrier of

S) -indexingf), and (iii) the operation symbols of S with-replacement(f, g) = rng((the operation

symbols of S) -indexingg). The following propositions are true:

(36) Let S_{1}, S_{2} be non void signatures, f be a function from the carrier of S_{1} into the carrier of S_{2}, and g be a function. Suppose f and g form morphism between S_{1} and S_{2}. Then f^{∗} · the arity of S_{1} = (the arity of S_{2}) · g.

(37) Suppose f and g form a replacement in S. Then (the carrier of S) -indexingf is a function from the carrier of S into the carrier of S with-replacement(f, g).

(38) Suppose f and g form a replacement in S. Let f′ be a function from the carrier of S into the carrier of S with-replacement(f, g). Suppose f′= (the carrier of S) -indexingf. Let g′ be a rng-retraction of (the operation symbols of S) -indexingg. Then the arity of S with-replacement(f, g) = f^{′∗}· the arity of S · g′.

(39) Suppose f and g form a replacement in S. Let g′ be a rng-retraction of (the operation symbols of S) -indexingg. Then the result sort of S with-replacement(f, g) = ((the carrier of S) -indexingf) · the result sort of S · g′.

(40) If f and g form morphism between S and S′, then S with-replacement(f, g) is a subsignature of S′.

(41) f and g form a replacement in S if and only if (the carrier of S) -indexingf and (the operation symbols of S) -indexingg form morphism between S and S with-replacement(f, g).

(42) Suppose domf ⊆ the carrier of S and domg ⊆ the operation symbols of S and f and g form a replacement in S. Then id_{the carrier of S}+·f and id_{the operation symbols of S}+·g form morphism be-

tween S and S with-replacement(f, g).

(43) Suppose domf = the carrier of S and domg = the operation symbols of S and f and g form a replacement in S. Then f and g form morphism between S and S with-replacement(f, g).

(44) If f and g form a replacement in S, then S with-replacement((the carrier of S) -indexingf, g) = S with-replacement(f, g).

(45) If f and g form a replacement in S, then S with-replacement(f, (the operation symbols of S) -indexingg) = S with-replacement(f, g).

3. Signature Extensions

Let S be a signature. A signature is called an extension of S if: (Def. 5) S is a subsignature of it.

The following propositions are true:

(46) For all signatures S, E holds S is a subsignature of E iff E is an extension of S.

(47) Every signature S is an extension of S.

(48) For every signature S_{1} and for every extension S_{2} of S_{1} holds every extension of S_{2} is an extension of S_{1}.

(49) For all non empty signatures S_{1}, S_{2} such that S_{1} ≈ S_{2} holds S_{1}+·S_{2} is an extension of S_{1}.

(50) For all non empty signatures S_{1}, S_{2} holds S_{1}+·S_{2} is an extension of S_{2}.

(51) Let S_{1}, S_{2}, S be non empty many sorted signatures and f_{1}, g_{1}, f_{2}, g_{2} be functions. Suppose f_{1} ≈ f_{2} and f_{1} and g_{1} form morphism between S_{1} and S and f_{2} and g_{2} form morphism between S_{2} and S. Then f_{1}+·f_{2} and g_{1}+·g_{2} form morphism between S_{1}+·S_{2} and S.

(52) Let S_{1}, S_{2}, E be non empty signatures. Then E is an extension of S_{1} and an extension of S_{2} if and only if S_{1} ≈ S_{2} and E is an extension of S_{1}+·S_{2}.

Let S be a non empty signature. One can check that every extension of S is non empty.

Let S be a non void signature. One can verify that every extension of S is non void.

One can prove the following proposition

(53) For all signatures S, T such that S is empty holds T is an extension of

S. Let S be a signature. One can check that there exists an extension of S

which is non empty, non void, and strict.

The following three propositions are true:

(54) Let S be a non void signature and E be an extension of S. Suppose f and g form a replacement in E. Then f and g form a replacement in S.

(55) Let S be a non void signature and E be an extension of S. Suppose f and g form a replacement in E. Then E with-replacement(f, g) is an extension of S with-replacement(f, g).

(56) Let S_{1}, S_{2} be non void signatures. Suppose S_{1} ≈ S_{2}. Let f, g be functions. If f and g form a replacement in S_{1}+·S_{2}, then (S_{1}+·S_{2}) with-replacement(f, g) = (S_{1} with-replacement(f, g))+·(S_{2} with-replacement(f, g)).

4. Algebras

Algebra is defined by: (Def. 6) There exists a non void signature S such that it is a feasible algebra over

S. Let S be a signature. An algebra is called an algebra of S if:

(Def. 7) There exists a non void extension E of S such that it is a feasible algebra

over E. One can prove the following propositions:

(57) For every non void signature S holds every feasible algebra over S is an algebra of S.

(58) For every signature S and for every extension E of S holds every algebra of E is an algebra of S.

(59) Let S be a signature, E be a non empty signature, and A be an algebra over E. Suppose A is an algebra of S. Then the carrier of S ⊆ the carrier of E and the operation symbols of S ⊆ the operation symbols of E.

(60) Let S be a non void signature, E be a non empty signature, and A be an algebra over E. Suppose A is an algebra of S. Let o be an operation symbol of S. Then (the characteristics of A)(o) is a function from (the sorts of A)^{#}(Arity(o)) into (the sorts of A)(the result sort of o).

(61) Let S be a non empty signature, A be an algebra of S, and E be a non empty many sorted signature. If A is an algebra over E, then A is an algebra over E+·S.

(62) Let S_{1}, S_{2} be non empty signatures and A be an algebra over S_{1}. Suppose A is an algebra over S_{2}. Then the carrier of S_{1} = the carrier of S_{2} and the operation symbols of S_{1} = the operation symbols of S_{2}.

(63) For every non void signature S and for every non-empty disjoint algebra A over S holds the sorts of A are one-to-one.

technical preliminaries to algebraic … 93

(64) Let S be a non void signature, A be a disjoint algebra over S, and C_{1}, C_{2} be components of the sorts of A. Then C_{1} = C_{2} or C_{1} misses C_{2}.

(65) Let S, S′ be non void signatures and A be a non-empty disjoint algebra over S. Suppose A is an algebra over S′. Then the many sorted signature of S = the many sorted signature of S′.

(66) Let S′ be a non void signature and A be a non-empty disjoint algebra over S. If A is an algebra of S′, then S is an extension of S′.

References

[1] Grzegorz Bancerek. Minimal signature for partial algebra. Formalized Mathematics, 5(3):405–414, 1996.

[2] Grzegorz Bancerek. Translations, endomorphisms, and stable equational theories. Formalized Mathematics, 5(4):553–564, 1996.

[3] Grzegorz Bancerek. Algebra of morphisms. Formalized Mathematics, 6(2):303–310, 1997.

[4] Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279–287, 1997.

[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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[10] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part 1. Formalized Mathematics, 2(5):683–687, 1991.

[11] Yatsuka Nakamura and Grzegorz Bancerek. Combining of circuits. Formalized Mathematics, 5(2):283–295, 1996.

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

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

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

[15] Zinaida Trybulec and Halina Święczkowska. 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] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

94 grzegorz bancerek

Multivariate Polynomials with Arbitrary Number of Variables^{1}

Piotr Rudnicki University of Alberta Edmonton

Andrzej Trybulec University of Białystok

Summary. The goal of this article is to define multivariate polynomials in arbitrary number of indeterminates and then to prove that they constitute a ring (over appropriate structure of coefficients).

The introductory section includes quite a number of auxiliary lemmas related to many different parts of the MML. The second section characterizes the sequence flattening operation, introduced in [7], but so far lacking theorems about its fundamental properties.

We first define formal power series in arbitrary number of variables. The auxiliary concept on which the construction of formal power series is based is the notion of a bag. A bag of a set X is a natural function on X which is zero almost everywhere. The elements of X play the role of formal variables and a bag gives their exponents thus forming a power product. Series are defined for an ordered set of variables (we use ordinal numbers). A series in o variables over a structure S is a function assigning an element of the carrier of S (coefficient) to each bag of o.

We define the operations of addition, complement and multiplication for formal power series and prove their properties which depend on assumed properties of the structure from which the coefficients are taken. (We would like to note that proving associativity of multiplication turned out to be technically complicated.)

Polynomial is defined as a formal power series with finite number of non zero coefficients. In conclusion, the ring of polynomials is defined.

MML Identifier: POLYNOM1.

The terminology and notation used in this paper are introduced in the following articles: [24], [23], [10], [35], [1], [3], [7], [6], [11], [31], [15], [25], [12], [13], [8],

1This work has been supported by NSERC Grant OGP9207 and NATO CRG 951368.

[38], [29], [22], [5], [18], [2], [30], [33], [4], [28], [9], [36], [37], [32], [19], [26], [34], [27], [16], [21], [20], [17], and [14].

1. Basics

The following propositions are true:

(1) For all natural numbers i, j holds ·_{ℕ}(i, j) = i · j.

(2) Let X be a set, A be a non empty set, F be a binary operation on A, f be a function from X into A, and x be an element of A. Then dom(F^{∘}(f, x)) = X.

(3) For all natural numbers a, b, c holds a − ′b − ′c = a − ′(b + c).

(4) For every set X and for every binary relation R such that fieldR ⊆ X holds R is a binary relation on X.

(5) Let K be a non empty loop structure and p_{1}, p_{2} be finite sequences of elements of the carrier of K. If domp_{1} = domp_{2}, then dom(p_{1} + p_{2}) = domp_{1}.

(6) For every function f and for all sets x, y holds rng(f +·(x, y)) ⊆ rngf ∪ {y}.

Let A, B be sets, let f be a function from A into B, let x be a set, and let y be an element of B. Then f +·(x, y) is a function from A into B.

Let X be a set, let f be a many sorted set indexed by X, and let x, y be sets. Then f +·(x, y) is a many sorted set indexed by X.

Next we state the proposition

(7) For every one-to-one function f holds \overline{\overline{(f qua set)}} = \overline{\overline{rngf}} . Let A be a non empty set, let F, G be binary operations on A, and let z, u

be elements of A. Observe that ⟨A, F, G, z, u⟩ is non empty.

Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then π_{i}p is an element of D.

Let X be a set and let S be a 1-sorted structure. (Def. 1) A function from X into the carrier of S is said to be a function from X

into S. Let X be a set. Note that there exists an order in X which is linear-order

and well-ordering.

The following propositions are true:

(8) Let X be a non empty set, A be a non empty finite subset of X, R be an order in X, and x be an element of X. Suppose x ∈ A and R linearly orders A and for every element y of X such that y ∈ A holds ⟨⟨x, y⟩⟩∈ R. Then π_{1} SgmX(R, A) = x.

(9) Let X be a non empty set, A be a non empty finite subset of X, R be an order in X, and x be an element of X. Suppose x ∈ A and R linearly orders A and for every element y of X such that y ∈ A holds ⟨⟨y, x⟩⟩∈ R. Then π_{len SgmX(R,A)} SgmX(R, A) = x.

Let X be a non empty set, let A be a non empty finite subset of X, and let R be linear-order order in X. One can verify that SgmX(R, A) is non empty and one-to-one.

Let us observe that ∅ is finite sequence yielding. Let us observe that there exists a finite sequence which is finite sequence

yielding.

Let F, G be finite sequence yielding finite sequences. Then F ^{⌢} G is a finite sequence yielding finite sequence.

Let D be a set. Note that every finite sequence of elements of D^{∗} is finite sequence yielding.

Let i be a natural number and let f be a finite sequence. Note that i ↦ f is finite sequence yielding.

Let us observe that every function which is finite sequence yielding is also function yielding.

Let F be a finite sequence yielding finite sequence and let x be a set. Note that F (x) is finite sequence-like.

Let F be a finite sequence. Observe that \overline{\overline{F}} is finite sequence-like. Let us observe that there exists a finite sequence which is cardinal yielding. We now state the proposition

(10) Let f be a function. Then f is cardinal yielding if and only if for every

set y such that y ∈ rngf holds y is a cardinal number. Let F, G be cardinal yielding finite sequences. Note that F ^{⌢} G is cardinal

yielding.

Let us note that every finite sequence of elements of ℕ is cardinal yielding. Let us observe that there exists a finite sequence of elements of ℕ which is

cardinal yielding.

Let D be a set and let F be a finite sequence of elements of D^{∗}. Then \overline{\overline{F}} is a cardinal yielding finite sequence of elements of ℕ.

Let F be a finite sequence of elements of ℕ and let i be a natural number. Observe that F ↾i is cardinal yielding.

We now state the proposition

(11) For every function F and for every set X holds \overline{\overline{F ↾X}} = \overline{\overline{F}} ↾X.

Let F be an empty function. One can verify that \overline{\overline{F}} is empty. Next we state two propositions:

(12) For every set p holds \overline{\overline{⟨p⟩}}= ⟨\overline{\overline{p}}⟩.

(13) For all finite sequences F, G holds \overline{\overline{F ^{⌢} G}} = \overline{\overline{F}} ^{⌢} \overline{\overline{G}}.

Let X be a set. Note that ε_{X} is finite sequence yielding. Let f be a finite sequence. Observe that ⟨f⟩ is finite sequence yielding. One can prove the following proposition

(14) Let f be a function. Then f is finite sequence yielding if and only if for

every set y such that y ∈ rngf holds y is a finite sequence. Let F, G be finite sequence yielding finite sequences. One can verify that

F ^{⌢} G is finite sequence yielding.

Next we state four propositions:

(15) Let L be a non empty loop structure and F be a finite sequence of elements of (the carrier of L)^{∗}. Then dom∑ F = domF.

(16) Let L be a non empty loop structure and F be a finite sequence of elements of (the carrier of L)^{∗}. Then ∑(ε_{(the carrier of L)^{∗}}) = ε_{(the carrier of L)}.

(17) For every non empty loop structure L and for every element p of (the carrier of L)^{∗} holds ⟨∑ p⟩ = ∑⟨p⟩.

(18) Let L be a non empty loop structure and F, G be finite sequences of

elements of (the carrier of L)^{∗}. Then ∑(F ^{⌢} G) = (∑ F)^{⌢} ∑ G. Let L be a non empty groupoid, let a be an element of the carrier of L,

and let p be a finite sequence of elements of the carrier of L. The functor a · p yielding a finite sequence of elements of the carrier of L is defined by:

(Def. 2) dom(a·p) = domp and for every set i such that i ∈ domp holds π_{i}(a·p) =

a · π_{i}p. The functor p · a yielding a finite sequence of elements of the carrier of L is defined as follows:

(Def. 3) dom(p·a) = domp and for every set i such that i ∈ domp holds π_{i}(p·a) =

π_{i}p · a. The following propositions are true:

(19) Let L be a non empty groupoid and a be an element of the carrier of L. Then a · ε_{(the carrier of L)} = ε_{(the carrier of L)}.

(20) Let L be a non empty groupoid and a be an element of the carrier of L. Then ε_{(the carrier of L)} · a = ε_{(the carrier of L)}.

(21) For every non empty groupoid L and for all elements a, b of the carrier of L holds a · ⟨b⟩ = ⟨a · b⟩.

(22) For every non empty groupoid L and for all elements a, b of the carrier of L holds ⟨b⟩ · a = ⟨b · a⟩.

(23) Let L be a non empty groupoid, a be an element of the carrier of L, and p, q be finite sequences of elements of the carrier of L. Then a · (p^{⌢} q) = (a · p)^{⌢} (a · q).

(24) Let L be a non empty groupoid, a be an element of the carrier of L, and p, q be finite sequences of elements of the carrier of L. Then (p^{⌢} q) · a =

(p · a)^{⌢} (q · a). We now state two propositions:

(25) Let L be an add-associative right zeroed right complementable leftdistributive non empty double loop structure and x be an element of the carrier of L. Then 0_{L} · x = 0_{L}.

(26) Let L be an add-associative right zeroed right complementable rightdistributive non empty double loop structure and x be an element of the carrier of L. Then x · 0_{L} = 0_{L}.

One can verify that every non empty multiplicative loop with zero structure which is non degenerated is also non trivial.

Let us mention that there exists a non empty strict multiplicative loop with zero structure which is unital.

Let us observe that there exists a non empty strict double loop structure which is Abelian, add-associative, right zeroed, right complementable, associative, commutative, distributive, unital, and non trivial.

Next we state three propositions:

(27) Let L be an add-associative right zeroed right complementable unital right-distributive non empty double loop structure. If 0_{L} = 1_{L}, then L is trivial.

(28) Let L be an add-associative right zeroed right complementable unital distributive non empty double loop structure, a be an element of the carrier of L, and p be a finite sequence of elements of the carrier of L. Then ∑(a · p) = a · ∑ p.

(29) Let L be an add-associative right zeroed right complementable unital distributive non empty double loop structure, a be an element of the carrier of L, and p be a finite sequence of elements of the carrier of L. Then ∑(p · a) = ∑ p · a.

2. Sequence Flattening

Let D be a set and let F be an empty finite sequence of elements of D^{∗}. Observe that Flat(F) is empty.

One can prove the following propositions:

(30) For every set D and for every finite sequence F of elements of D^{∗} holds lenFlat(F) = ∑ \overline{\overline{F}} .

(31) Let D, E be sets, F be a finite sequence of elements of D^{∗}, and G be a finite sequence of elements of E^{∗}. If \overline{\overline{F}} = \overline{\overline{G}}, then lenFlat(F) = lenFlat(G).

(32) Let D be a set, F be a finite sequence of elements of D^{∗}, and k be a set. Suppose k ∈ domFlat(F). Then there exist natural numbers i, j such that i ∈ domF and j ∈ domF (i) and k = ∑ \overline{\overline{F ↾(i − ′1)}} + j and F (i)(j) = Flat(F)(k).

(33) Let D be a set, F be a finite sequence of elements of D^{∗}, and i, j be natural numbers. If i ∈ domF and j ∈ domF (i), then ∑ \overline{\overline{F ↾(i − ′1)}} + j ∈ domFlat(F) and F (i)(j) = Flat(F)(∑ \overline{\overline{F ↾(i − ′1)}} + j).

(34) Let L be an add-associative right zeroed right complementable non empty loop structure and F be a finite sequence of elements of (the carrier of L)^{∗}. Then ∑ Flat(F) = ∑ ∑ F.

(35) Let X, Y be non empty sets, f be a finite sequence of elements of X^{∗}, and v be a function from X into Y. Then (domf ⟼→ v) ∘ f is a finite sequence of elements of Y^{∗}.

(36) Let X, Y be non empty sets, f be a finite sequence of elements of X^{∗}, and v be a function from X into Y. Then there exists a finite sequence F of elements of Y^{∗} such that F = (domf ⟼→ v) ∘ f and v · Flat(f) = Flat(F).

3. Functions Yielding Natural Numbers

Let us note that ∅ is natural-yielding. One can check that there exists a function which is natural-yielding. Let f be a natural-yielding function and let x be a set. Then f(x) is a natural

number.

Let f be a natural-yielding function, let x be a set, and let n be a natural number. Observe that f +·(x, n) is natural-yielding.

Let X be a set. One can check that every function from X into ℕ is naturalyielding.

Let X be a set. Observe that there exists a many sorted set indexed by X which is natural-yielding.

Let X be a set and let b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. The functor b_{1} + b_{2} yields a many sorted set indexed by X and is defined as follows:

(Def. 5)^{2} For every set x holds (b_{1} + b_{2})(x) = b_{1}(x) + b_{2}(x). Let us note that the functor b_{1} + b_{2} is commutative. The functor b_{1} − ′b_{2} yields a many sorted set indexed by X and is defined by:

(Def. 6) For every set x holds (b_{1} − ′b_{2})(x) = b_{1}(x) − ′b_{2}(x). Next we state two propositions:

2The definition (Def. 4) has been removed.

(37) Let X be a set and b, b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. If for every set x such that x ∈ X holds b(x) = b_{1}(x) + b_{2}(x), then b = b_{1} + b_{2}.

(38) Let X be a set and b, b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. If for every set x such that x ∈ X holds b(x) = b_{1}(x) − ′b_{2}(x), then b = b_{1} − ′b_{2}.

Let X be a set and let b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. Observe that b_{1} + b_{2} is natural-yielding and b_{1} − ′b_{2} is natural-yielding.

The following two propositions are true:

(39) For every set X and for all natural-yielding many sorted sets b_{1}, b_{2}, b_{3} indexed by X holds (b_{1} + b_{2}) + b_{3} = b_{1} + (b_{2} + b_{3}).

(40) For every set X and for all natural-yielding many sorted sets b, c, d indexed by X holds b − ′c − ′d = b − ′(c + d).

4. The Support of a Function

Let f be a function. The functor supportf is defined as follows: (Def. 7) For every set x holds x ∈ supportf iff f(x) ≠ 0.

One can prove the following proposition

(41) For every function f holds supportf ⊆ domf.

Let f be a function. We say that f is finite-support if and only if: (Def. 8) supportf is finite.

We introduce f has finite-support as a synonym of f is finite-support. Let us mention that ∅ is finite-support. Let us note that every function which is finite is also finite-support. Let us observe that there exists a function which is natural-yielding, finite-

support, and non empty.

Let f be a finite-support function. Observe that supportf is finite. Let X be a set. Note that there exists a function from X into ℕ which is

finite-support.

Let f be a finite-support function and let x, y be sets. Observe that f +·(x, y) is finite-support.

Let X be a set. One can verify that there exists a many sorted set indexed by X which is natural-yielding and finite-support.

One can prove the following propositions:

(42) For every set X and for all natural-yielding many sorted sets b_{1}, b_{2} indexed by X holds support(b_{1} + b_{2}) = supportb_{1} ∪ supportb_{2}.

(43) For every set X and for all natural-yielding many sorted sets b_{1}, b_{2} indexed by X holds support(b_{1} − ′b_{2}) ⊆ supportb_{1}.

Let X be a non empty set, let S be a zero structure, and let f be a function from X into S. The functor Supportf yielding a subset of X is defined by:

(Def. 9) For every element x of X holds x ∈ Supportf iff f(x) ≠ 0_{S}.

Let X be a non empty set, let S be a zero structure, and let p be a function from X into S. We say that p is finite-Support if and only if:

(Def. 10) Supportp is finite.

We introduce p has finite-Support as a synonym of p is finite-Support.

5. Bags

Let X be a set. A bag of X is a natural-yielding finite-support many sorted set indexed by X.

Let X be a finite set. Observe that every many sorted set indexed by X is finite-support.

Let X be a set and let b_{1}, b_{2} be bag of X. Note that b_{1} + b_{2} is finite-support and b_{1} − ′b_{2} is finite-support.

The following proposition is true

(44) For every set X holds X ⟼→ 0 is a bag of X.

Let n be an ordinal number and let p, q be bag of n. The predicate p < q is defined as follows:

(Def. 11) There exists an ordinal number k such that p(k) < q(k) and for every

ordinal number l such that l ∈ k holds p(l) = q(l). Let us note that the predicate p < q is antisymmetric.

Next we state the proposition

(45) For every ordinal number n and for all bag p, q, r of n such that p < q

and q < r holds p < r. Let n be an ordinal number and let p, q be bag of n. The predicate p q is

defined as follows: (Def. 12) p < q or p = q.

Let us note that the predicate p q is reflexive.

