FORMALIZED MATHEMATICS Volume 7, Number 2, 1998 University of Białystok

The Lawson Topology^{1}

Grzegorz Bancerek University of Białystok

Summary. The article includes definitions, lemmas and theorems 1.1–1.7, 1.9, 1.10 presented in Chapter III of [9, pp. 142–146].

MML Identifier: WAYBEL19.

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

1. Lower Topology

Let T be a non empty FR-structure. We say that T is lower if and only if: (Def. 1) {− ↑x : x ranges over elements of T } is a prebasis of T.

Let us note that every non empty reflexive topological space-like FR-structure which is trivial is also lower.

One can verify that there exists a top-lattice which is lower, trivial, complete, and strict.

We now state the proposition

(1) For every non empty relational structure L_{1} holds there exists a strict correct topological augmentation of L_{1} which is lower.

We now state the proposition

(2) Let L_{2}, L_{3} be topological space-like lower non empty FR-structures. Suppose the relational structure of L_{2} = the relational structure of L_{3}. Then the topology of L_{2} = the topology of L_{3}.

1This work has been partially supported by KBN grant 8 T11C 018 12 and NATO CRG 951368 grant.

Let R be a non empty relational structure. The functor ω(R) yielding a family of subsets of R is defined by:

(Def. 2) For every lower correct topological augmentation T of R holds ω(R) =

the topology of T. Next we state a number of propositions:

(3) Let R_{1}, R_{2} be non empty relational structures. Suppose the relational structure of R_{1} = the relational structure of R_{2}. Then ω(R_{1}) = ω(R_{2}).

(4) For every lower non empty FR-structure T and for every point x of T holds − ↑x is open and ↑x is closed.

(5) For every transitive lower non empty FR-structure T and for every subset A of T such that A is open holds A is lower.

(6) For every transitive lower non empty FR-structure T and for every subset A of T such that A is closed holds A is upper.

(7) Let T be a non empty topological space-like FR-structure. Then T is lower if and only if {− ↑F; F ranges over subsets of T: F is finite} is a basis of T.

(8) Let S, T be lower complete top-lattices and f be a map from S into T. Suppose that for every non empty subset X of S holds f preserves inf of X. Then f is continuous.

(9) Let S, T be lower complete top-lattices and f be a map from S into T. If f is infs-preserving, then f is continuous.

(10) Let T be a lower complete top-lattice, B_{1} be a prebasis of T, and F be a non empty filtered subset of T. Suppose that for every subset A of T such that A ∈ B_{1} and inf F ∈ A holds F meets A. Then inf F ∈ \overline{F} .

(11) Let S, T be lower complete top-lattices and f be a map from S into T. If f is continuous, then f is filtered-infs-preserving.

(12) Let S, T be lower complete top-lattices and f be a map from S into T. Suppose f is continuous and for every finite subset X of S holds f preserves inf of X. Then f is infs-preserving.

(13) Let T be a lower topological space-like reflexive transitive non empty

FR-structure and x be a point of T. Then \overline{\{x\}} = ↑x. A top-poset is a topological space-like reflexive transitive antisymmetric FR-

structure.

One can check that every non empty top-poset which is lower is also T_{0}. Let R be a lower-bounded non empty relational structure. One can verify

that every topological augmentation of R is lower-bounded.

We now state four propositions:

(14) Let S, T be non empty relational structures, s be an element of S, and t be an element of T. Then − ↑⟨⟨s, t⟩⟩= [: − ↑s, the carrier of T :]∪[: the carrier of S, − ↑t:].

(15) Let S, T be lower-bounded non empty posets, S′ be a lower correct topological augmentation of S, and T′ be a lower correct topological augmentation of T. Then ω([: S, T :]) = the topology of [: S′, (T′ qua non empty topological space) :].

(16) Let S, T be lower lower-bounded non empty top-posets. Then ω([: S, (T qua poset) :]) = the topology of [: S, (T qua non empty topological space) :].

(17) Let T, T_{2} be lower complete top-lattices. Suppose T_{2} is a topological augmentation of [: T, (T qua lattice) :]. Let f be a map from T_{2} into T. If f = ⊓_{T} , then f is continuous.

2. Refinements Revisited

The scheme TopInd deals with a top-lattice A and and states that:

For every subset A of A such that A is open holds P[A] provided the following conditions are met:

• There exists a prebasis K of A such that for every subset A of A such that A ∈ K holds P[A],

• For every family F of subsets of A such that for every subset A of A such that A ∈ F holds P[A] holds P[⋃ F],

• For all subsets A_{1}, A_{2} of A such that P[A_{1}] and P[A_{2}] holds P[A_{1} ∩ A_{2}], and

• P[Ω_{A}]. One can prove the following proposition

(18) Let L_{2}, L_{3} be up-complete antisymmetric non empty reflexive relational

structures. Suppose that

(i) the relational structure of L_{2} = the relational structure of L_{3}, and (ii) for every element x of L_{2} holds ↓↓x is directed and non empty.

If L_{2} satisfies axiom of approximation, then L_{3} satisfies axiom of approximation.

Let T be a continuous non empty poset. One can verify that every topological augmentation of T is continuous.

The following propositions are true:

(19) Let T, S be topological spaces, R be a refinement of T and S, and W be a subset of R. If W ∈ the topology of T or W ∈ the topology of S, then W is open.

(20) Let T, S be topological spaces, R be a refinement of T and S, V be a subset of T, and W be a subset of R. If W = V, then if V is open, then W is open.

(21) Let T, S be topological spaces. Suppose the carrier of T = the carrier of S. Let R be a refinement of T and S, V be a subset of T, and W be a subset of R. If W = V, then if V is closed, then W is closed.

(22) Let T be a non empty topological space and K, O be sets such that

K ⊆ O and O ⊆ the topology of T. Then

(i) if K is a basis of T, then O is a basis of T, and (ii) if K is a prebasis of T, then O is a prebasis of T.

(23) Let T_{1}, T_{2} be non empty topological spaces. Suppose the carrier of T_{1} = the carrier of T_{2}. Let T be a refinement of T_{1} and T_{2}, B_{2} be a prebasis of T_{1}, and B_{3} be a prebasis of T_{2}. Then B_{2} ∪ B_{3} is a prebasis of T.

(24) Let T_{1}, S_{1}, T_{2}, S_{2} be non empty topological spaces, R_{1} be a refinement of T_{1} and S_{1}, R_{2} be a refinement of T_{2} and S_{2}, f be a map from T_{1} into T_{2}, g be a map from S_{1} into S_{2}, and h be a map from R_{1} into R_{2}. Suppose h = f and h = g. If f is continuous and g is continuous, then h is continuous.

(25) Let T be a non empty topological space, K be a prebasis of T, N be a net in T, and p be a point of T. Suppose that for every subset A of T such that p ∈ A and A ∈ K holds N is eventually in A. Then p ∈ LimN.

(26) Let T be a non empty topological space, N be a net in T, and S be a subset of T. If N is eventually in S, then LimN ⊆ \overline{S}.

(27) Let R be a non empty relational structure and X be a non empty subset of R. Then the mapping of ⟨X; id⟩ = id_{X} and the mapping of ⟨X^{op}; id⟩ = id_{X}.

(28) For every reflexive antisymmetric non empty relational structure R and for every element x of R holds ↑x ∩ ↓x = {x}.

3. Lawson Topology

Let T be a reflexive non empty FR-structure. We say that T is Lawson if and only if:

(Def. 3) ω(T) ∪ σ(T) is a prebasis of T.

Next we state the proposition

(29) Let R be a complete lattice, L_{1} be a lower correct topological augmentation of R, S be a Scott topological augmentation of R, and T be a correct topological augmentation of R. Then T is Lawson if and only if T is a refinement of S and L_{1}.

Let R be a complete lattice. One can check that there exists a topological augmentation of R which is Lawson, strict, and correct.

Let us observe that there exists a top-lattice which is Scott, complete, and strict and there exists a complete strict top-lattice which is Lawson and continuous.

We now state three propositions:

(30) For every Lawson complete top-lattice T holds σ(T) ∪ {− ↑x : x ranges over elements of T } is a prebasis of T.

(31) Let T be a Lawson complete top-lattice. Then σ(T) ∪ {W \ ↑x; W ranges over subsets of T, x ranges over elements of T: W ∈ σ(T)} is a prebasis of T.

(32) Let T be a Lawson complete top-lattice. Then {W \ ↑F; W ranges over subsets of T, F ranges over subsets of T: W ∈ σ(T) ∧ F is finite} is a basis of T.

Let T be a complete lattice. The functor λ(T) yields a family of subsets of T and is defined as follows:

(Def. 4) For every Lawson correct topological augmentation S of T holds λ(T) =

the topology of S. We now state a number of propositions:

(33) For every complete lattice R holds λ(R) = UniCl(FinMeetCl(σ(R) ∪ ω(R))).

(34) Let R be a complete lattice, T be a lower correct topological augmentation of R, S be a Scott correct topological augmentation of R, and M be a refinement of S and T. Then λ(R) = the topology of M.

(35) For every lower up-complete top-lattice T and for every subset A of T such that A is open holds A has the property (S).

(36) For every Lawson complete top-lattice T and for every subset A of T such that A is open holds A has the property (S).

(37) Let S be a Scott complete top-lattice, T be a Lawson correct topological augmentation of S, and A be a subset of S. If A is open, then for every subset C of T such that C = A holds C is open.

(38) Let T be a Lawson complete top-lattice and x be an element of T. Then ↑x is closed and ↓x is closed and {x} is closed.

(39) For every Lawson complete top-lattice T and for every element x of T holds − ↑x is open and − ↓x is open and − {x} is open.

(40) For every Lawson complete continuous top-lattice T and for every element x of T holds ↑↑x is open and − ↑↑x is closed.

(41) Let S be a Scott complete top-lattice, T be a Lawson correct topological augmentation of S, and A be an upper subset of T. If A is open, then for every subset C of S such that C = A holds C is open.

(42) Let T be a Lawson complete top-lattice and A be a lower subset of T.

168 grzegorz bancerek

Then A is closed if and only if A is closed under directed sups.

(43) For every Lawson complete top-lattice T and for every non empty filtered subset F of T holds Lim⟨F^{op}; id⟩ = {inf F }.

Let us observe that every complete top-lattice which is Lawson is also T_{1} and compact.

Let us observe that every complete continuous top-lattice which is Lawson is also Hausdorff.

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

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

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

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

[9] 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.

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

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

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

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

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

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

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

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

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

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

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

Received June 21, 1998

Kernel Projections and Quotient Lattices

Piotr Rudnicki^{1} University of Alberta Edmonton

Summary. This article completes the Mizar formalization of Chapter I, Section 2 from [12]. After presenting some preliminary material (not all of which is later used in this article) we give the proof of theorem 2.7 (i), p.60. We do not follow the hint from [12] suggesting using the equations 2.3, p. 58. The proof is taken directly from the definition of continuous lattice. The goal of the last section is to prove the correspondence between the set of all congruences of a continuous lattice and the set of all kernel operators of the lattice which preserve directed sups (Corollary 2.13).

MML Identifier: WAYBEL20.

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

1. Preliminaries

The following two propositions are true:

(1) For every set X and for every subset S of △_{X} holds π_{1}(S) = π_{2}(S).

(2) For all non empty sets X, Y and for every function f from X into Y holds [: f, f :]^{−1} (△_{Y}) is an equivalence relation of X.

Let L_{1}, L_{2}, T_{1}, T_{2} be relational structures, let f be a map from L_{1} into T_{1}, and let g be a map from L_{2} into T_{2}. Then [: f, g:] is a map from [: L_{1}, L_{2} :] into [: T_{1}, T_{2} :].

One can prove the following propositions:

1This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

(3) For all functions f, g and for every set X holds π_{1}([: f, g:]^{∘}X) ⊆ f^{∘}π_{1}(X) and π_{2}([: f, g:]^{∘}X) ⊆ g^{∘}π_{2}(X).

(4) For all functions f, g and for every set X such that X ⊆ [: domf, domg:] holds π_{1}([: f, g:]^{∘}X) = f^{∘}π_{1}(X) and π_{2}([: f, g:]^{∘}X) = g^{∘}π_{2}(X).

(5) For every non empty antisymmetric relational structure S such that inf ∅ exists in S holds S is upper-bounded.

(6) For every non empty antisymmetric relational structure S such that sup ∅ exists in S holds S is lower-bounded.

(7) Let L_{1}, L_{2} be antisymmetric non empty relational structures and D be a subset of [: L_{1}, L_{2} :]. If inf D exists in [: L_{1}, L_{2} :], then inf D = ⟨⟨inf π_{1}(D), inf π_{2}(D)⟩⟩.

(8) Let L_{1}, L_{2} be antisymmetric non empty relational structures and D be a subset of [: L_{1}, L_{2} :]. If sup D exists in [: L_{1}, L_{2} :], then supD = ⟨⟨supπ_{1}(D), supπ_{2}(D)⟩⟩.

(9) Let L_{1}, L_{2}, T_{1}, T_{2} be antisymmetric non empty relational structures, f be a map from L_{1} into T_{1}, and g be a map from L_{2} into T_{2}. Suppose f is infs-preserving and g is infs-preserving. Then [: f, g:] is infs-preserving.

(10) Let L_{1}, L_{2}, T_{1}, T_{2} be antisymmetric reflexive non empty relational structures, f be a map from L_{1} into T_{1}, and g be a map from L_{2} into T_{2}. Suppose f is filtered-infs-preserving and g is filtered-infs-preserving. Then [: f, g:] is filtered-infs-preserving.

(11) Let L_{1}, L_{2}, T_{1}, T_{2} be antisymmetric non empty relational structures, f be a map from L_{1} into T_{1}, and g be a map from L_{2} into T_{2}. Suppose f is sups-preserving and g is sups-preserving. Then [: f, g:] is sups-preserving.

(12) Let L_{1}, L_{2}, T_{1}, T_{2} be antisymmetric reflexive non empty relational structures, f be a map from L_{1} into T_{1}, and g be a map from L_{2} into T_{2}. Suppose f is directed-sups-preserving and g is directed-sups-preserving. Then [: f, g:] is directed-sups-preserving.

(13) Let L be an antisymmetric non empty relational structure and X be a subset of [: L, L:]. Suppose X ⊆ △_{the carrier of L} and inf X exists in [: L, L:]. Then inf X ∈ △_{the carrier of L}.

(14) Let L be an antisymmetric non empty relational structure and X be a subset of [: L, L:]. Suppose X ⊆ △_{the carrier of L} and sup X exists in [: L, L:]. Then supX ∈ △_{the carrier of L}.

(15) Let L, M be non empty relational structures. If L and M are isomorphic and L is reflexive, then M is reflexive.

(16) Let L, M be non empty relational structures. If L and M are isomorphic and L is transitive, then M is transitive.

(17) Let L, M be non empty relational structures. Suppose L and M are isomorphic and L is antisymmetric. Then M is antisymmetric.

(18) Let L, M be non empty relational structures. If L and M are isomorphic and L is complete, then M is complete.

(19) Let L be a non empty transitive relational structure and k be a map from L into L. If k is infs-preserving, then k^{∘} is infs-preserving.

(20) Let L be a non empty transitive relational structure and k be a map from L into L. If k is filtered-infs-preserving, then k^{∘} is filtered-infs-preserving.

(21) Let L be a non empty transitive relational structure and k be a map from L into L. If k is sups-preserving, then k^{∘} is sups-preserving.

(22) Let L be a non empty transitive relational structure and k be a map from L into L. If k is directed-sups-preserving, then k^{∘} is directed-sups-preserving.

(23) Let S, T be reflexive antisymmetric non empty relational structures and f be a map from S into T. If f is directed-sups-preserving, then f is monotone.

(24) Let S, T be reflexive antisymmetric non empty relational structures and f be a map from S into T. If f is filtered-infs-preserving, then f is monotone.

(25) Let S, T be non empty relational structures and f be a map from S into T. Suppose f is monotone. Let X be a subset of S. If X is filtered, then f^{∘}X is filtered.

(26) Let L_{1}, L_{2}, L_{3} be non empty relational structures, f be a map from L_{1} into L_{2}, and g be a map from L_{2} into L_{3}. Suppose f is infs-preserving and g is infs-preserving. Then g · f is infs-preserving.

(27) Let L_{1}, L_{2}, L_{3} be non empty reflexive antisymmetric relational structures, f be a map from L_{1} into L_{2}, and g be a map from L_{2} into L_{3}. Suppose f is filtered-infs-preserving and g is filtered-infs-preserving. Then g · f is filtered-infs-preserving.

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

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

2. Some Remarks on Lattice Product

We now state several propositions:

(30) Let I be a non empty set and J be a relational structure yielding nonempty many sorted set indexed by I. Suppose that for every element i of I holds J(i) is a lower-bounded antisymmetric relational structure. Then ∏ J is lower-bounded.

(31) Let I be a non empty set and J be a relational structure yielding nonempty many sorted set indexed by I. Suppose that for every element i of I holds J(i) is an upper-bounded antisymmetric relational structure. Then ∏ J is upper-bounded.

(32) Let I be a non empty set and J be a relational structure yielding nonempty many sorted set indexed by I. Suppose that for every element i of I holds J(i) is a lower-bounded antisymmetric relational structure. Let i be an element of I. Then ⊥_{∏ J}(i) = ⊥_{J(i)}.

(33) Let I be a non empty set and J be a relational structure yielding nonempty many sorted set indexed by I. Suppose that for every element i of I holds J(i) is an upper-bounded antisymmetric relational structure. Let i be an element of I. Then ⊤_{∏ J}(i) = ⊤_{J(i)}.

(34) 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 continuous complete lattice. Then ∏ J is continuous.

3. Kernel Projections and Quotient Lattices

We now state the proposition

(35) Let L, T be continuous complete lattices, g be a CLHomomorphism of L, T, and S be a subset of the carrier of [: L, L:]. Suppose S = [: g, g:]^{−1} (△_{the carrier of T}). Then sub(S) is a continuous subframe of [: L, L:].

Let L be a relational structure and let R be a subset of the carrier of [: L, L:]. Let us assume that R is an equivalence relation of the carrier of L. The functor EqRel(R) yields an equivalence relation of the carrier of L and is defined by:

(Def. 1) EqRel(R) = R.

Let L be a non empty relational structure and let R be a subset of [: L, L:]. We say that R is a continuous lattice congruence if and only if:

(Def. 2) R is an equivalence relation of the carrier of L and sub(R) is a continuous subframe of [: L, L:].

We now state the proposition

(36) Let L be a complete lattice and R be a non empty subset of [: L, L:]. Suppose R is a continuous lattice congruence. Let x be an element of the carrier of L. Then ⟨⟨inf([x]_{EqRel(R)}), x⟩⟩∈ R.

Let L be a complete lattice and let R be a non empty subset of [: L, L:]. Let us assume that R is a continuous lattice congruence. The kernel operation of R yields a kernel map from L into L and is defined by:

(Def. 3) For every element x of L holds (the kernel operation of R)(x) =

inf([x]_{EqRel(R)}). Next we state three propositions:

(37) Let L be a complete lattice and R be a non empty subset of [: L, L:].

Suppose R is a continuous lattice congruence. Then

(i) the kernel operation of R is directed-sups-preserving, and (ii) R = [: the kernel operation of R, the kernel operation of

R:]^{−1} (△_{the carrier of L}).

(38) Let L be a continuous complete lattice, R be a subset of [: L, L:], and k be a kernel map from L into L. Suppose k is directed-sups-preserving and R = [: k, k:]^{−1} (△_{the carrier of L}). Then there exists a continuous complete strict lattice L_{4} such that

(i) the carrier of L_{4} = ClassesEqRel(R), (ii) the internal relation of L_{4} = {⟨⟨[x]_{EqRel(R)}, [y]_{EqRel(R)}⟩⟩; x ranges over

elements of L, y ranges over elements of L: k(x) k(y)}, and (iii) for every map g from L into L_{4} such that for every element x of L holds

g(x) = [x]_{EqRel(R)} holds g is a CLHomomorphism of L, L_{4}.

(39) Let L be a continuous complete lattice and R be a subset of [: L, L:].

Suppose that

(i) R is an equivalence relation of the carrier of L, and (ii) there exists a continuous complete lattice L_{4} such that the carrier of

L_{4} = ClassesEqRel(R) and for every map g from L into L_{4} such that for every element x of L holds g(x) = [x]_{EqRel(R)} holds g is a CLHomomorphism of L, L_{4}. Then sub(R) is a continuous subframe of [: L, L:].

Let L be a non empty reflexive relational structure. Observe that there exists a map from L into L which is directed-sups-preserving and kernel.

Let L be a non empty reflexive relational structure and let k be a kernel map from L into L. The kernel congruence of k yields a non empty subset of [: L, L:] and is defined by:

(Def. 4) The kernel congruence of k = [: k, k:]^{−1} (△_{the carrier of L}).

We now state two propositions:

(40) Let L be a non empty reflexive relational structure and k be a kernel map from L into L. Then the kernel congruence of k is an equivalence relation of the carrier of L.

(41) Let L be a continuous complete lattice and k be a directed-sups-preserving kernel map from L into L. Then the kernel congruence of k is a continuous lattice congruence.

Let L be a continuous complete lattice and let R be a non empty subset of [: L, L:]. Let us assume that R is a continuous lattice congruence. The functor L/_{R} yielding a continuous complete strict lattice is defined by:

(Def. 5) The carrier of ^{L}/_{R} = ClassesEqRel(R) and for all elements x, y of ^{L}/_{R}

holds x y iff ⌈−⌉_{L}x ⌈−⌉_{L}y. The following propositions are true:

(42) Let L be a continuous complete lattice and R be a non empty subset of [: L, L:]. Suppose R is a continuous lattice congruence. Let x be a set. Then x is an element of ^{L}/_{R} if and only if there exists an element y of L such that x = [y]_{EqRel(R)}.

(43) Let L be a continuous complete lattice and R be a non empty subset of [: L, L:]. Suppose R is a continuous lattice congruence. Then R = the kernel congruence of the kernel operation of R.

(44) Let L be a continuous complete lattice and k be a directed-sups-preserving kernel map from L into L. Then k = the kernel operation of the kernel congruence of k.

(45) Let L be a continuous complete lattice and p be a projection map from L into L. Suppose p is infs-preserving. Then Imp is a continuous lattice and Imp is infs-inheriting.

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

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

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

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

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

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

kernel projections and quotient lattices 175

[12] 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.

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

[16] Robert Milewski. Completely-irreducible elements. Formalized Mathematics, 7(1):9–12, 1998.

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

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

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

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

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

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

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

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

[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.

176 piotr rudnicki

Lawson Topology in Continuous Lattices^{1}

Grzegorz Bancerek University of Białystok

Summary. The article completes Mizar formalization of Section 1 of Chapter III of [9, pp. 145–147].

MML Identifier: WAYBEL21.

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

1. Semilattice Homomorphism and Inheritance

Let S, T be semilattices. Let us assume that if S is upper-bounded, then T is upper-bounded. A map from S into T is said to be a semilattice morphism from S into T if:

(Def. 1) For every finite subset X of S holds it preserves inf of X.

Let S, T be semilattices. One can check that every map from S into T which is meet-preserving is also monotone.

Let S be a semilattice and let T be an upper-bounded semilattice. One can check that every semilattice morphism from S into T is meet-preserving.

Next we state a number of propositions:

(1) For all upper-bounded semilattices S, T and for every semilattice morphism f from S into T holds f(⊤_{S}) = ⊤_{T} .

(2) Let S, T be semilattices and f be a map from S into T. Suppose f is meet-preserving. Let X be a finite non empty subset of S. Then f preserves inf of X.