The following propositions are true:

(46) For every ordinal number n and for all bag p, q, r of n such that p q and q r holds p r.

(47) For every ordinal number n and for all bag p, q, r of n such that p < q and q r holds p < r.

(48) For every ordinal number n and for all bag p, q, r of n such that p q and q < r holds p < r.

(49) For every ordinal number n and for all bag p, q of n holds p q or q p.

Let X be a set and let d, b be bag of X. We say that d divides b if and only if:

(Def. 13) For every set k holds d(k) b(k).

Let us note that the predicate d divides b is reflexive.

One can prove the following propositions:

(50) For every set n and for all bag d, b of n such that for every set k such that k ∈ n holds d(k) b(k) holds d divides b.

(51) For every ordinal number n and for all bag b_{1}, b_{2} of n such that b_{1} divides b_{2} holds (b_{2} − ′b_{1}) + b_{1} = b_{2}.

(52) For every set X and for all bag b_{1}, b_{2} of X holds (b_{2} + b_{1}) − ′b_{1} = b_{2}.

(53) For every ordinal number n and for all bag d, b of n such that d divides b holds d b.

(54) For every set n and for all bag b, b_{1}, b_{2} of n such that b = b_{1} + b_{2} holds

b_{1} divides b. Let X be a set. The functor BagsX is defined as follows:

(Def. 14) For every set x holds x ∈ BagsX iff x is a bag of X. Let X be a set. Then BagsX is a subset of BagsX. One can prove the following proposition

(55) Bags∅ = {∅}.

Let X be a set. Note that BagsX is non empty.

Let X be a set and let B be a non empty subset of BagsX. We see that the element of B is a bag of X.

Let n be a set, let L be a non empty 1-sorted structure, let p be a function from Bagsn into L, and let x be a bag of n. Then p(x) is an element of L.

Let X be a set. The functor EmptyBagX yielding an element of BagsX is defined by:

(Def. 15) EmptyBagX = X ⟼→ 0.

The following propositions are true:

(56) For all sets X, x holds (EmptyBagX)(x) = 0.

(57) For every set X and for every bag b of X holds b + EmptyBagX = b.

(58) For every set X and for every bag b of X holds b − ′EmptyBagX = b.

(59) For every set X and for every bag b of X holds EmptyBagX − ′b = EmptyBagX.

(60) For every set X and for every bag b of X holds b − ′b = EmptyBagX.

(61) For every set n and for all bag b_{1}, b_{2} of n such that b_{1} divides b_{2} and b_{2} − ′b_{1} = EmptyBagn holds b_{2} = b_{1}.

(62) For every set n and for every bag b of n such that b divides EmptyBagn holds EmptyBagn = b.

(63) For every set n and for every bag b of n holds EmptyBagn divides b.

(64) For every ordinal number n and for every bag b of n holds EmptyBagn b.

Let n be an ordinal number. The functor BagOrdern yields an order in Bagsn and is defined as follows:

(Def. 16) For all bag p, q of n holds ⟨⟨p, q⟩⟩∈ BagOrdern iff p q.

Let n be an ordinal number. Note that BagOrdern is linear-order. Let X be a set and let f be a function from X into ℕ. The functor NatMinorf

yielding a subset of ℕ^{X} is defined by the condition (Def. 17).

(Def. 17) Let g be a natural-yielding many sorted set indexed by X. Then g ∈ NatMinorf if and only if for every set x such that x ∈ X holds g(x) f(x).

Next we state the proposition

(65) For every set X and for every function f from X into ℕ holds f ∈ NatMinorf.

Let X be a set and let f be a function from X into ℕ. Observe that NatMinorf is non empty and functional.

Let X be a set and let f be a function from X into ℕ. One can verify that every element of NatMinorf is natural-yielding.

The following proposition is true

(66) For every set X and for every finite-support function f from X into ℕ holds NatMinorf ⊆ BagsX.

Let X be a set and let f be a finite-support function from X into ℕ. Then supportf is an element of FinX.

The following proposition is true

(67) For every non empty set X and for every finite-support function f from X into ℕ holds \overline{\overline{NatMinorf}} = ·_{ℕ}-∑_{support f}(+_{ℕ})^{∘}(f, 1).

Let X be a set and let f be a finite-support function from X into ℕ. One can verify that NatMinorf is finite.

Let n be an ordinal number and let b be a bag of n. The functor divisorsb yields a finite sequence of elements of Bagsn and is defined by the condition (Def. 18).

(Def. 18) There exists a non empty finite subset S of Bagsn such that divisorsb =

SgmX(BagOrdern, S) and for every bag p of n holds p ∈ S iff p divides b. Let n be an ordinal number and let b be a bag of n. One can check that

divisorsb is non empty and one-to-one.

The following four propositions are true:

(68) Let n be an ordinal number, i be a natural number, and b be a bag of n. If i ∈ domdivisorsb, then π_{i} divisorsb qua element of Bagsn divides b.

(69) For every ordinal number n and for every bag b of n holds π_{1} divisorsb = EmptyBagn and π_{len divisors b} divisorsb = b.

(70) Let n be an ordinal number, i be a natural number, and b, b_{1}, b_{2} be bag of n. If i > 1 and i < lendivisorsb, then π_{i} divisorsb ≠ EmptyBagn and π_{i} divisorsb ≠ b.

(71) For every ordinal number n holds divisorsEmptyBagn = ⟨EmptyBagn⟩. Let n be an ordinal number and let b be a bag of n. The functor decompb

yields a finite sequence of elements of (Bagsn)^{2} and is defined as follows: (Def. 19) domdecompb = domdivisorsb and for every natural number i and for

every bag p of n such that i ∈ domdecompb and p = π_{i} divisorsb holds π_{i} decompb = ⟨p, b − ′p⟩.

One can prove the following propositions:

(72) Let n be an ordinal number, i be a natural number, and b be a bag of n. If i ∈ domdecompb, then there exist bag b_{1}, b_{2} of n such that π_{i} decompb = ⟨b_{1}, b_{2}⟩ and b = b_{1} + b_{2}.

(73) Let n be an ordinal number and b, b_{1}, b_{2} be bag of n. If b = b_{1} + b_{2}, then there exists a natural number i such that i ∈ domdecompb and π_{i} decompb = ⟨b_{1}, b_{2}⟩.

(74) Let n be an ordinal number, i be a natural number, and b, b_{1}, b_{2} be bag of

n. If i ∈ domdecompb and π_{i} decompb = ⟨b_{1}, b_{2}⟩, then b_{1} = π_{i} divisorsb. Let n be an ordinal number and let b be a bag of n. Note that decompb is

non empty one-to-one and finite sequence yielding.

Let n be an ordinal number and let b be an element of Bagsn. One can verify that decompb is non empty one-to-one and finite sequence yielding.

Next we state four propositions:

(75) For every ordinal number n and for every bag b of n holds π_{1} decompb = ⟨EmptyBagn, b⟩ and π_{len decomp b} decompb = ⟨b, EmptyBagn⟩.

(76) Let n be an ordinal number, i be a natural number, and b, b_{1}, b_{2} be bag of n. If i > 1 and i < lendecompb and π_{i} decompb = ⟨b_{1}, b_{2}⟩, then b_{1} ≠ EmptyBagn and b_{2} ≠ EmptyBagn.

(77) For every ordinal number n holds decompEmptyBagn = ⟨⟨EmptyBagn, EmptyBagn⟩⟩.

(78) Let n be an ordinal number, b be a bag of n, and f, g be finite sequences

of elements of ((Bagsn)^{3})^{∗}. Suppose that

(i) domf = domdecompb, (ii) domg = domdecompb, (iii) for every natural number k such that k ∈ domf holds f(k) =

(decomp(π_{1}π_{k} decompb qua element of Bagsn))^{⌢}(lendecomp(π_{1}π_{k} decomp b qua element of Bagsn) ↦ ⟨π_{2}π_{k} decompb⟩), and (iv) for every natural number k such that k ∈ domg holds g(k) =

(lendecomp(π_{2}π_{k} decompb qua element of Bagsn) ↦ ⟨π_{1}π_{k} decompb⟩)^{⌢} decomp(π_{2}π_{k} decompb qua element of Bagsn). Then there exists a permutation p of domFlat(f) such that Flat(g) = Flat(f) · p.

6. Formal Power Series

Let X be a set and let S be a 1-sorted structure. (Def. 20) A function from BagsX into S is said to be a Series of X, S.

Let n be a set, let L be a right zeroed non empty loop structure, and let p, q be Series of n, L. The functor p + q yielding a Series of n, L is defined as follows:

(Def. 21) For every bag x of n holds (p + q)(x) = p(x) + q(x).

One can prove the following proposition

(79) Let n be a set, L be a right zeroed non empty loop structure, and p, q

be Series of n, L. Then Supportp + q ⊆ Supportp ∪ Supportq. Let n be a set, let L be an Abelian right zeroed non empty loop structure,

and let p, q be Series of n, L. Let us notice that the functor p+q is commutative.

Next we state the proposition

(80) Let n be a set, L be an add-associative right zeroed non empty double

loop structure, and p, q, r be Series of n, L. Then (p + q) + r = p + (q + r). Let n be a set, let L be an add-associative right zeroed right complementable

non empty loop structure, and let p be a Series of n, L. The functor − p yields a Series of n, L and is defined by:

(Def. 22) For every bag x of n holds (− p)(x) = − p(x).

Let n be a set, let L be an add-associative right zeroed right complementable non empty loop structure, and let p, q be Series of n, L. The functor p−q yields a Series of n, L and is defined by:

(Def. 23) p − q = p + − q.

Let n be a set and let S be a non empty zero structure. The functor 0 (n, S) yields a Series of n, S and is defined by:

(Def. 24) 0 (n, S) = Bagsn ⟼→ 0_{S}.

One can prove the following propositions:

(81) For every set n and for every non empty zero structure S and for every bag b of n holds (0 (n, S))(b) = 0_{S}.

(82) For every set n and for every right zeroed non empty loop structure L and for every Series p of n, L holds p + 0 (n, L) = p.

Let n be a set and let L be a unital non empty multiplicative loop with zero structure. The functor 1 (n, L) yielding a Series of n, L is defined as follows:

(Def. 25) 1 (n, L) = 0 (n, L) +·(EmptyBagn, 1_{L}).

We now state two propositions:

(83) Let n be a set, L be an add-associative right zeroed right complementable non empty loop structure, and p be a Series of n, L. Then p−p = 0 (n, L).

(84) Let n be a set and L be a unital non empty multiplicative loop with zero structure. Then (1 (n, L))(EmptyBagn) = 1_{L} and for every bag b of n such that b ≠ EmptyBagn holds (1 (n, L))(b) = 0_{L}.

Let n be an ordinal number, let L be an add-associative right complementable right zeroed non empty double loop structure, and let p, q be Series of n, L. The functor p ∗ q yields a Series of n, L and is defined by the condition (Def. 26).

(Def. 26) Let b be a bag of n. Then there exists a finite sequence s of elements of

the carrier of L such that

(i) (p ∗ q)(b) = ∑ s, (ii) lens = lendecompb, and (iii) for every natural number k such that k ∈ doms there exist bag b_{1}, b_{2}

of n such that π_{k} decompb = ⟨b_{1}, b_{2}⟩ and π_{k}s = p(b_{1}) · q(b_{2}). One can prove the following two propositions:

(85) Let n be an ordinal number, L be an Abelian add-associative right zeroed right complementable distributive associative non empty double loop structure, and p, q, r be Series of n, L. Then p ∗ (q + r) = p ∗ q + p ∗ r.

(86) Let n be an ordinal number, L be an Abelian add-associative right zeroed right complementable unital distributive associative non empty double loop structure, and p, q, r be Series of n, L. Then (p ∗ q) ∗ r = p ∗ (q ∗ r).

Let n be an ordinal number, let L be an Abelian add-associative right zeroed right complementable commutative non empty double loop structure, and let p, q be Series of n, L. Let us note that the functor p ∗ q is commutative.

One can prove the following three propositions:

(87) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and p be a Series of n, L. Then p ∗ 0 (n, L) = 0 (n, L).

(88) Let n be an ordinal number, L be an add-associative right complementable right zeroed distributive unital non trivial non empty double loop structure, and p be a Series of n, L. Then p ∗ 1 (n, L) = p.

(89) Let n be an ordinal number, L be an add-associative right complementable right zeroed distributive unital non trivial non empty double loop structure, and p be a Series of n, L. Then 1 (n, L) ∗ p = p.

7. Polynomials

Let n be a set and let S be a non empty zero structure. Note that there exists a Series of n, S which is finite-Support.

Let n be an ordinal number and let S be a non empty zero structure. A Polynomial of n, S is a finite-Support Series of n, S.

Let n be an ordinal number, let L be a right zeroed non empty loop structure, and let p, q be Polynomial of n, L. Observe that p + q is finite-Support.

Let n be an ordinal number, let L be an add-associative right zeroed right complementable non empty loop structure, and let p be a Polynomial of n, L. Note that − p is finite-Support.

Let n be a natural number, let L be an add-associative right zeroed right complementable non empty loop structure, and let p, q be Polynomial of n, L. Note that p − q is finite-Support.

Let n be an ordinal number and let S be a non empty zero structure. Observe that 0 (n, S) is finite-Support.

Let n be an ordinal number and let L be an add-associative right zeroed right complementable unital right-distributive non trivial non empty double loop structure. Observe that 1 (n, L) is finite-Support.

Let n be an ordinal number, let L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and let p, q be Polynomial of n, L. One can check that p ∗ q is finite-Support.

8. The Ring of Polynomials

Let n be an ordinal number and let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure. The functor Polynom-Ring(n, L) yields a strict non empty double loop structure and is defined by the conditions (Def. 27).

(Def. 27)(i) For every set x holds x ∈ the carrier of Polynom-Ring(n, L) iff x is a

Polynomial of n, L, (ii) for all elements x, y of Polynom-Ring(n, L) and for all Polynomial p, q

of n, L such that x = p and y = q holds x + y = p + q, (iii) for all elements x, y of Polynom-Ring(n, L) and for all Polynomial p, q

of n, L such that x = p and y = q holds x · y = p ∗ q, (iv) 0_{Polynom-Ring (n,L)} = 0 (n, L), and

(v) 1_{Polynom-Ring (n,L)} = 1 (n, L).

Let n be an ordinal number and let L be an Abelian right zeroed addassociative right complementable unital distributive non trivial non empty double loop structure. One can check that Polynom-Ring(n, L) is Abelian.

multivariate polynomials … 109

Let n be an ordinal number and let L be an add-associative right zeroed right complementable unital distributive non trivial non empty double loop structure. Observe that Polynom-Ring(n, L) is add-associative.

Let n be an ordinal number and let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure. Note that Polynom-Ring(n, L) is right zeroed.

Let n be an ordinal number and let L be a right complementable right zeroed add-associative unital distributive non trivial non empty double loop structure. Observe that Polynom-Ring(n, L) is right complementable.

Let n be an ordinal number and let L be an Abelian add-associative right zeroed right complementable commutative unital distributive non trivial non empty double loop structure. Note that Polynom-Ring(n, L) is commutative.

Let n be an ordinal number and let L be an Abelian add-associative right zeroed right complementable unital distributive associative non trivial non empty double loop structure. Note that Polynom-Ring(n, L) is associative.

Let n be an ordinal number and let L be a right zeroed Abelian addassociative right complementable unital distributive associative non trivial non empty double loop structure. One can check that Polynom-Ring(n, L) is unital and right-distributive.

References

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

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

[3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

[5] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.

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

[7] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91–101, 1993.

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

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

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

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

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

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

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

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

[16] 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.

[17] Andrzej Kondracki. The chinese remainder theorem. Formalized Mathematics, 6(4):573– 577, 1997.

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

110 piotr rudnicki and andrzej trybulec

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

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

[21] Beata Madras. On the concept of the triangulation. Formalized Mathematics, 5(3):457– 462, 1996.

[22] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339– 345, 1996.

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

[24] Wojciech Skaba and Michał Muzalewski. From double loops to fields. Formalized Mathematics, 2(1):185–191, 1991.

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

[26] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

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

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

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

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

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

[33] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.

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

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

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

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

[38] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205–211, 1992.

Continuous Lattices between T_{0} Spaces^{1}

Grzegorz Bancerek University of Białystok

Summary. Formalization of [17, pp. 128–130], chapter II, section 4 (4.1 – 4.9).

MML Identifier: WAYBEL26.

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

Let I be a set and let J be a relational structure yielding many sorted set indexed by I. We introduce I -prod_{POS} J as a synonym of ∏ J.

Let I be a set and let J be a relational structure yielding nonempty many sorted set indexed by I. One can check that I -prod_{POS} J is constituted functions.

Let I be a set and let J be a topological space yielding nonempty many sorted set indexed by I. We introduce I -prod_{TOP} J as a synonym of ∏ J.

Let X, Y be non empty topological spaces. The functor [X → Y] yields a non empty strict relational structure and is defined as follows:

(Def. 1) [X → Y] = [X → ΩY].

Let X, Y be non empty topological spaces. Observe that [X → Y] is reflexive transitive and constituted functions.

Let X be a non empty topological space and let Y be a non empty T_{0} topological space. Observe that [X → Y] is antisymmetric.

We now state three propositions:

(1) Let X, Y be non empty topological spaces and a be a set. Then a is an element of [X → Y] if and only if a is a continuous map from X into ΩY.

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

(2) Let X, Y be non empty topological spaces and a be a set. Then a is an element of [X → Y] if and only if a is a continuous map from X into Y.

(3) Let X, Y be non empty topological spaces, a, b be elements of [X → Y], and f, g be maps from X into ΩY. If a = f and b = g, then a b iff f g.

Let X, Y be non empty topological spaces, let x be a point of X, and let A be a subset of the carrier of ([X → Y]). Then π_{x}A is a subset of ΩY.

Let X, Y be non empty topological spaces, let x be a set, and let A be a non empty subset of the carrier of ([X → Y]). Observe that π_{x}A is non empty.

We now state three propositions:

(4) Ω(the Sierpiński space) is a topological augmentation of 2_{⊆}^{1} .

(5) Let X be a non empty topological space. Then there exists a map f from ⟨the topology of X, ⊆⟩ into [X → the Sierpiński space] such that f is isomorphic and for every open subset V of X holds f(V) = χ_{V,the carrier of X}.

(6) Let X be a non empty topological space. Then ⟨the topology of X, ⊆⟩ and [X → the Sierpiński space] are isomorphic.

Let X, Y, Z be non empty topological spaces and let f be a continuous map from Y into Z. The functor [X → f] yields a map from [X → Y] into [X → Z] and is defined by:

(Def. 2) For every continuous map g from X into Y holds ([X → f])(g) = f · g. The functor [f → X] yields a map from [Z → X] into [Y → X] and is defined by:

(Def. 3) For every continuous map g from Z into X holds ([f → X])(g) = g · f. The following propositions are true:

(7) Let X be a non empty topological space and Y be a monotone convergence T_{0}-space. Then [X → Y] is a directed-sups-inheriting relational substructure of (ΩY)^{the carrier of X}.

(8) For every non empty topological space X and for every monotone convergence T_{0}-space Y holds [X → Y] is up-complete.

(9) For all non empty topological spaces X, Y, Z and for every continuous map f from Y into Z holds [X → f] is monotone.

(10) Let X, Y be non empty topological spaces and f be a continuous map from Y into Y. If f is idempotent, then [X → f] is idempotent.

(11) For all non empty topological spaces X, Y, Z and for every continuous map f from Y into Z holds [f → X] is monotone.

(12) Let X, Y be non empty topological spaces and f be a continuous map from Y into Y. If f is idempotent, then [f → X] is idempotent.

(13) Let X, Y, Z be non empty topological spaces, f be a continuous map from Y into Z, x be an element of X, and A be a subset of [X → Y].

continuous lattices between t_{0} spaces 113

Then π_{x}([X → f])^{∘}A = f^{∘}π_{x}A.

(14) Let X be a non empty topological space, Y, Z be monotone convergence T_{0}-spaces, and f be a continuous map from Y into Z. Then [X → f] is directed-sups-preserving.

(15) Let X, Y, Z be non empty topological spaces, f be a continuous map from Y into Z, x be an element of Y, and A be a subset of [Z → X]. Then π_{x}([f → X])^{∘}A = π_{f(x)}A.

(16) Let Y, Z be non empty topological spaces, X be a monotone convergence T_{0}-space, and f be a continuous map from Y into Z. Then [f → X] is directed-sups-preserving.

(17) Let X, Z be non empty topological spaces and Y be a non empty subspace of Z. Then [X → Y] is a full relational substructure of [X → Z].

(18) Let Z be a monotone convergence T_{0}-space, Y be a non empty subspace of Z, and f be a continuous map from Z into Y. Suppose f is a retraction. Then ΩY is a directed-sups-inheriting relational substructure of ΩZ.

(19) Let X be a non empty topological space, Z be a monotone convergence T_{0}-space, Y be a non empty subspace of Z, and f be a continuous map from Z into Y. If f is a retraction, then [X → f] is a retraction of [X → Z] into [X → Y].

(20) Let X be a non empty topological space, Z be a monotone convergence T_{0}-space, and Y be a non empty subspace of Z. If Y is a retract of Z, then [X → Y] is a retract of [X → Z].

(21) Let X, Y, Z be non empty topological spaces and f be a continuous map from Y into Z. If f is a homeomorphism, then [X → f] is isomorphic.

(22) Let X, Y, Z be non empty topological spaces. If Y and Z are homeomorphic, then [X → Y] and [X → Z] are isomorphic.

(23) Let X be a non empty topological space, Z be a monotone convergence T_{0}-space, and Y be a non empty subspace of Z. Suppose Y is a retract of Z and [X → Z] is complete and continuous. Then [X → Y] is complete and continuous.

(24) Let X be a non empty topological space and Y, Z be monotone convergence T_{0}-spaces. Suppose Y is a topological retract of Z and [X → Z] is complete and continuous. Then [X → Y] is complete and continuous.

(25) Let Y be a non trivial T_{0}-space. Suppose Y is not a T_{1} space. Then the Sierpiński space is a topological retract of Y.

(26) Let X be a non empty topological space and Y be a non trivial T_{0}-space. If [X → Y] has l.u.b.’s, then Y is not a T_{1} space.

One can check that the Sierpiński space is non trivial and monotone convergence.

One can verify that there exists a T_{0}-space which is injective, monotone convergence, and non trivial.

The following propositions are true:

(27) Let X be a non empty topological space and Y be a monotone convergence non trivial T_{0}-space. If [X → Y] is complete and continuous, then ⟨the topology of X, ⊆⟩ is continuous.

(28) Let X be a non empty topological space, x be a point of X, and Y be a monotone convergence T_{0}-space. Then there exists a directed-sups-preserving projection map F from [X → Y] into [X → Y] such that

(i) for every continuous map f from X into Y holds F (f) = X ⟼→ f(x), and

(ii) there exists a continuous map h from X into X such that h = X ⟼→ x

and F = [h → Y].

(29) Let X be a non empty topological space and Y be a monotone convergence T_{0}-space. If [X → Y] is complete and continuous, then ΩY is complete and continuous.

(30) Let X be a non empty 1-sorted structure, I be a non empty set, J be a topological space yielding nonempty many sorted set indexed by I, f be a map from X into I -prod_{TOP} J, and i be an element of I. Then (commute(f))(i) = proj(J, i) · f.

(31) For every 1-sorted structure S and for every set M holds the support of M ⟼→ S = M ⟼→ the carrier of S.

(32) Let X, Y be non empty topological spaces, M be a non empty set, and f be a continuous map from X into M -prod_{TOP}(M ⟼→ Y). Then commute(f) is a function from M into the carrier of ([X → Y]).

(33) For all non empty topological spaces X, Y holds the carrier of ([X → Y]) ⊆ (the carrier of Y)^{the carrier of X}.

(34) Let X, Y be non empty topological spaces, M be a non empty set, and f be a function from M into the carrier of ([X → Y]). Then commute(f) is a continuous map from X into M -prod_{TOP}(M ⟼→ Y).

(35) Let X be a non empty topological space and M be a non empty set. Then there exists a map F from [X → M -prod_{TOP}(M ⟼→ the Sierpiński space)] into M -prod_{POS}(M ⟼→ ([X → the Sierpiński space])) such that F is isomorphic and for every continuous map f from X into M -prod_{TOP}(M ⟼→ the Sierpiński space) holds F (f) = commute(f).

(36) Let X be a non empty topological space and M be a non empty set. Then [X → M -prod_{TOP}(M ⟼→ the Sierpiński space)] and M -prod_{POS}(M ⟼→ ([X → the Sierpiński space])) are isomorphic.

(37) Let X be a non empty topological space. Suppose ⟨the topology of X, ⊆⟩ is continuous. Let Y be an injective T_{0}-space. Then [X → Y] is complete

and continuous. Let us observe that there exists a top-lattice which is non trivial, complete,

and Scott.

Next we state the proposition

(38) Let X be a non empty topological space and L be a non trivial complete Scott top-lattice. Then [X → L] is complete and continuous if and only if ⟨the topology of X, ⊆⟩ is continuous and L is continuous.

Let f be a function. Observe that Uniondisjointf is relation-like. Let f be a function. The functor G_{f} yields a binary relation and is defined

as follows: (Def. 4) G_{f} = (Uniondisjointf)^{⌣}.

In the sequel x, y are sets and f is a function. We now state three propositions:

(39) ⟨⟨x, y⟩⟩∈ G_{f} iff x ∈ domf and y ∈ f(x).

(40) For every finite set X holds π_{1}(X) is finite and π_{2}(X) is finite.

(41) Let X, Y be non empty topological spaces, S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩, and f be a map from X into S. If G_{f} is an open subset of [: X, Y :], then f is continuous.

Let W be a binary relation and let X be a set. The functor Θ_{X}(W) yielding a function is defined by:

(Def. 5) domΘ_{X}(W) = X and for every x such that x ∈ X holds (Θ_{X}(W))(x) =

W^{∘}{x}. One can prove the following proposition

(42) For every binary relation W and for every set X such that domW ⊆ X

holds G_{Θ_{X}(W)} = W. Let X, Y be topological spaces. Observe that every subset of the carrier of

[: X, Y :] is relation-like and every element of the topology of [: X, Y :] is relationlike.

Next we state four propositions:

(43) Let X, Y be non empty topological spaces, W be an open subset of [: X, Y :], and x be a point of X. Then W^{∘}{x} is an open subset of Y.

(44) Let X, Y be non empty topological spaces, S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩, and W be an open subset of [: X, Y :]. Then Θ_{the carrier of X}(W) is a continuous map from X into S.

(45) Let X, Y be non empty topological spaces, S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩, and W_{1}, W_{2} be open subsets of [: X, Y :]. Suppose W_{1} ⊆ W_{2}. Let f_{1}, f_{2} be elements of [X → S]. If f_{1} = Θ_{the carrier of X}(W_{1}) and f_{2} = Θ_{the carrier of X}(W_{2}), then f_{1} f_{2}.

116 grzegorz bancerek

(46) Let X, Y be non empty topological spaces and S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩. Then there exists a map F from ⟨the topology of [: X, Y :], ⊆⟩ into [X → S] such that F is monotone and for every open subset W of [: X, Y :] holds F (W) = Θ_{the carrier of X}(W).

References

[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537– 541, 1990.

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

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

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

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

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

[7] Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1):35–43, 1998.

[8] Grzegorz Bancerek. Retracts and inheritance. Formalized Mathematics, 9(1):77–85, 2001.

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

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

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

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

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

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

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

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

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

[18] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

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

[20] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57–62, 1998.

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

[22] Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269–277, 1997.

[23] Artur Korniłowicz and Jarosław Gryko. Injective spaces. Part II. Formalized Mathematics, 9(1):41–47, 2001.

[24] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

[25] Beata Madras. Products of many sorted algebras. Formalized Mathematics, 5(1):55–60, 1996.

[26] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.

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

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

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

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

[31] Andrzej Trybulec. Scott topology. Formalized Mathematics, 6(2):311–319, 1997.

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

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

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

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

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

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

118 grzegorz bancerek

Predicate Calculus for Boolean Valued Functions. Part VI

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC14.

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

1. Preliminaries

In this paper Y denotes a non empty set. We now state several propositions:

(1) For every element z of Y and for all partitions P_{1}, P_{2} of Y holds EqClass(z, P_{1} ∧ P_{2}) = EqClass(z, P_{1}) ∩ EqClass(z, P_{2}).

(2) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ⋀ G = A ∧ B.

(3) Let G be a subset of PARTITIONS(Y) and B, C, D be partitions of Y. Suppose G is a coordinate and G = {B, C, D} and B ≠ C and C ≠ D and D ≠ B. Then ⋀ G = B ∧ C ∧ D.

(4) Let G be a subset of PARTITIONS(Y) and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then CompF(A, G) = B ∧ C and CompF(B, G) = C ∧ A and CompF(C, G) = A ∧ B.

(5) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(A, G) = B ∧ C ∧ D.

(6) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(B, G) = A ∧ C ∧ D.

(7) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(C, G) = A ∧ B ∧ D.

(8) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(D, G) = A ∧ C ∧ B.

2. Predicate Calculus

We adopt the following rules: a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C are partitions of Y.

One can prove the following propositions:

(9) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(10) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{∀_{a,C} G,A}G,B}G = ∀_{∀_{∀_{a,C} G,B}G,A}G.

(11) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{∃_{a,C} G,A}G,B}G = ∀_{∀_{∃_{a,C} G,B}G,A}G.

(12) Let G be a subset of PARTITIONS(Y), B, C, D be partitions of Y, h be a function, and b, c, d be sets. Suppose B ≠ C and C ≠ D and D ≠ B and h = (B⟼→ . b)+·(C⟼→ . c)+·(D⟼→ . d). Then domh = {B, C, D} and h(B) = b and h(C) = c and h(D) = d and rngh = {h(B), h(C), h(D)}.

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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[6] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[7] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

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

Received October 19, 1999 122 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part VII

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC15.

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

In this paper Y is a non empty set. Next we state a number of propositions:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C be partitions of Y, and z, u be elements of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A and EqClass(z, C) = EqClass(u, C). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∀_{a,C} G,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∃_{a,C} G,B}G,A}G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{∀_{a,C} G,A}G,B}G = ∃_{∃_{∀_{a,C} G,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{∃_{a,C} G,A}G,B}G = ∃_{∃_{∃_{a,C} G,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∀_{∀_{a,C} G,B}G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∀_{∃_{a,C} G,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∀_{a,C} G,B}G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∃_{a,C} G,B}G,A}G.

(17) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∀_{a,C} G,B}G,A}G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∃_{a,C} G,B}G,A}G.

(19) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∀_{a,C} G,B}G,A}G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∃_{a,C} G,B}G,A}G.

(21) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∀_{a,C} G,B}G,A}G.

(22) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∃_{a,C} G,B}G,A}G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

126 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part VIII

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC16.

The terminology and notation used here are introduced in the following articles: [1], [2], [3], [4], and [5].

In this paper Y is a non empty set. We now state a number of propositions:

(1) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(7) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B, C of Y holds ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(8) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B, C of Y holds ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(16) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B, C of Y holds ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[4] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

130 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part IX

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC17.

The terminology and notation used in this paper are introduced in the following papers: [1], [2], [3], [4], and [5].

In this paper Y is a non empty set. The following propositions are true:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[4] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

134 shunichi kobayashi

Asymptotic Notation. Part I: Theory^{1}

Richard Krueger University of Alberta Edmonton

Piotr Rudnicki University of Alberta Edmonton

Paul Shelley University of Alberta Edmonton

Summary. The widely used textbook by Brassard and Bratley [2] includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79–97). We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety. A more detailed report on this experiment will be available separately. This article presents the development of notions and a follow-up article [9] includes examples and solutions to problems. The preliminaries introduce a number of properties of real sequences, some operations on real sequences, and a characterization of convergence. The remaining sections in this article correspond to sections of Chapter 3 of [2]. Section 2 defines the O notation and proves the threshold, maximum, and limit rules. Section 3 introduces the Ω and Θ notations and their analogous rules. Conditional asymptotic notation is defined in Section 4 where smooth functions are also discussed. Section 5 defines some operations on asymptotic notation (we have decided not to introduce the asymptotic notation for functions of several variables as it is a straightforward generalization of notions for unary functions).

MML Identifier: ASYMPT 0.

The terminology and notation used in this paper have been introduced in the following articles: [13], [11], [3], [4], [8], [1], [10], [5], [14], [7], [6], and [12].

1This work has been supported by NSERC Grant OGP9207.

1. Preliminaries

In this paper c, d denote real numbers and n, N denote natural numbers.

In this article we present several logical schemes. The scheme FinSegRng1 deals with natural numbers A, B, a non empty set C, and a unary functor F yielding an element of C, and states that:

{F(i); i ranges over natural numbers: A i ∧ i B} is a finite non empty subset of C

provided the parameters meet the following requirement:

• A B.

The scheme FinImInit1 deals with a natural number A, a non empty set B, and a unary functor F yielding an element of B, and states that:

{F(n); n ranges over natural numbers: n A} is a finite non empty subset of B

for all values of the parameters.

The scheme FinImInit2 deals with a natural number A, a non empty set B, and a unary functor F yielding an element of B, and states that:

{F(n); n ranges over natural numbers: n < A} is a finite non empty subset of B

provided the parameters meet the following requirement:

• A > 0.

Let c be a real number. We say that c is positive if and only if: (Def. 1) c > 0.

We say that c is negative if and only if: (Def. 2) c < 0.

We say that c is logbase if and only if: (Def. 3) c > 0 and c ≠ 1.

One can check the following observations:

∗ there exists a real number which is positive,

∗ there exists a real number which is negative,

∗ there exists a real number which is logbase,

∗ there exists a real number which is non negative,

∗ there exists a real number which is non positive, and

∗ there exists a real number which is non logbase.

Let f be a sequence of real numbers. We say that f is eventually-nonnegative if and only if:

(Def. 4) There exists N such that for every n such that n N holds f(n) 0.

We say that f is positive if and only if: (Def. 5) For every n holds f(n) > 0.

We say that f is eventually-positive if and only if: (Def. 6) There exists N such that for every n such that n N holds f(n) > 0.

We say that f is eventually-nonzero if and only if: (Def. 7) There exists N such that for every n such that n N holds f(n) ≠ 0.

We say that f is eventually-nondecreasing if and only if: (Def. 8) There exists N such that for every n such that n N holds f(n)

f(n + 1). Let us mention that there exists a sequence of real numbers which is eventually-

nonnegative, eventually-nonzero, positive, eventually-positive, and eventuallynondecreasing.

One can verify the following observations:

∗ every sequence of real numbers which is positive is also eventuallypositive,

∗ every sequence of real numbers which is eventually-positive is also eventuallynonnegative and eventually-nonzero, and

∗ every sequence of real numbers which is eventually-nonnegative and eventually-nonzero is also eventually-positive.

Let f, g be eventually-nonnegative sequences of real numbers. Note that f + g is eventually-nonnegative.

Let f be a sequence of real numbers and let c be a real number. The functor c + f yields a sequence of real numbers and is defined by:

(Def. 9) For every n holds (c + f)(n) = c + f(n).

We introduce f + c as a synonym of c + f.

Let f be an eventually-nonnegative sequence of real numbers and let c be a positive real number. One can check that c f is eventually-nonnegative.

Let f be an eventually-nonnegative sequence of real numbers and let c be a non negative real number. Note that c + f is eventually-nonnegative.

Let f be an eventually-nonnegative sequence of real numbers and let c be a positive real number. One can check that c + f is eventually-positive.

Let f, g be sequences of real numbers. The functor max(f, g) yielding a sequence of real numbers is defined as follows:

(Def. 10) For every n holds (max(f, g))(n) = max(f(n), g(n)).

Let us notice that the functor max(f, g) is commutative.

Let f be a sequence of real numbers and let g be an eventually-nonnegative sequence of real numbers. One can check that max(f, g) is eventually-nonnegative.

Let f be a sequence of real numbers and let g be an eventually-positive sequence of real numbers. One can verify that max(f, g) is eventually-positive.

Let f, g be sequences of real numbers. We say that g majorizes f if and only if:

(Def. 11) There exists N such that for every n such that n N holds f(n) g(n).

The following propositions are true:

(1) Let f be a sequence of real numbers and N be a natural number. Suppose that for every n such that n N holds f(n) f(n + 1). Let n, m be natural numbers. If N n and n m, then f(n) f(m).

(2) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) ≠ 0, then g/f is convergent and lim(g/f) = (lim(f/g))^{−1} .

(3) For every eventually-nonnegative sequence f of real numbers such that f is convergent holds 0 limf.

(4) Let f, g be sequences of real numbers. If f is convergent and g is convergent and g majorizes f, then limf limg.

(5) Let f be a sequence of real numbers and g be an eventually-nonzero sequence of real numbers. If f/g is divergent to +∞, then g/f is convergent and lim(g/f) = 0.

2. A Notation for ”the order of”

Let f be an eventually-nonnegative sequence of real numbers. The functor O(f) yielding a non empty set of functions from ℕ to ℝ is defined by:

(Def. 12) O(f) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,N} (c > 0 ∧ ⋀_{n} (n N ⇒

t(n) c · f(n) ∧ t(n) 0))}. The following propositions are true:

(6) Let x be a set and f be an eventually-nonnegative sequence of real numbers. Suppose x ∈ O(f). Then x is an eventually-nonnegative sequence of real numbers.

(7) Let f be a positive sequence of real numbers and t be an eventuallynonnegative sequence of real numbers. Then t ∈ O(f) if and only if there exists c such that c > 0 and for every n holds t(n) c · f(n).

(8) Let f be an eventually-positive sequence of real numbers, t be an eventually-nonnegative sequence of real numbers, and N be a natural number. Suppose t ∈ O(f) and for every n such that n N holds f(n) > 0. Then there exists c such that c > 0 and for every n such that n N holds t(n) c · f(n).

(9) For all eventually-nonnegative sequences f, g of real numbers holds O(f + g) = O(max(f, g)).

(10) For every eventually-nonnegative sequence f of real numbers holds f ∈ O(f).

(11) For all eventually-nonnegative sequences f, g of real numbers such that f ∈ O(g) holds O(f) ⊆ O(g).

(12) For all eventually-nonnegative sequences f, g, h of real numbers such that f ∈ O(g) and g ∈ O(h) holds f ∈ O(h).

(13) Let f be an eventually-nonnegative sequence of real numbers and c be a positive real number. Then O(f) = O(c f).

(14) Let c be a non negative real number and x, f be eventually-nonnegative sequences of real numbers. If x ∈ O(f), then x ∈ O(c + f).

(15) For all eventually-positive sequences f, g of real numbers such that f/g is convergent and lim(f/g) > 0 holds O(f) = O(g).

(16) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) = 0, then f ∈ O(g) and g ∉ O(f).

(17) Let f, g be eventually-positive sequences of real numbers. If f/g is divergent to +∞, then f ∉ O(g) and g ∈ O(f).

3. Other Asymptotic Notation

Let f be an eventually-nonnegative sequence of real numbers. The functor Ω(f) yielding a non empty set of functions from ℕ to ℝ is defined by:

(Def. 13) Ω(f) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{d,N} (d > 0 ∧ ⋀_{n} (n N ⇒

t(n) d · f(n) ∧ t(n) 0))}. The following propositions are true:

(18) Let x be a set and f be an eventually-nonnegative sequence of real numbers. Suppose x ∈ Ω(f). Then x is an eventually-nonnegative sequence of real numbers.

(19) For all eventually-nonnegative sequences f, g of real numbers holds f ∈ Ω(g) iff g ∈ O(f).

(20) For every eventually-nonnegative sequence f of real numbers holds f ∈ Ω(f).

(21) For all eventually-nonnegative sequences f, g, h of real numbers such that f ∈ Ω(g) and g ∈ Ω(h) holds f ∈ Ω(h).

(22) For all eventually-positive sequences f, g of real numbers such that f/g is convergent and lim(f/g) > 0 holds Ω(f) = Ω(g).

(23) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) = 0, then g ∈ Ω(f) and f ∉ Ω(g).

(24) Let f, g be eventually-positive sequences of real numbers. If f/g is divergent to +∞, then g ∉ Ω(f) and f ∈ Ω(g).

(25) Let f, t be positive sequences of real numbers. Then t ∈ Ω(f) if and only if there exists d such that d > 0 and for every n holds d · f(n) t(n).

(26) For all eventually-nonnegative sequences f, g of real numbers holds Ω(f+ g) = Ω(max(f, g)).

Let f be an eventually-nonnegative sequence of real numbers. The functor Θ(f) yielding a non empty set of functions from ℕ to ℝ is defined as follows:

(Def. 14) Θ(f) = O(f) ∩ Ω(f).

Next we state several propositions:

(27) Let f be an eventually-nonnegative sequence of real numbers. Then Θ(f) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,d,N} (c > 0 ∧ d > 0 ∧ ⋀_{n} (n N ⇒ d · f(n) t(n) ∧ t(n) c · f(n)))}.

(28) For every eventually-nonnegative sequence f of real numbers holds f ∈ Θ(f).

(29) For all eventually-nonnegative sequences f, g of real numbers such that f ∈ Θ(g) holds g ∈ Θ(f).

(30) For all eventually-nonnegative sequences f, g, h of real numbers such that f ∈ Θ(g) and g ∈ Θ(h) holds f ∈ Θ(h).

(31) Let f, t be positive sequences of real numbers. Then t ∈ Θ(f) if and only if there exist c, d such that c > 0 and d > 0 and for every n holds d · f(n) t(n) and t(n) c · f(n).

(32) For all eventually-nonnegative sequences f, g of real numbers holds Θ(f+ g) = Θ(max(f, g)).

(33) For all eventually-positive sequences f, g of real numbers such that f/g is convergent and lim(f/g) > 0 holds f ∈ Θ(g).

(34) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) = 0, then f ∈ O(g) and f ∉ Θ(g).

(35) Let f, g be eventually-positive sequences of real numbers. If f/g is divergent to +∞, then f ∈ Ω(g) and f ∉ Θ(g).

4. Conditional Asymptotic Notation

Let f be an eventually-nonnegative sequence of real numbers and let X be a set. The functor O(f|X) yields a non empty set of functions from ℕ to ℝ and is defined as follows:

(Def. 15) O(f|X) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,N} (c > 0 ∧ ⋀_{n} (n

N ∧ n ∈ X ⇒ t(n) c · f(n) ∧ t(n) 0))}. Let f be an eventually-nonnegative sequence of real numbers and let X be

a set. The functor Ω(f|X) yields a non empty set of functions from ℕ to ℝ and is defined by:

(Def. 16) Ω(f|X) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{d,N} (d > 0 ∧ ⋀_{n} (n

N ∧ n ∈ X ⇒ t(n) d · f(n) ∧ t(n) 0))}. Let f be an eventually-nonnegative sequence of real numbers and let X be

a set. The functor Θ(f|X) yielding a non empty set of functions from ℕ to ℝ is defined by the condition (Def. 17).

(Def. 17) Θ(f|X) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,d,N} (c > 0 ∧ d > 0 ∧

⋀_{n} (n N ∧ n ∈ X ⇒ d · f(n) t(n) ∧ t(n) c · f(n)))}. Next we state the proposition

(36) For every eventually-nonnegative sequence f of real numbers and for

every set X holds Θ(f|X) = O(f|X) ∩ Ω(f|X). Let f be a sequence of real numbers and let b be a natural number. The

functor f_{b} yielding a sequence of real numbers is defined by: (Def. 18) For every n holds f_{b}(n) = f(b · n).

Let f be an eventually-nonnegative sequence of real numbers and let b be a natural number. We say that f is smooth w.r.t. b if and only if:

(Def. 19) f is eventually-nondecreasing and f_{b} ∈ O(f).

Let f be an eventually-nonnegative sequence of real numbers. We say that f is smooth if and only if:

(Def. 20) For every natural number b such that b 2 holds f is smooth w.r.t. b.

We now state four propositions:

(37) Let f be an eventually-nonnegative sequence of real numbers. Given a natural number b such that b 2 and f is smooth w.r.t. b. Then f is smooth.

(38) Let f be an eventually-nonnegative sequence of real numbers, t be an eventually-nonnegative eventually-nondecreasing sequence of real numbers, and b be a natural number. Suppose f is smooth and b 2 and t ∈ O(f|{b^{n} : n ranges over natural numbers}). Then t ∈ O(f).

(39) Let f be an eventually-nonnegative sequence of real numbers, t be an eventually-nonnegative eventually-nondecreasing sequence of real numbers, and b be a natural number. Suppose f is smooth and b 2 and t ∈ Ω(f|{b^{n} : n ranges over natural numbers}). Then t ∈ Ω(f).

(40) Let f be an eventually-nonnegative sequence of real numbers, t be an eventually-nonnegative eventually-nondecreasing sequence of real numbers, and b be a natural number. Suppose f is smooth and b 2 and t ∈ Θ(f|{b^{n} : n ranges over natural numbers}). Then t ∈ Θ(f).

5. Operations on Asymptotic Notation

Let X be a non empty set and let F, G be non empty sets of functions from X to ℝ. The functor F + G yields a non empty set of functions from X to ℝ and is defined by the condition (Def. 21).

(Def. 21) F +G = {t; t ranges over elements of ℝ^{X}: ⋁_{f,g: element of ℝ^{X}} (f ∈ F ∧ g ∈ G ∧ ⋀_{n: element of X} t(n) = f(n) + g(n))}.

142 richard krueger et al.

Let X be a non empty set and let F, G be non empty sets of functions from X to ℝ. The functor max(F, G) yields a non empty set of functions from X to ℝ and is defined by the condition (Def. 22).

(Def. 22) max(F, G) = {t; t ranges over elements of ℝ^{X}: ⋁_{f,g: element of ℝ^{X}} (f ∈ F ∧ g ∈ G ∧ ⋀_{n: element of X} t(n) = max(f(n), g(n)))}.

Next we state two propositions:

(41) For all eventually-nonnegative sequences f, g of real numbers holds O(f) + O(g) = O(f + g).

(42) For all eventually-nonnegative sequences f, g of real numbers holds max(O(f), O(g)) = O(max(f, g)).