1Partially supported by NATO Grant CRG 951368, NSERC OGP 9207 grant and KBN grant 8 T11C 018 12.

(3) Let S, T be upper-bounded semilattices and f be a meet-preserving map from S into T. If f(⊤_{S}) = ⊤_{T} , then f is a semilattice morphism from S into T.

(4) Let S, T be semilattices and f be a map from S into T. Suppose f is meet-preserving and for every filtered non empty subset X of S holds f preserves inf of X. Let X be a non empty subset of S. Then f preserves inf of X.

(5) Let S, T be semilattices and f be a map from S into T. Suppose f is infs-preserving. Then f is a semilattice morphism from S into T.

(6) Let S_{1}, T_{1}, S_{2}, T_{2} be non empty relational structures. Suppose that (i) the relational structure of S_{1} = the relational structure of S_{2}, and (ii) the relational structure of T_{1} = the relational structure of T_{2}.

Let f_{1} be a map from S_{1} into T_{1} and f_{2} be a map from S_{2} into T_{2} such that f_{1} = f_{2}. Then

(iii) if f_{1} is infs-preserving, then f_{2} is infs-preserving, and (iv) if f_{1} is directed-sups-preserving, then f_{2} is directed-sups-preserving.

(7) Let S_{1}, T_{1}, S_{2}, T_{2} be non empty relational structures. Suppose that (i) the relational structure of S_{1} = the relational structure of S_{2}, and (ii) the relational structure of T_{1} = the relational structure of T_{2}.

Let f_{1} be a map from S_{1} into T_{1} and f_{2} be a map from S_{2} into T_{2} such that f_{1} = f_{2}. Then

(iii) if f_{1} is sups-preserving, then f_{2} is sups-preserving, and (iv) if f_{1} is filtered-infs-preserving, then f_{2} is filtered-infs-preserving.

(8) Let T be a complete lattice and S be an infs-inheriting full non empty relational substructure of T. Then incl(S, T) is infs-preserving.

(9) Let T be a complete lattice and S be a sups-inheriting full non empty relational substructure of T. Then incl(S, T) is sups-preserving.

(10) 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.

(11) Let T be a complete lattice and S be a filtered-infs-inheriting full non empty relational substructure of T. Then incl(S, T) is filtered-infs-preserving.

(12) Let T_{1}, T_{2}, R be relational structures and S be a relational substructure

of T_{1}. Suppose that

(i) the relational structure of T_{1} = the relational structure of T_{2}, and (ii) the relational structure of S = the relational structure of R.

Then R is a relational substructure of T_{2} and if S is full, then R is a full relational substructure of T_{2}.

(13) Every non empty relational structure T is an infs-inheriting supsinheriting full relational substructure of T.

Let T be a complete lattice. Observe that there exists a continuous subframe of T which is complete.

We now state a number of propositions:

(14) Let T be a semilattice and S be a full non empty relational substructure of T. Then S is meet-inheriting if and only if for every finite non empty subset X of S holds ⌈−⌉_{T} X ∈ the carrier of S.

(15) Let T be a sup-semilattice and S be a full non empty relational substructure of T. Then S is join-inheriting if and only if for every finite non empty subset X of S holds ⨆_{T} X ∈ the carrier of S.

(16) Let T be an upper-bounded semilattice and S be a meet-inheriting full non empty relational substructure of T. Suppose ⊤_{T} ∈ the carrier of S and S is filtered-infs-inheriting. Then S is infs-inheriting.

(17) Let T be a lower-bounded sup-semilattice and S be a join-inheriting full non empty relational substructure of T. Suppose ⊥_{T} ∈ the carrier of S and S is directed-sups-inheriting. Then S is sups-inheriting.

(18) Let T be a complete lattice and S be a full non empty relational substructure of T. If S is infs-inheriting, then S is complete.

(19) Let T be a complete lattice and S be a full non empty relational substructure of T. If S is sups-inheriting, then S is complete.

(20) Let T_{1}, T_{2} be non empty relational structures, S_{1} be a non empty full relational substructure of T_{1}, and S_{2} be a non empty full relational substructure of T_{2}. Suppose that

(i) the relational structure of T_{1} = the relational structure of T_{2}, and (ii) the carrier of S_{1} = the carrier of S_{2}.

If S_{1} is infs-inheriting, then S_{2} is infs-inheriting.

(21) Let T_{1}, T_{2} be non empty relational structures, S_{1} be a non empty full relational substructure of T_{1}, and S_{2} be a non empty full relational substructure of T_{2}. Suppose that

(i) the relational structure of T_{1} = the relational structure of T_{2}, and (ii) the carrier of S_{1} = the carrier of S_{2}.

If S_{1} is sups-inheriting, then S_{2} is sups-inheriting.

(22) Let T_{1}, T_{2} be non empty relational structures, S_{1} be a non empty full relational substructure of T_{1}, and S_{2} be a non empty full relational substructure of T_{2}. Suppose that

(i) the relational structure of T_{1} = the relational structure of T_{2}, and (ii) the carrier of S_{1} = the carrier of S_{2}.

If S_{1} is directed-sups-inheriting, then S_{2} is directed-sups-inheriting.

(23) Let T_{1}, T_{2} be non empty relational structures, S_{1} be a non empty full relational substructure of T_{1}, and S_{2} be a non empty full relational substructure of T_{2}. Suppose that

(i) the relational structure of T_{1} = the relational structure of T_{2}, and (ii) the carrier of S_{1} = the carrier of S_{2}.

If S_{1} is filtered-infs-inheriting, then S_{2} is filtered-infs-inheriting.

2. Nets and Limits

The following proposition is true

(24) Let S, T be non empty topological spaces, N be a net in S, and f be a map from S into T. If f is continuous, then f^{∘} LimN ⊆ Lim(f · N).

Let T be a non empty relational structure and let N be a non empty net structure over T. Let us observe that N is antitone if and only if:

(Def. 2) For all elements i, j of N such that i j holds N(i) N(j).

Let T be a non empty reflexive relational structure and let x be an element of T. Observe that ⟨{x}^{op}; id⟩ is transitive directed monotone and antitone.

Let T be a non empty reflexive relational structure. Note that there exists a net in T which is monotone, antitone, reflexive, and strict.

Let T be a non empty relational structure and let F be a non empty subset of T. Note that ⟨F^{op}; id⟩ is antitone.

Let S, T be non empty reflexive relational structures, let f be a monotone map from S into T, and let N be an antitone non empty net structure over S. Note that f · N is antitone.

We now state a number of propositions:

(25) Let S be a complete lattice and N be a net in S. Then {⌈−⌉_{S}{N(i); i ranges over elements of the carrier of N: i j} : j ranges over elements of the carrier of N} is a directed non empty subset of S.

(26) Let S be a non empty poset and N be a monotone reflexive net in S. Then {⌈−⌉_{S}{N(i); i ranges over elements of the carrier of N: i j} : j ranges over elements of the carrier of N} is a directed non empty subset of S.

(27) Let S be a non empty 1-sorted structure, N be a non empty net structure over S, and X be a set. If rng(the mapping of N) ⊆ X, then N is eventually in X.

(28) For every inf-complete non empty poset R and for every non empty filtered subset F of R holds lim inf⟨F^{op}; id⟩ = inf F.

(29) Let S, T be inf-complete non empty posets, X be a non empty filtered subset of S, and f be a monotone map from S into T. Then lim inf(f · ⟨X^{op}; id⟩) = inf(f^{∘}X).

(30) Let S, T be non empty top-posets, X be a non empty filtered subset of S, f be a monotone map from S into T, and Y be a non empty filtered subset of T. If Y = f^{∘}X, then f · ⟨X^{op}; id⟩ is a subnet of ⟨Y^{op}; id⟩.

(31) Let S, T be non empty top-posets, X be a non empty filtered subset of S, f be a monotone map from S into T, and Y be a non empty filtered subset of T. If Y = f^{∘}X, then Lim⟨Y^{op}; id⟩ ⊆ Lim(f · ⟨X^{op}; id⟩).

(32) Let S be a non empty reflexive relational structure and D be a non empty subset of S. Then the mapping of NetStr(D) = id_{D} and the carrier of NetStr(D) = D and NetStr(D) is a full relational substructure of S.

(33) Let S, T be up-complete non empty posets, f be a monotone map from S into T, and D be a non empty directed subset of S. Then lim inf(f · NetStr(D)) = sup(f^{∘}D).

(34) Let S be a non empty reflexive relational structure, D be a non empty directed subset of S, and i, j be elements of NetStr(D). Then i j if and only if (NetStr(D))(i) (NetStr(D))(j).

(35) For every Lawson complete top-lattice T and for every directed non

empty subset D of T holds supD ∈ LimNetStr(D). Let T be a non empty 1-sorted structure, let N be a net in T, and let M

be a non empty net structure over T. Let us assume that M is a subnet of N. A map from M into N is said to be a embedding of M into N if it satisfies the conditions (Def. 3).

(Def. 3)(i) The mapping of M = (the mapping of N) · it, and

(ii) for every element m of N there exists an element n of M such that for

every element p of M such that n p holds m it(p). One can prove the following propositions:

(36) Let T be a non empty 1-sorted structure, N be a net in T, M be a non empty subnet of N, e be a embedding of M into N, and i be an element of M. Then M(i) = N(e(i)).

(37) For every complete lattice T and for every net N in T and for every subnet M of N holds lim inf N lim inf M.

(38) Let T be a complete lattice, N be a net in T, M be a subnet of N, and e be a embedding of M into N. Suppose that for every element i of N and for every element j of M such that e(j) i there exists an element j′ of M such that j′ j and N(i) M(j′). Then lim inf N = lim inf M.

(39) Let T be a non empty relational structure, N be a net in T, and M be a non empty full structure of a subnet of N. Suppose that for every element i of N there exists an element j of N such that j i and j ∈ the carrier of M. Then M is a subnet of N and incl(M, N) is a embedding of M into N.

(40) Let T be a non empty relational structure, N be a net in T, and i be an element of N. Then N↾i is a subnet of N and incl(N↾i, N) is a embedding of N↾i into N.

(41) For every complete lattice T and for every net N in T and for every element i of N holds lim inf(N↾i) = lim inf N.

(42) Let T be a non empty relational structure, N be a net in T, and X be a set. Suppose N is eventually in X. Then there exists an element i of N such that N(i) ∈ X and rng(the mapping of N↾i) ⊆ X.

(43) Let T be a Lawson complete top-lattice and N be an eventually-filtered net in T. Then rng(the mapping of N) is a filtered non empty subset of T.

(44) For every Lawson complete top-lattice T and for every eventually-filtered net N in T holds LimN = {inf N}.

3. Lawson Topology Revisited

One can prove the following propositions:

(45) Let S, T be Lawson complete top-lattices and f be a meet-preserving map from S into T. Then f is continuous if and only if the following conditions are satisfied:

(i) f is directed-sups-preserving, and (ii) for every non empty subset X of S holds f preserves inf of X.

(46) Let S, T be Lawson complete top-lattices and f be a semilattice morphism from S into T. Then f is continuous if and only if f is infs-preserving and directed-sups-preserving.

Let S, T be non empty relational structures and let f be a map from S into T. We say that f is liminfs-preserving if and only if:

(Def. 4) For every net N in S holds f(lim inf N) = lim inf(f · N).

One can prove the following propositions:

(47) Let S, T be Lawson complete top-lattices and f be a semilattice morphism from S into T. Then f is continuous if and only if f is liminfspreserving.

(48) Let T be a Lawson complete continuous top-lattice and S be a meetinheriting full non empty relational substructure of T. Suppose ⊤_{T} ∈ the carrier of S and there exists a subset X of T such that X = the carrier of S and X is closed. Then S is infs-inheriting.

(49) Let T be a Lawson complete continuous top-lattice and S be a full non empty relational substructure of T. Given a subset X of T such that X = the carrier of S and X is closed. Then S is directed-sups-inheriting.

(50) Let T be a Lawson complete continuous top-lattice and S be an infsinheriting directed-sups-inheriting full non empty relational substructure of T. Then there exists a subset X of T such that X = the carrier of S and X is closed.

(51) Let T be a Lawson complete continuous top-lattice, S be an infsinheriting directed-sups-inheriting full non empty relational substructure of T, and N be a net in T. If N is eventually in the carrier of S, then lim inf N ∈ the carrier of S.

(52) Let T be a Lawson complete continuous top-lattice and S be a meet-

inheriting full non empty relational substructure of T. Suppose that

(i) ⊤_{T} ∈ the carrier of S, and (ii) for every net N in T such that rng(the mapping of N) ⊆ the carrier of

S holds lim inf N ∈ the carrier of S. Then S is infs-inheriting.

(53) Let T be a Lawson complete continuous top-lattice and S be a full non empty relational substructure of T. Suppose that for every net N in T such that rng(the mapping of N) ⊆ the carrier of S holds lim inf N ∈ the carrier of S. Then S is directed-sups-inheriting.

(54) Let T be a Lawson complete continuous top-lattice, S be a meetinheriting full non empty relational substructure of T, and X be a subset of T. Suppose X = the carrier of S and ⊤_{T} ∈ X. Then X is closed if and only if for every net N in T such that N is eventually in X holds lim inf N ∈ X.

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

[6] Grzegorz Bancerek. The Lawson topology. Formalized Mathematics, 7(2):163–168, 1998.

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

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

[9] 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.

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

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

184 grzegorz bancerek

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

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

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

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

Representation Theorem for Free Continuous Lattices

Piotr Rudnicki^{1} University of Alberta Edmonton

Summary. We present the Mizar formalization of theorem 4.17, Chapter I from [11]: a free continuous lattice with m generators is isomorphic to the lattice of filters of 2^{X} (\overline{\overline{X}} = m) which is freely generated by \{↑ x : x ∈ X\} (the set of ultrafilters).

MML Identifier: WAYBEL22.

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

1. Preliminaries

The following propositions are true:

(1) For every upper-bounded semilattice L and for every non empty directed subset F of ⟨Filt(L), ⊆⟩ holds supF = ⋃ F.

(2) Let L, S, T be complete non empty posets, f be a CLHomomorphism of L, S, and g be a CLHomomorphism of S, T. Then g · f is a CLHomomorphism of L, T.

(3) For every non empty relational structure L holds id_{L} is infs-preserving.

(4) For every non empty relational structure L holds id_{L} is directed-sups-preserving.

(5) For every complete non empty poset L holds id_{L} is a CLHomomorphism of L, L.

1This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

(6) For every upper-bounded non empty poset L with g.l.b.’s holds ⟨Filt(L), ⊆⟩ is a continuous subframe of 2_{⊆}^{the carrier of L}.

Let L be an upper-bounded non empty poset with g.l.b.’s. Observe that ⟨Filt(L), ⊆⟩ is continuous.

Let L be an upper-bounded non empty poset. One can check that every element of the carrier of ⟨Filt(L), ⊆⟩ is non empty.

2. Free Generators of Continuous Lattices

Let S be a continuous complete non empty poset and let A be a set. We say that A is a set of free generators of S if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let T be a continuous complete non empty poset and f be a function from A into the carrier of T. Then there exists a CLHomomorphism h of S, T such that h↾A = f and for every CLHomomorphism h′ of S, T such that h′↾A = f holds h′= h.

Next we state two propositions:

(7) Let S be a continuous complete non empty poset and A be a set. If A is a set of free generators of S, then A is a subset of S.

(8) Let S be a continuous complete non empty poset and A be a set. Suppose A is a set of free generators of S. Let h′ be a CLHomomorphism of S, S. If h′↾A = id_{A}, then h′= id_{S}.

3. Representation Theorem for Free Continuous Lattices

In the sequel X is a set, F is a filter of 2_{⊆}^{X}, x is an element of 2_{⊆}^{X}, and z is an element of X.

Let us consider X. The fixed ultrafilters of X is a family of subsets of 2_{⊆}^{X} and is defined as follows:

(Def. 2) The fixed ultrafilters of X = {↑x : ⋁_{z} x = {z}}.

One can prove the following three propositions:

(9) The fixed ultrafilters of X ⊆ Filt(2_{⊆}^{X}).

(10) \overline{\overline{the fixed ultrafilters of X}} = \overline{\overline{X}} .

(11) F = ⨆_{(⟨Filt(2_{⊆}^{X}),⊆⟩)}{⌈−⌉_{(⟨Filt(2_{⊆}^{X}),⊆⟩)}{↑x : ⋁_{z} (x = {z} ∧ z ∈ Y)}; Y ranges

over subsets of X: Y ∈ F }. Let us consider X, let L be a continuous complete non empty poset, and let f

be a function from the fixed ultrafilters of X into the carrier of L. The extension of f to homomorphism is a map from ⟨Filt(2_{⊆}^{X}), ⊆⟩ into L and is defined by the condition (Def. 3).

(Def. 3) Let F_{1} be an element of the carrier of (⟨Filt(2_{⊆}^{X}), ⊆⟩). Then (the extension of f to homomorphism)(F_{1}) = ⨆_{L}{⌈−⌉_{L}{f(↑x) : ⋁_{z} (x = {z} ∧ z ∈ Y)}; Y ranges over subsets of X: Y ∈ F_{1}}.

One can prove the following propositions:

(12) Let L be a continuous complete non empty poset and f be a function from the fixed ultrafilters of X into the carrier of L. Then the extension of f to homomorphism is monotone.

(13) Let L be a continuous complete non empty poset and f be a function from the fixed ultrafilters of X into the carrier of L. Then (the extension of f to homomorphism)(⊤_{⟨Filt(2_{⊆}^{X}),⊆⟩}) = ⊤_{L}.

Let us consider X, let L be a continuous complete non empty poset, and let f be a function from the fixed ultrafilters of X into the carrier of L. Observe that the extension of f to homomorphism is directed-sups-preserving.

Let us consider X, let L be a continuous complete non empty poset, and let f be a function from the fixed ultrafilters of X into the carrier of L. Note that the extension of f to homomorphism is infs-preserving.

The following propositions are true:

(14) Let L be a continuous complete non empty poset and f be a function from the fixed ultrafilters of X into the carrier of L. Then (the extension of f to homomorphism)↾(the fixed ultrafilters of X) = f.

(15) Let L be a continuous complete non empty poset, f be a function from the fixed ultrafilters of X into the carrier of L, and h be a CLHomomorphism of ⟨Filt(2_{⊆}^{X}), ⊆⟩, L. Suppose h↾the fixed ultrafilters of X = f. Then h = the extension of f to homomorphism.

(16) The fixed ultrafilters of X is a set of free generators of ⟨Filt(2_{⊆}^{X}), ⊆⟩.

(17) Let L, M be continuous complete lattices and F, G be sets. Suppose F is a set of free generators of L and G is a set of free generators of M and \overline{\overline{F}} = \overline{\overline{G}}. Then L and M are isomorphic.

(18) Let L be a continuous complete lattice and G be a set. Suppose G is a set of free generators of L and \overline{\overline{G}} = \overline{\overline{X}} . Then L and ⟨Filt(2_{⊆}^{X}), ⊆⟩ are isomorphic.

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.

188 piotr rudnicki

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

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

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

[10] Mariusz Giero. More on products of many sorted algebras. Formalized Mathematics, 5(4):621–626, 1996.

[11] 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.

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

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

[14] Robert Milewski. Completely-irreducible elements. Formalized Mathematics, 7(1):9–12, 1998.

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

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

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

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

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

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

Oriented Chains

Yatsuka Nakamura Shinshu University Nagano

Piotr Rudnicki University of Alberta Edmonton

Summary. In [5] we introduced a number of notions about vertex sequences associated with undirected chains of edges in graphs. In this article, we introduce analogous concepts for oriented chains and use them to prove properties of cutting and glueing of oriented chains, and the existence of a simple oriented chain in an oriented chain.

MML Identifier: GRAPH 4.

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

1. Oriented Vertex Sequences

For simplicity, we adopt the following rules: p, q denote finite sequences, m, n denote natural numbers, G denotes a graph, x, y, v, v_{1}, v_{2}, v_{3}, v_{4} denote elements of the vertices of G, e denotes a set, and X denotes a set.

Let us consider G, let us consider x, y, and let us consider e. We say that e orientedly joins x, y if and only if:

(Def. 1) (The source of G)(e) = x and (the target of G)(e) = y.

We now state the proposition

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

Let us consider G and let x, y be elements of the vertices of G. We say that x, y are orientedly incident if and only if:

(Def. 2) There exists a set v such that v ∈ the edges of G and v orientedly joins x, y.

One can prove the following proposition

(2) If e orientedly joins v_{1}, v_{2} and e orientedly joins v_{3}, v_{4}, then v_{1} = v_{3} and v_{2} = v_{4}.

We follow the rules: v_{5}, v_{6}, v_{7} are finite sequences of elements of the vertices of G and c, c_{1}, c_{2} are oriented chains of G.

We now state the proposition

(3) ε is an oriented chain of G. Let us consider G. Observe that there exists a chain of G which is empty

and oriented.

Let us consider G, X. The functor G -SVSetX yields a set and is defined by:

(Def. 3) G -SVSetX = {v : ⋁_{e: element of the edges of G} (e ∈ X ∧ v = (the source

of G)(e))}. Let us consider G, X. The functor G -TVSetX yielding a set is defined by:

(Def. 4) G -TVSetX = {v : ⋁_{e: element of the edges of G} (e ∈ X ∧ v = (the target

of G)(e))}. Next we state the proposition

(4) If X = ∅, then G -SVSetX = ∅ and G -TVSetX = ∅. Let us consider G, v_{5} and let c be a finite sequence. We say that v_{5} is oriented

vertex seq of c if and only if: (Def. 5) lenv_{5} = lenc + 1 and for every n such that 1 n and n lenc holds

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

(5) If v_{5} is oriented vertex seq of c, then v_{5} is vertex sequence of c.

(6) If v_{5} is oriented vertex seq of c, then G -SVSetrngc ⊆ rngv_{5}.

(7) If v_{5} is oriented vertex seq of c, then G -TVSetrngc ⊆ rngv_{5}.

(8) If c ≠ ε and v_{5} is oriented vertex seq of c, then rngv_{5} ⊆ (G -SVSetrngc)∪ (G -TVSetrngc).

2. Cutting and Glueing of Oriented Chains

One can prove the following propositions:

(9) ⟨v⟩ is oriented vertex seq of ε.

(10) There exists v_{5} such that v_{5} is oriented vertex seq of c.

(11) If c ≠ ε and v_{6} is oriented vertex seq of c and v_{7} is oriented vertex seq of c, then v_{6} = v_{7}.

Let us consider G, c. Let us assume that c ≠ ε. The functor oriented-vertex-seqc yielding a finite sequence of elements of the vertices of G is defined as follows:

(Def. 6) oriented-vertex-seqc is oriented vertex seq of c.

Next we state several propositions:

(12) If v_{5} is oriented vertex seq of c and c_{1} = c↾ Segn and v_{6} = v_{5}↾ Seg(n+1), then v_{6} is oriented vertex seq of c_{1}.

(13) If 1 m and m n and n lenc and q = ⟨c(m), …, c(n)⟩, then q is an oriented chain of G.

(14) Suppose 1 m and m n and n lenc and c_{1} = ⟨c(m), …, c(n)⟩ and v_{5} is oriented vertex seq of c and v_{6} = ⟨v_{5}(m), …, v_{5}(n + 1)⟩. Then v_{6} is oriented vertex seq of c_{1}.

(15) Suppose v_{6} is oriented vertex seq of c_{1} and v_{7} is oriented vertex seq of c_{2} and v_{6}(lenv_{6}) = v_{7}(1). Then c_{1} ^{⌢} c_{2} is an oriented chain of G.