Let F, G be non empty sets of functions from ℕ to ℝ. The functor F^{G} yielding a non empty set of functions from ℕ to ℝ is defined by the condition (Def. 23).

(Def. 23) F^{G} = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{f,g: element of ℝ^{ℕ}} ⋁_{N : element of ℕ} (f ∈ F ∧ g ∈ G ∧ ⋀_{n: element of ℕ} (n N ⇒ t(n) = f(n)^{g(n)}))}.

References

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

[2] Gilles Brassard and Paul Bratley. Fundamentals of Algorithmics. Prentice Hall, 1996.

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

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

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

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

[9] Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics, 9(1):143–154, 2001.

[10] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

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

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

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

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

Asymptotic Notation. Part II: Examples and Problems^{1}

Richard Krueger University of Alberta Edmonton

Piotr Rudnicki University of Alberta Edmonton

Paul Shelley University of Alberta Edmonton

Summary. The widely used textbook by Brassard and Bratley [2] includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79–97). We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety. This article is a follow-up to [11] in which we introduced the basic notions and general theory. This article presents a Mizar formalization of examples and solutions to problems from Chapter 3 of [2] (some of the examples and solved problems are also in [11]). Not all problems have been solved as some required solutions not amenable for formalization.

MML Identifier: ASYMPT 1.

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

1. Examples from the Text

We adopt the following rules: c, e denote real numbers, k, n, m, N, n_{1}, M denote natural numbers, and x denotes a set.

One can prove the following two propositions:

1This work has been supported by NSERC Grant OGP9207.

(1) Let t, t_{1} be sequences of real numbers. Suppose that

(i) t(0) = 0, (ii) for every n such that n > 0 holds t(n) = (12 · n^{3} · log_{2} n − 5 · n^{2}) +

(log_{2} n)^{2} + 36, (iii) t_{1}(0) = 0, and (iv) for every n such that n > 0 holds t_{1}(n) = n^{3} · log_{2} n.

Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = t and s_{1} = t_{1} and s ∈ O(s_{1}).

(2) Let a, b be logbase real numbers and f, g be sequences of real numbers. Suppose a > 1 and b > 1 and f(0) = 0 and for every n such that n > 0 holds f(n) = log_{a} n and g(0) = 0 and for every n such that n > 0 holds g(n) = log_{b} n. Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = f and s_{1} = g and O(s) = O(s_{1}).

Let a, b, c be real numbers. The functor {a^{b·n+c)}}_{n∈ℕ} yields a sequence of real numbers and is defined by:

(Def. 1) ({a^{b·n+c)}}_{n∈ℕ})(n) = a^{b·n+c}.

Let a be a positive real number and let b, c be real numbers. One can verify that {a^{b·n+c)}}_{n∈ℕ} is eventually-positive.

The following proposition is true

(3) For all positive real numbers a, b such that a < b holds {b^{1·n+0)}}_{n∈ℕ} ∉ O({a^{1·n+0)}}_{n∈ℕ}).

The sequence {log_{2} n}_{n∈ℕ} of real numbers is defined as follows: (Def. 2) {log_{2} n}_{n∈ℕ}(0) = 0 and for every n such that n > 0 holds

{log_{2} n}_{n∈ℕ}(n) = log_{2} n. Let a be a real number. The functor {n^{a}}_{n∈ℕ} yielding a sequence of real

numbers is defined as follows: (Def. 3) {n^{a}}_{n∈ℕ}(0) = 0 and for every n such that n > 0 holds {n^{a}}_{n∈ℕ}(n) = n^{a}.

Let us mention that {log_{2} n}_{n∈ℕ} is eventually-positive. Let a be a real number. Observe that {n^{a}}_{n∈ℕ} is eventually-positive. We now state several propositions:

(4) Let f, g be eventually-nonnegative sequences of real numbers. Then O(f) ⊆ O(g) and O(f) ≠ O(g) if and only if f ∈ O(g) and f ∉ Ω(g).

(5) O(\{log_{2} n\}_{n∈ℕ}) ⊆ O(\{n^{( \frac{1}{2} )}\}_{n∈ℕ}) and O(\{log_{2} n\}_{n∈ℕ}) ≠ O(\{n^{( \frac{1}{2} )}\}_{n∈ℕ}).

(6) \{n^{( \frac{1}{2} )}\}_{n∈ℕ} ∈ Ω(\{log_{2} n\}_{n∈ℕ}) and \{log_{2} n\}_{n∈ℕ} ∉ Ω(\{n^{( \frac{1}{2} )}\}_{n∈ℕ}).

(7) For every sequence f of real numbers and for every natural number k such that for every n holds f(n) = ∑_{κ=0}^{n}({n^{k}}_{n∈ℕ})(κ) holds f ∈ Θ({n^{(k+1)}}_{n∈ℕ}).

(8) Let f be a sequence of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{log_{2} n}. Then there exists an eventuallypositive sequence s of real numbers such that s = f and s is not smooth.

Let b be a real number. The functor {b}_{n∈ℕ} yields a sequence of real numbers and is defined as follows:

(Def. 4) {b}_{n∈ℕ} = ℕ ⟼→ b.

Let us note that {1}_{n∈ℕ} is eventually-nonnegative. One can prove the following proposition

(9) Let f be an eventually-nonnegative sequence of real numbers. Then there exists a non empty set F of functions from ℕ to ℝ such that F = {{n^{1}}_{n∈ℕ}} and f ∈ F^{O({1}_{n∈ℕ} )} iff there exist N, c, k such that c > 0 and for every n such that n N holds 1 f(n) and f(n) c · {n^{k}}_{n∈ℕ}(n).

2. Problem 3.1

One can prove the following proposition

(10) For every sequence f of real numbers such that for every n holds f(n) = (3 · 10^{6} − 18 · 10^{3} · n) + 27 · n^{2} holds f ∈ O({n^{2}}_{n∈ℕ}).

3. Problem 3.5

We now state three propositions:

(11) {n^{2}}_{n∈ℕ} ∈ O({n^{3}}_{n∈ℕ}).

(12) {n^{2}}_{n∈ℕ} ∉ Ω({n^{3}}_{n∈ℕ}).

(13) There exists an eventually-positive sequence s of real numbers such that

s = {2^{1·n+1)}}_{n∈ℕ} and {2^{1·n+0)}}_{n∈ℕ} ∈ Θ(s). Let a be a natural number. The functor {(n + a)!}_{n∈ℕ} yielding a sequence

of real numbers is defined by: (Def. 5) {(n + a)!}_{n∈ℕ}(n) = (n + a)!.

Let a be a natural number. Observe that {(n+a)!}_{n∈ℕ} is eventually-positive. We now state the proposition

(14) {(n + 0)!}_{n∈ℕ} ∉ Θ({(n + 1)!}_{n∈ℕ}).

146 richard krueger et al.

4. Problem 3.6

The following proposition is true

(15) For every sequence f of real numbers such that f ∈ O({n^{1}}_{n∈ℕ}) holds

f f ∈ O({n^{2}}_{n∈ℕ}).

5. Problem 3.7

We now state the proposition

(16) There exists an eventually-positive sequence s of real numbers such that

s = {2^{1·n+0)}}_{n∈ℕ} and 2 {n^{1}}_{n∈ℕ} ∈ O({n^{1}}_{n∈ℕ}) and {2^{2·n+0)}}_{n∈ℕ} ∉ O(s).

6. Problem 3.8

One can prove the following proposition

(17) If log_{2} 3 < \frac{159}{100}, then \{n^{(log_{2} 3)}\}_{n∈ℕ} ∈ O(\{n^{( \frac{159}{100} )}\}_{n∈ℕ}) and \{n^{(log_{2} 3)}\}_{n∈ℕ} ∉

Ω(\{n^{( \frac{159}{100} )}\}_{n∈ℕ}) and \{n^{(log_{2} 3)}\}_{n∈ℕ} ∉ Θ(\{n^{( \frac{159}{100} )}\}_{n∈ℕ}).

7. Problem 3.11

We now state the proposition

(18) Let f, g be sequences of real numbers. Suppose for every n holds f(n) = n mod 2 and for every n holds g(n) = (n + 1) mod 2. Then there exist eventually-nonnegative sequences s, s_{1} of real numbers such that s = f and s_{1} = g and s ∉ O(s_{1}) and s_{1} ∉ O(s).

8. Problem 3.19

We now state two propositions:

(19) For all eventually-nonnegative sequences f, g of real numbers holds O(f) = O(g) iff f ∈ Θ(g).

(20) For all eventually-nonnegative sequences f, g of real numbers holds f ∈ Θ(g) iff Θ(f) = Θ(g).

9. Problem 3.21

The following propositions are true:

(21) Let e be a real number and f be a sequence of real numbers. Suppose 0 < e and f(0) = 0 and for every n such that n > 0 holds f(n) = n·log_{2} n. Then there exists an eventually-positive sequence s of real numbers such that s = f and O(s) ⊆ O({n^{(1+e)}}_{n∈ℕ}) and O(s) ≠ O({n^{(1+e)}}_{n∈ℕ}).

(22) Let e be a real number and g be a sequence of real numbers. Suppose 0 < e and e < 1 and g(0) = 0 and g(1) = 0 and for every n such that n > 1 holds g(n) = \frac{n^{2}}{log_{2} n}. Then there exists an eventually-positive sequence s of real numbers such that s = g and O({n^{(1+e)}}_{n∈ℕ}) ⊆ O(s) and O({n^{(1+e)}}_{n∈ℕ}) ≠ O(s).

(23) Let f be a sequence of real numbers. Suppose f(0) = 0 and f(1) = 0 and for every n such that n > 1 holds f(n) = \frac{n^{2}}{log_{2} n}. Then there exists an eventually-positive sequence s of real numbers such that s = f and O(s) ⊆ O({n^{8}}_{n∈ℕ}) and O(s) ≠ O({n^{8}}_{n∈ℕ}).

(24) Let g be a sequence of real numbers. Suppose that for every n holds g(n) = ((n^{2} − n) + 1)^{4}. Then there exists an eventually-positive sequence s of real numbers such that s = g and O({n^{8}}_{n∈ℕ}) = O(s).

(25) Let e be a real number. Suppose 0 < e and e < 1. Then there exists an eventually-positive sequence s of real numbers such that s = {1 + e^{1·n+0)}}_{n∈ℕ} and O({n^{8}}_{n∈ℕ}) ⊆ O(s) and O({n^{8}}_{n∈ℕ}) ≠ O(s).

10. Problem 3.22

One can prove the following propositions:

(26) Let f, g be sequences of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{log_{2} n} and g(0) = 0 and for every n such that n > 0 holds g(n) = n^{\sqrt{n}}. Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = f and s_{1} = g and O(s) ⊆ O(s_{1}) and O(s) ≠ O(s_{1}).

(27) Let f be a sequence of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{\sqrt{n}}. Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = f and s_{1} = {2^{1·n+0)}}_{n∈ℕ} and O(s) ⊆ O(s_{1}) and O(s) ≠ O(s_{1}).

(28) There exist eventually-positive sequences s, s_{1} of real numbers such that s = {2^{1·n+0)}}_{n∈ℕ} and s_{1} = {2^{1·n+1)}}_{n∈ℕ} and O(s) = O(s_{1}).

(29) There exist eventually-positive sequences s, s_{1} of real numbers such that s = {2^{1·n+0)}}_{n∈ℕ} and s_{1} = {2^{2·n+0)}}_{n∈ℕ} and O(s) ⊆ O(s_{1}) and O(s) ≠ O(s_{1}).

(30) There exists an eventually-positive sequence s of real numbers such that s = {2^{2·n+0)}}_{n∈ℕ} and O(s) ⊆ O({(n + 0)!}_{n∈ℕ}) and O(s) ≠ O({(n + 0)!}_{n∈ℕ}).

(31) O({(n + 0)!}_{n∈ℕ}) ⊆ O({(n + 1)!}_{n∈ℕ}) and O({(n + 0)!}_{n∈ℕ}) ≠ O({(n + 1)!}_{n∈ℕ}).

(32) Let g be a sequence of real numbers. Suppose g(0) = 0 and for every n such that n > 0 holds g(n) = n^{n}. Then there exists an eventually-positive sequence s of real numbers such that s = g and O({(n + 1)!}_{n∈ℕ}) ⊆ O(s) and O({(n + 1)!}_{n∈ℕ}) ≠ O(s).

11. Problem 3.23

One can prove the following proposition

(33) Let given n. Suppose n 1. Let f be a sequence of real numbers and k be a natural number. If for every n holds f(n) = ∑_{κ=0}^{n}({n^{k}}_{n∈ℕ})(κ), then f(n) \frac{n^{k+1}}{k+1} .

12. Problem 3.24

One can prove the following proposition

(34) Let f, g be sequences of real numbers. Suppose g(0) = 0 and for every n such that n > 0 holds g(n) = n · log_{2} n and for every n holds f(n) = log_{2}(n!). Then there exists an eventually-nonnegative sequence s of real numbers such that s = g and f ∈ Θ(s).

13. Problem 3.26

The following proposition is true

(35) Let f be an eventually-nondecreasing eventually-nonnegative sequence of real numbers and t be a sequence of real numbers. Suppose that for every n holds if n mod 2 = 0, then t(n) = 1 and if n mod 2 = 1, then t(n) = n. Then t ∉ Θ(f).

14. Problem 3.28

Let f be a function from ℕ into ℝ^{∗} and let n be a natural number. Then f(n) is a finite sequence of elements of ℝ.

Let n be a natural number and let a, b be positive real numbers. The functor Prob28(n, a, b) yields a real number and is defined by:

(Def. 6)(i) Prob28(n, a, b) = 0 if n = 0,

(ii) there exists a natural number l and there exists a function p_{28} from ℕ into ℝ^{∗} such that l+1 = n and Prob28(n, a, b) = π_{n}p_{28}(l) and p_{28}(0) = ⟨a⟩ and for every natural number n there exists a natural number n_{1} such that n_{1} = ⌈\frac{n+1+1}{2}⌉ and p_{28}(n + 1) = p_{28}(n)^{⌢} ⟨4 · π_{n_{1}}p_{28}(n) + b · (n + 1 + 1)⟩, otherwise.

Let a, b be positive real numbers. The functor {Prob28(n, a, b)}_{n∈ℕ} yields a sequence of real numbers and is defined by:

(Def. 7) ({Prob28(n, a, b)}_{n∈ℕ})(n) = Prob28(n, a, b).

The following proposition is true

(36) For all positive real numbers a, b holds {Prob28(n, a, b)}_{n∈ℕ} is eventually-nondecreasing.

15. Problem 3.30

The non empty subset {2^{n} : n ∈ ℕ} of ℕ is defined by: (Def. 8) {2^{n} : n ∈ ℕ} = {2^{n} : n ranges over natural numbers}.

Next we state three propositions:

(37) Let f be a sequence of real numbers. Suppose that for every n holds if n ∈ {2^{n} : n ∈ ℕ}, then f(n) = n and if n ∉ {2^{n} : n ∈ ℕ}, then f(n) = 2^{n}. Then f ∈ Θ({n^{1}}_{n∈ℕ}|{2^{n} : n ∈ ℕ}) and f ∉ Θ({n^{1}}_{n∈ℕ}) and {n^{1}}_{n∈ℕ} is smooth and f is not eventually-nondecreasing.

(38) Let f, g be sequences of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{2^{^{⌊}^{log _{2} n}^{⌋}}} and g(0) = 0 and for every n such that n > 0 holds g(n) = n^{n}. Then there exists an eventually-positive sequence s of real numbers such that

(i) s = g, (ii) f ∈ Θ(s|{2^{n} : n ∈ ℕ}), (iii) f ∉ Θ(s), (iv) f is eventually-nondecreasing,

(v) s is eventually-nondecreasing, and (vi) s is not smooth w.r.t. 2.

(39) Let g be a sequence of real numbers. Suppose that for every n holds if n ∈ {2^{n} : n ∈ ℕ}, then g(n) = n and if n ∉ {2^{n} : n ∈ ℕ}, then g(n) = n^{2}. Then there exists an eventually-positive sequence s of real numbers such that s = g and {n^{1}}_{n∈ℕ} ∈ Θ(s|{2^{n} : n ∈ ℕ}) and {n^{1}}_{n∈ℕ} ∉ Θ(s) and s_{2} ∈ O(s) and {n^{1}}_{n∈ℕ} is eventually-nondecreasing and s is not eventuallynondecreasing.

16. Problem 3.31

Let x be a natural number. The functor x¡ yielding a natural number is defined as follows:

(Def. 9)(i) There exists n such that n! x and x < (n + 1)! and x¡ = n! if x ≠ 0, (ii) x¡ = 0, otherwise. Next we state the proposition

(40) Let f be a sequence of real numbers. Suppose that for every n holds f(n) = n¡. Then there exists an eventually-positive sequence s of real numbers such that s = f and f is eventually-nondecreasing and for every n holds f(n) {n^{1}}_{n∈ℕ}(n) and s is not smooth.

17. Problem 3.34

Let us mention that {n^{1}}_{n∈ℕ} − {1}_{n∈ℕ} is eventually-positive. One can prove the following proposition

(41) Θ({n^{1}}_{n∈ℕ} − {1}_{n∈ℕ}) + Θ({n^{1}}_{n∈ℕ}) = Θ({n^{1}}_{n∈ℕ}).

18. Problem 3.35

One can prove the following proposition

(42) There exists a non empty set F of functions from ℕ to ℝ such that F = {{n^{1}}_{n∈ℕ}} and for every n holds {n^{(− 1)}}_{n∈ℕ}(n) {n^{1}}_{n∈ℕ}(n) and {n^{(− 1)}}_{n∈ℕ} ∉ F^{O({1}_{n∈ℕ} )}.

19. Addition

The following proposition is true

(43) Let c be a non negative real number and x, f be eventually-nonnegative sequences of real numbers. Given e, N such that e > 0 and for every n such that n N holds f(n) e. If x ∈ O(c + f), then x ∈ O(f).

20. Potentatially Useful

The following propositions are true:

(44) 2^{2} = 4.

(45) 2^{3} = 8.

(46) 2^{4} = 16.

(47) 2^{5} = 32.

(48) 2^{6} = 64.

(49) 2^{12} = 4096.

(50) For every n such that n 3 holds n^{2} > 2 · n + 1.

(51) For every n such that n 10 holds 2^{n−1} > (2 · n)^{2}.

(52) For every n such that n 9 holds (n + 1)^{6} < 2 · n^{6}.

(53) For every n such that n 30 holds 2^{n} > n^{6}.

(54) For every real number x such that x > 9 holds 2^{x} > (2 · x)^{2}.

(55) There exists N such that for every n such that n N holds \sqrt{n}−log_{2} n > 1.

(56) For all real numbers a, b, c such that a > 0 and c > 0 and c ≠ 1 holds a^{b} = c^{b·log_{c} a}.

(57) (4 + 1)! = 120.

(58) 5^{5} = 3125.

(59) 4^{4} = 256.

(60) For every n holds (n^{2} − n) + 1 > 0.

(61) For every n such that n 2 holds n! > 1.

(62) For all n_{1}, n such that n n_{1} holds n! n_{1}!.

(63) For every k such that k 1 there exists n such that n! k and k < (n + 1)! and for every m such that m! k and k < (m + 1)! holds m = n.

(64) For every n such that n 2 holds ⌈\frac{n}{2}⌉ < n.

(65) For every n such that n 3 holds n! > n.

(66) For all natural numbers m, n such that m > 0 holds m^{n} is a natural number.

(67) For every n such that n 2 holds 2^{n} > n + 1.

(68) Let a be a logbase real number and f be a sequence of real numbers. Suppose a > 1 and f(0) = 0 and for every n such that n > 0 holds f(n) = log_{a} n. Then f is eventually-positive.

(69) For all eventually-nonnegative sequences f, g of real numbers holds f ∈ O(g) and g ∈ O(f) iff O(f) = O(g).

(70) For all real numbers a, b, c such that 0 < a and a b and c 0 holds a^{c} b^{c}.

(71) For every n such that n 4 holds 2 · n + 3 < 2^{n}.

(72) For every n such that n 6 holds (n + 1)^{2} < 2^{n}.

(73) For every real number c such that c > 6 holds c^{2} < 2^{c}.

(74) Let e be a positive real number and f be a sequence of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = log_{2}(n^{e}). Then f/{n^{e}}_{n∈ℕ} is convergent and lim(f/{n^{e}}_{n∈ℕ}) = 0.

(75) For every real number e such that e > 0 holds {log_{2} n}_{n∈ℕ}/{n^{e}}_{n∈ℕ} is convergent and lim({log_{2} n}_{n∈ℕ}/{n^{e}}_{n∈ℕ}) = 0.

(76) For every sequence f of real numbers and for every N such that for every n such that n N holds f(n) 0 holds ∑_{κ=0}^{N} f(κ) 0.

(77) For all sequences f, g of real numbers and for every N such that for every n such that n N holds f(n) g(n) holds ∑_{κ=0}^{N} f(κ) ∑_{κ=0}^{N} g(κ).

(78) Let f be a sequence of real numbers and b be a real number. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = b. Let N be a natural number. Then ∑_{κ=0}^{N} f(κ) = b · N.

(79) For all sequences f, g of real numbers and for all natural numbers N, M holds ∑_{κ=N+1}^{M} f(κ) + f(N + 1) = ∑_{κ=N+1+1}^{M} f(κ).

(80) Let f, g be sequences of real numbers, M be a natural number, and given N. Suppose N M + 1. If for every n such that M + 1 n and n N holds f(n) g(n), then ∑_{κ=N+1}^{M} f(κ) ∑_{κ=N+1}^{M} g(κ).

(81) For every n holds ⌈\frac{n}{2}⌉ n.

(82) Let f be a sequence of real numbers, b be a real number, and N be a natural number. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = b. Let M be a natural number. Then ∑_{κ=N+1}^{M} f(κ) = b · (N − M).

(83) Let f, g be sequences of real numbers, N be a natural number, and c be a real number. Suppose f is convergent and limf = c and for every n such that n N holds f(n) = g(n). Then g is convergent and limg = c.

(84) For every n such that n 1 holds (n^{2} − n) + 1 n^{2}.

(85) For every n such that n 1 holds n^{2} 2 · ((n^{2} − n) + 1).

(86) For every real number e such that 0 < e and e < 1 there exists N such that for every n such that n N holds n·log_{2}(1 + e)−8·log_{2} n > 8·log_{2} n.

(87) For every n such that n 10 holds \frac{2^{2·n}}{n!} < \frac{1}{2^{n−9}} .

(88) For every n such that n 3 holds 2 · (n − 2) n − 1.

(89) For every real number c such that c 0 holds c^{\frac{1}{2}} = \sqrt{c}.

(90) There exists N such that for every n such that n N holds n − \sqrt{n} · log_{2} n > \frac{n}{2}.

(91) For every sequence s of real numbers such that for every n holds s(n) = (1 + \frac{1}{n+1})^{n+1} holds s is non-decreasing.

(92) For every n such that n 1 holds (\frac{n+1}{n})^{n} (\frac{n+2}{n+1})^{n+1}.

(93) For all k, n such that k n holds \binom{n}{k} \frac{\binom{n+1}{k}}{n+1}.

(94) For every sequence f of real numbers such that for every n holds f(n) = log_{2}(n!) and for every n holds f(n) = ∑_{κ=0}^{n}({log_{2} n}_{n∈ℕ})(κ).

(95) For every n such that n 4 holds n · log_{2} n 2 · n.

(96) Let a, b be positive real numbers. Then Prob28(0, a, b) = 0 and Prob28(1, a, b) = a and for every n such that n 2 there exists n_{1} such that n_{1} = ⌈\frac{n}{2}⌉ and Prob28(n, a, b) = 4 · Prob28(n_{1}, a, b) + b · n.

(97) For every n such that n 2 holds n^{2} > n + 1.

(98) For every n such that n 1 holds 2^{n+1} − 2^{n} > 1.

(99) For every n such that n 2 holds 2^{n} − 1 ∉ {2^{n} : n ∈ ℕ}.

(100) For all n, k such that k 1 and n! k and k < (n + 1)! holds k¡ = n!.

(101) For all real numbers a, b, c such that a > 1 and b a and c 1 holds log_{a} c log_{b} c.

References

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

[2] Gilles Brassard and Paul Bratley. Fundamentals of Algorithmics. Prentice Hall, 1996.

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

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

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

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

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

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

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

[10] Elżbieta Kraszewska and Jan Popiołek. Series in Banach and Hilbert Spaces. Formalized Mathematics, 2(5):695–699, 1991.

154 richard krueger et al.

[11] Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part I: Theory. Formalized Mathematics, 9(1):135–142, 2001.

[12] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

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

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

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

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

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

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

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

Predicate Calculus for Boolean Valued Functions. Part X

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC18.

The notation and terminology used here are introduced in the following articles: [1], [2], [3], [4], and [5].

In this paper Y is a non empty set. One can prove the following propositions:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∃_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∃_{a,B}G,A}G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∃_{a,B}G,A} G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∃_{a,B}G,A} G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[4] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

Predicate Calculus for Boolean Valued Functions. Part XI

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC19.

The terminology and notation used in this paper have been introduced in the following articles: [1], [2], [3], [4], and [5].

For simplicity, we adopt the following rules: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C are partitions of Y.

One can prove the following propositions:

(1) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(2) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(3) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(4) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(5) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∀_{¬a,B} G,A}G.

(6) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∀_{¬a,B} G,A}G.

(7) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(8) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(9) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(10) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(11) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(12) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(13) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(14) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(15) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(16) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(17) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(18) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(19) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(21)^{1} If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and

C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(22) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(23) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(24) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(25) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(26) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

1The proposition (20) has been removed.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[4] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

160 shunichi kobayashi

Four Variable Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC20.

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

1. Preliminaries

For simplicity, we follow the rules: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C, D are partitions of Y.

One can prove the following propositions:

(1) Let h be a function and A′, B′, C′, D′be sets. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′.

(2) Let A, B, C, D be sets, h be a function, and A′, B′, C′, D′ be sets. If h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(A⟼→ . A′), then domh = {A, B, C, D}.

(3) For every function h and for all sets A′, B′, C′, D′ such that G = {A, B, C, D} and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(A⟼→ . A′) holds rngh = {h(A), h(B), h(C), h(D)}.

(4) Let z, u be elements of Y and h be a function. Suppose G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then EqClass(u, B ∧ C ∧ D) ∩ EqClass(z, A) ≠ ∅.

(5) Let z, u be elements of Y. Suppose G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D and EqClass(z, C ∧ D) = EqClass(u, C ∧ D). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Four Variable Predicate Calculus

Next we state a number of propositions:

(6) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(7) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(8) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(9) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(10) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{a,B}G,A}G = ∃_{∃_{a,A}G,B}G.

(11) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(12) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(13) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(14) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(15) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(16) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(17) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(18) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(19) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(20) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(21) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(22) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(23) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(24) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(25) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(26) If G is a coordinate and G = {A, B, C, D} and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(27) If G is a coordinate and G = {A, B, C, D} and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(28) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(29) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(30) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(31) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(32) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(33) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(34) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(35) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(36) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(37) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(38) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(39) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(40) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(41) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[6] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

four variable predicate calculus for boolean … 165

[7] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

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

166 shunichi kobayashi

Four Variable Predicate Calculus for Boolean Valued Functions. Part II

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC21.

The notation and terminology used here have been introduced in the following papers: [1], [2], [4], [3], and [5].

For simplicity, we use the following convention: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C, D are partitions of Y.

Next we state a number of propositions:

(1) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(2) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

(3) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(4) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

168 shunichi kobayashi

(5) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∃_{a,B}G,A}G.

(6) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∃_{a,B}G,A}G.

(7) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(8) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(9) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(10) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(11) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(12) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(13) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∃_{a,B}G,A} G.

(14) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∃_{a,B}G,A} G.

(15) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(16) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(17) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(18) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(19) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∀_{¬a,B} G,A}G.

(20) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∀_{¬a,B} G,A}G.

(21) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(22) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(23) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(24) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(25) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(26) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(27) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(28) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(29) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(30) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(31) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐

∃_{¬∃_{a,B}G,A} G.

(32) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(33) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(34) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(35) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(36) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(37) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(38) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(39) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(40) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

References

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

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[3] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

Function Spaces in the Category of Directed Suprema Preserving Maps^{1}

Grzegorz Bancerek University of Białystok

Adam Naumowicz University of Białystok

Summary. Formalization of [15, pp. 115–117], chapter II, section 2 (2.5 – 2.10).

MML Identifier: WAYBEL27.

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

1. Currying, Uncurrying and Commuting Functions

Let F be a function. We say that F is uncurrying if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) For every set x such that x ∈ domF holds x is a function yielding

function, and (ii) for every function f such that f ∈ domF holds F (f) = uncurryf.

We say that F is currying if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) For every set x such that x ∈ domF holds x is a function and π_{1}(x)

is a binary relation, and (ii) for every function f such that f ∈ domF holds F (f) = curryf.

We say that F is commuting if and only if the conditions (Def. 3) are satisfied.

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

(Def. 3)(i) For every set x such that x ∈ domF holds x is a function yielding

function, and (ii) for every function f such that f ∈ domF holds F (f) = commute(f). Let us note that every function which is empty is also uncurrying, currying,

and commuting.

Let us mention that there exists a function which is uncurrying, currying, and commuting.

Let F be an uncurrying function and let X be a set. Observe that F ↾X is uncurrying.

Let F be a currying function and let X be a set. Note that F ↾X is currying. The following propositions are true:

(1) Let X, Y, Z, D be sets. Suppose D ⊆ (Z^{Y})^{X}. Then there exists a many sorted set F indexed by D such that F is uncurrying and rngF ⊆ Z^{[: X, Y :]}.

(2) Let X, Y, Z, D be sets. Suppose D ⊆ Z^{[: X, Y :]}. Then there exists a many sorted set F indexed by D such that F is currying and if if Y = ∅, then X = ∅, then rngF ⊆ (Z^{Y})^{X}.

Let X, Y, Z be sets. Note that there exists a many sorted set indexed by (Z^{Y})^{X} which is uncurrying and there exists a many sorted set indexed by Z^{[: X, Y :]} which is currying.

Next we state several propositions:

(3) Let A, B be non empty sets, C be a set, and f, g be commuting functions. If domf ⊆ (C^{B})^{A} and rngf ⊆ domg, then g · f = id_{dom f}.

(4) Let B be a non empty set, A, C be sets, f be an uncurrying function, and g be a currying function. If domf ⊆ (C^{B})^{A} and rngf ⊆ domg, then g · f = id_{dom f}.

(5) Let A, B, C be sets, f be a currying function, and g be an uncurrying function. If domf ⊆ C^{[: A, B:]} and rngf ⊆ domg, then g · f = id_{dom f}.

(6) For every function yielding function f and for all sets i, A such that i ∈ domcommute(f) holds (commute(f))(i)^{∘}A ⊆ π_{i}f^{∘}A.

(7) Let f be a function yielding function and i, A be sets. If for every function g such that g ∈ f^{∘}A holds i ∈ domg, then π_{i}f^{∘}A ⊆ (commute(f))(i)^{∘}A.

(8) For all sets X, Y and for every function f such that rngf ⊆ Y^{X} and for all sets i, A such that i ∈ X holds (commute(f))(i)^{∘}A = π_{i}f^{∘}A.

(9) For every function f and for all sets i, A such that [: A, {i}:] ⊆ domf holds π_{i}(curryf)^{∘}A = f^{∘}[: A, {i}:].

Let X be a set and let Y be a non empty functional set. One can verify that every function from X into Y is function yielding.

Let T be a constituted functions 1-sorted structure. Observe that the carrier of T is functional.

Let X be a set and let L be a non empty relational structure. One can check that L^{X} is constituted functions.

One can verify that there exists a lattice which is constituted functions, complete, and strict and there exists a 1-sorted structure which is constituted functions and non empty.

Let T be a constituted functions non empty relational structure. Note that every non empty relational substructure of T is constituted functions.

Next we state four propositions:

(10) Let S, T be complete lattices, f be an idempotent map from T into T, and h be a map from S into Imf. Then f · h = h.

(11) Let S be a non empty relational structure and T, T_{1} be non empty relational structures. Suppose T is a relational substructure of T_{1}. Let f be a map from S into T and f_{1} be a map from S into T_{1}. If f is monotone and f = f_{1}, then f_{1} is monotone.

(12) Let S be a non empty relational structure and T, T_{1} be non empty relational structures. Suppose T is a full relational substructure of T_{1}. Let f be a map from S into T and f_{1} be a map from S into T_{1}. If f_{1} is monotone and f = f_{1}, then f is monotone.

(13) For every set X and for every subset V of X holds (χ_{V,X})^{−1} ({1}) = V and (χ_{V,X})^{−1} ({0}) = X \ V.

2. Maps of Power Posets

Let X be a non empty set, let T be a non empty relational structure, let f be an element of T^{X}, and let x be an element of X. Then f(x) is an element of T.

Next we state several propositions:

(14) Let X be a non empty set, T be a non empty relational structure, and f, g be elements of T^{X}. Then f g if and only if for every element x of X holds f(x) g(x).

(15) Let X be a set and L, S be non empty relational structures. Suppose the relational structure of L = the relational structure of S. Then L^{X} = S^{X}.

(16) Let S_{1}, S_{2}, T_{1}, T_{2} be non empty topological spaces. Suppose that (i) the topological structure of S_{1} = the topological structure of S_{2}, and (ii) the topological structure of T_{1} = the topological structure of T_{2}.

Then [S_{1} → T_{1}] = [S_{2} → T_{2}].

(17) Let X be a set. Then there exists a map f from 2_{⊆}^{X} into (2_{⊆}^{1} )^{X} such that f is isomorphic and for every subset Y of X holds f(Y) = χ_{Y,X}.

(18) For every set X holds 2_{⊆}^{X} and (2_{⊆}^{1} )^{X} are isomorphic.

(19) Let X, Y be non empty sets, T be a non empty poset, S_{1} be a full non empty relational substructure of (T^{X})^{Y}, S_{2} be a full non empty relational substructure of (T^{Y})^{X}, and F be a map from S_{1} into S_{2}. If F is commuting, then F is monotone.

(20) Let X, Y be non empty sets, T be a non empty poset, S_{1} be a full non empty relational substructure of (T^{Y})^{X}, S_{2} be a full non empty relational substructure of T^{[: X, Y :]}, and F be a map from S_{1} into S_{2}. If F is uncurrying, then F is monotone.

(21) Let X, Y be non empty sets, T be a non empty poset, S_{1} be a full non empty relational substructure of (T^{Y})^{X}, S_{2} be a full non empty relational substructure of T^{[: X, Y :]}, and F be a map from S_{2} into S_{1}. If F is currying, then F is monotone.

3. Posets of Directed Suprema Preserving Maps

Let S be a non empty relational structure and let T be a non empty reflexive antisymmetric relational structure. The functor UPS(S, T) yielding a strict relational structure is defined by the conditions (Def. 4).

(Def. 4)(i) UPS(S, T) is a full relational substructure of T^{the carrier of S}, and

(ii) for every set x holds x ∈ the carrier of UPS(S, T) iff x is a directed-

sups-preserving map from S into T. Let S be a non empty relational structure and let T be a non empty refle-

xive antisymmetric relational structure. Observe that UPS(S, T) is non empty reflexive antisymmetric and constituted functions.

Let S be a non empty relational structure and let T be a non empty poset. One can verify that UPS(S, T) is transitive.

We now state the proposition

(22) Let S be a non empty relational structure and T be a non empty reflexive antisymmetric relational structure. Then the carrier of UPS(S, T) ⊆ (the carrier of T)^{the carrier of S}.

Let S be a non empty relational structure, let T be a non empty reflexive antisymmetric relational structure, let f be an element of UPS(S, T), and let s be an element of S. Then f(s) is an element of T.

Next we state three propositions:

(23) Let S be a non empty relational structure, T be a non empty reflexive antisymmetric relational structure, and f, g be elements of UPS(S, T). Then f g if and only if for every element s of S holds f(s) g(s).

(24) For all complete Scott top-lattices S, T holds UPS(S, T) = SCMaps(S, T).

(25) Let S, S′ be non empty relational structures and T, T′ be non empty

reflexive antisymmetric relational structures. Suppose that

(i) the relational structure of S = the relational structure of S′, and (ii) the relational structure of T = the relational structure of T′.

Then UPS(S, T) = UPS(S′, T′). Let S, T be complete lattices. Note that UPS(S, T) is complete. The following propositions are true:

(26) Let S, T be complete lattices. Then UPS(S, T) is a sups-inheriting relational substructure of T^{the carrier of S}.

(27) For all complete lattices S, T and for every subset A of UPS(S, T) holds

supA = ⨆_{(T^{the carrier of S})} A. Let S_{1}, S_{2}, T_{1}, T_{2} be non empty reflexive antisymmetric relational structu-

res and let f be a map from S_{1} into S_{2}. Let us assume that f is directedsups-preserving. Let g be a map from T_{1} into T_{2}. Let us assume that g is directed-sups-preserving. The functor UPS(f, g) yields a map from UPS(S_{2}, T_{1}) into UPS(S_{1}, T_{2}) and is defined by:

(Def. 5) For every directed-sups-preserving map h from S_{2} into T_{1} holds

(UPS(f, g))(h) = g · h · f. Next we state a number of propositions:

(28) Let S_{1}, S_{2}, S_{3}, T_{1}, T_{2}, T_{3} be non empty posets, f_{1} be a directed-sups-preserving map from S_{2} into S_{3}, f_{2} be a directed-sups-preserving map from S_{1} into S_{2}, g_{1} be a directed-sups-preserving map from T_{1} into T_{2}, and g_{2} be a directed-sups-preserving map from T_{2} into T_{3}. Then UPS(f_{2}, g_{2}) · UPS(f_{1}, g_{1}) = UPS(f_{1} · f_{2}, g_{2} · g_{1}).

(29) For all non empty reflexive antisymmetric relational structures S, T holds UPS(id_{S}, id_{T}) = id_{UPS(S,T)}.

(30) Let S_{1}, S_{2}, T_{1}, T_{2} be complete lattices, f be a directed-sups-preserving map from S_{1} into S_{2}, and g be a directed-sups-preserving map from T_{1} into T_{2}. Then UPS(f, g) is directed-sups-preserving.

(31) Ω(the Sierpiński space) is Scott.

(32) For every complete Scott top-lattice S holds [S → the Sierpiński space] = UPS(S, 2_{⊆}^{1} ).

(33) Let S be a complete lattice. Then there exists a map F from UPS(S, 2_{⊆}^{1} ) into ⟨σ(S), ⊆⟩ such that F is isomorphic and for every directed-sups-preserving map f from S into 2_{⊆}^{1} holds F (f) = f^{−1} ({1}).

(34) For every complete lattice S holds UPS(S, 2_{⊆}^{1} ) and ⟨σ(S), ⊆⟩ are isomorphic.

(35) Let S_{1}, S_{2}, T_{1}, T_{2} be complete lattices, f be a map from S_{1} into S_{2}, and g be a map from T_{1} into T_{2}. If f is isomorphic and g is isomorphic, then UPS(f, g) is isomorphic.

(36) Let S_{1}, S_{2}, T_{1}, T_{2} be complete lattices. Suppose S_{1} and S_{2} are isomorphic and T_{1} and T_{2} are isomorphic. Then UPS(S_{2}, T_{1}) and UPS(S_{1}, T_{2}) are isomorphic.

(37) Let S, T be complete lattices and f be a directed-sups-preserving projection map from T into T. Then ImUPS(id_{S}, f) = UPS(S, Imf).

(38) Let X be a non empty set, S, T be non empty posets, f be a directedsups-preserving map from S into T^{X}, and i be an element of X. Then (commute(f))(i) is a directed-sups-preserving map from S into T.

(39) Let X be a non empty set, S, T be non empty posets, and f be a directedsups-preserving map from S into T^{X}. Then commute(f) is a function from X into the carrier of UPS(S, T).

(40) Let X be a non empty set, S, T be non empty posets, and f be a function from X into the carrier of UPS(S, T). Then commute(f) is a directed-sups-preserving map from S into T^{X}.

(41) For every non empty set X and for all non empty posets S, T holds there exists a map from UPS(S, T^{X}) into UPS(S, T)^{X} which is commuting and isomorphic.

(42) For every non empty set X and for all non empty posets S, T holds UPS(S, T^{X}) and (UPS(S, T))^{X} are isomorphic.

(43) For all continuous complete lattices S, T holds UPS(S, T) is continuous.

(44) For all algebraic complete lattices S, T holds UPS(S, T) is algebraic.

(45) Let R, S, T be complete lattices and f be a directed-sups-preserving map from R into UPS(S, T). Then uncurryf is a directed-sups-preserving map from [: R, S:] into T.

(46) Let R, S, T be complete lattices and f be a directed-sups-preserving map from [: R, S:] into T. Then curryf is a directed-sups-preserving map from R into UPS(S, T).

(47) For all complete lattices R, S, T holds there exists a map from UPS(R, UPS(S, T)) into UPS([: R, S:], T) which is uncurrying and isomorphic.

References

[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537– 541, 1990.

[2] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

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

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

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

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

function spaces in the category of directed … 177

[8] Grzegorz Bancerek. Continuous lattices between T_{0} spaces. Formalized Mathematics, 9(1):111–117, 2001.

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

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

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

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

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

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

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

[16] Adam Grabowski. On the category of posets. Formalized Mathematics, 5(4):501–505, 1996.

[17] Adam Grabowski. Scott-continuous functions. Formalized Mathematics, 7(1):13–18, 1998.

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

[19] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57–62, 1998.

[20] Artur Korniłowicz. Cartesian products of relations and relational structures. Formalized Mathematics, 6(1):145–152, 1997.

[21] Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269–277, 1997.

[22] Artur Korniłowicz and Jarosław Gryko. Injective spaces. Part II. Formalized Mathematics, 9(1):41–47, 2001.

[23] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

[24] Beata Madras. Products of many sorted algebras. Formalized Mathematics, 5(1):55–60, 1996.

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

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

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

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

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

[30] Andrzej Trybulec. Scott topology. Formalized Mathematics, 6(2):311–319, 1997.

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

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

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

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

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

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

178 grzegorz bancerek and adam naumowicz

Property of Complex Functions

Takashi Mitsuishi Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. This article introduces properties of complex function, calculations of them, boundedness and constant.

MML Identifier: CFUNCT 1.

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

1. Definitions of Complex Functions

For simplicity, we adopt the following convention: X, Y are sets, C is a non empty set, c is an element of C, f, f_{1}, f_{2}, f_{3}, g, g_{1} are partial functions from C to ℂ, p is a real number, and r, q are elements of ℂ.

A Complex is an element of ℂ.

Let us consider C, f_{1}, f_{2}. The functor \frac{f_{1}}{f_{2}} yields a partial function from C to ℂ and is defined as follows:

(Def. 1) dom(\frac{f_{1}}{f_{2}} ) = domf_{1} ∩ (domf_{2} \ f_{2}^{−1} (\{0_{ℂ}\})) and for every c such that c ∈ dom(\frac{f_{1}}{f_{2}} ) holds (\frac{f_{1}}{f_{2}} )_{c} = (f_{1})_{c} · ((f_{2})_{c})^{−1} .

Let us consider C, f. The functor \frac{1}{f} yields a partial function from C to ℂ and is defined by:

(Def. 2) dom(\frac{1}{f}) = domf \f^{−1} (\{0_{ℂ}\}) and for every c such that c ∈ dom(\frac{1}{f}) holds

(\frac{1}{f})_{c} = (f_{c})^{−1} . Next we state a number of propositions:

(3)^{1} dom(f_{1} + f_{2}) = domf_{1} ∩ domf_{2} and for every c such that c ∈ dom(f_{1} +

f_{2}) holds (f_{1} + f_{2})_{c} = (f_{1})_{c} + (f_{2})_{c}.

(4) dom(f_{1}−f_{2}) = domf_{1}∩domf_{2} and for every c such that c ∈ dom(f_{1}−f_{2}) holds (f_{1} − f_{2})_{c} = (f_{1})_{c} − (f_{2})_{c}.

(5) dom(f_{1} f_{2}) = domf_{1} ∩ domf_{2} and for every c such that c ∈ dom(f_{1} f_{2}) holds (f_{1} f_{2})_{c} = (f_{1})_{c} · (f_{2})_{c}.

(6) dom(\frac{f_{1}}{f_{2}} ) = domf_{1} ∩ (domf_{2} \ f_{2}^{−1} (\{0_{ℂ}\})) and for every c such that c ∈ dom(\frac{f_{1}}{f_{2}} ) holds (\frac{f_{1}}{f_{2}} )_{c} = (f_{1})_{c} · ((f_{2})_{c})^{−1} .

(7) dom(r f) = domf and for every c such that c ∈ dom(r f) holds (r f)_{c} = r · f_{c}.

(9)^{2} dom(− f) = domf and for every c such that c ∈ dom(− f) holds (− f)_{c} =

− f_{c}.

(10) dom(\frac{1}{f}) = domf \f^{−1} (\{0_{ℂ}\}) and for every c such that c ∈ dom(\frac{1}{f}) holds (\frac{1}{f})_{c} = (f_{c})^{−1} .

(15)^{3} dom(\frac{1}{g}) ⊆ domg and domg ∩ (domg \ g^{−1} (\{0_{ℂ}\})) = domg \ g^{−1} (\{0_{ℂ}\}).

(16) dom(f_{1} f_{2}) \ (f_{1} f_{2})^{−1} ({0_{ℂ}}) = (domf_{1} \ f_{1}^{−1} ({0_{ℂ}})) ∩ (domf_{2} \ f_{2}^{−1} ({0_{ℂ}})).

(17) If c ∈ dom(\frac{1}{f}), then f_{c} ≠ 0_{ℂ}.

(18) (\frac{1}{f})^{−1} (\{0_{ℂ}\}) = ∅.

(19) |f|^{−1} ({0}) = f^{−1} ({0_{ℂ}}) and (− f)^{−1} ({0_{ℂ}}) = f^{−1} ({0_{ℂ}}).

(20) dom(_{\frac{1}{\frac{1}{f}}}) = dom(f↾ dom(\frac{1}{f})).