(16) Suppose v_{6} is oriented vertex seq of c_{1} and v_{7} is oriented vertex seq of c_{2} and v_{6}(lenv_{6}) = v_{7}(1) and c = c_{1} ^{⌢} c_{2} and v_{5} = v_{6} ⌢⌢ v_{7}. Then v_{5} is oriented vertex seq of c.

3. Oriented Simple Chains in Oriented Chains

Let us consider G and let I_{1} be an oriented chain of G. We say that I_{1} is Simple if and only if the condition (Def. 7) is satisfied.

(Def. 7) There exists v_{5} such that v_{5} is oriented vertex seq of I_{1} and for all n, m such that 1 n and n < m and m lenv_{5} and v_{5}(n) = v_{5}(m) holds n = 1 and m = lenv_{5}.

Let us consider G. Note that there exists an oriented chain of G which is Simple.

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

Next we state two propositions:

(17) Every oriented simple chain of G is an oriented chain of G.

(18) For every oriented chain q of G holds q↾ Segn is an oriented chain of G. In the sequel s_{1} is an oriented simple chain of G. Next we state several propositions:

(19) s_{1}↾ Segn is an oriented simple chain of G.

(20) For every oriented chain s′_{1} of G such that s′_{1} = s_{1} holds s′_{1} is Simple.

(21) Every Simple oriented chain of G is an oriented simple chain of G.

(22) Suppose c is not Simple and v_{5} is oriented vertex seq of c. Then there exists a FinSubsequence f_{1} of c and there exists a FinSubsequence f_{2} of v_{5} and there exist c_{1}, v_{6} such that lenc_{1} < lenc and v_{6} is oriented vertex seq of c_{1} and lenv_{6} < lenv_{5} and v_{5}(1) = v_{6}(1) and v_{5}(lenv_{5}) = v_{6}(lenv_{6}) and Seqf_{1} = c_{1} and Seqf_{2} = v_{6}.

(23) Suppose v_{5} is oriented vertex seq of c. Then there exists a FinSubsequence f_{1} of c and there exists a FinSubsequence f_{2} of v_{5} and there exist s_{1}, v_{6} such that Seqf_{1} = s_{1} and Seqf_{2} = v_{6} and v_{6} is oriented vertex seq of s_{1} and v_{5}(1) = v_{6}(1) and v_{5}(lenv_{5}) = v_{6}(lenv_{6}).

Let us consider G. Observe that every oriented chain of G which is empty is also oriented.

Next we state three propositions:

(24) If p is an oriented path of G, then p↾ Segn is an oriented path of G.

(25) s_{1} is an oriented path of G.

(26) Let c_{1} be a finite sequence. Then

(i) c_{1} is a Simple oriented chain of G iff c_{1} is an oriented simple chain of G, and

(ii) if c_{1} is an oriented simple chain of G, then c_{1} is an oriented path of G.

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] Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.

[5] Yatsuka Nakamura and Piotr Rudnicki. Vertex sequences induced by chains. Formalized Mathematics, 5(3):297–304, 1996.

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

[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.

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

Graph Theoretical Properties of Arcs in the Plane and Fashoda Meet Theorem

Yatsuka Nakamura Shinshu University Nagano

Summary. We define a graph on an abstract set, edges of which are pairs of any two elements. For any finite sequence of a plane, we give a definition of nodic, which means that edges by a finite sequence are crossed only at terminals. If the first point and the last point of a finite sequence differs, simpleness as a chain and nodic condition imply unfoldedness and s.n.c. condition. We generalize Goboard Theorem, proved by us before, to a continuous case. We call this Fashoda Meet Theorem, which was taken from Fashoda incident of 100 years ago.

MML Identifier: JGRAPH 1.

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

1. A Graph by Cartesian Product

For simplicity, we adopt the following convention: G denotes a graph, v_{1} denotes a finite sequence of elements of the vertices of G, I_{1} denotes an oriented chain of G, n, m, k, i, j denote natural numbers, and r, r_{1}, r_{2} denote real numbers.

Next we state four propositions:

(1) \frac{0}{r} = 0.

(2) \sqrt{r_{1}^{2} + r_{2}^{2}} |r_{1}| + |r_{2}|.

(3) |r_{1}| \sqrt{r_{1}^{2} + r_{2}^{2}} and |r_{2}| \sqrt{r_{1}^{2} + r_{2}^{2}}.

(4) Let given v_{1}. Suppose I_{1} is Simple and v_{1} is oriented vertex seq of I_{1}. Let given n, m. If 1 n and n < m and m lenv_{1} and v_{1}(n) = v_{1}(m), then n = 1 and m = lenv_{1}.

Let X be a set. The functor PGraphX yields a multi graph structure and is defined by:

(Def. 1) PGraphX = ⟨X, [: X, X:], π_{1}(X × X), π_{2}(X × X)⟩.

We now state two propositions:

(5) For every non empty set X holds PGraphX is a graph.

(6) For every non empty set X holds the vertices of PGraphX = X. Let f be a finite sequence. The functor PairFf yielding a finite sequence is

defined by: (Def. 2) lenPairFf = lenf − ′1 and for every natural number i such that 1 i

and i < lenf holds (PairFf)(i) = ⟨⟨f(i), f(i + 1)⟩⟩. In the sequel X is a non empty set. Let X be a non empty set. Then PGraphX is a graph. The following propositions are true:

(7) Every finite sequence of elements of X is a finite sequence of elements of the vertices of PGraphX.

(8) For every finite sequence f of elements of X holds PairFf is a finite sequence of elements of the edges of PGraphX.

Let X be a non empty set and let f be a finite sequence of elements of X. Then PairFf is a finite sequence of elements of the edges of PGraphX.

We now state two propositions:

(9) Let n be a natural number and f be a finite sequence of elements of X. If 1 n and n lenPairFf, then (PairFf)(n) ∈ the edges of PGraphX.

(10) For every finite sequence f of elements of X holds PairFf is an oriented

chain of PGraphX. Let X be a non empty set and let f be a finite sequence of elements of X.

Then PairFf is an oriented chain of PGraphX.

The following proposition is true

(11) Let f be a finite sequence of elements of X and f_{1} be a finite sequence of elements of the vertices of PGraphX. If lenf 1 and f = f_{1}, then f_{1} is oriented vertex seq of PairFf.

2. Shortcuts of Finite Sequences in Plane

Let X be a non empty set and let f, g be finite sequences of elements of X. We say that g is Shortcut of f if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) f(1) = g(1), (ii) f(lenf) = g(leng), and (iii) there exists a FinSubsequence f_{2} of PairFf and there exists a FinSub-

sequence f_{3} of f and there exists an oriented simple chain s_{1} of PGraphX and there exists a finite sequence g_{1} of elements of the vertices of PGraphX such that Seqf_{2} = s_{1} and Seqf_{3} = g and g_{1} = g and g_{1} is oriented vertex seq of s_{1}.

We now state four propositions:

(12) For all finite sequences f, g of elements of X such that g is Shortcut of f holds 1 leng and leng lenf.

(13) Let f be a finite sequence of elements of X. Suppose lenf 1. Then there exists a finite sequence g of elements of X such that g is Shortcut of f.

(14) For all finite sequences f, g of elements of X such that g is Shortcut of f holds rngPairFg ⊆ rngPairFf.

(15) Let f, g be finite sequences of elements of X. Suppose f(1) ≠ f(lenf) and g is Shortcut of f. Then g is one-to-one and rngPairFg ⊆ rngPairFf and g(1) = f(1) and g(leng) = f(lenf).

Let us consider n and let I_{1} be a finite sequence of elements of E_{T}^{n}. We say that I_{1} is nodic if and only if the condition (Def. 4) is satisfied.

(Def. 4) Let given i, j. Suppose L(I_{1}, i) ∩ L(I_{1}, j) ≠ ∅. Then L(I_{1}, i) ∩ L(I_{1}, j) = {I_{1}(i)} but I_{1}(i) = I_{1}(j) or I_{1}(i) = I_{1}(j + 1) or L(I_{1}, i) ∩ L(I_{1}, j) = {I_{1}(i+1)} but I_{1}(i+1) = I_{1}(j) or I_{1}(i+1) = I_{1}(j+1) or L(I_{1}, i) = L(I_{1}, j).

One can prove the following propositions:

(16) For every finite sequence f of elements of E_{T}^{2} such that f is s.n.c. holds f is s.c.c..

(17) For every finite sequence f of elements of E_{T}^{2} such that f is s.c.c. and L(f, 1) ∩ L(f, lenf − ′1) = ∅ holds f is s.n.c..

(18) For every finite sequence f of elements of E_{T}^{2} such that f is nodic and PairFf is Simple holds f is s.c.c..

(19) For every finite sequence f of elements of E_{T}^{2} such that f is nodic and PairFf is Simple and f(1) ≠ f(lenf) holds f is s.n.c..

(20) For all points p_{1}, p_{2}, p_{3} of E_{T}^{n} such that there exists a set x such that x ≠ p_{2} and x ∈ L(p_{1}, p_{2}) ∩ L(p_{2}, p_{3}) holds p_{1} ∈ L(p_{2}, p_{3}) or p_{3} ∈ L(p_{1}, p_{2}).

(21) Let f be a finite sequence of elements of E_{T}^{2}. Suppose f is s.n.c. and L(f, 1) ∩ L(f, 1 + 1) ⊆ {π_{1+1}f} and L(f, lenf − ′2) ∩ L(f, lenf − ′1) ⊆ {π_{len f− ′1}f}. Then f is unfolded.

(22) For every finite sequence f of elements of X such that PairFf is Simple and f(1) ≠ f(lenf) holds f is one-to-one and lenf ≠ 1.

(23) For every finite sequence f of elements of X such that f is one-to-one and lenf > 1 holds PairFf is Simple and f(1) ≠ f(lenf).

(24) Let f be a finite sequence of elements of E_{T}^{2}. If f is nodic and PairFf is Simple and f(1) ≠ f(lenf), then f is unfolded.

(25) Let f, g be finite sequences of elements of E_{T}^{2} and given i. Suppose g is Shortcut of f and 1 i and i + 1 leng. Then there exists a natural number k_{1} such that 1 k_{1} and k_{1} + 1 lenf and π_{k_{1}}f = π_{i}g and π_{k_{1}+1}f = π_{i+1}g and f(k_{1}) = g(i) and f(k_{1} + 1) = g(i + 1).

(26) For all finite sequences f, g of elements of E_{T}^{2} such that g is Shortcut of f holds rngg ⊆ rngf.

(27) For all finite sequences f, g of elements of E_{T}^{2} such that g is Shortcut of f holds L̃(g) ⊆ L̃(f).

(28) Let f, g be finite sequences of elements of E_{T}^{2}. If f is special and g is Shortcut of f, then g is special.

(29) Let f be a finite sequence of elements of E_{T}^{2}. Suppose f is special and 2 lenf and f(1) ≠ f(lenf). Then there exists a finite sequence g of elements of E_{T}^{2} such that 2 leng and g is special and one-to-one and L̃(g) ⊆ L̃(f) and f(1) = g(1) and f(lenf) = g(leng) and rngg ⊆ rngf.

(30) Let f_{1}, f_{4} be finite sequences of elements of E_{T}^{2}. Suppose that (i) f_{1} is special, (ii) f_{4} is special, (iii) 2 lenf_{1}, (iv) 2 lenf_{4},

(v) f_{1}(1) ≠ f_{1}(lenf_{1}), (vi) f_{4}(1) ≠ f_{4}(lenf_{4}), (vii) X-coordinate(f_{1}) lies between (X-coordinate(f_{1}))(1) and

(X-coordinate(f_{1}))(lenf_{1}), (viii) X-coordinate(f_{4}) lies between (X-coordinate(f_{1}))(1) and

(X-coordinate(f_{1}))(lenf_{1}), (ix) Y-coordinate(f_{1}) lies between (Y-coordinate(f_{4}))(1) and

(Y-coordinate(f_{4}))(lenf_{4}), and

(x) Y-coordinate(f_{4}) lies between (Y-coordinate(f_{4}))(1) and (Y-coordinate(f_{4}))(lenf_{4}). Then L̃(f_{1}) ∩ L̃(f_{4}) ≠ ∅.

3. Norm of Points in E_{T}^{n}

The following proposition is true

(31) For all real numbers a, b, r_{1}, r_{2} such that a r_{1} and r_{1} b and a r_{2}

and r_{2} b holds |r_{1} − r_{2}| b − a. Let us consider n and let p be a point of E_{T}^{n}. The functor |p| yields a real

number and is defined by: (Def. 5) For every element w of R^{n} such that p = w holds |p| = |w|.

In the sequel p, p_{1}, p_{2} are points of E_{T}^{n}.

We now state a number of propositions:

(32) |0_{E_{T}^{n}}| = 0.

(33) If |p| = 0, then p = 0_{E_{T}^{n}}.

(34) |p| 0.

(35) |− p| = |p|.

(36) |r · p| = |r| · |p|.

(37) |p_{1} + p_{2}| |p_{1}| + |p_{2}|.

(38) |p_{1} − p_{2}| |p_{1}| + |p_{2}|.

(39) |p_{1}| − |p_{2}| |p_{1} + p_{2}|.

(40) |p_{1}| − |p_{2}| |p_{1} − p_{2}|.

(41) |p_{1} − p_{2}| = 0 iff p_{1} = p_{2}.

(42) If p_{1} ≠ p_{2}, then |p_{1} − p_{2}| > 0.

(43) |p_{1} − p_{2}| = |p_{2} − p_{1}|.

(44) |p_{1} − p_{2}| |p_{1} − p| + |p − p_{2}|.

(45) For all points x_{1}, x_{2} of E^{n} such that x_{1} = p_{1} and x_{2} = p_{2} holds |p_{1}−p_{2}| = ρ(x_{1}, x_{2}).

(46) For every point p of E_{T}^{2} holds |p|^{2} = |p_{1}|^{2} + |p_{2}|^{2}.

(47) For every point p of E_{T}^{2} holds |p| = \sqrt{|p_{1}|^{2} + |p_{2}|^{2}}.

(48) For every point p of E_{T}^{2} holds |p| |p_{1}| + |p_{2}|.

(49) For all points p_{1}, p_{2} of E_{T}^{2} holds |p_{1}−p_{2}| |(p_{1})_{1}−(p_{2})_{1}|+|(p_{1})_{2}−(p_{2})_{2}|.

(50) For every point p of E_{T}^{2} holds |p_{1}| |p| and |p_{2}| |p|.

(51) For all points p_{1}, p_{2} of E_{T}^{2} holds |(p_{1})_{1} − (p_{2})_{1}| |p_{1} − p_{2}| and |(p_{1})_{2} − (p_{2})_{2}| |p_{1} − p_{2}|.

(52) If p ∈ L(p_{1}, p_{2}), then there exists r such that 0 r and r 1 and p = (1 − r) · p_{1} + r · p_{2}.

(53) If p ∈ L(p_{1}, p_{2}), then |p − p_{1}| |p_{1} − p_{2}| and |p − p_{2}| |p_{1} − p_{2}|.

4. Extended Goboard Theorem and Fashoda Meet Theorem

In the sequel M denotes a metric space. Next we state several propositions:

(54) For all subsets P, Q of M_{top} such that P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact holds dist_{min}^{min}(P, Q) 0.

(55) Let P, Q be subsets of M_{top}. Suppose P ≠ ∅ and P is compact and Q ≠ ∅ and Q is compact. Then P ∩ Q = ∅ if and only if dist_{min}^{min}(P, Q) > 0.

(56) Let f be a finite sequence of elements of E_{T}^{2} and a, c, d be real numbers.

Suppose that

(i) 1 lenf, (ii) X-coordinate(f) lies between (X-coordinate(f))(1) and

(X-coordinate(f))(lenf), (iii) Y-coordinate(f) lies between c and d, (iv) a > 0, and

(v) for every i such that 1 i and i + 1 lenf holds |π_{i}f − π_{i+1}f| < a. Then there exists a finite sequence g of elements of E_{T}^{2} such that

(vi) g is special, (vii) g(1) = f(1), (viii) g(leng) = f(lenf), (ix) leng lenf,

(x) X-coordinate(g) lies between (X-coordinate(f))(1) and (X-coordinate(f))(lenf),

(xi) Y-coordinate(g) lies between c and d, (xii) for every j such that j ∈ domg there exists k such that k ∈ domf and

|π_{j}g − π_{k}f| < a, and (xiii) for every j such that 1 j and j + 1 leng holds |π_{j}g − π_{j+1}g| < a.

(57) Let f be a finite sequence of elements of E_{T}^{2} and a, c, d be real numbers.

Suppose that

(i) 1 lenf, (ii) Y-coordinate(f) lies between (Y-coordinate(f))(1) and

(Y-coordinate(f))(lenf), (iii) X-coordinate(f) lies between c and d, (iv) a > 0, and

(v) for every i such that 1 i and i + 1 lenf holds |π_{i}f − π_{i+1}f| < a. Then there exists a finite sequence g of elements of E_{T}^{2} such that

(vi) g is special, (vii) g(1) = f(1), (viii) g(leng) = f(lenf), (ix) leng lenf, (x) Y-coordinate(g) lies between (Y-coordinate(f))(1) and

(Y-coordinate(f))(lenf), (xi) X-coordinate(g) lies between c and d, (xii) for every j such that j ∈ domg there exists k such that k ∈ domf and

|π_{j}g − π_{k}f| < a, and (xiii) for every j such that 1 j and j + 1 leng holds |π_{j}g − π_{j+1}g| < a.

(58) For every subset P of the carrier of E_{T}^{2} and for all points p_{1}, p_{2} of E_{T}^{2} such that P is an arc from p_{1} to p_{2} holds p_{1} ≠ p_{2}.

(59) For every finite sequence f of elements of E_{T}^{2} such that 1 lenf holds lenX-coordinate(f) = lenf and (X-coordinate(f))(1) = (π_{1}f)_{1} and (X-coordinate(f))(lenf) = (π_{len f}f)_{1}.

(60) For every finite sequence f of elements of E_{T}^{2} such that 1 lenf holds lenY-coordinate(f) = lenf and (Y-coordinate(f))(1) = (π_{1}f)_{2} and (Y-coordinate(f))(lenf) = (π_{len f}f)_{2}.

(61) For every finite sequence f of elements of E_{T}^{2} and for every i such that i ∈ domf holds (X-coordinate(f))(i) = (π_{i}f)_{1} and (Y-coordinate(f))(i) = (π_{i}f)_{2}.

(62) Let P, Q be non empty subsets of the carrier of E_{T}^{2} and p_{1}, p_{2}, q_{1}, q_{2} be

points of E_{T}^{2}. Suppose that

(i) P is an arc from p_{1} to p_{2}, (ii) Q is an arc from q_{1} to q_{2}, (iii) for every point p of E_{T}^{2} such that p ∈ P holds (p_{1})_{1} p_{1} and p_{1} (p_{2})_{1}, (iv) for every point p of E_{T}^{2} such that p ∈ Q holds (p_{1})_{1} p_{1} and p_{1} (p_{2})_{1},

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

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

Then P ∩ Q ≠ ∅. In the sequel X, Y are non empty topological spaces. We now state three propositions:

(63) Let f be a map from X into Y, P be a non empty subset of the carrier of Y, and f_{1} be a map from X into Y ↾P. If f = f_{1} and f is continuous, then f_{1} is continuous.

(64) Let f be a map from X into Y and P be a non empty subset of the carrier of Y. Suppose X is compact and Y is a T_{2} space and f is continuous and one-to-one and P = rngf. Then there exists a map f_{1} from X into Y ↾P such that f = f_{1} and f_{1} is a homeomorphism.

(65) Let f, g be maps from 𝕀 into E_{T}^{2}, a, b, c, d be real numbers, and O, I be

points of 𝕀. Suppose that

(i) O = 0, (ii) I = 1, 200 yatsuka nakamura

(iii) f is continuous and one-to-one, (iv) g is continuous and one-to-one,

(v) f(O)_{1} = a, (vi) f(I)_{1} = b, (vii) g(O)_{2} = c, (viii) g(I)_{2} = d, and (ix) for every point r of 𝕀 holds a f(r)_{1} and f(r)_{1} b and a g(r)_{1} and

g(r)_{1} b and c f(r)_{2} and f(r)_{2} d and c g(r)_{2} and g(r)_{2} d. Then rngf ∩ rngg ≠ ∅.

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Received August 21, 1998 202 yatsuka nakamura

Algebraic Group on Fixed-length Bit Integer and its Adaptation to IDEA

Cryptography

Yasushi Fuwa Shinshu University Nagano Yoshinori Fujisawa Shinshu University Nagano

Summary. In this article, an algebraic group on fixed-length bit integer is constructed and its adaptation to IDEA cryptography is discussed. In the first section, we present some selected theorems on integers. In the continuous section, we construct an algebraic group on fixed-length integer. In the third section, operations of IDEA Cryptograms are defined and some theorems on these operations are proved. In the fourth section, we define sequences of IDEA Cryptogram’s operations and discuss their nature. Finally, we make a model of IDEA Cryptogram and prove that the ciphertext that is encrypted by IDEA encryption algorithm can be decrypted by the IDEA decryption algorithm.

MML Identifier: IDEA 1.

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

1. Some Selected Theorems on Integers

We adopt the following rules: i, j, k, n are natural numbers and x, y, z are tuples of n and Boolean.

Next we state several propositions:

(1) For all i, j, k such that j is prime and i < j and k < j and i ≠ 0 there exists a natural number a such that a · i mod j = k and a < j.

(2) For all natural numbers n, k_{1}, k_{2} such that n ≠ 0 and k_{1}modn = k_{2}modn and k_{1} k_{2} there exists a natural number t such that k_{2} − k_{1} = n · t.

(3) For all natural numbers a, b holds a − b a.

(4) For all natural numbers b_{1}, b_{2}, c such that b_{2} c holds b_{2} − b_{1} c.

(5) For all natural numbers a, b, c such that 0 < a and 0 < b and a < c and b < c and c is prime holds a · b mod c ≠ 0.

(6) For every non empty natural number n holds the n-th power of 2 ≠ 1.

2. Basic Operators of IDEA Cryptograms

Let us consider n. The functor ZEROn yielding a tuple of n and Boolean is defined by:

(Def. 1) ZEROn = n ↦ false.

Let us consider n and let x, y be tuples of n and Boolean. The functor x ⊕ y yields a tuple of n and Boolean and is defined by:

(Def. 2) For every i such that i ∈ Segn holds π_{i}(x ⊕ y) = π_{i}x ⊕ π_{i}y. The following propositions are true:

(7) For all n, x holds x ⊕ x = ZEROn.

(8) For all n, x, y holds x ⊕ y = y ⊕ x.

Let us consider n and let x, y be tuples of n and Boolean. Let us observe that the functor x ⊕ y is commutative.

One can prove the following propositions:

(9) For all n, x holds ZEROn ⊕ x = x.

(10) For all n, x, y, z holds (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z). Let us consider n and let i be a natural number. We say that i is expressible

by n if and only if: (Def. 3) i < the n-th power of 2.

The following proposition is true

(11) For every n holds n -BinarySequence(0) = ZEROn. Let us consider n and let i, j be natural numbers. The functor

ADD MOD(i, j, n) yields a natural number and is defined by: (Def. 4) ADD MOD(i, j, n) = (i + j) mod (the n-th power of 2).

Let us consider n and let i be a natural number. Let us assume that i is expressible by n. The functor NEG N(i, n) yielding a natural number is defined by:

(Def. 5) NEG N(i, n) = (the n-th power of 2)−i.

Let us consider n and let i be a natural number. Let us assume that i is expressible by n. The functor NEG MOD(i, n) yielding a natural number is defined as follows:

(Def. 6) NEG MOD(i, n) = NEG N(i, n) mod (the n-th power of 2).