(21) If r ≠ 0_{ℂ}, then (r f)^{−1} ({0_{ℂ}}) = f^{−1} ({0_{ℂ}}).

2. Basic Properties of Operations

The following propositions are true:

(22) (f_{1} + f_{2}) + f_{3} = f_{1} + (f_{2} + f_{3}).

(23) (f_{1} f_{2}) f_{3} = f_{1} (f_{2} f_{3}).

(24) (f_{1} + f_{2}) f_{3} = f_{1} f_{3} + f_{2} f_{3}.

(25) f_{3} (f_{1} + f_{2}) = f_{3} f_{1} + f_{3} f_{2}.

(26) r (f_{1} f_{2}) = (r f_{1}) f_{2}.

(27) r (f_{1} f_{2}) = f_{1} (r f_{2}).

(28) (f_{1} − f_{2}) f_{3} = f_{1} f_{3} − f_{2} f_{3}.

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

2The proposition (8) has been removed.

3The propositions (11)–(14) have been removed.

(29) f_{3} f_{1} − f_{3} f_{2} = f_{3} (f_{1} − f_{2}).

(30) r (f_{1} + f_{2}) = r f_{1} + r f_{2}.

(31) (r · q) f = r (q f).

(32) r (f_{1} − f_{2}) = r f_{1} − r f_{2}.

(33) f_{1} − f_{2} = (− 1_{ℂ}) (f_{2} − f_{1}).

(34) f_{1} − (f_{2} + f_{3}) = f_{1} − f_{2} − f_{3}.

(35) 1_{ℂ} f = f.

(36) f_{1} − (f_{2} − f_{3}) = (f_{1} − f_{2}) + f_{3}.

(37) f_{1} + (f_{2} − f_{3}) = (f_{1} + f_{2}) − f_{3}.

(38) |f_{1} f_{2}| = |f_{1}| |f_{2}|.

(39) |r f| = |r| |f|.

(40) − f = (− 1_{ℂ}) f.

(41) − − f = f.

(42) f_{1} − f_{2} = f_{1} + − f_{2}.

(43) f_{1} − − f_{2} = f_{1} + f_{2}.

(44) _{\frac{1}{\frac{1}{f}}} = f↾ dom(\frac{1}{f}).

(45) \frac{1}{f_{1} f_{2}} = \frac{1}{f_{1}} \frac{1}{f_{2}} .

(46) If r ≠ 0_{ℂ}, then \frac{1}{r f} = r^{−1} \frac{1}{f} .

(47) 1_{ℂ} ≠ 0_{ℂ}.

(48) (− 1_{ℂ})^{−1} = − 1_{ℂ}.

(49) \frac{1}{− f} = (− 1_{ℂ}) \frac{1}{f} .

(50) \frac{1}{|f|} = |\frac{1}{f} |.

(51) \frac{f}{g} = f \frac{1}{g}.

(52) r \frac{g}{f} = \frac{r g}{f} .

(53) \frac{f}{g} g = f↾ dom(\frac{1}{g}).

(54) \frac{f}{g} \frac{f_{1}}{g_{1}} = \frac{f f_{1}}{g g_{1}} .

(55) _{\frac{1}{\frac{f_{1}}{f_{2}}}} = \frac{f_{2}↾ dom( \frac{1}{f_{2}} )}{f_{1}}.

(56) g \frac{f_{1}}{f_{2}} = \frac{g f_{1}}{f_{2}} .

(57) _{\frac{g}{\frac{f_{1}}{f_{2}}}} = \frac{g (f_{2}↾ dom( \frac{1}{f_{2}} ))}{f_{1}}.

(58) − \frac{f}{g} = \frac{− f}{g} and \frac{f}{− g} = − \frac{f}{g} .

(59) \frac{f_{1}}{f} + \frac{f_{2}}{f} = \frac{f_{1}+f_{2}}{f} and \frac{f_{1}}{f} − \frac{f_{2}}{f} = \frac{f_{1}−f_{2}}{f} .

(60) \frac{f_{1}}{f} + \frac{g_{1}}{g} = \frac{f_{1} g+g_{1} f}{f g} .

(61) \frac{\frac{f}{g}}{\frac{f_{1}}{g_{1}}} = \frac{f (g_{1}↾ dom( \frac{1}{g_{1}} ))}{g f_{1}}.

(62) \frac{f_{1}}{f} − \frac{g_{1}}{g} = \frac{f_{1} g−g_{1} f}{f g} .

(63) |\frac{f_{1}}{f_{2}} | = \frac{|f_{1}|}{|f_{2}|}.

(64) (f_{1}+f_{2})↾X = f_{1}↾X+f_{2}↾X and (f_{1}+f_{2})↾X = f_{1}↾X+f_{2} and (f_{1}+f_{2})↾X = f_{1} + f_{2}↾X.

(65) (f_{1} f_{2})↾X = (f_{1}↾X) (f_{2}↾X) and (f_{1} f_{2})↾X = (f_{1}↾X) f_{2} and (f_{1} f_{2})↾X = f_{1} (f_{2}↾X).

(66) (− f)↾X = − f↾X and \frac{1}{f} ↾X = \frac{1}{f↾X} and |f|↾X = |f↾X|.

(67) (f_{1}−f_{2})↾X = f_{1}↾X−f_{2}↾X and (f_{1}−f_{2})↾X = f_{1}↾X−f_{2} and (f_{1}−f_{2})↾X = f_{1} − f_{2}↾X.

(68) \frac{f_{1}}{f_{2}} ↾X = \frac{f_{1}↾X}{f_{2}↾X} and \frac{f_{1}}{f_{2}} ↾X = \frac{f_{1}↾X}{f_{2}} and \frac{f_{1}}{f_{2}} ↾X = \frac{f_{1}}{f_{2}↾X} .

(69) (r f)↾X = r (f↾X).

3. Total Partial Functions from a Domain, to Complex

We now state a number of propositions: (70)(i) f_{1} is total and f_{2} is total iff f_{1} + f_{2} is total, (ii) f_{1} is total and f_{2} is total iff f_{1} − f_{2} is total, and (iii) f_{1} is total and f_{2} is total iff f_{1} f_{2} is total.

(71) f is total iff r f is total.

(72) f is total iff − f is total.

(73) f is total iff |f| is total.

(74) \frac{1}{f} is total iff f^{−1} (\{0_{ℂ}\}) = ∅ and f is total.

(75) f_{1} is total and f_{2}^{−1} (\{0_{ℂ}\}) = ∅ and f_{2} is total iff \frac{f_{1}}{f_{2}} is total.

(76) If f_{1} is total and f_{2} is total, then (f_{1}+f_{2})_{c} = (f_{1})_{c}+(f_{2})_{c} and (f_{1}−f_{2})_{c} = (f_{1})_{c} − (f_{2})_{c} and (f_{1} f_{2})_{c} = (f_{1})_{c} · (f_{2})_{c}.

(77) If f is total, then (r f)_{c} = r · f_{c}.

(78) If f is total, then (− f)_{c} = − f_{c} and |f|(c) = |f_{c}|.

(79) If \frac{1}{f} is total, then (\frac{1}{f})_{c} = (f_{c})^{−1} .

(80) If f_{1} is total and \frac{1}{f_{2}} is total, then (\frac{f_{1}}{f_{2}} )_{c} = (f_{1})_{c} · ((f_{2})_{c})^{−1} .

4. Bounded and Constant Partial Functions from a Domain, to Complex

Let us consider C, f, Y. We say that f is bounded on Y if and only if: (Def. 3) |f| is bounded on Y.

The following propositions are true:

(81) f is bounded on Y iff there exists a real number p such that for every c such that c ∈ Y ∩ domf holds |f_{c}| p.

(82) If Y ⊆ X and f is bounded on X, then f is bounded on Y.

(83) If X ∩ domf = ∅, then f is bounded on X.

(84) If f is bounded on Y, then r f is bounded on Y.

(85) |f| is lower bounded on X.

(86) If f is bounded on Y, then |f| is bounded on Y and − f is bounded on Y.

(87) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} + f_{2} is bounded on X ∩ Y.

(88) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} f_{2} is bounded on X ∩ Y and f_{1} − f_{2} is bounded on X ∩ Y.

(89) If f is bounded on X and bounded on Y, then f is bounded on X ∪ Y.

(90) Suppose f_{1} is a constant on X and f_{2} is a constant on Y. Then f_{1} + f_{2} is a constant on X ∩ Y and f_{1} − f_{2} is a constant on X ∩ Y and f_{1} f_{2} is a constant on X ∩ Y.

(91) If f is a constant on Y, then q f is a constant on Y.

(92) If f is a constant on Y, then |f| is a constant on Y and − f is a constant on Y.

(93) If f is a constant on Y, then f is bounded on Y.

(94) If f is a constant on Y, then for every r holds r f is bounded on Y and − f is bounded on Y and |f| is bounded on Y.

(95) If f_{1} is bounded on X and f_{2} is a constant on Y, then f_{1} +f_{2} is bounded on X ∩ Y.

(96) Suppose f_{1} is bounded on X and f_{2} is a constant on Y. Then f_{1} − f_{2} is bounded on X ∩ Y and f_{2} − f_{1} is bounded on X ∩ Y and f_{1} f_{2} is bounded on X ∩ Y.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

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

184 takashi mitsuishi et al.

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

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

[5] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

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

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

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

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

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

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

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

Property of Complex Sequence and Continuity of Complex Function

Takashi Mitsuishi Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. This article introduces properties of complex sequence and continuity of complex function. The first section shows convergence of complex sequence and constant complex sequence. In the next section, definition of continuity of complex function and properties of continuous complex function are shown.

MML Identifier: CFCONT 1.

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

1. Complex Sequence

For simplicity, we adopt the following rules: n, m, k denote natural numbers, x denotes a set, X, X_{1} denote sets, g, x_{0}, x_{1}, x_{2} denote elements of ℂ, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6} denote complex sequences, Y denotes a subset of ℂ, f, f_{1}, f_{2}, h, h_{1}, h_{2} denote partial functions from ℂ to ℂ, r, s denote real numbers, and N_{1} denotes an increasing sequence of naturals.

Let us consider h, s_{3}. Let us assume that rngs_{3} ⊆ domh. The functor h · s_{3} yielding a complex sequence is defined by:

(Def. 1) h · s_{3} = (h qua function) ·(s_{3}).

Let us consider f, x_{0}. We say that f is continuous in x_{0} if and only if: (Def. 2) x_{0} ∈ domf and for every s_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}).

One can prove the following propositions: (2)^{1} s_{4} = s_{5} − s_{6} iff for every n holds s_{4}(n) = s_{5}(n) − s_{6}(n).

(3) rng(s_{3} ↑ n) ⊆ rngs_{3}.

(4) If rngs_{3} ⊆ domf, then s_{3}(n) ∈ domf.

(5) x ∈ rngs_{3} iff there exists n such that x = s_{3}(n).

(6) s_{3}(n) ∈ rngs_{3}.

(7) If s_{4} is a subsequence of s_{3}, then rngs_{4} ⊆ rngs_{3}.

(8) If s_{4} is a subsequence of s_{3} and s_{3} is non-zero, then s_{4} is non-zero.

(9) (s_{4} + s_{5}) N_{1} = s_{4} N_{1} + s_{5} N_{1} and (s_{4} − s_{5}) N_{1} = s_{4} N_{1} − s_{5} N_{1} and (s_{4} s_{5}) N_{1} = s_{4} N_{1} (s_{5} N_{1}).

(10) (g s_{3}) N_{1} = g (s_{3} N_{1}).

(11) (− s_{3}) N_{1} = − s_{3} N_{1} and |s_{3}| · N_{1} = |s_{3} N_{1}|.

(12) (s_{3} N_{1})^{−1} = s_{3}^{−1} N_{1}.

(13) (s_{4}/s_{3}) N_{1} = (s_{4} N_{1})/(s_{3} N_{1}).

(14) If for every n holds s_{3}(n) ∈ Y, then rngs_{3} ⊆ Y.

(15) If rngs_{3} ⊆ domh, then h · s_{3} = (h qua function) ·(s_{3}).

(16) If rngs_{3} ⊆ domf, then (f · s_{3})(n) = f_{s_{3}(n)}.

(17) If rngs_{3} ⊆ domf, then (f · s_{3}) ↑ n = f · (s_{3} ↑ n).

(18) If rngs_{3} ⊆ domh_{1} ∩ domh_{2}, then (h_{1} + h_{2}) · s_{3} = h_{1} · s_{3} + h_{2} · s_{3} and (h_{1} − h_{2}) · s_{3} = h_{1} · s_{3} − h_{2} · s_{3} and (h_{1} h_{2}) · s_{3} = (h_{1} · s_{3}) (h_{2} · s_{3}).

(19) If rngs_{3} ⊆ domh, then (g h) · s_{3} = g (h · s_{3}).

(20) If rngs_{3} ⊆ domh, then − h · s_{3}= (− h) · s_{3}.

(21) If rngs_{3} ⊆ dom(\frac{1}{h}), then h · s_{3} is non-zero.

(22) If rngs_{3} ⊆ dom(\frac{1}{h}), then \frac{1}{h} · s_{3} = (h · s_{3})^{−1} .

(23) If rngs_{3} ⊆ domh, then ℜ((h · s_{3}) N_{1}) = ℜ(h · (s_{3} N_{1})).

(24) If rngs_{3} ⊆ domh, then ℑ((h · s_{3}) N_{1}) = ℑ(h · (s_{3} N_{1})).

(25) If rngs_{3} ⊆ domh, then (h · s_{3}) N_{1} = h · (s_{3} N_{1}).

(26) If rngs_{4} ⊆ domh and s_{5} is a subsequence of s_{4}, then h·s_{5} is a subsequence of h · s_{4}.

(27) If h is total, then (h · s_{3})(n) = h_{s_{3}(n)}.

(28) If h is total, then h · (s_{3} ↑ n) = (h · s_{3}) ↑ n.

(29) If h_{1} is total and h_{2} is total, then (h_{1} + h_{2}) · s_{3} = h_{1} · s_{3} + h_{2} · s_{3} and (h_{1} − h_{2}) · s_{3} = h_{1} · s_{3} − h_{2} · s_{3} and (h_{1} h_{2}) · s_{3} = (h_{1} · s_{3}) (h_{2} · s_{3}).

(30) If h is total, then (g h) · s_{3} = g (h · s_{3}).

(31) If rngs_{3} ⊆ dom(h↾X), then (h↾X) · s_{3} = h · s_{3}.

1The proposition (1) has been removed.

(32) If rngs_{3} ⊆ dom(h↾X) and if rngs_{3} ⊆ dom(h↾Y) or X ⊆ Y, then (h↾X) · s_{3} = (h↾Y) · s_{3}.

(33) If rngs_{3} ⊆ dom(h↾X) and h^{−1} (\{0_{ℂ}\}) = ∅, then (\frac{1}{h}↾X) · s_{3} = ((h↾X) ·

s_{3})^{−1} . Let f be a function. We say that f is constant if and only if:

(Def. 3) For all sets n_{1}, n_{2} such that n_{1} ∈ domf and n_{2} ∈ domf holds f(n_{1}) =

f(n_{2}). Let us consider s_{3}. Let us observe that s_{3} is constant if and only if:

(Def. 4) There exists g such that for every n holds s_{3}(n) = g.

Next we state a number of propositions:

(34) s_{3} is constant iff there exists g such that rngs_{3} = {g}.

(35) s_{3} is constant iff for every n holds s_{3}(n) = s_{3}(n + 1).

(36) s_{3} is constant iff for all n, k holds s_{3}(n) = s_{3}(n + k).

(37) s_{3} is constant iff for all n, m holds s_{3}(n) = s_{3}(m).

(38) s_{3} ↑ k is a subsequence of s_{3}.

(39) If s_{4} is a subsequence of s_{3} and s_{3} is convergent, then s_{4} is convergent.

(40) If s_{4} is a subsequence of s_{3} and s_{3} is convergent, then lims_{4} = lims_{3}.

(41) If s_{3} is convergent and there exists k such that for every n such that k n holds s_{4}(n) = s_{3}(n), then s_{4} is convergent.

(42) If s_{3} is convergent and there exists k such that for every n such that k n holds s_{4}(n) = s_{3}(n), then lims_{3} = lims_{4}.

(43) If s_{3} is convergent, then s_{3} ↑ k is convergent and lim(s_{3} ↑ k) = lims_{3}.

(44) If s_{3} is convergent and there exists k such that s_{3} = s_{4} ↑ k, then s_{4} is convergent.

(45) If s_{3} is convergent and there exists k such that s_{3} = s_{4} ↑k, then lims_{4} = lims_{3}.

(46) If s_{3} is convergent and lims_{3} ≠ 0_{ℂ}, then there exists k such that s_{3} ↑ k is non-zero.

(47) If s_{3} is convergent and lims_{3} ≠ 0_{ℂ}, then there exists s_{4} which is a subsequence of s_{3} and non-zero.

(48) If s_{3} is constant, then s_{3} is convergent.

(49) If s_{3} is constant and g ∈ rngs_{3} or s_{3} is constant and there exists n such that s_{3}(n) = g, then lims_{3} = g.

(50) If s_{3} is constant, then for every n holds lims_{3} = s_{3}(n).

(51) If s_{3} is convergent and lims_{3} ≠ 0_{ℂ}, then for every s_{4} such that s_{4} is a subsequence of s_{3} and non-zero holds lim(s_{4}^{−1} ) = (lims_{3})^{−1} .

(52) If s_{3} is constant and s_{4} is convergent, then lim(s_{3} + s_{4}) = s_{3}(0) + lims_{4} and lim(s_{3} − s_{4}) = s_{3}(0) − lims_{4} and lim(s_{4} − s_{3}) = lims_{4} − s_{3}(0) and

lim(s_{3} s_{4}) = s_{3}(0) · lims_{4}. The scheme CompSeqChoice concerns and states that:

There exists s_{1} such that for every n holds P[n, s_{1}(n)] provided the following condition is satisfied:

• For every n there exists g such that P[n, g].

2. Continuity of Complex Sequence

We now state several propositions:

(53) f is continuous in x_{0} if and only if the following conditions are satisfied: (i) x_{0} ∈ domf, and (ii) for every s_{1} such that rngs_{1} ⊆ domf and s_{1} is convergent and lims_{1} =

x_{0} and for every n holds s_{1}(n) ≠ x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}).

(54) f is continuous in x_{0} if and only if the following conditions are satisfied: (i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

x_{1} such that x_{1} ∈ domf and |x_{1} − x_{0}| < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(55) Suppose f_{1} is continuous in x_{0} and f_{2} is continuous in x_{0}. Then f_{1} + f_{2} is continuous in x_{0} and f_{1} − f_{2} is continuous in x_{0} and f_{1} f_{2} is continuous in x_{0}.

(56) If f is continuous in x_{0}, then g f is continuous in x_{0}.

(57) If f is continuous in x_{0}, then − f is continuous in x_{0}.

(58) If f is continuous in x_{0} and f_{x_{0}} ≠ 0_{ℂ}, then \frac{1}{f} is continuous in x_{0}.

(59) If f_{1} is continuous in x_{0} and (f_{1})_{x_{0}} ≠ 0_{ℂ} and f_{2} is continuous in x_{0}, then

\frac{f_{2}}{f_{1}} is continuous in x_{0}. Let us consider f, X. We say that f is continuous on X if and only if:

(Def. 5) X ⊆ domf and for every x_{0} such that x_{0} ∈ X holds f↾X is continuous

in x_{0}. One can prove the following propositions:

(60) Let given X, f. Then f is continuous on X if and only if the following

conditions are satisfied:

(i) X ⊆ domf, and (ii) for every s_{1} such that rngs_{1} ⊆ X and s_{1} is convergent and lims_{1} ∈ X

holds f · s_{1} is convergent and f_{lim s_{1}} = lim(f · s_{1}).

(61) f is continuous on X if and only if the following conditions are satisfied: (i) X ⊆ domf, and (ii) for all x_{0}, r such that x_{0} ∈ X and 0 < r there exists s such that 0 < s

and for every x_{1} such that x_{1} ∈ X and |x_{1} − x_{0}| < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(62) f is continuous on X iff f↾X is continuous on X.

(63) If f is continuous on X and X_{1} ⊆ X, then f is continuous on X_{1}.

(64) If x_{0} ∈ domf, then f is continuous on {x_{0}}.

(65) Let given X, f_{1}, f_{2}. Suppose f_{1} is continuous on X and f_{2} is continuous on X. Then f_{1} + f_{2} is continuous on X and f_{1} − f_{2} is continuous on X and f_{1} f_{2} is continuous on X.

(66) Let given X, X_{1}, f_{1}, f_{2}. Suppose f_{1} is continuous on X and f_{2} is continuous on X_{1}. Then f_{1} + f_{2} is continuous on X ∩ X_{1} and f_{1} − f_{2} is continuous on X ∩ X_{1} and f_{1} f_{2} is continuous on X ∩ X_{1}.

(67) For all g, X, f such that f is continuous on X holds g f is continuous on X.

(68) If f is continuous on X, then − f is continuous on X.

(69) If f is continuous on X and f^{−1} (\{0_{ℂ}\}) = ∅, then \frac{1}{f} is continuous on X.

(70) If f is continuous on X and (f↾X)^{−1} (\{0_{ℂ}\}) = ∅, then \frac{1}{f} is continuous on X.

(71) If f_{1} is continuous on X and f_{1}^{−1} ({0_{ℂ}}) = ∅ and f_{2} is continuous on X, then \frac{f_{2}}{f_{1}} is continuous on X.

(72) If f is total and for all x_{1}, x_{2} holds f_{x_{1}+x_{2}} = f_{x_{1}} + f_{x_{2}} and there exists

x_{0} such that f is continuous in x_{0}, then f is continuous on ℂ. Let us consider X. We say that X is compact if and only if:

(Def. 6) For every s_{1} such that rngs_{1} ⊆ X there exists s_{2} such that s_{2} is a

subsequence of s_{1} and convergent and lims_{2} ∈ X. One can prove the following propositions:

(73) For every f such that domf is compact and f is continuous on domf holds rngf is compact.

(74) If Y ⊆ domf and Y is compact and f is continuous on Y, then f^{∘}Y is compact.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

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

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

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

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

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

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

[8] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized Mathematics, 9(1):179–184, 2001.

190 takashi mitsuishi et al.

[9] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.

[10] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Serieses. Formalized Mathematics, 6(3):403–410, 1997.

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

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

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

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

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

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

Scalar Multiple of Riemann Definite Integral

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. The goal of this article is to prove a scalar multiplicity of Riemann definite integral. Therefore, we defined a scalar product to the subset of real space, and we proved some relating lemmas. At last, we proved a scalar multiplicity of Riemann definite integral. As a result, a linearity of Riemann definite integral was proven by unifying the previous article [7].

MML Identifier: INTEGRA2.

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

1. Lemmas of Finite Sequence

We adopt the following rules: r, x, y are real numbers, i, j are natural numbers, and p is a finite sequence of elements of ℝ.

The following proposition is true

(1) For every closed-interval subset A of ℝ and for every x holds x ∈ A iff inf A x and x supA.

Let I_{1} be a finite sequence of elements of ℝ. We say that I_{1} is non-decreasing if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let n be a natural number. Suppose n ∈ domI_{1} and n + 1 ∈ domI_{1}. Let

r, s be real numbers. If r = I_{1}(n) and s = I_{1}(n + 1), then r s. One can verify that there exists a finite sequence of elements of ℝ which is

non-decreasing.

The following three propositions are true:

(2) Let p be a non-decreasing finite sequence of elements of ℝ and given i, j. If i ∈ domp and j ∈ domp and i j, then p(i) p(j).

(3) Let given p. Then there exists a non-decreasing finite sequence q of elements of ℝ such that p and q are fiberwise equipotent.

(4) Let D be a non empty set, f be a finite sequence of elements of D, and k_{1}, k_{2}, k_{3} be natural numbers. If 1 k_{1} and k_{3} lenf and k_{1} k_{2} and k_{2} < k_{3}, then (mid(f, k_{1}, k_{2}))^{⌢} mid(f, k_{2} + 1, k_{3}) = mid(f, k_{1}, k_{3}).

2. Scalar Product of Real Subset

Let X be a subset of ℝ and let r be a real number. The functor r · X yields a subset of ℝ and is defined as follows:

(Def. 2) r · X = {r · x : x ∈ X}. The following propositions are true:

(5) Let X, Y be non empty sets and f be a partial function from X to ℝ. If f is upper bounded on X and Y ⊆ X, then f↾Y is upper bounded on Y.

(6) Let X, Y be non empty sets and f be a partial function from X to ℝ. If f is lower bounded on X and Y ⊆ X, then f↾Y is lower bounded on Y.

(7) For every non empty subset X of ℝ holds r · X is non empty.

(8) For every subset X of ℝ holds r · X = {r · x : x ∈ X}.

(9) For every non empty subset X of ℝ such that X is upper bounded and 0 r holds r · X is upper bounded.

(10) For every non empty subset X of ℝ such that X is upper bounded and r 0 holds r · X is lower bounded.

(11) For every non empty subset X of ℝ such that X is lower bounded and 0 r holds r · X is lower bounded.

(12) For every non empty subset X of ℝ such that X is lower bounded and r 0 holds r · X is upper bounded.

(13) For every non empty subset X of ℝ such that X is upper bounded and 0 r holds sup(r · X) = r · supX.

(14) For every non empty subset X of ℝ such that X is upper bounded and r 0 holds inf(r · X) = r · supX.

(15) For every non empty subset X of ℝ such that X is lower bounded and 0 r holds inf(r · X) = r · inf X.

(16) For every non empty subset X of ℝ such that X is lower bounded and r 0 holds sup(r · X) = r · inf X.

3. Scalar Multiple of Integral

The following propositions are true:

(17) For every non empty set X and for every partial function f from X to ℝ such that f is total holds rng(r f) = r · rngf.

(18) For all non empty sets X, Z and for every partial function f from X to ℝ holds rng(r (f↾Z)) = r · rng(f↾Z).

(19) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (upper sum setr f)(D) r · inf rngf · vol(A).

(20) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (upper sum setr f)(D) r · suprngf · vol(A).

(21) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (lower sum setr f)(D) r · suprngf · vol(A).

(22) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (lower sum setr f)(D) r · inf rngf · vol(A).

(23) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is upper bounded on A and total and r 0. Then (upper volume(r f, D))(i) = r · (upper volume(f, D))(i).

(24) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is upper bounded on A and total and r 0. Then (lower volume(r f, D))(i) = r · (upper volume(f, D))(i).

(25) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is lower bounded on A and total and r 0. Then (lower volume(r f, D))(i) = r · (lower volume(f, D))(i).

(26) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is lower bounded on A and total and r 0. Then (upper volume(r f, D))(i) = r · (lower volume(f, D))(i).

(27) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is upper bounded on A and total and r 0, then upper sum(r f, D) = r · upper sum(f, D).

(28) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is upper bounded on A and total and r 0, then lower sum(r f, D) = r · upper sum(f, D).

(29) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is lower bounded on A and total and r 0, then lower sum(r f, D) = r · lower sum(f, D).

(30) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is lower bounded on A and total and r 0, then upper sum(r f, D) = r · lower sum(f, D).

(31) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. Suppose f is total and bounded on A and f is integrable on A. Then r f is integrable on A and integralr f = r · integralf.

4. Monotoneity of Integral

One can prove the following propositions:

(32) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. Suppose f is total and bounded on A and f is integrable on A and for every x such that x ∈ A holds f(x) 0. Then integralf 0.

(33) Let A be a closed-interval subset of ℝ and f, g be partial functions from

A to ℝ. Suppose that

(i) f is total and bounded on A, (ii) f is integrable on A, (iii) g is total and bounded on A, and (iv) g is integrable on A.

Then f − g is integrable on A and integralf − g = integralf − integralg.

(34) Let A be a closed-interval subset of ℝ and f, g be partial functions from

A to ℝ. Suppose that

(i) f is total and bounded on A, (ii) f is integrable on A, (iii) g is total and bounded on A, (iv) g is integrable on A, and

(v) for every x such that x ∈ A holds f(x) g(x). Then integralf integralg.

5. Definition of Division Sequence

Next we state two propositions:

(35) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is total and bounded on A, then rngupper sum setf is lower bounded.

(36) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is total and bounded on A, then rnglower sum setf is upper bounded.

Let A be a closed-interval subset of ℝ. A DivSequence of A is a function from ℕ into divsA.

Let A be a closed-interval subset of ℝ and let T be a DivSequence of A. The functor δ_{T} yielding a sequence of real numbers is defined by:

(Def. 3) For every i holds δ_{T} (i) = δ_{T (i)}.

Let A be a closed-interval subset of ℝ, let f be a partial function from A to ℝ, and let T be a DivSequence of A. The functor upper sum(f, T) yields a sequence of real numbers and is defined by:

(Def. 4) For every i holds (upper sum(f, T))(i) = upper sum(f, T (i)). The functor lower sum(f, T) yields a sequence of real numbers and is defined as follows:

(Def. 5) For every i holds (lower sum(f, T))(i) = lower sum(f, T (i)).

The following propositions are true:

(37) Let A be a closed-interval subset of ℝ and D_{1}, D_{2} be elements of divsA. If D_{1} D_{2}, then for every j such that j ∈ domD_{2} there exists i such that i ∈ domD_{1} and divset(D_{2}, j) ⊆ divset(D_{1}, i).

(38) For all finite non empty subsets X, Y of ℝ such that X ⊆ Y holds maxX maxY.

(39) For all finite non empty subsets X, Y of ℝ such that there exists y such that y ∈ Y and maxX y holds maxX maxY.

(40) For all closed-interval subsets A, B of ℝ such that A ⊆ B holds vol(A) vol(B).

(41) For every closed-interval subset A of ℝ and for all elements D_{1}, D_{2} of divsA such that D_{1} D_{2} holds δ_{(D_{1})} δ_{(D_{2})}.

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.

196 noboru endou et al.

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

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

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

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

[7] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93–102, 1999.

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

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

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

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

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

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

[14] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Formalized Mathematics, 5(2):167–172, 1996.

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

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

Darboux’s Theorem

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article, we have proved the Darboux’s theorem. This theorem is important to prove the Riemann integrability. We can replace an upper bound and a lower bound of a function which is the definition of Riemann integration with convergence of sequence by Darboux’s theorem.

MML Identifier: INTEGRA3.

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

1. Lemmas of Division

We adopt the following convention: x, y are real numbers, i, j, k are natural numbers, and p, q are finite sequences of elements of ℝ.

The following propositions are true:

(1) Let A be a closed-interval subset of ℝ and D be an element of divsA. If vol(A) ≠ 0, then there exists i such that i ∈ domD and vol(divset(D, i)) > 0.

(2) Let A be a closed-interval subset of ℝ, D be an element of divsA, and given x. If x ∈ A, then there exists j such that j ∈ domD and x ∈ divset(D, j).

(3) Let A be a closed-interval subset of ℝ and D_{1}, D_{2} be elements of divsA. Then there exists an element D of divsA such that D_{1} D and D_{2} D and rngD = rngD_{1} ∪ rngD_{2}.

(4) Let A be a closed-interval subset of ℝ and D, D_{1} be elements of divsA. Suppose δ_{(D_{1})} < minrngupper volume(χ_{A,A}, D). Let given x, y, i. If i ∈ domD_{1} and x ∈ rngD ∩ divset(D_{1}, i) and y ∈ rngD ∩ divset(D_{1}, i), then x = y.

(5) For all p, q such that rngp = rngq and p is increasing and q is increasing holds p = q.

(6) Let A be a closed-interval subset of ℝ, D, D_{1} be elements of divsA, and given i, j. Suppose D D_{1} and i ∈ domD and j ∈ domD and i j. Then indx(D_{1}, D, i) indx(D_{1}, D, j) and indx(D_{1}, D, i) ∈ domD_{1} and indx(D_{1}, D, j) ∈ domD_{1}.

(7) Let A be a closed-interval subset of ℝ, D, D_{1} be elements of divsA, and given i, j. Suppose D D_{1} and i ∈ domD and j ∈ domD and i < j. Then indx(D_{1}, D, i) < indx(D_{1}, D, j) and indx(D_{1}, D, i) ∈ domD_{1} and indx(D_{1}, D, j) ∈ domD_{1}.

(8) For every closed-interval subset A of ℝ and for every element D of divsA holds δ_{D} 0.

(9) Let A be a closed-interval subset of ℝ, g be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and lenD_{1} 2 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪{x} and g is total and bounded on A. Then ∑ lower volume(g, D_{2}) − ∑ lower volume(g, D_{1}) (suprngg − inf rngg) · δ_{(D_{1})}.

(10) Let A be a closed-interval subset of ℝ, g be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and lenD_{1} 2 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪{x} and g is total and bounded on A. Then ∑ upper volume(g, D_{1}) − ∑ upper volume(g, D_{2}) (suprngg − inf rngg) · δ_{(D_{1})}.

(11) Let A be a closed-interval subset of ℝ, D be an element of divsA, r be a real number, and i, j be natural numbers. Suppose i ∈ domD and j ∈ domD and i j and r < (mid(D, i, j))(1). Then there exists a closed-interval subset B of ℝ such that r = inf B and supB = (mid(D, i, j))(lenmid(D, i, j)) and lenmid(D, i, j) = (j − i) + 1 and mid(D, i, j) is a DivisionPoint of B.

(12) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and vol(A) ≠ 0 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪ {x} and f is total and bounded on A and x > inf A. Then ∑ lower volume(f, D_{2})−∑ lower volume(f, D_{1}) (suprngf −inf rngf)· δ_{(D_{1})}.

(13) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and vol(A) ≠ 0 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪ {x} and f is total and bounded on A and x > inf A. Then ∑ upper volume(f, D_{1})−∑ upper volume(f, D_{2}) (suprngf−inf rngf)· δ_{(D_{1})}.

(14) Let A be a closed-interval subset of ℝ, D_{1}, D_{2} be elements of divsA, r be a real number, and i, j be natural numbers. Suppose i ∈ domD_{1} and j ∈ domD_{1} and i j and D_{1} D_{2} and r < (mid(D_{2}, indx(D_{2}, D_{1}, i), indx(D_{2}, D_{1}, j)))(1). Then there exists a closed-interval subset B of ℝ and there exist elements M_{1}, M_{2} of divsB such that r = inf B and supB = M_{2}(lenM_{2}) and supB = M_{1}(lenM_{1}) and M_{1} M_{2} and M_{1} = mid(D_{1}, i, j) and M_{2} = mid(D_{2}, indx(D_{2}, D_{1}, i), indx(D_{2}, D_{1}, j)).

(15) Let A be a closed-interval subset of ℝ, D be an element of divsA, and given x. If x ∈ rngD, then D(1) x and x D(lenD).

(16) Let p be a finite sequence of elements of ℝ and given i, j, k. Suppose p is increasing and i ∈ domp and j ∈ domp and k ∈ domp and p(i) p(k) and p(k) p(j). Then p(k) ∈ rngmid(p, i, j).

(17) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D be an element of divsA, and given i. If f is total and bounded on A and i ∈ domD, then inf rng(f↾ divset(D, i)) suprngf.

(18) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D be an element of divsA, and given i. If f is total and bounded on A and i ∈ domD, then suprng(f↾ divset(D, i)) inf rngf.

2. Darboux’s Theorem

The following two propositions are true:

(19) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and T be a DivSequence of A. Suppose f is total and bounded on A and δ_{T} is convergent to 0 and vol(A) ≠ 0. Then lower sum(f, T) is convergent and limlower sum(f, T) = lower integralf.

(20) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and T be a DivSequence of A. Suppose f is total and bounded on A and δ_{T} is convergent to 0 and vol(A) ≠ 0. Then upper sum(f, T) is convergent and limupper sum(f, T) = upper integralf.

References

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

200 noboru endou et al.

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

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

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

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

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

[7] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93–102, 1999.

[8] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191–196, 2001.

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

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

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

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

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

[14] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[15] Piotr Rudnicki. The for (going up) macro instruction. Formalized Mathematics, 7(1):107– 114, 1998.

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

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

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

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

Five Variable Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC22.

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

1. Preliminaries

For simplicity, we follow the rules: Y denotes a non empty set, a denotes an element of BVF(Y), G denotes a subset of PARTITIONS(Y), and A, B, C, D, E denote partitions of Y.

One can prove the following propositions:

(1) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(A, G) = B ∧ C ∧ D ∧ E.

(2) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(B, G) = A ∧ C ∧ D ∧ E.

(3) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(C, G) = A ∧ B ∧ D ∧ E.

(4) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(D, G) = A ∧ B ∧ C ∧ E.

(5) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(E, G) = A ∧ B ∧ C ∧ D.

(6) Let A, B, C, D, E be sets, h be a function, and A′, B′, C′, D′, E′ be sets. Suppose A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′.

(7) Let A, B, C, D, E be sets, h be a function, and A′, B′, C′, D′, E′ be sets. Suppose A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E}.

(8) Let A, B, C, D, E be sets, h be a function, and A′, B′, C′, D′, E′ be sets. Suppose A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E)}.

(9) Let G be a subset of PARTITIONS(Y), A, B, C, D, E be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then EqClass(u, B ∧ C ∧ D ∧ E) ∩ EqClass(z, A) ≠ ∅.

(10) Let G be a subset of PARTITIONS(Y), A, B, C, D, E be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and EqClass(z, C ∧ D ∧ E) = EqClass(u, C ∧ D ∧ E). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Predicate Calculus

One can prove the following propositions:

(11) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(12) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(13) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(14) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(15) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(16) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(17) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(18) ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(19) ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(20) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(22)^{1} Suppose that 204 shunichi kobayashi

1The proposition (21) has been removed.

G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(23) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ¬∀_{∀_{a,A}G,B}G = ∃_{¬∀_{a,B}G,A} G.

(24) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ¬∀_{∀_{a,A}G,B}G = ∃_{∃_{¬a,B} G,A}G.

(25) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(26) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[6] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[7] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

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

Six Variable Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC23.

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

1. Preliminaries

For simplicity, we follow the rules: Y denotes a non empty set, a denotes an element of BVF(Y), G denotes a subset of PARTITIONS(Y), and A, B, C, D, E, F denote partitions of Y.

We now state a number of propositions:

(1) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(A, G) = B ∧ C ∧ D ∧ E ∧ F.

(2) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(B, G) = A ∧ C ∧ D ∧ E ∧ F.

(3) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(C, G) = A ∧ B ∧ D ∧ E ∧ F.

(4) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(D, G) = A ∧ B ∧ C ∧ E ∧ F.

(5) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(E, G) = A ∧ B ∧ C ∧ D ∧ F.

(6) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(F, G) = A ∧ B ∧ C ∧ D ∧ E.

(7) Let A, B, C, D, E, F be sets, h be a function, and A′, B′, C′, D′, E′, F′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′ and h(F) = F′.

(8) Let A, B, C, D, E, F be sets, h be a function, and A′, B′, C′, D′, E′, F′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F }.

(9) Let A, B, C, D, E, F be sets, h be a function, and A′, B′, C′, D′, E′, F′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(A⟼→ . A′).

Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F)}.

(10) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then EqClass(u, B ∧ C ∧ D ∧ E ∧ F) ∩ EqClass(z, A) ≠ ∅.

(11) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and EqClass(z, C ∧ D ∧ E ∧ F) = EqClass(u, C ∧ D ∧ E ∧ F). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Predicate Calculus

The following propositions are true:

(12) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(13) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(14) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(15) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(16) Suppose that 208 shunichi kobayashi

G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(17) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(18) ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(19) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(20) ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(21) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[6] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[7] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

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

The Construction and Computation of for-loop Programs for SCMPDS^{1}

Jing-Chao Chen Shanghai Jiaotong University

Piotr Rudnicki University of Alberta

Summary. This article defines two for-loop statements for SCMPDS. One is called for-up, which corresponds to ”for (i=x; i<0; i+=n) S” in C language. Another is called for-down, which corresponds to ”for (i=x; i>0; i-=n) S”. Here, we do not present their unconditional halting (called parahalting) property, because we have not found that there exists a useful for-loop statement with unconditional halting, and the proof of unconditional halting is much simpler than that of conditional halting. It is hard to formalize all halting conditions, but some cases can be formalized. We choose loop invariants as halting conditions to prove halting problem of for-up/down statements. When some variables (except the loop control variable) keep undestroyed on a set for the loop invariant, and the loop body is halting for this condition, the corresponding for-up/down is halting and computable under this condition. The computation of for-loop statements can be realized by evaluating its body. At the end of the article, we verify for-down statements by two examples for summing.

MML Identifier: SCMPDS 7.

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

1. Preliminaries

For simplicity, we adopt the following convention: x is a set, n is a natural number, a is a Int position, i, j, k are instructions of SCMPDS, s, s_{1}, s_{2} are

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

states of SCMPDS, l_{1}, l are instructions-locations of SCMPDS, and I, J, K are Program-block.

We now state a number of propositions:

(1) For every state s of SCMPDS and for all natural numbers m, n such that IC_{s} = insposm holds ICplusConst(s, n − m) = insposn.

(2) For all finite partial states P, Q of SCMPDS such that P ⊆ Q holds ProgramPart(P) ⊆ ProgramPart(Q).

(3) For all programmed finite partial states P, Q of SCMPDS and for every natural number k such that P ⊆ Q holds Shift(P, k) ⊆ Shift(Q, k).

(4) If IC_{s} = inspos0, then Initialized(s) = s.

(5) If IC_{s} = inspos0, then s+·Initialized(I) = s+·I.

(6) (Computation(s))(n)↾the instruction locations of SCMPDS = s↾the instruction locations of SCMPDS.

(7) Let s_{1}, s_{2} be states of SCMPDS. Suppose IC_{(s_{1})} = IC_{(s_{2})} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM} and s_{1}↾the instruction locations of SCMPDS = s_{2}↾the instruction locations of SCMPDS. Then s_{1} = s_{2}.

(8) l ∈ domI iff l ∈ domInitialized(I).

(9) If x ∈ domI, then I(x) = (s+·(I+·Start-At(l)))(x).

(10) If l_{1} ∈ domI, then (s+·Initialized(I))(l_{1}) = I(l_{1}).

(11) (s+·(I+·Start-At(l)))(a) = s(a).

(12) (s+·Start-At(l_{1}))(IC_{SCMPDS} ) = l_{1}.

(13) card(I;i) = cardI + 1.

(14) (I;i;j)(insposcardI) = i.

(15) (i;I;j);k = i;(I;j;k).

(16) Shift(J, cardI) ⊆ I;J;K.

(17) I ⊆ stopI;J.

(18) If l_{1} ∈ domI, then (Shift(stopI, n))(l_{1} + n) = (Shift(I, n))(l_{1} + n).

(19) If cardI > 0, then (Shift(stopI, n))(insposn) = (Shift(I, n))(insposn).

(20) For every state s of SCMPDS and for every instruction i of SCMPDS such that InsCode(i) ∈ {0, 4, 5, 6} holds Exec(i, s)↾Data-Loc_{SCM} = s↾Data-Loc_{SCM}.

(21) For all states s, s_{3} of SCMPDS holds (s+·s_{3}↾the instruction locations of SCMPDS)↾Data-Loc_{SCM} = s↾Data-Loc_{SCM}.

(22) For every instruction i of SCMPDS holds rngLoad(i) = {i}.

(23) If IC_{(s_{1})} = IC_{(s_{2})} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}, then IC_{Exec(i,s_{1})} = IC_{Exec(i,s_{2})} and Exec(i, s_{1})↾Data-Loc_{SCM} = Exec(i, s_{2})↾Data-Loc_{SCM}.

(24) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and Initialized(stopI) ⊆ s_{1} and Initialized(stopI) ⊆ s_{2} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

(25) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and halting on s_{1} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let k be a natural number. Then (Computation(s_{1}+·Initialized(stopI)))(k) and (Computation(s_{2}+·Initialized(stopI)))(k) are equal outside the instruction locations of SCMPDS and CurInstr((Computation(s_{1}+·Initialized (stopI)))(k)) = CurInstr((Computation(s_{2}+·Initialized(stopI)))(k)).

(26) Let I be a Program-block. Suppose that (i) I is closed on s_{1} and halting on s_{1}, (ii) Initialized(stopI) ⊆ s_{1}, (iii) Initialized(stopI) ⊆ s_{2}, and (iv) s_{1} and s_{2} are equal outside the instruction locations of SCMPDS.

Let k be a natural number. Then (Computation(s_{1}))(k) and (Computation(s_{2}))(k) are equal outside the instruction locations of SCMPDS and CurInstr((Computation(s_{1}))(k)) = CurInstr((Computation(s_{2}))(k)).

(27) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and halting on s_{1} and Initialized(stopI) ⊆ s_{1} and Initialized(stopI) ⊆ s_{2} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then LifeSpan(s_{1}) = LifeSpan(s_{2}).

(28) Let I be a Program-block. Suppose that (i) I is closed on s_{1} and halting on s_{1}, (ii) Initialized(stopI) ⊆ s_{1}, (iii) Initialized(stopI) ⊆ s_{2}, and (iv) s_{1} and s_{2} are equal outside the instruction locations of SCMPDS.

Then LifeSpan(s_{1}) = LifeSpan(s_{2}) and Result(s_{1}) and Result(s_{2}) are equal outside the instruction locations of SCMPDS.

(29) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and halting on s_{1} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then LifeSpan(s_{1}+·Initialized(stopI)) = LifeSpan(s_{2}+·Initialized(stopI)) and Result(s_{1}+·Initialized(stopI)) and Result(s_{2}+·Initialized(stopI)) are equal outside the instruction locations of SCMPDS.

(30) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose

that

(i) I is closed on s_{1} and halting on s_{1}, (ii) Initialized(stopI) ⊆ s_{1}, (iii) Initialized(stopI) ⊆ s_{2}, and (iv) there exists a natural number k such that (Computation(s_{1}))(k) and

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

Let I be a Program-block. One can check that Initialized(I) is initial. The following propositions are true:

(31) Let s be a state of SCMPDS, I be a Program-block, and a be a Int position. If I is halting on s, then (IExec(I, s))(a) = (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI)))(a).