We now state several propositions:

(12) For all n, i such that i is expressible by n holds ADD MOD(i, NEG MOD(i, n), n) = 0.

(13) For all n, i, j holds ADD MOD(i, j, n) = ADD MOD(j, i, n).

(14) For all n, i such that i is expressible by n holds ADD MOD(0, i, n) = i.

(15) For all n, i, j, k holds ADD MOD(ADD MOD(i, j, n), k, n) = ADD MOD(i, ADD MOD(j, k, n), n).

(16) For all n, i, j holds ADD MOD(i, j, n) is expressible by n.

(17) For all n, i such that i is expressible by n holds NEG MOD(i, n) is

expressible by n. Let us consider n and let i be a natural number. The functor

ChangeVal 1(i, n) yields a natural number and is defined by:

(Def. 7) ChangeVal 1(i, n) = \{ \begin{matrix}the n-th power of 2, if i = 0, \\ i, otherwise. \end{matrix}

We now state two propositions:

(18) For all n, i such that i is expressible by n holds ChangeVal 1(i, n) the n-th power of 2 and ChangeVal 1(i, n) > 0.

(19) Let n, a_{1}, a_{2} be natural numbers. Suppose a_{1} is expressible by n and a_{2} is expressible by n and ChangeVal 1(a_{1}, n) = ChangeVal 1(a_{2}, n). Then a_{1} = a_{2}.

Let us consider n and let i be a natural number. The functor ChangeVal 2(i, n) yields a natural number and is defined as follows:

(Def. 8) ChangeVal 2(i, n) = \{ \begin{matrix}0, if i = the n-th power of 2, \\ i, otherwise. \end{matrix}

We now state two propositions:

(20) For all n, i such that i is expressible by n holds ChangeVal 2(i, n) is expressible by n.

(21) For all natural numbers n, a_{1}, a_{2} such that a_{1} ≠ 0 and a_{2} ≠ 0 and

ChangeVal 2(a_{1}, n) = ChangeVal 2(a_{2}, n) holds a_{1} = a_{2}. Let us consider n and let i, j be natural numbers. The functor

MUL MOD(i, j, n) yields a natural number and is defined as follows: (Def. 9) MUL MOD(i, j, n) = ChangeVal 2(ChangeVal 1(i, n)·

ChangeVal 1(j, n) mod ((the n-th power of 2)+1), n). Let n be a non empty natural number and let i be a natural number. Let

us assume that i is expressible by n and (the n-th power of 2)+1 is prime. The functor INV MOD(i, n) yielding a natural number is defined as follows:

(Def. 10) MUL MOD(i, INV MOD(i, n), n) = 1 and INV MOD(i, n) is expressible by n.

The following propositions are true:

(22) For all n, i, j holds MUL MOD(i, j, n) = MUL MOD(j, i, n).

(23) For all n, i such that i is expressible by n holds MUL MOD(1, i, n) = i.

(24) Let given n, i, j, k. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) i is expressible by n, (iii) j is expressible by n, and (iv) k is expressible by n.

Then MUL MOD(MUL MOD(i, j, n), k, n) = MUL MOD(i, MUL MOD(j, k, n), n).

(25) For all n, i, j holds MUL MOD(i, j, n) is expressible by n.

(26) If ChangeVal 2(i, n) = 1, then i = 1.

3. Operations of IDEA Cryptograms

Let us consider n and let m, k be finite sequences of elements of ℕ. The functor IDEAoperationA(m, k, n) yielding a finite sequence of elements of ℕ is defined by the conditions (Def. 11).

(Def. 11)(i) lenIDEAoperationA(m, k, n) = lenm, and

(ii) for every natural number i such that i ∈ domm holds if i = 1, then (IDEAoperationA(m, k, n))(i) = MUL MOD(m(1), k(1), n) and if i = 2, then (IDEAoperationA(m, k, n))(i) = ADD MOD(m(2), k(2), n) and if i = 3, then (IDEAoperationA(m, k, n))(i) = ADD MOD(m(3), k(3), n) and if i = 4, then (IDEAoperationA(m, k, n))(i) = MUL MOD(m(4), k(4), n) and if i ≠ 1 and i ≠ 2 and i ≠ 3 and i ≠ 4, then (IDEAoperationA(m, k, n))(i) = m(i).

In the sequel m, k, k_{1}, k_{2} denote finite sequences of elements of ℕ. Let n be a non empty natural number and let m, k be finite sequences of

elements of ℕ. The functor IDEAoperationB(m, k, n) yielding a finite sequence of elements of ℕ is defined by the conditions (Def. 12).

(Def. 12)(i) lenIDEAoperationB(m, k, n) = lenm, and

(ii) for every natural number i such that i ∈ domm holds if i = 1, then (IDEAoperationB(m, k, n))(i) = Absval((n -BinarySequence(m(1))) ⊕ (n -BinarySequence(MUL MOD(ADD MOD(MUL MOD(Absval ((n -BinarySequence(m(1))) ⊕ (n -BinarySequence(m(3)))), k(5), n), Absval((n -BinarySequence(m(2)))⊕(n -BinarySequence(m(4)))), n), k(6), n)))) and if i = 2, then (IDEAoperationB(m, k, n))(i) = Absval((n -BinarySequence(m(2))) ⊕ (n -BinarySequence(ADD MOD(MUL MOD(Absval((n -BinarySequence (m(1))) ⊕ (n -BinarySequence(m(3)))), k(5), n), MUL MOD(ADD MOD (MUL MOD(Absval((n -BinarySequence (m(1)))⊕(n -BinarySequence(m(3)))), k(5), n), Absval((n -BinarySequence(m (2))) ⊕ (n -BinarySequence(m(4)))), n), k(6), n), n)))) and if i = 3, then (IDEAoperationB(m, k, n))(i) = Absval((n -BinarySequence(m(3))) ⊕ (n -BinarySequence(MUL MOD(ADD MOD(MUL MOD(Absval ((n -BinarySequence(m(1)))⊕(n -BinarySequence(m(3)))), k(5), n), Absval ((n -BinarySequence(m(2))) ⊕ (n -BinarySequence(m(4)))), n), k(6), n)))) and if i = 4, then (IDEAoperationB(m, k, n))(i) = Absval((n -BinarySequence(m(4))) ⊕ (n -BinarySequence (ADD MOD(MUL MOD(Absval((n -BinarySequence(m(1)))⊕ (n -BinarySequence(m(3)))), k(5), n), MUL MOD(ADD MOD(MUL MOD (Absval((n -BinarySequence(m(1)))⊕(n -BinarySequence(m(3)))), k(5), n), Absval((n -BinarySequence(m(2)))⊕(n -BinarySequence(m(4)))), n), k(6), n), n)))) and if i ≠ 1 and i ≠ 2 and i ≠ 3 and i ≠ 4, then (IDEAoperationB(m, k, n))(i) = m(i).

Let m be a finite sequence of elements of ℕ. The functor IDEAoperationCm yields a finite sequence of elements of ℕ and is defined as follows:

(Def. 13) lenIDEAoperationCm = lenm and for every natural number i such that i ∈ domm holds (IDEAoperationCm)(i) = (i = 2 → m(3), (i = 3 → m(2), m(i))).

The following propositions are true:

(27) Let given n, m, k. Suppose lenm 4. Then (i) (IDEAoperationA(m, k, n))(1) is expressible by n, (ii) (IDEAoperationA(m, k, n))(2) is expressible by n, (iii) (IDEAoperationA(m, k, n))(3) is expressible by n, and (iv) (IDEAoperationA(m, k, n))(4) is expressible by n.

(28) Let n be a non empty natural number and given m, k. Suppose lenm 4.

Then

(i) (IDEAoperationB(m, k, n))(1) is expressible by n, (ii) (IDEAoperationB(m, k, n))(2) is expressible by n, (iii) (IDEAoperationB(m, k, n))(3) is expressible by n, and (iv) (IDEAoperationB(m, k, n))(4) is expressible by n.

(29) Let given m. Suppose that (i) lenm 4, (ii) m(1) is expressible by n, (iii) m(2) is expressible by n, (iv) m(3) is expressible by n, and

(v) m(4) is expressible by n. Then

(vi) (IDEAoperationCm)(1) is expressible by n, (vii) (IDEAoperationCm)(2) is expressible by n, (viii) (IDEAoperationCm)(3) is expressible by n, and (ix) (IDEAoperationCm)(4) is expressible by n.

(30) Let n be a non empty natural number and given m, k_{1}, k_{2}. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n,

(v) m(3) is expressible by n, (vi) m(4) is expressible by n, (vii) k_{1}(1) is expressible by n, (viii) k_{1}(2) is expressible by n, (ix) k_{1}(3) is expressible by n,

(x) k_{1}(4) is expressible by n, (xi) k_{2}(1) = INV MOD(k_{1}(1), n), (xii) k_{2}(2) = NEG MOD(k_{1}(2), n), (xiii) k_{2}(3) = NEG MOD(k_{1}(3), n), and (xiv) k_{2}(4) = INV MOD(k_{1}(4), n).

Then IDEAoperationA(IDEAoperationA(m, k_{1}, n), k_{2}, n) = m.

(31) Let n be a non empty natural number and given m, k_{1}, k_{2}. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n,

(v) m(3) is expressible by n, (vi) m(4) is expressible by n, (vii) k_{1}(1) is expressible by n, (viii) k_{1}(2) is expressible by n, (ix) k_{1}(3) is expressible by n,

(x) k_{1}(4) is expressible by n, (xi) k_{2}(1) = INV MOD(k_{1}(1), n), (xii) k_{2}(2) = NEG MOD(k_{1}(3), n), (xiii) k_{2}(3) = NEG MOD(k_{1}(2), n), and (xiv) k_{2}(4) = INV MOD(k_{1}(4), n).

Then IDEAoperationA(IDEAoperationCIDEAoperationA (IDEAoperationCm, k_{1}, n), k_{2}, n) = m.

(32) Let n be a non empty natural number and given m, k_{1}, k_{2}. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n, (v) m(3) is expressible by n, (vi) m(4) is expressible by n, (vii) k_{1}(5) is expressible by n, (viii) k_{1}(6) is expressible by n, (ix) k_{2}(5) = k_{1}(5), and

(x) k_{2}(6) = k_{1}(6).

Then IDEAoperationB(IDEAoperationB(m, k_{1}, n), k_{2}, n) = m.

(33) For every m such that lenm 4 holds IDEAoperationCIDEAoperationCm = m.

4. Sequences of IDEA Cryptogram’s Operations

The set MESSAGES is defined by: (Def. 14) MESSAGES = ℕ^{∗}.

Let us mention that MESSAGES is non empty.

Let us mention that every element of MESSAGES is function-like and relation-like.

Let us note that every element of MESSAGES is finite sequence-like.

Let n be a non empty natural number and let us consider k. The functor IDEA P(k, n) yielding a function from MESSAGES into MESSAGES is defined as follows:

(Def. 15) For every m holds (IDEA P(k, n))(m) = IDEAoperationA

(IDEAoperationCIDEAoperationB(m, k, n), k, n). Let n be a non empty natural number and let us consider k. The func-

tor IDEA Q(k, n) yields a function from MESSAGES into MESSAGES and is defined as follows:

(Def. 16) For every m holds (IDEA Q(k, n))(m) = IDEAoperationB

(IDEAoperationA(IDEAoperationCm, k, n), k, n). Let r, l_{1} be natural numbers, let n be a non empty natural number, and let

K_{1} be a matrix over ℕ of dimension l_{1} × 6. The functor IDEA P F(K_{1}, n, r) yielding a finite sequence is defined as follows:

(Def. 17) lenIDEA P F(K_{1}, n, r) = r and for every i such that i ∈ domIDEA P F(K_{1}, n, r) holds (IDEA P F(K_{1}, n, r))(i) = IDEA P(Line(K_{1}, i), n).

Let r, l_{1} be natural numbers, let n be a non empty natural number, and let K_{1} be a matrix over ℕ of dimension l_{1} × 6. One can verify that IDEA P F(K_{1}, n, r) is function yielding.

Let r, l_{1} be natural numbers, let n be a non empty natural number, and let K_{1} be a matrix over ℕ of dimension l_{1} × 6. The functor IDEA Q F(K_{1}, n, r) yielding a finite sequence is defined as follows:

(Def. 18) lenIDEA Q F(K_{1}, n, r) = r and for every i such that i ∈ domIDEA Q F(K_{1}, n, r) holds (IDEA Q F(K_{1}, n, r))(i) = IDEA Q(Line(K_{1}, (r − ′i) + 1), n).

Let r, l_{1} be natural numbers, let n be a non empty natural number, and let K_{1} be a matrix over ℕ of dimension l_{1} × 6. Observe that IDEA Q F(K_{1}, n, r) is function yielding.

Let us consider k, n. The functor IDEA PS(k, n) yields a function from MESSAGES into MESSAGES and is defined as follows:

(Def. 19) For every m holds (IDEA PS(k, n))(m) = IDEAoperationA(m, k, n).

Let us consider k, n. The functor IDEA QS(k, n) yields a function from MESSAGES into MESSAGES and is defined as follows:

(Def. 20) For every m holds (IDEA QS(k, n))(m) = IDEAoperationA(m, k, n).

Let n be a non empty natural number and let us consider k. The functor IDEA PE(k, n) yielding a function from MESSAGES into MESSAGES is defined by:

(Def. 21) For every m holds (IDEA PE(k, n))(m) = IDEAoperationA

(IDEAoperationB(m, k, n), k, n). Let n be a non empty natural number and let us consider k. The func-

tor IDEA QE(k, n) yielding a function from MESSAGES into MESSAGES is defined by:

(Def. 22) For every m holds (IDEA QE(k, n))(m) = IDEAoperationB

(IDEAoperationA(m, k, n), k, n). We now state a number of propositions:

(34) Let n be a non empty natural number and given m, k_{1}, k_{2}. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n,

(v) m(3) is expressible by n, (vi) m(4) is expressible by n, (vii) k_{1}(1) is expressible by n, (viii) k_{1}(2) is expressible by n, (ix) k_{1}(3) is expressible by n,

(x) k_{1}(4) is expressible by n, (xi) k_{1}(5) is expressible by n, (xii) k_{1}(6) is expressible by n, (xiii) k_{2}(1) = INV MOD(k_{1}(1), n), (xiv) k_{2}(2) = NEG MOD(k_{1}(3), n), (xv) k_{2}(3) = NEG MOD(k_{1}(2), n), (xvi) k_{2}(4) = INV MOD(k_{1}(4), n), (xvii) k_{2}(5) = k_{1}(5), and (xviii) k_{2}(6) = k_{1}(6).

Then (IDEA Q(k_{2}, n) · IDEA P(k_{1}, n))(m) = m.

(35) Let n be a non empty natural number and given m, k_{1}, k_{2}. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n,

(v) m(3) is expressible by n, (vi) m(4) is expressible by n, (vii) k_{1}(1) is expressible by n, (viii) k_{1}(2) is expressible by n, (ix) k_{1}(3) is expressible by n,

(x) k_{1}(4) is expressible by n, (xi) k_{2}(1) = INV MOD(k_{1}(1), n), (xii) k_{2}(2) = NEG MOD(k_{1}(2), n), (xiii) k_{2}(3) = NEG MOD(k_{1}(3), n), and (xiv) k_{2}(4) = INV MOD(k_{1}(4), n).

Then (IDEA QS(k_{2}, n) · IDEA PS(k_{1}, n))(m) = m.

(36) Let n be a non empty natural number and given m, k_{1}, k_{2}. Suppose that (i) (the n-th power of 2)+1 is prime, (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n,

(v) m(3) is expressible by n, (vi) m(4) is expressible by n, (vii) k_{1}(1) is expressible by n, (viii) k_{1}(2) is expressible by n, (ix) k_{1}(3) is expressible by n,

(x) k_{1}(4) is expressible by n, (xi) k_{1}(5) is expressible by n, (xii) k_{1}(6) is expressible by n, (xiii) k_{2}(1) = INV MOD(k_{1}(1), n), (xiv) k_{2}(2) = NEG MOD(k_{1}(2), n), (xv) k_{2}(3) = NEG MOD(k_{1}(3), n), (xvi) k_{2}(4) = INV MOD(k_{1}(4), n), (xvii) k_{2}(5) = k_{1}(5), and (xviii) k_{2}(6) = k_{1}(6).

Then (IDEA QE(k_{2}, n) · IDEA PE(k_{1}, n))(m) = m.

(37) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and k be a natural number. Then IDEA P F(K_{1}, n, k + 1) = (IDEA P F(K_{1}, n, k))^{⌢} ⟨IDEA P(Line(K_{1}, k + 1), n)⟩.

(38) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and k be a natural number. Then IDEA Q F(K_{1}, n, k + 1) = ⟨IDEA Q(Line(K_{1}, k + 1), n)⟩ ^{⌢} IDEA Q F(K_{1}, n, k).

(39) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and k be a natural number. Then IDEA P F(K_{1}, n, k) is a composable finite sequence.

(40) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and k be a natural number. Then IDEA Q F(K_{1}, n, k) is a composable finite sequence.

(41) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and k be a natural number. If k ≠ 0, then IDEA P F(K_{1}, n, k) is a composable sequence from MESSAGES into MESSAGES.

(42) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and k be a natural number. If k ≠ 0, then IDEA Q F(K_{1}, n, k) is a composable sequence from MESSAGES into MESSAGES.

(43) Let n be a non empty natural number, M be a finite sequence of elements of ℕ, and given m, k. Suppose M = (IDEA P(k, n))(m) and lenm 4. Then

(i) lenM 4, (ii) M(1) is expressible by n, (iii) M(2) is expressible by n, (iv) M(3) is expressible by n, and

(v) M(4) is expressible by n.

(44) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and r be a natural number. Then rngcompose_{MESSAGES} IDEA P F(K_{1}, n, r) ⊆ MESSAGES and domcompose_{MESSAGES} IDEA P F(K_{1}, n, r) = MESSAGES.

(45) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, and r be a natural number. Then rngcompose_{MESSAGES} IDEA Q F(K_{1}, n, r) ⊆ MESSAGES and domcompose_{MESSAGES} IDEA Q F(K_{1}, n, r) = MESSAGES.

(46) Let n be a non empty natural number, m be a finite sequence of elements of ℕ, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, r be a natural number, and M be a finite sequence of elements of ℕ. If M = (compose_{MESSAGES} IDEA P F(K_{1}, n, r))(m) and lenm 4, then lenM 4.

(47) Let n be a non empty natural number, l_{1} be a natural number, K_{1} be a matrix over ℕ of dimension l_{1} × 6, r be a natural number, M be a finite sequence of elements of ℕ, and given m. Suppose that

(i) M = (compose_{MESSAGES} IDEA P F(K_{1}, n, r))(m), (ii) lenm 4, (iii) m(1) is expressible by n, (iv) m(2) is expressible by n,

(v) m(3) is expressible by n, and (vi) m(4) is expressible by n.

Then (vii) lenM 4, (viii) M(1) is expressible by n, (ix) M(2) is expressible by n,

(x) M(3) is expressible by n, and (xi) M(4) is expressible by n.

5. Modeling of IDEA Cryptogram

One can prove the following propositions:

(48) Let n be a non empty natural number, l_{1} be a natural number, K_{2}, K_{3} be matrices over ℕ of dimension l_{1} × 6, r be a natural number, and given m. Suppose that

(i) l_{1} r, (ii) (the n-th power of 2)+1 is prime, (iii) lenm 4, (iv) m(1) is expressible by n,

(v) m(2) is expressible by n, (vi) m(3) is expressible by n, (vii) m(4) is expressible by n, and (viii) for every natural number i such that i r holds (K_{2})_{i,1} is expressible

by n and (K_{2})_{i,2} is expressible by n and (K_{2})_{i,3} is expressible by n and (K_{2})_{i,4} is expressible by n and (K_{2})_{i,5} is expressible by n and (K_{2})_{i,6} is expressible by n and (K_{3})_{i,1} is expressible by n and (K_{3})_{i,2} is expressible by n and (K_{3})_{i,3} is expressible by n and (K_{3})_{i,4} is expressible by n and (K_{3})_{i,5} is expressible by n and (K_{3})_{i,6} is expressible by n and (K_{3})_{i,1} = INV MOD((K_{2})_{i,1}, n) and (K_{3})_{i,2} = NEG MOD((K_{2})_{i,3}, n) and (K_{3})_{i,3} = NEG MOD((K_{2})_{i,2}, n) and (K_{3})_{i,4} = INV MOD((K_{2})_{i,4}, n) and (K_{2})_{i,5} = (K_{3})_{i,5} and (K_{2})_{i,6} = (K_{3})_{i,6}. Then (compose_{MESSAGES} ((IDEA P F(K_{2}, n, r))^{⌢}IDEA Q F(K_{3}, n, r)))(m) = m.

(49) Let n be a non empty natural number, l_{1} be a natural number, K_{2}, K_{3} be matrices over ℕ of dimension l_{1} × 6, r be a natural number, k_{3}, k_{4}, k_{5}, k_{6} be finite sequences of elements of ℕ, and given m. Suppose that

(i) l_{1} r, (ii) (the n-th power of 2)+1 is prime, (iii) lenm 4, (iv) m(1) is expressible by n,

(v) m(2) is expressible by n, (vi) m(3) is expressible by n, (vii) m(4) is expressible by n, (viii) for every natural number i such that i r holds (K_{2})_{i,1} is expressible

by n and (K_{2})_{i,2} is expressible by n and (K_{2})_{i,3} is expressible by n and (K_{2})_{i,4} is expressible by n and (K_{2})_{i,5} is expressible by n and (K_{2})_{i,6} is expressible by n and (K_{3})_{i,1} is expressible by n and (K_{3})_{i,2} is expressible by n and (K_{3})_{i,3} is expressible by n and (K_{3})_{i,4} is expressible by n and (K_{3})_{i,5} is expressible by n and (K_{3})_{i,6} is expressible by n and (K_{3})_{i,1} = INV MOD((K_{2})_{i,1}, n) and (K_{3})_{i,2} = NEG MOD((K_{2})_{i,3}, n) and (K_{3})_{i,3} = NEG MOD((K_{2})_{i,2}, n) and (K_{3})_{i,4} = INV MOD((K_{2})_{i,4}, n) and (K_{2})_{i,5} = (K_{3})_{i,5} and (K_{2})_{i,6} = (K_{3})_{i,6},

(ix) k_{3}(1) is expressible by n,

(x) k_{3}(2) is expressible by n, (xi) k_{3}(3) is expressible by n, (xii) k_{3}(4) is expressible by n, (xiii) k_{4}(1) = INV MOD(k_{3}(1), n), (xiv) k_{4}(2) = NEG MOD(k_{3}(2), n), (xv) k_{4}(3) = NEG MOD(k_{3}(3), n), (xvi) k_{4}(4) = INV MOD(k_{3}(4), n), (xvii) k_{5}(1) is expressible by n, (xviii) k_{5}(2) is expressible by n, (xix) k_{5}(3) is expressible by n, (xx) k_{5}(4) is expressible by n, (xxi) k_{5}(5) is expressible by n, (xxii) k_{5}(6) is expressible by n, (xxiii) k_{6}(1) = INV MOD(k_{5}(1), n), (xxiv) k_{6}(2) = NEG MOD(k_{5}(2), n), (xxv) k_{6}(3) = NEG MOD(k_{5}(3), n), (xxvi) k_{6}(4) = INV MOD(k_{5}(4), n), (xxvii) k_{6}(5) = k_{5}(5), and

algebraic group on fixed-length bit integer … 215

(xxviii) k_{6}(6) = k_{5}(6). Then (IDEA QS(k_{4}, n) · (compose_{MESSAGES} IDEA Q F(K_{3}, n, r)· (IDEA QE(k_{6}, n) · (IDEA PE(k_{5}, n) · (compose_{MESSAGES} IDEA P F (K_{2}, n, r) · IDEA PS(k_{3}, n))))))(m) = m.