(32) Let s be a state of SCMPDS, I be a parahalting Program-block, and a be a Int position. Then (IExec(I, s))(a) = (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI)))(a).

(33) Let I be a Program-block and i be a natural number. If Initialized(stopI) ⊆ s and I is closed on s and halting on s and i < LifeSpan(s), then IC_{(Computation (s))(i)} ∈ domI.

(34) Let I be a shiftable Program-block. Suppose Initialized(stopI) ⊆ s_{1} and I is closed on s_{1} and halting on s_{1}. Let n be a natural number. Suppose Shift(I, n) ⊆ s_{2} and cardI > 0 and IC_{(s_{2})} = insposn and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let i be a natural number. If i < LifeSpan(s_{1}), then IC_{(Computation (s_{1}))(i)} +n = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

(35) For every No-StopCode Program-block I such that Initialized(stopI) ⊆ s and I is halting on s and cardI > 0 holds LifeSpan(s) > 0.

(36) Let I be a No-StopCode shiftable Program-block. Suppose Initialized (stopI) ⊆ s_{1} and I is closed on s_{1} and halting on s_{1}. Let n be a natural number. Suppose Shift(I, n) ⊆ s_{2} and cardI > 0 and IC_{(s_{2})} = insposn and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then IC_{(Computation (s_{2}))(LifeSpan (s_{1}))} = insposcardI + n and (Computation(s_{1})) (LifeSpan(s_{1}))↾Data-Loc_{SCM} = (Computation(s_{2}))(LifeSpan(s_{1}))↾Data-Loc_{SCM}.

(37) Let s be a state of SCMPDS, I be a Program-block, and n be a natural number. If IC_{(Computation (s+·Initialized (I)))(n)} = inspos0, then (Computation(s+·Initialized(I)))(n)+·Initialized(I) = (Computation(s+·Initialized(I)))(n).

(38) Let I be a Program-block, J be a Program-block, and k be a natural number. Suppose I is closed on s and halting on s and k LifeSpan(s+·Initialized(stopI)). Then (Computation(s+·Initialized (stopI)))(k) and (Computation(s+·((I;J)+·Start-At(inspos0))))(k) are

equal outside the instruction locations of SCMPDS.

(39) Let I, J be Program-block and k be a natural number. Suppose I ⊆ J and I is closed on s and halting on s and k LifeSpan(s+·Initialized(stopI)). Then (Computation(s+·Initialized(J)))(k) and (Computation(s+·Initialized(stopI)))(k) are equal outside the instruction locations of SCMPDS.

(40) Let I, J be Program-block and k be a natural number. Suppose k LifeSpan(s+·Initialized(stopI)) and I ⊆ J and I is closed on s and halting on s. Then IC_{(Computation (s+·Initialized (J)))(k)} ∈ domstopI.

(41) Let I, J be Program-block. Suppose I ⊆ J and I is closed on s and halting on s. Then CurInstr((Computation(s+·Initialized(J))) (LifeSpan(s+·Initialized(stopI)))) = halt_{SCMPDS} or IC_{(Computation (s+·Initialized (J)))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(42) Let I, J be Program-block. Suppose I is halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then J is closed on (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI))) and halting on (Computation(s+·Initialized(stopI))) (LifeSpan(s+·Initialized(stopI))).

(43) Let I be a Program-block and J be a shiftable Program-block. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then I;J is closed on s and I;J is halting on s.

(44) Let I be a No-StopCode Program-block and J be a Programblock. If I ⊆ J and I is closed on s and halting on s, then IC_{(Computation (s+·Initialized (J)))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(45) Let I be a Program-block, s be a state of SCMPDS, and k be a natural number. If I is halting on s and k < LifeSpan(s+·Initialized(stopI)), then CurInstr((Computation(s+·Initialized(stopI)))(k)) ≠ halt_{SCMPDS} .

(46) Let I, J be Program-block, s be a state of SCMPDS, and k be a natural number. Suppose I is closed on s and halting on s and k < LifeSpan(s+·Initialized(stopI)). Then CurInstr((Computation(s+· Initialized(stopI;J)))(k)) ≠ halt_{SCMPDS} .

(47) Let I be a No-StopCode Program-block and J be a shiftable Program-block. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then LifeSpan(s+·Initialized(stopI;J)) = LifeSpan(s+·Initialized(stopI)) + LifeSpan(Result(s+·Initialized(stopI))+·Initialized(stopJ)).

(48) Let I be a No-StopCode Program-block and J be a shiftable Programblock. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then IExec(I;J, s) = IExec(J, IExec(I, s))+·Start-At(IC_{IExec(J,IExec(I,s))} + cardI).

(49) Let I be a No-StopCode Program-block and J be a shiftable Programblock. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then (IExec(I;J, s))(a) = (IExec(J, IExec(I, s)))(a).

(50) Let I be a No-StopCode Program-block and j be a parahalting shiftable instruction of SCMPDS. If I is closed on s and halting on s, then (IExec(I;j, s))(a) = (Exec(j, IExec(I, s)))(a).

2. The Construction of for-up loop Program

Let a be a Int position, let i be an integer, let n be a natural number, and let I be a Program-block. The functor for-up(a, i, n, I) yielding a Program-block is defined by:

(Def. 1) for-up(a, i, n, I) = ((a, i) >= 0 goto cardI + 3);I; AddTo(a, i, n); goto (− (cardI + 2)).

3. The Computation of for-up loop Program

We now state several propositions:

(51) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then cardfor-up(a, i, n, I) = cardI + 3.

(52) Let a be a Int position, i be an integer, n, m be natural numbers, and I be a Program-block. Then m < cardI + 3 if and only if insposm ∈ domfor-up(a, i, n, I).

(53) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then (for-up(a, i, n, I))(inspos0) = (a, i) >= 0 goto cardI + 3 and (for-up(a, i, n, I))(insposcardI + 1) = AddTo(a, i, n) and (for-up(a, i, n, I))(insposcardI + 2) = goto (− (cardI + 2)).

(54) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then for-up(a, i, n, I) is closed on s and for-up(a, i, n, I) is halting on s.

(55) Let s be a state of SCMPDS, I be a Program-block, a, c be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IExec(for-up(a, i, n, I), s) = s+·Start-At(insposcardI + 3).

(56) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IC_{IExec(for-up(a,i,n,I),s)} = insposcardI + 3.

(57) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then (IExec(for-up(a, i, n, I), s))(b) = s(b).

(58) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) < 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then for-up(a, i, n, I) is closed on s and for-up(a, i, n, I) is halting on s.

(59) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) < 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then IExec(for-up(a, i, n, I), s) = IExec(for-up(a, i, n, I), IExec(I; AddTo(a, i, n), s)).

Let I be a shiftable Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Observe that for-up(a, i, n, I) is shiftable.

Let I be a No-StopCode Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Note that for-up(a, i, n, I) is No-StopCode.

4. The Construction of for-down loop Program

Let a be a Int position, let i be an integer, let n be a natural number, and let I be a Program-block. The functor for − down (a, i, n, I) yielding a Programblock is defined as follows:

(Def. 2) for − down (a, i, n, I) = ((a, i) <= 0 goto cardI + 3);I; AddTo(a, i, − n); goto (− (cardI + 2)).

5. The Computation of for-down loop Program

One can prove the following propositions:

(60) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then cardfor − down (a, i, n, I) = cardI + 3.

(61) Let a be a Int position, i be an integer, n, m be natural numbers, and I be a Program-block. Then m < cardI + 3 if and only if insposm ∈ domfor − down (a, i, n, I).

(62) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then (for − down (a, i, n, I))(inspos0) = (a, i) <= 0 goto cardI + 3 and (for − down (a, i, n, I))(insposcardI + 1) = AddTo(a, i, − n) and (for − down (a, i, n, I))(insposcardI + 2) = goto (− (cardI + 2)).

(63) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then for − down (a, i, n, I) is closed on s and for − down (a, i, n, I) is halting on s.

(64) Let s be a state of SCMPDS, I be a Program-block, a, c be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IExec(for − down (a, i, n, I), s) = s+·Start-At(insposcardI + 3).

(65) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IC_{IExec(for−down (a,i,n,I),s)} = insposcardI + 3.

(66) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then (IExec(for − down (a, i, n, I), s))(b) = s(b).

(67) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) > 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then for − down (a, i, n, I) is closed on s and for − down (a, i, n, I) is halting on s.

(68) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) > 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then IExec(for − down (a, i, n, I), s) = IExec(for − down (a, i, n, I), IExec(I; AddTo(a, i, − n), s)).

Let I be a shiftable Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Observe that for − down (a, i, n, I) is shiftable.

Let I be a No-StopCode Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Note that for − down (a, i, n, I) is No-StopCode.

6. Two Examples for Summing

Let n be a natural number. The functor sumn yielding a Program-block is defined as follows:

(Def. 3) sumn = (GBP:=0);((GBP)_{2}:=n);((GBP)_{3}:=0); for − down (GBP, 2, 1,

Load(AddTo(GBP, 3, 1))). Next we state three propositions:

(69) For every state s of SCMPDS such that s(GBP) = 0 holds for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))) is closed on s and for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))) is halting on s.

(70) Let s be a state of SCMPDS and n be a natural number. If s(GBP) = 0 and s(intpos2) = n and s(intpos3) = 0, then (IExec(for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))), s))(intpos3) = n.

(71) For every state s of SCMPDS and for every natural number n holds

(IExec(sumn, s))(intpos3) = n. Let s_{4}, c_{1}, r_{1}, p_{1}, p_{2} be natural numbers. The functor sum(s_{4}, c_{1}, r_{1}, p_{1}, p_{2})

yields a Program-block and is defined as follows: (Def. 4) sum(s_{4}, c_{1}, r_{1}, p_{1}, p_{2}) = ((intposs_{4})_{r_{1}}:=0);(intposp_{1}:=p_{2});

for − down (intposs_{4}, c_{1}, 1, AddTo(intposs_{4}, r_{1}, intposp_{2}, 0); AddTo(intposp_{1}, 0, 1)).

Next we state three propositions:

(72) Let s be a state of SCMPDS and s_{4}, c_{2}, r_{1}, p_{1}, p_{3} be natural numbers. Suppose s(intposs_{4}) > s_{4} and c_{2} < r_{1} and s(intposp_{1}) = p_{3} and s(intposs_{4}) + r_{1} < p_{1} and p_{1} < p_{3} and p_{3} < s(intposp_{3}). Then for − down (intposs_{4}, c_{2}, 1, AddTo(intposs_{4}, r_{1}, intposp_{3}, 0); AddTo(intposp_{1}, 0, 1)) is closed on s and for − down (intposs_{4}, c_{2}, 1, AddTo(intposs_{4}, r_{1}, intposp_{3}, 0); AddTo(intposp_{1}, 0, 1)) is halting on s.

(73) Let s be a state of SCMPDS, s_{4}, c_{2}, r_{1}, p_{1}, p_{3} be natural numbers, and f be a finite sequence of elements of ℕ. Suppose that s(intposs_{4}) > s_{4} and c_{2} < r_{1} and s(intposp_{1}) = p_{3} and s(intposs_{4})+r_{1} < p_{1} and p_{1} < p_{3} and p_{3} < s(intposp_{3}) and s(DataLoc(s(intposs_{4}), r_{1})) = 0 and lenf = s(DataLoc(s(intposs_{4}), c_{2})) and for every natural number k such that k < lenf holds f(k + 1) = s(DataLoc(s(intposp_{3}), k)). Then (IExec(for − down (intposs_{4}, c_{2}, 1, AddTo(intposs_{4}, r_{1}, intposp_{3}, 0); AddTo(intposp_{1}, 0, 1)), s))(DataLoc(s(intposs_{4}), r_{1})) = ∑ f.

(74) Let s be a state of SCMPDS, s_{4}, c_{2}, r_{1}, p_{1}, p_{3} be natural numbers, and f be a finite sequence of elements of ℕ. Suppose that s(intposs_{4}) > s_{4} and c_{2} < r_{1} and s(intposs_{4}) + r_{1} < p_{1} and p_{1} < p_{3} and p_{3} < s(intposp_{3}) and lenf = s(DataLoc(s(intposs_{4}), c_{2})) and for every natural number k such that k < lenf holds f(k + 1) = s(DataLoc(s(intposp_{3}), k)). Then (IExec(sum(s_{4}, c_{2}, r_{1}, p_{1}, p_{3}), s)) (DataLoc(s(intposs_{4}), r_{1})) = ∑ f.

References

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

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

the construction and computation of for-loop … 219

[3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

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

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

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

[8] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[9] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Formalized Mathematics, 8(1):193–199, 1999.

[10] Jing-Chao Chen. Computation of two consecutive program blocks for SCMPDS. Formalized Mathematics, 8(1):211–217, 1999.

[11] Jing-Chao Chen. The construction and computation of conditional statements for SCMPDS. Formalized Mathematics, 8(1):219–234, 1999.

[12] Jing-Chao Chen. The construction and shiftability of program blocks for SCMPDS. Formalized Mathematics, 8(1):201–210, 1999.

[13] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

[14] Jing-Chao Chen. Recursive Euclide algorithm. Formalized Mathematics, 9(1):1–4, 2001.

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

[16] Andrzej Kondracki. The chinese remainder theorem. Formalized Mathematics, 6(4):573– 577, 1997.

[17] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Formalized Mathematics, 3(2):151–160, 1992.

[18] Yatsuka Nakamura and Andrzej Trybulec. On a mathematical model of programs. Formalized Mathematics, 3(2):241–250, 1992.

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

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

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

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

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

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

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

220 jing-chao chen and piotr rudnicki

Predicate Calculus for Boolean Valued Functions. Part XII

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC24.

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

1. Preliminaries

For simplicity, we adopt the following convention: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M, N are partitions of Y, and x, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9} are sets.

The following propositions are true:

(1) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(A, G) = B ∧ C ∧ D ∧ E ∧ F ∧ J.

(2) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(B, G) = A ∧ C ∧ D ∧ E ∧ F ∧ J.

(3) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(C, G) = A ∧ B ∧ D ∧ E ∧ F ∧ J.

(4) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(D, G) = A ∧ B ∧ C ∧ E ∧ F ∧ J.

(5) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(E, G) = A ∧ B ∧ C ∧ D ∧ F ∧ J.

(6) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(F, G) = A ∧ B ∧ C ∧ D ∧ E ∧ J.

(7) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(J, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F.

(8) Let A, B, C, D, E, F, J be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (A⟼→ . A′). Then h(A) = A′and h(B) = B′and h(C) = C′and h(D) = D′

and h(E) = E′ and h(F) = F′ and h(J) = J′.

(9) Let A, B, C, D, E, F, J be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (A⟼→ . A′). Then domh = {A, B, C, D, E, F, J}.

(10) Let A, B, C, D, E, F, J be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F), h(J)}.

(11) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then EqClass(u, B ∧ C ∧ D ∧ E ∧ F ∧ J) ∩ EqClass(z, A) ≠ ∅.

(12) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and EqClass(z, C ∧D ∧E ∧F ∧J) = EqClass(u, C ∧D ∧E ∧F ∧J). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

(13) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(A, G) = B ∧C ∧D ∧E ∧F ∧J ∧M.

(14) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(B, G) = A∧C ∧D ∧E ∧F ∧J ∧M.

(15) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(C, G) = A∧B ∧D ∧E ∧F ∧J ∧M.

(16) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(D, G) = A∧B ∧C ∧E ∧F ∧J ∧M.

(17) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(E, G) = A∧B ∧C ∧D ∧F ∧J ∧M.

(18) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(F, G) = A∧B ∧C ∧D ∧E ∧J ∧M.

(19) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(J, G) = A∧B ∧C ∧D ∧E ∧F ∧M.

(20) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(M, G) = A∧B ∧C ∧D ∧E ∧F ∧J.

(21) Let A, B, C, D, E, F, J, M be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (M⟼→ . M′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′ and h(F) = F′ and h(J) = J′ and h(M) = M′.

(22) Let A, B, C, D, E, F, J, M be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (M⟼→ . M′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F, J, M}.

(23) Let A, B, C, D, E, F, J, M be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (M⟼→ . M′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F), h(J), h(M)}.

(24) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then EqClass(u, B∧C∧D∧E∧F ∧J∧M)∩EqClass(z, A) ≠ ∅.

(25) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and EqClass(z, C ∧ D ∧ E ∧ F ∧ J ∧ M) = EqClass(u, C ∧ D ∧ E ∧ F ∧ J ∧ M). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

The scheme UI10 deals with a set A, a set B, a set C, a set D, a set E, a set F, a set G, a set H, a set I, a set J , and and states that:

P[A, B, C, D, E, F, G, H, I, J] provided the following condition is satisfied:

• For all sets x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} holds P[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}].

Let us consider x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}. The functor {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} yielding a set is defined as fol-

lows: (Def. 1) x ∈ {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} iff x = x_{1} or x = x_{2} or x = x_{3} or

x = x_{4} or x = x_{5} or x = x_{6} or x = x_{7} or x = x_{8} or x = x_{9}. We now state a number of propositions:

(26) x ∈ {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} iff x = x_{1} or x = x_{2} or x = x_{3} or x = x_{4} or x = x_{5} or x = x_{6} or x = x_{7} or x = x_{8} or x = x_{9}.

(27) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}} ∪ {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(28) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}} ∪ {x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(29) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}} ∪ {x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(30) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}} ∪ {x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(31) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} ∪ {x_{6}, x_{7}, x_{8}, x_{9}}.

(32) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} ∪ {x_{7}, x_{8}, x_{9}}.

(33) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}} ∪ {x_{8}, x_{9}}.

(34) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}} ∪ {x_{9}}.

(35) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(A, G) = B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N.

(36) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(B, G) = A ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N.

(37) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(C, G) = A ∧ B ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N.

(38) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(D, G) = A ∧ B ∧ C ∧ E ∧ F ∧ J ∧ M ∧ N.

(39) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(E, G) = A ∧ B ∧ C ∧ D ∧ F ∧ J ∧ M ∧ N.

(40) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(F, G) = A ∧ B ∧ C ∧ D ∧ E ∧ J ∧ M ∧ N.

(41) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(J, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ M ∧ N.

(42) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(M, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ N.

(43) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(N, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M.

(44) Let A, B, C, D, E, F, J, M, N be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′, N′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (M⟼→ . M′)+·(N⟼→ . N′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′ and h(F) = F′ and h(J) = J′ and h(M) = M′ and h(N) = N′.

(45) Let A, B, C, D, E, F, J, M, N be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′, N′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (M⟼→ . M′)+·(N⟼→ . N′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F, J, M, N}.

(46) Let A, B, C, D, E, F, J, M, N be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′, N′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+· (M⟼→ . M′)+·(N⟼→ . N′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F), h(J), h(M), h(N)}.

(47) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M, N be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then EqClass(u, B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N) ∩ EqClass(z, A) ≠ ∅.

(48) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M, N be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and EqClass(z, C∧D∧E∧F ∧J∧M∧N) = EqClass(u, C∧D∧E∧F ∧J∧M∧N). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Predicate Calculus

We now state a number of propositions:

(49) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(50) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(51) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(52) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(53) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(54) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(55) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(56) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(57) ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(58) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J

and F ≠ J. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(59) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(60) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(61) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(62) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(63) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(64) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(66)^{1} Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(67) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(68) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(69) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(70) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

1The proposition (65) has been removed.

(71) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(72) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(73) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(74) ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(75) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(76) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[6] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

[7] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

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

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

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

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

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

Index of MML Identifiers

ALGSPEC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ASYMPT 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135 ASYMPT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 BVFUNC10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 BVFUNC11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 BVFUNC12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 BVFUNC13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 BVFUNC14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 BVFUNC15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 BVFUNC16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 BVFUNC17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 BVFUNC18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 BVFUNC19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 BVFUNC20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 BVFUNC21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 BVFUNC22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 BVFUNC23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 BVFUNC24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 CFCONT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185 CFUNCT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179 CONLAT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 INTEGRA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 INTEGRA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 IRRAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 JORDAN9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 JORDAN10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 POLYNOM1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 RADIX 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 SCMPDS 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209 SCMP GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 WAYBEL24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 WAYBEL25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 WAYBEL26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 WAYBEL27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 YELLOW14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 YELLOW15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 YELLOW16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Contents Formaliz. Math. 9 (1)

Recursive Euclide Algorithm

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Scott-Continuous Functions. Part II

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Some Properties of Isomorphism between Relational Structures.

On the Product of Topological Spaces

By Jarosław Gryko and Artur Korniłowicz  . . . . . . . . . . . . . 13

Cages - the External Approximation of Jordan’s Curve

By Czesław Byliński and Mariusz Żynel  . . . . . . . . . . . . . . . . . . 19

Components and Basis of Topological Spaces

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Properties of the External Approximation of Jordan’s Curve

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Irrationality of e

By Freek Wiedijk  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Injective Spaces. Part II

By Artur Korniłowicz and Jarosław Gryko  . . . . . . . . . . . . . 41

Propositional Calculus for Boolean Valued Functions. Part VI

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Predicate Calculus for Boolean Valued Functions. Part III

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 51

A Characterization of Concept Lattices. Dual Concept Lattices

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Predicate Calculus for Boolean Valued Functions. Part IV

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 61

Predicate Calculus for Boolean Valued Functions. Part V

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 65

Continued on inside back cover

Definitions of Radix-2^{k} Signed-Digit Number and its Adder Algo-

rithm

By Yoshinori Fujisawa and Yasushi Fuwa  . . . . . . . . . . . . . . . . . 71

Retracts and Inheritance

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Technical Preliminaries to Algebraic Specifications

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Multivariate Polynomials with Arbitrary Number of Variables

By Piotr Rudnicki and Andrzej Trybulec  . . . . . . . . . . . . . . . . 95

Continuous Lattices between T_{0} Spaces

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Predicate Calculus for Boolean Valued Functions. Part VI

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Predicate Calculus for Boolean Valued Functions. Part VII

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Predicate Calculus for Boolean Valued Functions. Part VIII

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Predicate Calculus for Boolean Valued Functions. Part IX

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Asymptotic Notation. Part I: Theory

By Richard Krueger et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Asymptotic Notation. Part II: Examples and Problems

By Richard Krueger et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Predicate Calculus for Boolean Valued Functions. Part X

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Predicate Calculus for Boolean Valued Functions. Part XI

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Four Variable Predicate Calculus for Boolean Valued Functions.

Part I

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Four Variable Predicate Calculus for Boolean Valued Functions.

Part II

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Function Spaces in the Category of Directed Suprema Preserving

Maps

By Grzegorz Bancerek and Adam Naumowicz  . . . . . . . . . . . 171

Property of Complex Functions

By Takashi Mitsuishi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Property of Complex Sequence and Continuity of Complex Func-

tion

By Takashi Mitsuishi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Scalar Multiple of Riemann Definite Integral

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Darboux’s Theorem

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Five Variable Predicate Calculus for Boolean Valued Functions.

Part I

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Six Variable Predicate Calculus for Boolean Valued Functions.

Part I

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

The Construction and Computation of for-loop Programs for SCMPDS

By Jing-Chao Chen and Piotr Rudnicki  . . . . . . . . . . . . . . . . . . 209

Predicate Calculus for Boolean Valued Functions. Part XII

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Continued on inside back cover