References

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

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

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

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

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

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

[9] Andrzej Kondracki. The chinese remainder theorem. Formalized Mathematics, 6(4):573– 577, 1997.

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

[11] Robert Milewski. Binary arithmetics. Binary sequences. Formalized Mathematics, 7(1):23–26, 1998.

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

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

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

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

216 yasushi fuwa and yoshinori fujisawa

The Definition and Basic Properties of Topological Groups

Artur Korniłowicz University of Białystok

MML Identifier: TOPGRP 1.

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

1. Preliminaries

For simplicity, we follow the rules: S denotes a 1-sorted structure, R denotes a non empty 1-sorted structure, X denotes a subset of the carrier of R, T denotes a non empty topological structure, and x denotes a set.

Let X, Y be sets. One can verify that every function from X into Y which is bijective is also one-to-one and onto and every function from X into Y which is one-to-one and onto is also bijective.

Let X be a set. Observe that there exists a function from X into X which is one-to-one and onto.

Next we state the proposition

(1) rng(id_{S}) = Ω_{S}.

Let R be a non empty 1-sorted structure. Note that (id_{R})^{−1} is one-to-one.

We now state two propositions:

(2) (id_{R})^{−1} = id_{R}.

(3) (id_{R})^{−1} (X) = X.

Let S be a 1-sorted structure. One can check that there exists a map from S into S which is one-to-one and onto.

2. On the Groups

We use the following convention: H denotes a non empty groupoid, P, Q, P_{1}, Q_{1} denote subsets of the carrier of H, and h denotes an element of the carrier of H.

The following propositions are true:

(4) If P ⊆ P_{1} and Q ⊆ Q_{1}, then P · Q ⊆ P_{1} · Q_{1}.

(5) If P ⊆ Q, then P · h ⊆ Q · h.

(6) If P ⊆ Q, then h · P ⊆ h · Q.

In the sequel G denotes a group, A, B denote subsets of the carrier of G, and a denotes an element of the carrier of G.

One can prove the following propositions:

(7) a ∈ A^{−1} iff a^{−1} ∈ A.

(8) (A^{−1} )^{−1} = A.

(9) A ⊆ B iff A^{−1} ⊆ B^{−1} .

(10) ·_{G}^{−1}^{∘}A = A^{−1} .

(11) ·_{G}^{−1}^{−1} (A) = A^{−1} .

(12) ·_{G}^{−1} is one-to-one.

(13) rng·_{G}^{−1} = the carrier of G.

Let G be a group. Observe that ·_{G}^{−1} is one-to-one and onto.

Next we state two propositions:

(14) ·_{G}^{−1}^{−1} = ·_{G}^{−1}.

(15) (The multiplication of H)^{∘}[: P, Q:] = P · Q. Let G be a non empty groupoid and let a be an element of the carrier of G.

The functor a · □ yielding a map from G into G is defined by: (Def. 1) For every element x of the carrier of G holds (a · □)(x) = a · x.

The functor □ · a yields a map from G into G and is defined as follows: (Def. 2) For every element x of the carrier of G holds (□ · a)(x) = x · a.

Let G be a group and let a be an element of the carrier of G. One can verify that a · □ is one-to-one and onto and □ · a is one-to-one and onto.

Next we state four propositions:

(16) (h · □)^{∘}P = h · P.

(17) (□ · h)^{∘}P = P · h.

(18) (a · □)^{−1} = a^{−1} · □.

(19) (□ · a)^{−1} = □ · a^{−1} .

3. On the Topological Spaces

Let T be a non empty topological structure. Observe that (id_{T})^{−1} is continuous.

Next we state the proposition

(20) id_{T} is a homeomorphism.

Let T be a non empty topological space and let p be a point of T. Observe that every neighbourhood of p is non empty.

Next we state the proposition

(21) For every non empty topological space T and for every point p of T holds

Ω_{T} is a neighbourhood of p. Let T be a non empty topological space and let p be a point of T. One can

check that there exists a neighbourhood of p which is non empty and open.

One can prove the following propositions:

(22) Let S, T be non empty topological spaces and f be a map from S into T. Suppose f is open. Let p be a point of S and P be a neighbourhood of p. Then there exists an open neighbourhood R of f(p) such that R ⊆ f^{∘}P.

(23) Let S, T be non empty topological spaces and f be a map from S into T. Suppose that for every point p of S and for every open neighbourhood P of p there exists a neighbourhood R of f(p) such that R ⊆ f^{∘}P. Then f is open.

(24) Let S, T be non empty topological structures and f be a map from S into T. Then f is a homeomorphism if and only if the following conditions are satisfied:

(i) domf = Ω_{S}, (ii) rngf = Ω_{T} , (iii) f is one-to-one, and (iv) for every subset P of T holds P is closed iff f^{−1} (P) is closed.

(25) Let S, T be non empty topological structures and f be a map from S into T. Then f is a homeomorphism if and only if the following conditions are satisfied:

(i) domf = Ω_{S}, (ii) rngf = Ω_{T} , (iii) f is one-to-one, and (iv) for every subset P of S holds P is open iff f^{∘}P is open.

(26) Let S, T be non empty topological structures and f be a map from S into T. Then f is a homeomorphism if and only if the following conditions are satisfied:

(i) domf = Ω_{S}, (ii) rngf = Ω_{T} , (iii) f is one-to-one, and (iv) for every subset P of T holds P is open iff f^{−1} (P) is open.

(27) Let S be a topological space, T be a non empty topological space, and f be a map from S into T. Then f is continuous if and only if for every subset P of the carrier of T holds f^{−1} (IntP) ⊆ Int(f^{−1} (P)).

Let T be a non empty topological space. One can verify that there exists a subset of T which is non empty and dense.

The following two propositions are true:

(28) Let S, T be non empty topological spaces, f be a map from S into T, and A be a dense subset of S. If f is a homeomorphism, then f^{∘}A is dense.

(29) Let S, T be non empty topological spaces, f be a map from S into T, and A be a dense subset of T. If f is a homeomorphism, then f^{−1} (A) is dense.

Let S, T be non empty topological structures. Observe that every map from S into T which is homeomorphism is also onto, one-to-one, continuous, and open.

Let T be a non empty topological structure. Observe that there exists a map from T into T which is homeomorphism.

Let T be a non empty topological structure and let f be homeomorphism map from T into T. Note that f^{−1} is homeomorphism.

4. The Group of Homoemorphisms

Let T be a non empty topological structure. A map from T into T is said to be a homeomorphism of T if:

(Def. 3) It is a homeomorphism.

Let T be a non empty topological structure. Then id_{T} is a homeomorphism of T.

Let T be a non empty topological structure. One can check that every homeomorphism of T is homeomorphism.

We now state two propositions:

(30) For every homeomorphism f of T holds f^{−1} is a homeomorphism of T.

(31) For all homeomorphisms f, g of T holds f · g is a homeomorphism of T. Let T be a non empty topological structure. The group of homeomorphisms

of T is a strict groupoid and is defined by the conditions (Def. 4). (Def. 4)(i) x ∈ the carrier of the group of homeomorphisms of T iff x is a

homeomorphism of T, and (ii) for all homeomorphisms f, g of T holds (the multiplication of the group

of homeomorphisms of T)(f, g) = g · f. Let T be a non empty topological structure. Note that the group of home-

omorphisms of T is non empty.

We now state the proposition

(32) Let f, g be homeomorphisms of T and a, b be elements of the group of

homeomorphisms of T. If f = a and g = b, then a · b = g · f. Let T be a non empty topological structure. Note that the group of home-

omorphisms of T is group-like and associative.

The following two propositions are true:

(33) id_{T} = 1_{the group of homeomorphisms of T} .

(34) Let f be a homeomorphism of T and a be an element of the group of

homeomorphisms of T. If f = a, then a^{−1} = f^{−1} . Let T be a non empty topological structure. We say that T is homogeneous

if and only if: (Def. 5) For all points p, q of T there exists a homeomorphism f of T such that

f(p) = q. Let us note that every non empty topological structure which is trivial is

also homogeneous.

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

One can prove the following two propositions:

(35) Let T be a homogeneous non empty topological space. If there exists a point p of T such that {p} is closed, then T is a T_{1} space.

(36) Let T be a homogeneous non empty topological space. Given a point p of T such that let A be a subset of T. Suppose A is open and p ∈ A. Then there exists a subset B of T such that p ∈ B and B is open and \overline{B} ⊆ A. Then T is a T_{3} space.

5. On the Topological Groups

We consider topological group structures as extensions of groupoid and topological structure as systems

⟨ a carrier, a multiplication, a topology ⟩, where the carrier is a set, the multiplication is a binary operation on the carrier, and the topology is a family of subsets of the carrier.

Let A be a non empty set, let R be a binary operation on A, and let T be a family of subsets of A. Note that ⟨A, R, T ⟩ is non empty.

Let x be a set, let R be a binary operation on {x}, and let T be a family of subsets of {x}. One can verify that ⟨{x}, R, T ⟩ is trivial.

Let us observe that every non empty groupoid which is trivial is also grouplike, associative, and commutative.

Let a be a set. Observe that {a}_{top} is trivial.

Let us note that there exists a topological group structure which is strict and non empty.

One can verify that there exists a non empty topological group structure which is strict, topological space-like, and trivial.

Let G be a group-like associative non empty topological group structure. Then ·_{G}^{−1} is a map from G into G.

Let G be a group-like associative non empty topological group structure. We say that G is inverse-continuous if and only if:

(Def. 6) ·_{G}^{−1} is continuous.

Let G be a topological space-like topological group structure. We say that G is continuous if and only if:

(Def. 7) For every map f from [: G, G:] into G such that f = the multiplication

of G holds f is continuous. One can verify that there exists a topological space-like group-like associative

non empty topological group structure which is strict, commutative, trivial, inverse-continuous, and continuous.

A semi topological group is a topological space-like group-like associative non empty topological group structure.

A topological group is an inverse-continuous continuous semi topological group.

Next we state several propositions:

(37) Let T be a continuous non empty topological space-like topological group structure, a, b be elements of the carrier of T, and W be a neighbourhood of a · b. Then there exists an open neighbourhood A of a and there exists an open neighbourhood B of b such that A · B ⊆ W.

(38) Let T be a topological space-like non empty topological group structure. Suppose that for all elements a, b of the carrier of T and for every neighbourhood W of a · b there exists a neighbourhood A of a and there exists a neighbourhood B of b such that A · B ⊆ W. Then T is continuous.

(39) Let T be an inverse-continuous semi topological group, a be an element of the carrier of T, and W be a neighbourhood of a^{−1} . Then there exists an open neighbourhood A of a such that A^{−1} ⊆ W.

(40) Let T be a semi topological group. Suppose that for every element a of the carrier of T and for every neighbourhood W of a^{−1} there exists a neighbourhood A of a such that A^{−1} ⊆ W. Then T is inverse-continuous.

(41) Let T be a topological group, a, b be elements of the carrier of T, and W be a neighbourhood of a·b^{−1} . Then there exists an open neighbourhood A of a and there exists an open neighbourhood B of b such that A·B^{−1} ⊆ W.

(42) Let T be a semi topological group. Suppose that for all elements a, b of the carrier of T and for every neighbourhood W of a · b^{−1} there exists a neighbourhood A of a and there exists a neighbourhood B of b such that A · B^{−1} ⊆ W. Then T is a topological group.

Let G be a continuous non empty topological space-like topological group structure and let a be an element of the carrier of G. One can check that a · □ is continuous and □ · a is continuous.

Next we state two propositions:

(43) Let G be a continuous semi topological group and a be an element of the carrier of G. Then a · □ is a homeomorphism of G.

(44) Let G be a continuous semi topological group and a be an element of

the carrier of G. Then □ · a is a homeomorphism of G. The following proposition is true

(45) For every inverse-continuous semi topological group G holds ·_{G}^{−1} is a

homeomorphism of G. One can verify that every semi topological group which is continuous is also

homogeneous.

The following two propositions are true:

(46) Let G be a continuous semi topological group, F be a closed subset of G, and a be an element of the carrier of G. Then F · a is closed.

(47) Let G be a continuous semi topological group, F be a closed subset of

G, and a be an element of the carrier of G. Then a · F is closed. We now state the proposition

(48) For every inverse-continuous semi topological group G and for every

closed subset F of G holds F^{−1} is closed. The following two propositions are true:

(49) Let G be a continuous semi topological group, O be an open subset of G, and a be an element of the carrier of G. Then O · a is open.

(50) Let G be a continuous semi topological group, O be an open subset of

G, and a be an element of the carrier of G. Then a · O is open. We now state the proposition

(51) For every inverse-continuous semi topological group G and for every open

subset O of G holds O^{−1} is open. The following two propositions are true:

(52) For every continuous semi topological group G and for all subsets A, O of G such that O is open holds O · A is open.

224 artur korniłowicz

(53) For every continuous semi topological group G and for all subsets A, O of G such that O is open holds A · O is open.

One can prove the following propositions:

(54) Let G be an inverse-continuous semi topological group, a be a point of G, and A be a neighbourhood of a. Then A^{−1} is a neighbourhood of a^{−1} .

(55) Let G be a topological group, a be a point of G, and A be a neighbourhood of a · a^{−1} . Then there exists an open neighbourhood B of a such that B · B^{−1} ⊆ A.

(56) For every inverse-continuous semi topological group G and for every dense subset A of G holds A^{−1} is dense.

We now state two propositions:

(57) Let G be a continuous semi topological group, A be a dense subset of G, and a be a point of G. Then a · A is dense.

(58) Let G be a continuous semi topological group, A be a dense subset of G, and a be a point of G. Then A · a is dense.

We now state two propositions:

(59) Let G be a topological group, B be a basis of 1_{G}, and M be a dense subset of G. Then {V · x; V ranges over subsets of the carrier of G, x ranges over points of G: V ∈ B ∧ x ∈ M} is a basis of G.

(60) Every topological group is a T_{3} space.

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

the definition and basic properties of … 225

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

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

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

226 artur korniłowicz

The Correspondence Between Lattices of Subalgebras of

Universal Algebras and Many Sorted Algebras

Adam Naumowicz University of Białystok

Agnieszka Julia Marasik Warsaw University of Technology

Summary. The main goal of this paper is to show some properties of subalgebras of universal algebras and many sorted algebras, and then the isomorphic correspondence between lattices of such subalgebras.

MML Identifier: MSSUBLAT.

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

1. Preliminaries

In this paper a denotes a set and i denotes a natural number. We now state several propositions:

(1) (□ ⟼→ a)(0) = ε.

(2) (□ ⟼→ a)(1) = ⟨a⟩.

(3) (□ ⟼→ a)(2) = ⟨a, a⟩.

(4) (□ ⟼→ a)(3) = ⟨a, a, a⟩.

(5) For every finite sequence f of elements of {0} holds f = i ↦ 0 iff lenf = i.

(6) For every finite sequence f such that f = (□ ⟼→ 0)(i) holds lenf = i.

2. Some Properties of Subalgebras of Universal and Many Sorted Algebras

We now state the proposition

(7) For all universal algebras U_{1}, U_{2} such that U_{1} is a subalgebra of U_{2} holds MSSign(U_{1}) = MSSign(U_{2}).

Let U_{0} be a universal algebra. One can verify that the characteristic of U_{0} is function yielding.

One can prove the following propositions:

(8) Let U_{1}, U_{2} be universal algebras. Suppose U_{1} is a subalgebra of U_{2}. Let B be a subset of MSAlg(U_{2}). Suppose B = the sorts of MSAlg(U_{1}). Let o be an operation symbol of MSSign(U_{2}) and a be an operation symbol of MSSign(U_{1}). If a = o, then Den(a, MSAlg(U_{1})) = Den(o, MSAlg(U_{2}))↾ Args(a, MSAlg(U_{1})).

(9) For all universal algebras U_{1}, U_{2} such that U_{1} is a subalgebra of U_{2} holds the sorts of MSAlg(U_{1}) are a subset of MSAlg(U_{2}).

(10) Let U_{1}, U_{2} be universal algebras. Suppose U_{1} is a subalgebra of U_{2}. Let B be a subset of MSAlg(U_{2}). If B = the sorts of MSAlg(U_{1}), then B is operations closed.

(11) Let U_{1}, U_{2} be universal algebras. Suppose U_{1} is a subalgebra of U_{2}. Let B be a subset of MSAlg(U_{2}). If B = the sorts of MSAlg(U_{1}), then the characteristics of MSAlg(U_{1}) = Opers(MSAlg(U_{2}), B).

(12) For all universal algebras U_{1}, U_{2} such that U_{1} is a subalgebra of U_{2} holds MSAlg(U_{1}) is a subalgebra of MSAlg(U_{2}).

(13) Let U_{1}, U_{2} be universal algebras. Suppose MSAlg(U_{1}) is a subalgebra of MSAlg(U_{2}). Then the carrier of U_{1} is a subset of the carrier of U_{2}.

(14) Let U_{1}, U_{2} be universal algebras. Suppose MSAlg(U_{1}) is a subalgebra of MSAlg(U_{2}). Let B be a non empty subset of the carrier of U_{2}. If B = the carrier of U_{1}, then B is operations closed.

(15) Let U_{1}, U_{2} be universal algebras. Suppose MSAlg(U_{1}) is a subalgebra of MSAlg(U_{2}). Let B be a non empty subset of the carrier of U_{2}. If B = the carrier of U_{1}, then the characteristic of U_{1} = Opers(U_{2}, B).

(16) For all universal algebras U_{1}, U_{2} such that MSAlg(U_{1}) is a subalgebra

of MSAlg(U_{2}) holds U_{1} is a subalgebra of U_{2}. In the sequel M_{1} is a segmental trivial non void non empty many sorted

signature and A is a non-empty algebra over M_{1}.

Next we state a number of propositions:

(17) For every non-empty subalgebra B of A holds the carrier of Alg_{1}(B) is a subset of the carrier of Alg_{1}(A).

(18) Let B be a non-empty subalgebra of A and S be a non empty subset of the carrier of Alg_{1}(A). If S = the carrier of Alg_{1}(B), then S is operations closed.

(19) Let B be a non-empty subalgebra of A and S be a non empty subset of the carrier of Alg_{1}(A). If S = the carrier of Alg_{1}(B), then the characteristic of Alg_{1}(B) = Opers(Alg_{1}(A), S).

(20) For every non-empty subalgebra B of A holds Alg_{1}(B) is a subalgebra of Alg_{1}(A).

(21) Let S be a non empty non void many sorted signature and A, B be algebras over S. Then A is a subalgebra of B if and only if A is a subalgebra of the algebra of B.

(22) For all universal algebras A, B holds signatureA = signatureB iff MSSign(A) = MSSign(B).

(23) Let A be a non-empty algebra over M_{1}. Suppose the carrier of M_{1} = {0}. Then MSSign(Alg_{1}(A)) = the many sorted signature of M_{1}.

(24) Let A, B be non-empty algebras over M_{1}. Suppose the carrier of M_{1} = {0} and Alg_{1}(A) = Alg_{1}(B). Then the algebra of A = the algebra of B.

(25) Let A be a non-empty algebra over M_{1}. If the carrier of M_{1} = {0}, then the sorts of A = the sorts of MSAlg(Alg_{1}(A)).

(26) For every non-empty algebra A over M_{1} such that the carrier of M_{1} = {0} holds MSAlg(Alg_{1}(A)) = the algebra of A.

(27) Let A be a universal algebra and B be a strict non-empty subalgebra of MSAlg(A). If the carrier of MSSign(A) = {0}, then Alg_{1}(B) is a subalgebra of A.

3. The Correspondence Between Lattices of Subalgebras of Universal and Many Sorted Algebras

We now state three propositions:

(28) Let A be a universal algebra, a_{1}, b_{1} be strict non-empty subalgebras of A, and a_{2}, b_{2} be strict non-empty subalgebras of MSAlg(A). Suppose a_{2} = MSAlg(a_{1}) and b_{2} = MSAlg(b_{1}). Then (the sorts of a_{2}) ∪ (the sorts of b_{2}) = 0⟼→ . ((the carrier of a_{1}) ∪ (the carrier of b_{1})).

(29) Let A be a universal algebra, a_{1}, b_{1} be strict non-empty subalgebras of A, and a_{2}, b_{2} be strict non-empty subalgebras of MSAlg(A). Suppose a_{2} = MSAlg(a_{1}) and b_{2} = MSAlg(b_{1}). Then (the sorts of a_{2}) ∩ (the sorts of b_{2}) = 0⟼→ . (the carrier of a_{1}) ∩ (the carrier of b_{1}).

230 adam naumowicz and agnieszka julia marasik

(30) Let A be a strict universal algebra, a_{1}, b_{1} be strict non-empty subalgebras of A, and a_{2}, b_{2} be strict non-empty subalgebras of MSAlg(A). If a_{2} = MSAlg(a_{1}) and b_{2} = MSAlg(b_{1}), then MSAlg(a_{1} ⨆ b_{1}) = a_{2} ⨆ b_{2}.

Let A be a universal algebra with constants. One can check that MSSign(A) is non void strict segmental and trivial and has constant operations.

One can prove the following proposition

(31) Let A be a universal algebra with constants, a_{1}, b_{1} be strict non-empty subalgebras of A, and a_{2}, b_{2} be strict non-empty subalgebras of MSAlg(A). If a_{2} = MSAlg(a_{1}) and b_{2} = MSAlg(b_{1}), then MSAlg(a_{1} ∩ b_{1}) = a_{2} ∩ b_{2}.

Let A be a quasi total universal algebra structure. One can verify that the universal algebra structure of A is quasi total.

Let A be a partial universal algebra structure. Observe that the universal algebra structure of A is partial.

Let A be a non-empty universal algebra structure. Note that the universal algebra structure of A is non-empty.

Let A be a universal algebra with constants. Note that the universal algebra structure of A has constants.

We now state the proposition

(32) Let A be a universal algebra with constants. Then the lattice of subalgebras of the universal algebra structure of A and the lattice of subalgebras of MSAlg(the universal algebra structure of A) are isomorphic.

References

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

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

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

[4] Ewa Burakowska. Subalgebras of many sorted algebra. Lattice of subalgebras. Formalized Mathematics, 5(1):47–54, 1996.

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

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

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

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

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

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

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

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

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

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

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

the correspondence between lattices of … 231

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

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

232 adam naumowicz and agnieszka julia marasik

Introduction to Concept Lattices

Christoph Schwarzweller University of Tübingen

Summary. In this paper we give Mizar formalization of concept lattices. Concept lattices stem from the so called formal concept analysis — a part of applied mathematics that brings mathematical methods into the field of data anylysis and knowledge processing. Our approach follows the one given in [8].

MML Identifier: CONLAT 1.

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

1. Formal Contexts

We consider 2-sorted as systems

⟨ objects, a Attributes ⟩, where the objects constitute a set and the Attributes is a set.

Let C be a 2-sorted. We say that C is empty if and only if: (Def. 1) The objects of C are empty and the Attributes of C is empty.

We say that C is quasi-empty if and only if: (Def. 2) The objects of C are empty or the Attributes of C is empty.

Let us note that there exists a 2-sorted which is strict and non empty and there exists a 2-sorted which is strict and non quasi-empty.

One can verify that there exists a 2-sorted which is strict, empty, and quasiempty.

We consider ContextStr as extensions of 2-sorted as systems

⟨ objects, a Attributes, a Information ⟩, where the objects constitute a set, the Attributes is a set, and the Information is a relation between the objects and the Attributes.

One can check that there exists a ContextStr which is strict and non empty and there exists a ContextStr which is strict and non quasi-empty.

A FormalContext is a non quasi-empty ContextStr.

Let C be a 2-sorted. (Def. 3) An element of the objects of C is said to be an object of C. (Def. 4) An element of the Attributes of C is said to be a Attribute of C.

Let C be a non quasi-empty 2-sorted. Note that the Attributes of C is non empty and the objects of C is non empty.

Let C be a non quasi-empty 2-sorted. One can check that there exists a subset of the objects of C which is non empty and there exists a subset of the Attributes of C which is non empty.

Let C be a FormalContext, let o be an object of C, and let a be a Attribute of C. We say that o is connected with a if and only if:

(Def. 5) ⟨⟨o, a⟩⟩∈ the Information of C.

We introduce o is not connected with a as an antonym of o is connected with a.

2. Derivation Operators

Let C be a FormalContext. The functor ObjectDerivationC yields a function from 2^{the objects of C} into 2^{the Attributes of C} and is defined by the condition (Def. 6).

(Def. 6) Let O be an element of 2^{the objects of C}. Then (ObjectDerivationC)(O) = {a; a ranges over Attribute of C: ⋀_{o: object of C} (o ∈ O ⇒ o is connected with a)}.

Let C be a FormalContext. The functor AttributeDerivationC yields a function from 2^{the Attributes of C} into 2^{the objects of C} and is defined by the condition (Def. 7).

(Def. 7) Let A be an element of 2^{the Attributes of C}. Then (AttributeDerivationC)(A) = {o; o ranges over objects of C: ⋀_{a: Attribute of C} (a ∈ A ⇒ o is connected with a)}.

The following propositions are true:

(1) Let C be a FormalContext and o be an object of C. Then (ObjectDerivationC)({o}) = {a; a ranges over Attribute of C: o is connected with a}.

(2) Let C be a FormalContext and a be a Attribute of C. Then (AttributeDerivationC)({a}) = {o; o ranges over objects of C: o is connected with a}.

(3) For every FormalContext C and for all subsets O_{1}, O_{2} of the objects of C such that O_{1} ⊆ O_{2} holds (ObjectDerivationC)(O_{2}) ⊆ (ObjectDerivationC)(O_{1}).

(4) For every FormalContext C and for all subsets A_{1}, A_{2} of the Attributes of C such that A_{1} ⊆ A_{2} holds (AttributeDerivationC)(A_{2}) ⊆ (AttributeDerivationC)(A_{1}).

(5) For every FormalContext C and for every subset O of the objects of C holds O ⊆ (AttributeDerivationC)((ObjectDerivationC)(O)).

(6) For every FormalContext C and for every subset A of the Attributes of C holds A ⊆ (ObjectDerivationC)((AttributeDerivationC)(A)).

(7) For every FormalContext C and for every subset O of the objects of C holds (ObjectDerivationC)(O) = (ObjectDerivationC) ((AttributeDerivationC)((ObjectDerivationC)(O))).

(8) For every FormalContext C and for every subset A of the Attributes of C holds (AttributeDerivationC)(A) = (AttributeDerivationC)((ObjectDerivationC)((AttributeDerivationC)(A))).

(9) Let C be a FormalContext, O be a subset of the objects of C, and A be a subset of the Attributes of C. Then O ⊆ (AttributeDerivationC)(A) if and only if [: O, A:] ⊆ the Information of C.

(10) Let C be a FormalContext, O be a subset of the objects of C, and A be a subset of the Attributes of C. Then A ⊆ (ObjectDerivationC)(O) if and only if [: O, A:] ⊆ the Information of C.

(11) Let C be a FormalContext, O be a subset of the objects of C, and A be a subset of the Attributes of C. Then O ⊆ (AttributeDerivationC)(A) if and only if A ⊆ (ObjectDerivationC)(O).

Let C be a FormalContext. The functor ϕ(C) yielding a map from 2_{⊆}^{the objects of C} into 2_{⊆}^{the Attributes of C} is defined by:

(Def. 8) ϕ(C) = ObjectDerivationC.

Let C be a FormalContext. The functor psiC yields a map from 2_{⊆}^{the Attributes of C} into 2_{⊆}^{the objects of C} and is defined as follows:

(Def. 9) psiC = AttributeDerivationC.

We now state the proposition

(12) For every FormalContext C holds ⟨⟨ϕ(C), psiC⟩⟩is a connection between

2_{⊆}^{the objects of C} and 2_{⊆}^{the Attributes of C}. Let P, R be non empty relational structures and let C_{1} be a connection

between P and R. We say that C_{1} is co-Galois if and only if the condition (Def. 10) is satisfied.

(Def. 10) There exists a map f from P into R and there exists a map g from R into P such that (i) C_{1} = ⟨⟨f, g⟩⟩, (ii) f is antitone, (iii) g is antitone, and (iv) for all elements p_{1}, p_{2} of P and for all elements r_{1}, r_{2} of R holds

p_{1} g(f(p_{1})) and r_{1} f(g(r_{1})). We now state several propositions:

(13) Let P, R be non empty posets, C_{1} be a connection between P and R, f be a map from P into R, and g be a map from R into P. Suppose C_{1} = ⟨⟨f, g⟩⟩. Then C_{1} is co-Galois if and only if for every element p of P and for every element r of R holds p g(r) iff r f(p).

(14) Let P, R be non empty posets and C_{1} be a connection between P and R. Suppose C_{1} is co-Galois. Let f be a map from P into R and g be a map from R into P. If C_{1} = ⟨⟨f, g⟩⟩, then f = f · (g · f) and g = g · (f · g).

(15) For every FormalContext C holds ⟨⟨ϕ(C), psiC⟩⟩ is co-Galois.

(16) For every FormalContext C and for all subsets O_{1}, O_{2} of the objects of C holds (ObjectDerivationC)(O_{1} ∪ O_{2}) = (ObjectDerivationC)(O_{1}) ∩ (ObjectDerivationC)(O_{2}).

(17) For every FormalContext C and for all subsets A_{1}, A_{2} of the Attributes of C holds (AttributeDerivationC)(A_{1} ∪ A_{2}) = (AttributeDerivationC)(A_{1}) ∩ (AttributeDerivationC)(A_{2}).

(18) For every FormalContext C holds (ObjectDerivationC)(∅) = the Attributes of C.

(19) For every FormalContext C holds (AttributeDerivationC)(∅) = the objects of C.

3. Formal Concepts

Let C be a 2-sorted. We introduce ConceptStr over C which are systems ⟨ a Extent, a Intent ⟩,

where the Extent is a subset of the objects of C and the Intent is a subset of the Attributes of C.

Let C be a 2-sorted and let C_{2} be a ConceptStr over C. We say that C_{2} is empty if and only if:

(Def. 11) The Extent of C_{2} is empty and the Intent of C_{2} is empty.

We say that C_{2} is quasi-empty if and only if: (Def. 12) The Extent of C_{2} is empty or the Intent of C_{2} is empty.

Let C be a non quasi-empty 2-sorted. Observe that there exists a ConceptStr over C which is strict and non empty and there exists a ConceptStr over C which is strict and quasi-empty.

Let C be an empty 2-sorted. Observe that every ConceptStr over C is empty. Let C be a quasi-empty 2-sorted. Observe that every ConceptStr over C is

quasi-empty.

Let C be a FormalContext and let C_{2} be a ConceptStr over C. We say that C_{2} is concept-like if and only if:

(Def. 13) (ObjectDerivationC)(the Extent of C_{2}) = the Intent of C_{2} and

(AttributeDerivationC)(the Intent of C_{2}) = the Extent of C_{2}. Let C be a FormalContext. One can check that there exists a ConceptStr

over C which is concept-like and non empty.

Let C be a FormalContext. A FormalConcept of C is a concept-like non empty ConceptStr over C.

Let C be a FormalContext. Note that there exists a FormalConcept of C which is strict.

Next we state four propositions:

(20) Let C be a FormalContext and O be a subset of the objects of C. Then

(i) ⟨(AttributeDerivationC)((ObjectDerivationC)(O)), (ObjectDerivationC)(O)⟩ is a FormalConcept of C, and

(ii) for every subset O′ of the objects of C and for every subset A′ of the Attributes of C such that ⟨O′, A′⟩ is a FormalConcept of C and O ⊆ O′ holds (AttributeDerivationC)((ObjectDerivationC)(O)) ⊆ O′.

(21) Let C be a FormalContext and O be a subset of the objects of C. Then there exists a subset A of the Attributes of C such that ⟨O, A⟩ is a FormalConcept of C if and only if (AttributeDerivationC)((ObjectDerivationC)(O)) = O.

(22) Let C be a FormalContext and A be a subset of the Attributes of C.

Then

(i) ⟨(AttributeDerivationC)(A), (ObjectDerivationC) ((AttributeDerivationC)(A))⟩ is a FormalConcept of C, and

(ii) for every subset O′ of the objects of C and for every subset A′ of the Attributes of C such that ⟨O′, A′⟩ is a FormalConcept of C and A ⊆ A′ holds (ObjectDerivationC)((AttributeDerivationC)(A)) ⊆ A′.

(23) Let C be a FormalContext and A be a subset of the Attributes of C. Then there exists a subset O of the objects of C such that ⟨O, A⟩ is a FormalConcept of C if and only if (ObjectDerivationC)((AttributeDerivationC)(A)) = A.

Let C be a FormalContext and let C_{2} be a ConceptStr over C. We say that C_{2} is universal if and only if:

(Def. 14) The Extent of C_{2} = the objects of C.

Let C be a FormalContext and let C_{2} be a ConceptStr over C. We say that C_{2} is co-universal if and only if:

(Def. 15) The Intent of C_{2} = the Attributes of C.

Let C be a FormalContext. Note that there exists a FormalConcept of C which is strict and universal and there exists a FormalConcept of C which is strict and co-universal.

Let C be a FormalContext. The functor Concept − with − all − Objects C yields a strict universal FormalConcept of C and is defined by the condition (Def. 16).

(Def. 16) There exists a subset O of the objects of C and there exists a subset A of the Attributes of C such that Concept − with − all − Objects C = ⟨O, A⟩ and O = (AttributeDerivationC)(∅) and A = (ObjectDerivationC)((AttributeDerivationC)(∅)).

Let C be a FormalContext. The functor Concept − with − all − Attributes C yielding a strict co-universal FormalConcept of C is defined by the condition (Def. 17).

(Def. 17) There exists a subset O of the objects of C and there exists a subset A of the Attributes of C such that Concept − with − all − Attributes C = ⟨O, A⟩ and O = (AttributeDerivationC)((ObjectDerivationC)(∅)) and A = (ObjectDerivationC)(∅).

One can prove the following propositions:

(24) Let C be a FormalContext. Then the Extent of Concept − with − all − Objects C = the objects of C and the Intent of Concept − with − all − Attributes C = the Attributes of C.

(25) Let C be a FormalContext and C_{2} be a FormalConcept of C. Then (i) if the Extent of C_{2} = ∅, then C_{2} is co-universal, and (ii) if the Intent of C_{2} = ∅, then C_{2} is universal.

(26) Let C be a FormalContext and C_{2} be a strict FormalConcept of C. Then

(i) if the Extent of C_{2} = ∅, then C_{2} = Concept − with − all − Attributes C, and

(ii) if the Intent of C_{2} = ∅, then C_{2} = Concept − with − all − Objects C.

(27) Let C be a FormalContext and C_{2} be a quasi-empty ConceptStr over C. Suppose C_{2} is a FormalConcept of C. Then C_{2} is universal or co-universal.

(28) Let C be a FormalContext and C_{2} be a quasi-empty ConceptStr over C. If C_{2} is a strict FormalConcept of C, then C_{2} = Concept − with − all − Objects C or C_{2} = Concept − with − all − Attributes C.

Let C be a FormalContext. A non empty set is called a Set of FormalConcepts of C if:

(Def. 18) For every set X such that X ∈ it holds X is a FormalConcept of C.

Let C be a FormalContext and let F_{1} be a Set of FormalConcepts of C. We see that the element of F_{1} is a FormalConcept of C.

Let C be a FormalContext and let C_{3}, C_{4} be FormalConcept of C. We say that C_{3} is SubConcept of C_{4} if and only if:

(Def. 19) The Extent of C_{3} ⊆ the Extent of C_{4}.

We introduce C_{4} is SuperConcept of C_{3} as a synonym of C_{3} is SubConcept of C_{4}.

One can prove the following propositions:

(29) Let C be a FormalContext and C_{3}, C_{4} be FormalConcept of C. Then C_{3} is SuperConcept of C_{4} if and only if C_{4} is SubConcept of C_{3}.

(30) Let C be a FormalContext and C_{3}, C_{4} be FormalConcept of C. Then C_{3} is SubConcept of C_{4} if and only if the Extent of C_{3} ⊆ the Extent of C_{4}.

(31) Let C be a FormalContext and C_{3}, C_{4} be FormalConcept of C. Then C_{3} is SubConcept of C_{4} if and only if the Intent of C_{4} ⊆ the Intent of C_{3}.

(32) Let C be a FormalContext and C_{3}, C_{4} be FormalConcept of C. Then C_{3} is SuperConcept of C_{4} if and only if the Extent of C_{4} ⊆ the Extent of C_{3}.

(33) Let C be a FormalContext and C_{3}, C_{4} be FormalConcept of C. Then C_{3} is SuperConcept of C_{4} if and only if the Intent of C_{3} ⊆ the Intent of C_{4}.

(34) Let C be a FormalContext and C_{2} be a FormalConcept of C. Then C_{2} is SubConcept of Concept − with − all − Objects C and Concept − with − all − Attributes C is SubConcept of C_{2}.

4. Concept Lattices

Let C be a FormalContext. The functor B − carrier C yielding a non empty set is defined by the condition (Def. 20).

(Def. 20) B − carrier C = {⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ is non empty ∧ (ObjectDerivationC)(E) = I ∧ (AttributeDerivationC)(I) = E}.

Let C be a FormalContext. Then B − carrier C is a Set of FormalConcepts of C.

Let C be a FormalContext. One can check that B − carrier C is non empty. One can prove the following proposition

(35) For every FormalContext C and for every set C_{2} holds C_{2} ∈ B − carrier C

iff C_{2} is a strict FormalConcept of C. Let C be a FormalContext. The functor B − meet C yields a binary operation

on B − carrier C and is defined by the condition (Def. 21). (Def. 21) Let C_{3}, C_{4} be strict FormalConcept 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 (B − meet C)(C_{3}, C_{4}) = ⟨O, A⟩ and O = (the Extent of C_{3}) ∩ (the Extent of C_{4}) and A = (ObjectDerivationC)((AttributeDerivationC)((the Intent of C_{3}) ∪ (the Intent of C_{4}))).

Let C be a FormalContext. The functor B − join C yielding a binary operation on B − carrier C is defined by the condition (Def. 22).

(Def. 22) Let C_{3}, C_{4} be strict FormalConcept 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 (B − join C)(C_{3}, C_{4}) = ⟨O, A⟩ and O = (AttributeDerivationC)((ObjectDerivationC)((the Extent of C_{3}) ∪ (the Extent of C_{4}))) and A = (the Intent of C_{3}) ∩ (the Intent of C_{4}).

One can prove the following propositions:

(36) For every FormalContext C and for all strict FormalConcept C_{3}, C_{4} of C holds (B − meet C)(C_{3}, C_{4}) = (B − meet C)(C_{4}, C_{3}).

(37) For every FormalContext C and for all strict FormalConcept C_{3}, C_{4} of C holds (B − join C)(C_{3}, C_{4}) = (B − join C)(C_{4}, C_{3}).

(38) For every FormalContext C and for all strict FormalConcept C_{3}, C_{4}, C_{5} of C holds (B − meet C)(C_{3}, (B − meet C)(C_{4}, C_{5})) = (B − meet C)((B − meet C)(C_{3}, C_{4}), C_{5}).

(39) For every FormalContext C and for all strict FormalConcept C_{3}, C_{4}, C_{5} of C holds (B − join C)(C_{3}, (B − join C)(C_{4}, C_{5})) = (B − join C)((B − join C)(C_{3}, C_{4}), C_{5}).

(40) For every FormalContext C and for all strict FormalConcept C_{3}, C_{4} of C holds (B − join C)((B − meet C)(C_{3}, C_{4}), C_{4}) = C_{4}.

(41) For every FormalContext C and for all strict FormalConcept C_{3}, C_{4} of C holds (B − meet C)(C_{3}, (B − join C)(C_{3}, C_{4})) = C_{3}.

(42) For every FormalContext C and for every strict FormalConcept C_{2} of C holds (B − meet C)(C_{2}, Concept − with − all − Objects C) = C_{2}.

(43) For every FormalContext C and for every strict FormalConcept C_{2} of C holds (B − join C)(C_{2}, Concept − with − all − Objects C) = Concept − with − all − Objects C.

(44) For every FormalContext C and for every strict FormalConcept C_{2} of C holds (B − join C)(C_{2}, Concept − with − all − Attributes C) = C_{2}.

(45) For every FormalContext C and for every strict FormalConcept C_{2} of C holds (B − meet C)(C_{2}, Concept − with − all − Attributes C) = Concept − with − all − Attributes C.

Let C be a FormalContext. The functor ConceptLatticeC yielding a strict non empty lattice structure is defined as follows:

(Def. 23) ConceptLatticeC = ⟨B − carrier C, B − join C, B − meet C⟩. The following proposition is true

introduction to concept lattices 241

(46) For every FormalContext C holds ConceptLatticeC is a lattice. Let C be a FormalContext. One can verify that ConceptLatticeC is strict

non empty and lattice-like.

Let C be a FormalContext and let S be a non empty subset of the carrier of ConceptLatticeC. We see that the element of S is a FormalConcept of C.

Let C be a FormalContext and let C_{2} be an element of the carrier of ConceptLatticeC. The functor C_{2}^{T} yielding a strict FormalConcept of C is defined as follows:

(Def. 24) C_{2}^{T} = C_{2}. One can prove the following two propositions:

(47) Let C be a FormalContext and C_{3}, C_{4} be elements of the carrier of ConceptLatticeC. Then C_{3} ⊑ C_{4} if and only if C_{3}^{T} is SubConcept of C_{4}^{T}.

(48) For every FormalContext C holds ConceptLatticeC is a complete lattice. Let C be a FormalContext. Observe that ConceptLatticeC is complete.

References

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

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

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

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

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

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

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

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

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

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

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

[12] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 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] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

242 christoph schwarzweller

A Theory of Partitions. Part I

Shunichi Kobayashi Shinshu University Nagano

Kui Jia Shinshu University Nagano

Summary. In this paper, we define join and meet operations between partitions. The properties of these operations are proved. Then we introduce the correspondence between partitions and equivalence relations which preserve join and meet operations. The properties of these relationships are proved.

MML Identifier: PARTIT1.

The notation and terminology used in this paper have been introduced in the following articles: [9], [6], [5], [2], [3], [1], [10], [4], [8], and [7].

1. Preliminaries

For simplicity, we use the following convention: Y is a non empty set, P_{1}, P_{2} are partitions of Y, A, B are subsets of Y, i is a natural number, x, y, x_{1}, x_{2}, z_{0} are sets, and X, V, d, t, S_{1}, S_{2} are sets.

The following proposition is true

(1) If X ∈ P_{1} and V ∈ P_{1} and X ⊆ V, then X = V.

Let us consider S_{1}, S_{2}. We introduce S_{1} ⋐ S_{2} and S_{2} ⋑ S_{1} as synonyms of S_{1} is finer than S_{2}.

We now state several propositions:

(2) For every partition P_{1} of Y holds P_{1} ⋑ P_{1}.

(3) ⋃(S_{1} \ {∅}) = ⋃ S_{1}.

(4) For all partitions P_{1}, P_{2} of Y such that P_{1} ⋑ P_{2} and P_{2} ⋑ P_{1} holds P_{2} ⊆ P_{1}.

(5) For all partitions P_{1}, P_{2} of Y such that P_{1} ⋑ P_{2} and P_{2} ⋑ P_{1} holds P_{1} = P_{2}.

(7)^{1} For all partitions P_{1}, P_{2} of Y such that P_{1} ⋑ P_{2} holds P_{1} is coarser than

P_{2}. Let us consider Y, let P_{1} be a partition of Y, and let b be a set. We say that

b is a dependent set of P_{1} if and only if: (Def. 1) There exists a set B such that B ⊆ P_{1} and B ≠ ∅ and b = ⋃ B.

Let us consider Y, let P_{1}, P_{2} be partitions of Y, and let b be a set. We say that b is a minimal dependent set of P_{1} and P_{2} if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) b is a dependent set of P_{1} and a dependent set of P_{2}, and

(ii) for every set d such that d ⊆ b and d is a dependent set of P_{1} and a

dependent set of P_{2} holds d = b. We now state several propositions:

(8) For all partitions P_{1}, P_{2} of Y such that P_{1} ⋑ P_{2} and for every set b such that b ∈ P_{1} holds b is a dependent set of P_{2}.

(9) For every partition P_{1} of Y holds Y is a dependent set of P_{1}.

(10) Let F be a family of subsets of Y. Suppose Intersect(F) ≠ ∅ and for every X such that X ∈ F holds X is a dependent set of P_{1}. Then Intersect(F) is a dependent set of P_{1}.

(11) Let X_{0}, X_{1} be subsets of Y. Suppose X_{0} is a dependent set of P_{1} and X_{1} is a dependent set of P_{1} and X_{0} meets X_{1}. Then X_{0} ∩ X_{1} is a dependent set of P_{1}.

(12) For every subset X of Y such that X is a dependent set of P_{1} and X ≠ Y holds X^{c} is a dependent set of P_{1}.

(13) For every element y of Y there exists a subset X of Y such that y ∈ X and X is a minimal dependent set of P_{1} and P_{2}.

(14) For every partition P of Y and for every element y of Y there exists a

subset A of Y such that y ∈ A and A ∈ P. Let Y be a non empty set. One can verify that every partition of Y is non

empty.

Let Y be a set. The functor PARTITIONS(Y) is defined by: (Def. 3) For every set x holds x ∈ PARTITIONS(Y) iff x is a partition of Y.

Let Y be a set. One can check that PARTITIONS(Y) is non empty.

2. Join and Meet Operation Between Partitions

Let us consider Y and let P_{1}, P_{2} be partitions of Y. The functor P_{1} ∧ P_{2} yielding a partition of Y is defined by:

1The proposition (6) has been removed.

(Def. 4) P_{1} ∧ P_{2} = P_{1} ⋒ P_{2} \ {∅}.

Let us observe that the functor P_{1} ∧ P_{2} is commutative.

One can prove the following propositions:

(15) For every partition P_{1} of Y holds P_{1} ∧ P_{1} = P_{1}.

(16) For all partitions P_{1}, P_{2}, P_{3} of Y holds P_{1} ∧ P_{2} ∧ P_{3} = P_{1} ∧ P_{2} ∧ P_{3}.

(17) For all partitions P_{1}, P_{2} of Y holds P_{1} ⋑ P_{1} ∧ P_{2}.

(18) For all partitions P_{1}, P_{2}, P_{3} of Y such that P_{1} ⋑ P_{2} and P_{2} ⋑ P_{3} holds

P_{1} ⋑ P_{3}. Let us consider Y and let P_{1}, P_{2} be partitions of Y. The functor P_{1} ∨ P_{2}

yielding a partition of Y is defined by: (Def. 5) For every d holds d ∈ P_{1} ∨ P_{2} iff d is a minimal dependent set of P_{1} and

P_{2}. Let us observe that the functor P_{1} ∨ P_{2} is commutative.

One can prove the following propositions:

(19) For all partitions P_{1}, P_{2} of Y holds P_{1} ⋐ P_{1} ∨ P_{2}.

(20) For every partition P_{1} of Y holds P_{1} ∨ P_{1} = P_{1}.

(21) For all partitions P_{1}, P_{3} of Y such that P_{1} ⋐ P_{3} and x ∈ P_{3} and z_{0} ∈ P_{1} and t ∈ x and t ∈ z_{0} holds z_{0} ⊆ x.

(22) For all partitions P_{1}, P_{2} of Y such that x ∈ P_{1} ∨ P_{2} and z_{0} ∈ P_{1} and t ∈ x and t ∈ z_{0} holds z_{0} ⊆ x.

3. Partitions and Equivalence Relations

We now state the proposition

(23) Let P_{1} be a partition of Y. Then there exists an equivalence relation R_{1} of Y such that for all x, y holds ⟨⟨x, y⟩⟩ ∈ R_{1} if and only if the following conditions are satisfied:

(i) x ∈ Y, (ii) y ∈ Y, and (iii) there exists A such that A ∈ P_{1} and x ∈ A and y ∈ A. Let us consider Y. The functor Rel(Y) yields a function and is defined by

the conditions (Def. 6). (Def. 6)(i) domRel(Y) = PARTITIONS(Y), and

(ii) for every x such that x ∈ PARTITIONS(Y) there exists an equivalence relation R_{1} of Y such that (Rel(Y))(x) = R_{1} and for all sets x_{1}, x_{2} holds ⟨⟨x_{1}, x_{2}⟩⟩ ∈ R_{1} iff x_{1} ∈ Y and x_{2} ∈ Y and there exists A such that A ∈ x and x_{1} ∈ A and x_{2} ∈ A.

Let Y be a non empty set and let P_{1} be a partition of Y. The functor ≡_{(P_{1})} yielding an equivalence relation of Y is defined as follows:

(Def. 7) ≡_{(P_{1})} = (Rel(Y))(P_{1}).

The following propositions are true:

(24) For all partitions P_{1}, P_{2} of Y holds P_{1} ⋐ P_{2} iff ≡_{(P_{1})} ⊆ ≡_{(P_{2})} .

(25) Let P_{1}, P_{2} be partitions of Y, p_{0}, x, y be sets, and f be a finite sequence

of elements of Y. Suppose that

(i) p_{0} ⊆ Y, (ii) x ∈ p_{0}, (iii) f(1) = x, (iv) f(lenf) = y,

(v) 1 lenf, (vi) for every i such that 1 i and i < lenf there exist sets p_{2}, p_{3}, u such

that p_{2} ∈ P_{1} and p_{3} ∈ P_{2} and f(i) ∈ p_{2} and u ∈ p_{2} and u ∈ p_{3} and f(i + 1) ∈ p_{3}, and

(vii) p_{0} is a dependent set of P_{1} and a dependent set of P_{2}.

Then y ∈ p_{0}.

(26) Let R_{2}, R_{3} be equivalence relations of Y, f be a finite sequence of ele-

ments of Y, and x, y be sets. Suppose that

(i) x ∈ Y, (ii) y ∈ Y, (iii) f(1) = x, (iv) f(lenf) = y,

(v) 1 lenf, and (vi) for every i such that 1 i and i < lenf there exists a set u such that

u ∈ Y and ⟨⟨f(i), u⟩⟩∈ R_{2} ∪ R_{3} and ⟨⟨u, f(i + 1)⟩⟩∈ R_{2} ∪ R_{3}.

Then ⟨⟨x, y⟩⟩∈ R_{2} ⊔ R_{3}.

(27) For all partitions P_{1}, P_{2} of Y holds ≡_{P_{1}∨P_{2}} = ≡_{(P_{1})} ⊔ ≡_{(P_{2})} .

(28) For all partitions P_{1}, P_{2} of Y holds ≡_{P_{1}∧P_{2}} = ≡_{(P_{1})} ∩ ≡_{(P_{2})} .

(29) For all partitions P_{1}, P_{2} of Y such that ≡_{(P_{1})} = ≡_{(P_{2})} holds P_{1} = P_{2}.

(30) For all partitions P_{1}, P_{2}, P_{3} of Y holds P_{1} ∨ P_{2} ∨ P_{3} = P_{1} ∨ P_{2} ∨ P_{3}.

(31) For all partitions P_{1}, P_{2} of Y holds P_{1} ∧ P_{1} ∨ P_{2} = P_{1}.

(32) For all partitions P_{1}, P_{2} of Y holds P_{1} ∨ P_{1} ∧ P_{2} = P_{1}.

(33) For all partitions P_{1}, P_{2}, P_{3} of Y such that P_{1} ⋐ P_{3} and P_{2} ⋐ P_{3} holds P_{1} ∨ P_{2} ⋐ P_{3}.

(34) For all partitions P_{1}, P_{2}, P_{3} of Y such that P_{1} ⋑ P_{3} and P_{2} ⋑ P_{3} holds

P_{1} ∧ P_{2} ⋑ P_{3}. Let us consider Y. The functor I(Y) yielding a partition of Y is defined as

follows: (Def. 8) I(Y) = SmallestPartition(Y).

a theory of partitions. part i 247

Let us consider Y. The functor O(Y) yielding a partition of Y is defined by: (Def. 9) O(Y) = {Y }.

The following propositions are true:

(35) I(Y) = {B : ⋁_{x: set} (B = {x} ∧ x ∈ Y)}.

(36) For every partition P_{1} of Y holds O(Y) ⋑ P_{1} and P_{1} ⋑ I(Y).

(37) ≡_{O(Y)} = ∇_{Y} .

(38) ≡_{I(Y)} = △_{Y} .

(39) I(Y) ⋐ O(Y).

(40) For every partition P_{1} of Y holds O(Y)∨P_{1} = O(Y) and O(Y)∧P_{1} = P_{1}.

(41) For every partition P_{1} of Y holds I(Y)∨P_{1} = P_{1} and I(Y)∧P_{1} = I(Y).

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

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

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

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

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

248 shunichi kobayashi and kui jia

A Theory of Boolean Valued Functions and Partitions

Shunichi Kobayashi Shinshu University Nagano

Kui Jia Shinshu University Nagano

Summary. In this paper, we define Boolean valued functions. Some of their algebraic properties are proved. We also introduce and examine the infimum and supremum of Boolean valued functions and their properties. In the last section, relations between Boolean valued functions and partitions are discussed.

MML Identifier: BVFUNC 1.

The terminology and notation used in this paper are introduced in the following papers: [4], [6], [1], [2], [3], and [5].

1. Boolean Operations

In this paper Y denotes a non empty set.

Let k, l be elements of Boolean. The functor k ⇒ l is defined by: (Def. 1) k ⇒ l = ¬k ∨ l.

The functor k ⇔ l is defined as follows: (Def. 2) k ⇔ l = ¬(k ⊕ l).

Let k, l be elements of Boolean. The predicate k ⋐ l is defined by: (Def. 3) k ⇒ l = true.

Let us note that the predicate k ⋐ l is reflexive.

One can prove the following three propositions:

(1) For all elements k, l of Boolean and for all natural numbers n_{1}, n_{2} such that k = n_{1} and l = n_{2} holds k ⋐ l iff n_{1} n_{2}.

(2) For all elements k, l of Boolean such that k ⋐ l and l ⋐ k holds k = l.

(3) For all elements k, l, m of Boolean such that k ⋐ l and l ⋐ m holds k ⋐ m.

2. Boolean Valued Functions

Let us consider Y. The functor BVF(Y) is defined by: (Def. 4) BVF(Y) = Boolean^{Y} .

Let Y be a non empty set. Observe that BVF(Y) is functional and non empty.

Let us consider Y, let a be an element of BVF(Y), and let x be an element of Y. The functor Pj(a, x) yields an element of Boolean and is defined by:

(Def. 5) Pj(a, x) = a(x).

Let us consider Y and let a, b be elements of BVF(Y). The functor a ∧ b yields an element of BVF(Y) and is defined by:

(Def. 6) For every element x of Y holds Pj(a ∧ b, x) = Pj(a, x) ∧ Pj(b, x).

Let us notice that the functor a ∧ b is commutative.

Let us consider Y and let a, b be elements of BVF(Y). The functor a ∨ b yields an element of BVF(Y) and is defined by:

(Def. 7) For every element x of Y holds Pj(a ∨ b, x) = Pj(a, x) ∨ Pj(b, x).

Let us notice that the functor a ∨ b is commutative.

Let us consider Y and let a be an element of BVF(Y). The functor ¬a yielding an element of BVF(Y) is defined as follows:

(Def. 8) For every element x of Y holds Pj(¬a, x) = ¬ Pj(a, x).

Let us consider Y and let a, b be elements of BVF(Y). The functor a ⊕ b yields an element of BVF(Y) and is defined as follows:

(Def. 9) For every element x of Y holds Pj(a ⊕ b, x) = Pj(a, x) ⊕ Pj(b, x).

Let us note that the functor a ⊕ b is commutative.

Let us consider Y and let a, b be elements of BVF(Y). The functor a ⇒ b yields an element of BVF(Y) and is defined by:

(Def. 10) For every element x of Y holds Pj(a ⇒ b, x) = ¬ Pj(a, x) ∨ Pj(b, x).

Let us consider Y and let a, b be elements of BVF(Y). The functor a ⇔ b yielding an element of BVF(Y) is defined as follows:

(Def. 11) For every element x of Y holds Pj(a ⇔ b, x) = ¬(Pj(a, x) ⊕ Pj(b, x)).

Let us observe that the functor a ⇔ b is commutative.

Let us consider Y. The functor false(Y) yielding an element of BVF(Y) is defined by:

(Def. 12) For every element x of Y holds Pj(false(Y), x) = false.

Let us consider Y. The functor true(Y) yielding an element of BVF(Y) is defined as follows:

(Def. 13) For every element x of Y holds Pj(true(Y), x) = true.

The following propositions are true:

(4) For every element a of BVF(Y) holds ¬¬a = a.

(5) For every element a of BVF(Y) holds ¬ true(Y) = false(Y) and ¬ false(Y) = true(Y).

(6) For all elements a, b of BVF(Y) holds a ∧ a = a.

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

(8) For every element a of BVF(Y) holds a ∧ false(Y) = false(Y).

(9) For every element a of BVF(Y) holds a ∧ true(Y) = a.

(10) For every element a of BVF(Y) holds a ∨ a = a.

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

(12) For every element a of BVF(Y) holds a ∨ false(Y) = a.

(13) For every element a of BVF(Y) holds a ∨ true(Y) = true(Y).

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

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

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

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

Let us consider Y and let a, b be elements of BVF(Y). The predicate a ⋐ b is defined by:

(Def. 14) For every element x of Y such that Pj(a, x) = true holds Pj(b, x) = true.

Let us note that the predicate a ⋐ b is reflexive.

The following four propositions are true:

(18) For all elements a, b, c of BVF(Y) holds if a ⋐ b and b ⋐ a, then a = b and if a ⋐ b and b ⋐ c, then a ⋐ c.

(19) For all elements a, b of BVF(Y) holds a ⇒ b = true(Y) iff a ⋐ b.

(20) For all elements a, b of BVF(Y) holds a ⇔ b = true(Y) iff a = b.

(21) For every element a of BVF(Y) holds false(Y) ⋐ a and a ⋐ true(Y).

3. Infimum and Supremum

Let us consider Y and let a be an element of BVF(Y). The functor INFa yields an element of BVF(Y) and is defined as follows:

(Def. 15) INFa = \{ \begin{matrix}true(Y), if for every element x of Y holds Pj(a, x) = true, \\ false(Y), otherwise. \end{matrix}

The functor SUPa yielding an element of BVF(Y) is defined by:

(Def. 16) SUPa = \{ \begin{matrix}false(Y), if for every element x of Y holds Pj(a, x) = false, \\ true(Y), otherwise. \end{matrix}

Next we state two propositions:

(22) For every element a of BVF(Y) holds ¬ INFa = SUP¬a and ¬ SUPa = INF¬a.

(23) INFfalse(Y) = false(Y) and INFtrue(Y) = true(Y) and

SUPfalse(Y) = false(Y) and SUPtrue(Y) = true(Y). Let us consider Y. Observe that false(Y) is constant. Let us consider Y. One can verify that true(Y) is constant. Let Y be a non empty set. Observe that there exists an element of BVF(Y)

which is constant.

We now state several propositions:

(24) For every constant element a of BVF(Y) holds a = false(Y) or a = true(Y).

(25) For every constant element d of BVF(Y) holds INFd = d and SUPd = d.

(26) For all elements a, b of BVF(Y) holds INF(a ∧ b) = INFa ∧ INFb and SUP(a ∨ b) = SUPa ∨ SUPb.

(27) For every element a of BVF(Y) and for every constant element d of BVF(Y) holds INF(d ⇒ a) = d ⇒ INFa and INF(a ⇒ d) = SUPa ⇒ d.

(28) For every element a of BVF(Y) and for every constant element d of BVF(Y) holds INF(d ∨ a) = d ∨ INFa and SUP(d ∧ a) = d ∧ SUPa and SUP(a ∧ d) = SUPa ∧ d.

(29) For every element a of BVF(Y) and for every element x of Y holds Pj(INFa, x) ⋐ Pj(a, x).

(30) For every element a of BVF(Y) and for every element x of Y holds Pj(a, x) ⋐ Pj(SUPa, x).

4. Boolean Valued Functions and Partitions

Let us consider Y, let a be an element of BVF(Y), and let P_{1} be a partition of Y. We say that a is dependent of P_{1} if and only if:

(Def. 17) For every set F such that F ∈ P_{1} and for all sets x_{1}, x_{2} such that x_{1} ∈ F

and x_{2} ∈ F holds a(x_{1}) = a(x_{2}). The following two propositions are true:

(31) For every element a of BVF(Y) holds a is dependent of I(Y).

(32) For every constant element a of BVF(Y) holds a is dependent of O(Y). Let us consider Y and let P_{1} be a partition of Y. We see that the element

of P_{1} is a subset of Y.

Let us consider Y, let x be an element of Y, and let P_{1} be a partition of Y. Then EqClass(x, P_{1}) is an element of P_{1}. We introduce Lift(x, P_{1}) as a synonym of EqClass(x, P_{1}).

Let us consider Y, let a be an element of BVF(Y), and let P_{1} be a partition of Y. The functor INF(a, P_{1}) yields an element of BVF(Y) and is defined by the condition (Def. 18).

(Def. 18) Let y be an element of Y. Then

(i) if for every element x of Y such that x ∈ EqClass(y, P_{1}) holds Pj(a, x) = true, then Pj(INF(a, P_{1}), y) = true, and

(ii) if it is not true that for every element x of Y such that x ∈

EqClass(y, P_{1}) holds Pj(a, x) = true, then Pj(INF(a, P_{1}), y) = false. Let us consider Y, let a be an element of BVF(Y), and let P_{1} be a partition

of Y. The functor SUP(a, P_{1}) yielding an element of BVF(Y) is defined by the condition (Def. 19).

(Def. 19) Let y be an element of Y. Then

(i) if there exists an element x of Y such that x ∈ EqClass(y, P_{1}) and Pj(a, x) = true, then Pj(SUP(a, P_{1}), y) = true, and

(ii) if it is not true that there exists an element x of Y such that x ∈

EqClass(y, P_{1}) and Pj(a, x) = true, then Pj(SUP(a, P_{1}), y) = false. Next we state a number of propositions:

(33) For every element a of BVF(Y) and for every partition P_{1} of Y holds INF(a, P_{1}) is dependent of P_{1}.

(34) For every element a of BVF(Y) and for every partition P_{1} of Y holds SUP(a, P_{1}) is dependent of P_{1}.

(35) For every element a of BVF(Y) and for every partition P_{1} of Y holds INF(a, P_{1}) ⋐ a.

(36) For every element a of BVF(Y) and for every partition P_{1} of Y holds a ⋐ SUP(a, P_{1}).

(37) For every element a of BVF(Y) and for every partition P_{1} of Y holds ¬ INF(a, P_{1}) = SUP(¬a, P_{1}).

(38) For every element a of BVF(Y) holds INF(a, O(Y)) = INFa.

(39) For every element a of BVF(Y) holds SUP(a, O(Y)) = SUPa.

(40) For every element a of BVF(Y) holds INF(a, I(Y)) = a.

(41) For every element a of BVF(Y) holds SUP(a, I(Y)) = a.

(42) For all elements a, b of BVF(Y) and for every partition P_{1} of Y holds INF(a ∧ b, P_{1}) = INF(a, P_{1}) ∧ INF(b, P_{1}).

(43) For all elements a, b of BVF(Y) and for every partition P_{1} of Y holds SUP(a ∨ b, P_{1}) = SUP(a, P_{1}) ∨ SUP(b, P_{1}).

254 shunichi kobayashi and kui jia

Let us consider Y and let f be an element of BVF(Y). The functor GPartf yields a partition of Y and is defined by:

(Def. 20) GPartf = {{x; x ranges over elements of Y: f(x) = true}, {x′; x′ ranges over elements of Y: f(x′) = false}} \ {∅}.

The following propositions are true:

(44) For every element a of BVF(Y) holds a is dependent of GParta.

(45) For every element a of BVF(Y) and for every partition P_{1} of Y such that a is dependent of P_{1} holds P_{1} is finer than GParta.

References

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

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

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

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

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

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

Trigonometric Functions and Existence of Circle Ratio

Yuguang Yang Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article, we defined sinus and cosine as the real part and the imaginary part of the exponential function on complex, and also give their series expression. Then we proved the differentiablity of sinus, cosine and the exponential function of real. Finally, we showed the existence of the circle ratio, and some formulas of sinus, cosine.

MML Identifier: SIN COS.

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

1. Some Definitions and Properties of Complex Sequence

For simplicity, we adopt the following rules: p, q, r, t_{1}, t_{2}, t_{3} are elements of ℝ, w, z, z_{1}, z_{2} are elements of ℂ, k, l, m, n are natural numbers, s_{1} is a complex sequence, and r_{1} is a sequence of real numbers.

Let m, k be natural numbers. Let us assume that k m. The functor PN(m, k) yielding an element of ℕ is defined by:

(Def. 1) PN(m, k) = m − k.

Let m, k be natural numbers. The functor CHK(m, k) yields an element of ℂ and is defined by:

(Def. 2) CHK(m, k) = \{ \begin{matrix}1_{ℂ}, if m k, \\ 0_{ℂ}, otherwise. \end{matrix}

The functor RHK(m, k) yields an element of ℝ and is defined as follows:

(Def. 3) RHK(m, k) = \{ \begin{matrix}1, if m k, \\ 0, otherwise. \end{matrix}

In this article we present several logical schemes. The scheme ExComplex CASE deals with a binary functor F yielding an element of ℂ, and states that:

For every k there exists s_{1} such that for every n holds if n k, then s_{1}(n) = F(k, n) and if n > k, then s_{1}(n) = 0_{ℂ}

for all values of the parameter.

The scheme ExReal CASE deals with a binary functor F yielding an element of ℝ, and states that:

For every k there exists r_{1} such that for every n holds if n k, then r_{1}(n) = F(k, n) and if n > k, then r_{1}(n) = 0

for all values of the parameter.

The complex sequence Prod complex n is defined by: (Def. 4) (Prod complex n)(0) = 1_{ℂ} and for every n holds (Prod complex n)(n +

1) = (Prod complex n)(n) · ((n + 1) + 0i). The sequence Prod real n of real numbers is defined by:

(Def. 5) (Prod real n)(0) = 1 and for every n holds (Prod real n)(n + 1) =

(Prod real n)(n) · (n + 1). Let n be a natural number. The functor n!c yields an element of ℂ and is

defined as follows: (Def. 6) n!c = (Prod complex n)(n).

Let n be a natural number. Then n! is a real number and it can be characterized by the condition:

(Def. 7) n! = (Prod real n)(n).

Let z be an element of ℂ. The functor z ExpSeq yields a complex sequence and is defined as follows:

(Def. 8) For every n holds z ExpSeq(n) = \frac{z_{ℕ}^{n}}{n!c}.

Let a be an element of ℝ. The functor a ExpSeq yielding a sequence of real numbers is defined as follows:

(Def. 9) For every n holds a ExpSeq(n) = \frac{a_{ℕ}^{n}}{n!} .

The following propositions are true:

(1) If 0 < n, then n + 0i ≠ 0_{ℂ} and 0!c = 1_{ℂ} and n!c ≠ 0_{ℂ} and n + 1!c = n!c · ((n + 1) + 0i).

(2) n! ≠ 0 and (n + 1)! = n! · (n + 1).

(3) For every k such that 0 < k holds PN(k, 1)!c·(k+0i) = k!c and for all m, k such that k m holds PN(m, k)!c·(((m+1)−k)+0i) = PN(m+1, k)!c.

Let n be a natural number. The functor Coef n yielding a complex sequence is defined by:

(Def. 10) For every natural number k holds if k n, then (Coef n)(k) =

\frac{n!c}{k!c·PN(n,k)!c} and if k > n, then (Coef n)(k) = 0_{ℂ}. Let n be a natural number. The functor Coef en yields a complex sequence

and is defined as follows: (Def. 11) For every natural number k holds if k n, then (Coef en)(k) =

\frac{1_{ℂ}}{k!c·PN(n,k)!c} and if k > n, then (Coef en)(k) = 0_{ℂ}. Let us consider s_{1}. The functor Sifts_{1} yielding a complex sequence is defined

as follows: (Def. 12) (Sifts_{1})(0) = 0_{ℂ} and for every natural number k holds (Sifts_{1})(k + 1) =

s_{1}(k). Let us consider n and let z, w be elements of ℂ. The functor Expan(n, z, w)

yields a complex sequence and is defined as follows: (Def. 13) For every natural number k holds if k n, then (Expan(n, z, w))(k) =

(Coef n)(k) · z_{ℕ}^{k} · w_{ℕ}^{PN(n,k)} and if n < k, then (Expan(n, z, w))(k) = 0_{ℂ}. Let us consider n and let z, w be elements of ℂ. The functor Expan e(n, z, w)

yielding a complex sequence is defined by: (Def. 14) For every natural number k holds if k n, then (Expan e(n, z, w))(k) =

(Coef en)(k) · z_{ℕ}^{k} · w_{ℕ}^{PN(n,k)} and if n < k, then (Expan e(n, z, w))(k) = 0_{ℂ}. Let us consider n and let z, w be elements of ℂ. The functor Alfa(n, z, w)

yielding a complex sequence is defined by: (Def. 15) For every natural number k holds if k n, then (Alfa(n, z, w))(k) =

z ExpSeq(k) · (∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(PN(n, k)) and if n < k, then (Alfa(n, z, w))(k) = 0_{ℂ}.

Let a, b be elements of ℝ and let n be a natural number. The functor Conj(n, a, b) yielding a sequence of real numbers is defined as follows:

(Def. 16) For every natural number k holds if k n, then (Conj(n, a, b))(k) = a ExpSeq(k)·((∑_{α=0}^{κ} b ExpSeq(α))_{κ∈ℕ}(n)−(∑_{α=0}^{κ} b ExpSeq(α))_{κ∈ℕ}(PN(n, k))) and if n < k, then (Conj(n, a, b))(k) = 0.

Let z, w be elements of ℂ and let n be a natural number. The functor Conj(n, z, w) yielding a complex sequence is defined by:

(Def. 17) For every natural number k holds if k n, then (Conj(n, z, w))(k) = z ExpSeq(k)·((∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n)−(∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(PN(n, k))) and if n < k, then (Conj(n, z, w))(k) = 0_{ℂ}.

The following propositions are true:

(4) z ExpSeq(n + 1) = \frac{z ExpSeq (n)·z}{(n+1)+0i} and z ExpSeq(0) = 1_{ℂ} and |z ExpSeq(n)| = |z| ExpSeq(n).

(5) If 0 < k, then (Sifts_{1})(k) = s_{1}(PN(k, 1)).

(6) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Sifts_{1})(α))_{κ∈ℕ}(k) + s_{1}(k).

(7) (z + w)_{ℕ}^{n} = (∑_{α=0}^{κ}(Expan(n, z, w))(α))_{κ∈ℕ}(n).

(8) Expan e(n, z, w) = \frac{1_{ℂ}}{n!c} Expan(n, z, w).

(9) \frac{(z+w)_{ℕ}^{n}}{n!c} = (∑_{α=0}^{κ}(Expan e(n, z, w))(α))_{κ∈ℕ}(n).

(10) 0_{ℂ} ExpSeq is absolutely summable and ∑(0_{ℂ} ExpSeq) = 1_{ℂ}. Let us consider z. One can verify that z ExpSeq is absolutely summable. Next we state a number of propositions:

(11) z ExpSeq(0) = 1_{ℂ} and (Expan(0, z, w))(0) = 1_{ℂ}.

(12) If l k, then (Alfa(k + 1, z, w))(l) = (Alfa(k, z, w))(l) + (Expan e(k + 1, z, w))(l).

(13) (∑_{α=0}^{κ}(Alfa(k + 1, z, w))(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Alfa(k, z, w))(α))_{κ∈ℕ}(k) + (∑_{α=0}^{κ}(Expan e(k + 1, z, w))(α))_{κ∈ℕ}(k).

(14) z ExpSeq(k) = (Expan e(k, z, w))(k).

(15) (∑_{α=0}^{κ} z + w ExpSeq(α))_{κ∈ℕ}(n) = (∑_{α=0}^{κ}(Alfa(n, z, w))(α))_{κ∈ℕ}(n).

(16) (∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k) · (∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(k) − (∑_{α=0}^{κ} z + w ExpSeq(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Conj(k, z, w))(α))_{κ∈ℕ}(k).

(17) |(∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k)| (∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(k) and (∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(k) ∑(|z| ExpSeq) and |(∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k)| ∑(|z| ExpSeq).

(18) 1 ∑(|z| ExpSeq).

(19) 0 |z| ExpSeq(n).

(20) |(∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(n) and if n m, then |(∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ} |z| ExpSeq(α))_{κ∈ℕ}(n).

(21) |(∑_{α=0}^{κ} | Conj(k, z, w)|(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ} | Conj(k, z, w)|(α))_{κ∈ℕ}(n).

(22) For every p such that p > 0 there exists n such that for every k such that n k holds |(∑_{α=0}^{κ} | Conj(k, z, w)|(α))_{κ∈ℕ}(k)| < p.

(23) For every s_{1} such that for every k holds s_{1}(k) = (∑_{α=0}^{κ}(Conj(k, z, w))(α))_{κ∈ℕ}(k) holds s_{1} is convergent and lims_{1} = 0_{ℂ}.

2. Definition of Exponential Function on Complex

The partial function exp from ℂ to ℂ is defined as follows: (Def. 18) domexp = ℂ and for every element z of ℂ holds (exp)(z) =

∑(z ExpSeq). Let us consider z. The functor expz yielding an element of ℂ is defined by:

(Def. 19) expz = (exp)(z).

The following proposition is true

(24) For all z_{1}, z_{2} holds expz_{1} + z_{2} = expz_{1} · expz_{2}.

3. Definition of Sinus, Cosine, and Exponential Function on ℝ

The partial function sin from ℝ to ℝ is defined as follows: (Def. 20) domsin = ℝ and for every real number d holds (sin)(d) = ℑ(∑(0 +

di ExpSeq)). Let us consider t_{1}. The functor sint_{1} yielding an element of ℝ is defined by:

(Def. 21) sint_{1} = (sin)(t_{1}).

Next we state the proposition

(25) sin is a function from ℝ into ℝ.

The partial function cos from ℝ to ℝ is defined by: (Def. 22) domcos = ℝ and for every real number d holds (cos)(d) = ℜ(∑(0 +

di ExpSeq)). Let us consider t_{1}. The functor cost_{1} yields an element of ℝ and is defined

by: (Def. 23) cost_{1} = (cos)(t_{1}).

One can prove the following propositions:

(26) cos is a function from ℝ into ℝ.

(27) domsin = ℝ and domcos = ℝ.

(28) exp0 + t_{1}i = cost_{1} + sint_{1}i.

(29) (exp0 + t_{1}i)^{∗} = exp− (0 + t_{1}i).

(30) | exp0 + t_{1}i| = 1 and | sint_{1}| 1 and | cost_{1}| 1.

(31) (cos)(t_{1})^{2}+(sin)(t_{1})^{2} = 1 and (cos)(t_{1})·(cos)(t_{1})+(sin)(t_{1})·(sin)(t_{1}) = 1.

(32) (cost_{1})^{2} + (sint_{1})^{2} = 1 and cost_{1} · cost_{1} + sint_{1} · sint_{1} = 1.

(33) (cos)(0) = 1 and (sin)(0) = 0 and (cos)(− t_{1}) = (cos)(t_{1}) and (sin)(− t_{1}) = − (sin)(t_{1}).

(34) cos0 = 1 and sin0 = 0 and cos− t_{1} = cost_{1} and sin− t_{1} = − sint_{1}. Let t_{1} be an element of ℝ. The functor t_{1} P sin yielding a sequence of real

numbers is defined by: (Def. 24) For every n holds t_{1} P sin(n) = \frac{((− 1)_{ℕ}^{n})·t_{1}_{ℕ}^{2·n+1}}{(2·n+1)!}.

Let t_{1} be an element of ℝ. The functor t_{1} P cos yielding a sequence of real numbers is defined by:

(Def. 25) For every n holds t_{1} P cos(n) = \frac{((− 1)_{ℕ}^{n})·t_{1}_{ℕ}^{2·n}}{(2·n)!}.

The following propositions are true:

(35) For all z, k holds z_{ℕ}^{2·k}= (z_{ℕ}^{k})_{ℕ}^{2} and z_{ℕ}^{2·k}= (z_{ℕ}^{2})_{ℕ}^{k}.

(36) For all k, t_{1} holds (0 + t_{1}i)_{ℕ}^{2·k}= ((− 1)_{ℕ}^{k}) · t_{1}_{ℕ}^{2·k}+ 0i and (0 + t_{1}i)_{ℕ}^{2·k+1}= 0 + (((− 1)_{ℕ}^{k}) · t_{1}_{ℕ}^{2·k+1})i.

(37) For every n holds n!c = n! + 0i.

(38) For all t_{1}, n holds (∑_{α=0}^{κ} t_{1} P sin(α))_{κ∈ℕ}(n) = (∑_{α=0}^{κ} ℑ(0 + t_{1}i ExpSeq)(α))_{κ∈ℕ}(2·n+1) and (∑_{α=0}^{κ} t_{1} P cos(α))_{κ∈ℕ}(n) = (∑_{α=0}^{κ} ℜ(0+ t_{1}i ExpSeq)(α))_{κ∈ℕ}(2 · n).

(39) For every t_{1} holds (∑_{α=0}^{κ} t_{1} P sin(α))_{κ∈ℕ} is convergent and ∑(t_{1} P sin) = ℑ(∑(0 + t_{1}i ExpSeq)) and (∑_{α=0}^{κ} t_{1} P cos(α))_{κ∈ℕ} is convergent and ∑(t_{1} P cos) = ℜ(∑(0 + t_{1}i ExpSeq)).

(40) For every t_{1} holds (cos)(t_{1}) = ∑(t_{1} P cos) and (sin)(t_{1}) = ∑(t_{1} P sin).

(41) For all p, t_{1}, r_{1} such that r_{1} is convergent and limr_{1} = t_{1} and for every n holds r_{1}(n) p holds t_{1} p.

(42) For all n, k, m such that n < k holds m! > 0 and n! k!.

(43) For all t_{1}, n, k such that 0 t_{1} and t_{1} 1 and n k holds t_{1}_{ℕ}^{k} t_{1}_{ℕ}^{n}.

(44) For all t_{1}, n holds (t_{1} + 0i)_{ℕ}^{n} = (t_{1}_{ℕ}^{n}) + 0i.

(45) For all t_{1}, n holds \frac{(t_{1}+0i)_{ℕ}^{n}}{n!c} = \frac{t_{1}_{ℕ}^{n}}{n!} + 0i.

(46) ℑ(∑(p + 0i ExpSeq)) = 0.

(47) (cos)(1) > 0 and (sin)(1) > 0 and (cos)(1) < (sin)(1).

(48) For every t_{1} holds t_{1} ExpSeq = ℜ(t_{1} + 0i ExpSeq).

(49) For every t_{1} holds t_{1} ExpSeq is summable and ∑(t_{1} ExpSeq) = ℜ(∑(t_{1}+ 0i ExpSeq)).

(50) For all p, q holds ∑(p + q ExpSeq) = ∑(p ExpSeq) · ∑(q ExpSeq).

The partial function exp from ℝ to ℝ is defined by: (Def. 26) domexp = ℝ and for every real number d holds (exp)(d) =

∑(d ExpSeq). Let us consider t_{1}. The functor expt_{1} yields an element of ℝ and is defined

as follows: (Def. 27) expt_{1} = (exp)(t_{1}).

We now state a number of propositions:

(51) domexp = ℝ.

(52) For every element d of ℝ holds (exp)(d) = ∑(d ExpSeq).

(53) For every t_{1} holds (exp)(t_{1}) = ℜ(∑(t_{1} + 0i ExpSeq)).

(54) expt_{1} + 0i = expt_{1} + 0i.

(55) expp + q = expp · expq.

(56) exp0 = 1.

(57) For every t_{1} such that t_{1} > 0 holds (exp)(t_{1}) 1.

(58) For every t_{1} such that t_{1} < 0 holds 0 < (exp)(t_{1}) and (exp)(t_{1}) 1.

(59) For every t_{1} holds (exp)(t_{1}) > 0.

(60) For every t_{1} holds expt_{1} > 0.

4. Differential of Sinus, Cosine, and Exponential Function

Let z be an element of ℂ. The functor z P dt yields a complex sequence and is defined as follows:

(Def. 28) For every n holds z P dt(n) = \frac{z_{ℕ}^{n+1}}{n+2!c}.

Let z be an element of ℂ. The functor z P t yielding a complex sequence is defined by:

(Def. 29) For every n holds z P t(n) = \frac{z_{ℕ}^{n}}{n+2!c}.

Next we state a number of propositions:

(61) For every z holds z P dt is absolutely summable.

(62) For every z holds z · ∑(z P dt) = ∑(z ExpSeq) − 1_{ℂ} − z.

(63) For every p such that p > 0 there exists r such that r > 0 and for every z such that |z| < r holds | ∑(z P dt)| < p.

(64) For all z, z_{1} holds ∑(z_{1} + z ExpSeq) − ∑(z_{1} ExpSeq) = ∑(z_{1} ExpSeq) · z + z · ∑(z P dt) · ∑(z_{1} ExpSeq).

(65) For all p, q holds (cos)(p + q) − (cos)(p) = − q · (sin)(p)− q · ℑ(∑(0 + qi P dt) · ((cos)(p) + (sin)(p)i)).

(66) For all p, q holds (sin)(p + q) − (sin)(p) = q · (cos)(p) + q · ℜ(∑(0 + qi P dt) · ((cos)(p) + (sin)(p)i)).

(67) For all p, q holds (exp)(p + q) − (exp)(p) = q · (exp)(p) + q · (exp)(p) · ℜ(∑(q + 0i P dt)).

(68) For every p holds cos is differentiable in p and (cos)′(p) = − (sin)(p).

(69) For every p holds sin is differentiable in p and (sin)′(p) = (cos)(p).

(70) For every p holds exp is differentiable in p and (exp)′(p) = (exp)(p).

(71) exp is differentiable on ℝ and for every t_{1} such that t_{1} ∈ ℝ holds (exp)′(t_{1}) = (exp)(t_{1}).

(72) cos is differentiable on ℝ and for every t_{1} such that t_{1} ∈ ℝ holds (cos)′(t_{1}) = − (sin)(t_{1}).

(73) sin is differentiable on ℝ and for every t_{1} holds (sin)′(t_{1}) = (cos)(t_{1}).

(74) For every t_{1} such that t_{1} ∈ [0, 1] holds 0 < (cos)(t_{1}) and (cos)(t_{1}) \frac{1}{2}.

(75) [0, 1] ⊆ dom(\frac{sin}{cos}) and ]0, 1[ ⊆ dom(\frac{sin}{cos}).

(76) \frac{sin}{cos} is continuous on [0, 1].

(77) For all t_{2}, t_{3} such that t_{2} ∈ ]0, 1[ and t_{3} ∈ ]0, 1[ and (\frac{sin}{cos})(t_{2}) = (\frac{sin}{cos})(t_{3}) holds t_{2} = t_{3}.

5. Existence of Circle Ratio

The element Pai of ℝ is defined as follows:

(Def. 30) (\frac{sin}{cos})(\frac{Pai}{4} ) = 1 and Pai ∈ ]0, 4[.

We now state the proposition

(78) (sin)(\frac{Pai}{4} ) = (cos)(\frac{Pai}{4} ).

6. Formulas of Sinus, Cosine

Next we state several propositions:

(79) (sin)(t_{2}+t_{3}) = (sin)(t_{2})·(cos)(t_{3})+(cos)(t_{2})·(sin)(t_{3}) and (cos)(t_{2}+t_{3}) = (cos)(t_{2}) · (cos)(t_{3}) − (sin)(t_{2}) · (sin)(t_{3}).

(80) sint_{2} + t_{3} = sint_{2} · cost_{3} + cost_{2} · sint_{3} and cost_{2} + t_{3} = cost_{2} · cost_{3} − sint_{2} · sint_{3}.

(81) (cos)(\frac{Pai}{2} ) = 0 and (sin)(\frac{Pai}{2} ) = 1 and (cos)(Pai) = − 1 and (sin)(Pai) = 0 and (cos)(Pai+\frac{Pai}{2} ) = 0 and (sin)(Pai+\frac{Pai}{2} ) = − 1 and (cos)(2 · Pai) = 1 and (sin)(2 · Pai) = 0.

(82) cos \frac{Pai}{2} = 0 and sin \frac{Pai}{2} = 1 and cosPai = − 1 and sinPai = 0 and cosPai+\frac{Pai}{2} = 0 and sinPai+\frac{Pai}{2} = − 1 and cos2·Pai = 1 and sin2·Pai = 0.

(83)(i) (sin)(t_{1} + 2 · Pai) = (sin)(t_{1}), (ii) (cos)(t_{1} + 2 · Pai) = (cos)(t_{1}), (iii) (sin)(\frac{Pai}{2} − t_{1}) = (cos)(t_{1}), (iv) (cos)(\frac{Pai}{2} − t_{1}) = (sin)(t_{1}),

(v) (sin)(\frac{Pai}{2} + t_{1}) = (cos)(t_{1}), (vi) (cos)(\frac{Pai}{2} + t_{1}) = − (sin)(t_{1}), (vii) (sin)(Pai+t_{1}) = − (sin)(t_{1}), and (viii) (cos)(Pai+t_{1}) = − (cos)(t_{1}).

(84) sint_{1} + 2 · Pai = sint_{1} and cost_{1} + 2 · Pai = cost_{1} and sin \frac{Pai}{2} − t_{1} = cost_{1} and cos \frac{Pai}{2} − t_{1} = sint_{1} and sin \frac{Pai}{2} + t_{1} = cost_{1} and cos \frac{Pai}{2} + t_{1} = − sint_{1} and sinPai+t_{1} = − sint_{1} and cosPai+t_{1} = − cost_{1}.

(85) For every t_{1} such that t_{1} ∈ ]0, \frac{Pai}{2} [ holds (cos)(t_{1}) > 0.

(86) For every t_{1} such that t_{1} ∈ ]0, \frac{Pai}{2} [ holds cost_{1} > 0. trigonometric functions and existence of … 263

References

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264 yuguang yang and yasunari shidama

Some Properties of Special Polygonal Curves

Andrzej Trybulec University of Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. In the paper some auxiliary theorems are proved, needed in the proof of the second part of the Jordan curve theorem for special polygons. They deal mostly with characteristic points of plane non empty compacts introduced in [5], operation mid introduced in [19] and the predicate “f is in the area of g” (f and g : finite sequences of points of the plane) introduced in [28].

MML Identifier: SPRECT 3.

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

1. Preliminaries

In this paper i, j, k, n are natural numbers. The following propositions are true:

(1) For all sets A, B, C such that A misses B holds A ∩ (B ∪ C) = A ∩ C.

(2) For all sets A, B, C, p such that A ⊆ B and B ∩ C = {p} and p ∈ A holds A ∩ C = {p}.

(3) For all real numbers q, r, s, t such that t 0 and t 1 and s = (1 − t) · q + t · r and q s and r < s holds t = 0.

(4) For all real numbers q, r, s, t such that t 0 and t 1 and s = (1 − t) · q + t · r and q s and r > s holds t = 0.

(5) If i − ′k j, then i j + k.

(6) If i j + k, then i − ′k j.

(7) If i j − ′k and k j, then i + k j.

(8) If j + k i, then k i − ′j.

(9) If k i and i < j, then i − ′k < j − ′k.

(10) If i < j and k < j, then i − ′k < j − ′k.

(11) Let D be a non empty set, f be a non empty finite sequence of elements of D, and g be a finite sequence of elements of D. Then π_{len(g^{⌢}f)}(g^{⌢} f) = π_{len f}f.

(12) For all sets a, b, c, d holds the indices of ( \begin{matrix}a & b \\ c & d \end{matrix} ) = \{⟨⟨1, 1⟩⟩, ⟨⟨1, 2⟩⟩, ⟨⟨2, 1⟩⟩, ⟨⟨2, 2⟩⟩}.

2. Euclidean Space

We now state four propositions:

(13) For all points p, q of E_{T}^{n} and for every real number r such that 0 < r and p = (1 − r) · p + r · q holds p = q.

(14) For all points p, q of E_{T}^{n} and for every real number r such that r < 1 and p = (1 − r) · q + r · p holds p = q.

(15) For all points p, q of E_{T}^{n} such that p = \frac{1}{2} · (p + q) holds p = q.

(16) For all points p, q, r of E_{T}^{n} such that q ∈ L(p, r) and r ∈ L(p, q) holds q = r.

3. Euclidean Plane

One can prove the following propositions:

(17) Let A be a non empty subset of E_{T}^{2}, p be an element of the carrier of E^{2}, and r be a real number. If A = Ball(p, r), then A is connected.

(18) For all subsets A, B of E_{T}^{2} such that A is open and B is a component of A holds B is open.

(19) For all points p, q, r of E_{T}^{2} such that L(p, q) is horizontal and r ∈ L(p, q) holds p_{2} = r_{2}.

(20) For all points p, q, r of E_{T}^{2} such that L(p, q) is vertical and r ∈ L(p, q) holds p_{1} = r_{1}.

(21) For all points p, q, r, s of E_{T}^{2} such that L(p, q) is horizontal and L(r, s) is horizontal and L(p, q) meets L(r, s) holds p_{2} = r_{2}.

(22) For all points p, q, r of E_{T}^{2} such that L(p, q) is vertical and L(q, r) is horizontal holds L(p, q) ∩ L(q, r) = {q}.

(23) For all points p, q, r, s of E_{T}^{2} such that L(p, q) is horizontal and L(s, r) is vertical and r ∈ L(p, q) holds L(p, q) ∩ L(s, r) = {r}.

4. Miscellaneous

In the sequel p, q denote points of E_{T}^{2} and G denotes a Go-board. Next we state two propositions:

(24) If 1 j and j k and k widthG and 1 i and i lenG, then (G_{i,j})_{2} (G_{i,k})_{2}.

(25) If 1 j and j widthG and 1 i and i k and k lenG, then

(G_{i,j})_{1} (G_{k,j})_{1}. In the sequel C denotes a subset of E_{T}^{2}. We now state a number of propositions:

(26) L(NW-cornerC, NE-cornerC) ⊆ L̃(SpStSeqC).

(27) N-mostC ⊆ L(NW-cornerC, NE-cornerC).

(28) For every non empty compact subset C of E_{T}^{2} holds N-minC ∈ L(NW-cornerC, NE-cornerC).

(29) L(NW-cornerC, NE-cornerC) is horizontal.

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

(31) Let g be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. Suppose π_{1}g ≠ p and (π_{1}g)_{1} = p_{1} or (π_{1}g)_{2} = p_{2} and g is a special sequence and L(p, π_{1}g) ∩ L̃(g) = {π_{1}g}. Then ⟨p⟩^{⌢} g is a special sequence.

(32) Let g be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. Suppose π_{len g}g ≠ p and (π_{len g}g)_{1} = p_{1} or (π_{len g}g)_{2} = p_{2} and g is a special sequence and L(p, π_{len g}g) ∩ L̃(g) = {π_{len g}g}. Then g ^{⌢} ⟨p⟩ is a special sequence.

(33) Let f be a S-sequence in ℝ^{2} and p be a point of E_{T}^{2}. If 1 < j and j lenf and p ∈ L̃(mid(f, 1, j)), then LE p, π_{j}f, L̃(f), π_{1}f, π_{len f}f.

(34) For every finite sequence h of elements of E_{T}^{2} such that i ∈ domh and j ∈ domh holds L̃(mid(h, i, j)) ⊆ L̃(h).

(35) If 1 i and i < j, then for every finite sequence f of elements of E_{T}^{2} such that j lenf holds L̃(mid(f, i, j)) = L(f, i) ∪ L̃(mid(f, i + 1, j)).

(36) Let f be a finite sequence of elements of E_{T}^{2}. If 1 i, then if i < j and j lenf, then L̃(mid(f, i, j)) = L̃(mid(f, i, j − ′1)) ∪ L(f, j − ′1).

(37) Let g be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. Suppose g is a special sequence and p_{1} = (π_{1}g)_{1} or p_{2} = (π_{1}g)_{2} and L(p, π_{1}g) ∩ L̃(g) = {π_{1}g} and p ≠ π_{1}g. Then ⟨p⟩^{⌢} g is a special sequence.

(38) Let f, g be finite sequences of elements of E_{T}^{2}. Suppose that (i) f is a special sequence, (ii) g is a special sequence, (iii) (π_{len f}f)_{1} = (π_{1}g)_{1} or (π_{len f}f)_{2} = (π_{1}g)_{2}, (iv) L̃(f) misses L̃(g),

(v) L(π_{len f}f, π_{1}g) ∩ L̃(f) = {π_{len f}f}, and (vi) L(π_{len f}f, π_{1}g) ∩ L̃(g) = {π_{1}g}.

Then f ^{⌢} g is a special sequence.

(39) For every S-sequence f in ℝ^{2} and for every point p of E_{T}^{2} such that p ∈ L̃(f) holds π_{1} ⇂ f, p = π_{1}f.

(40) Let f be a S-sequence in ℝ^{2} and p, q be points of E_{T}^{2}. If 1 j and j < lenf and p ∈ L(f, j) and q ∈ L(π_{j}f, p), then LE q, p, L̃(f), π_{1}f, π_{len f}f.

5. Special Circular Sequences

Next we state the proposition

(41) For every non constant standard special circular sequence f holds

LeftComp(f) is open and RightComp(f) is open. Let f be a non constant standard special circular sequence. One can verify

the following observations:

∗ L̃(f) is non vertical and non horizontal,

∗ LeftComp(f) is region, and

∗ RightComp(f) is region. One can prove the following propositions:

(42) For every non constant standard special circular sequence f holds RightComp(f) misses L̃(f).

(43) For every non constant standard special circular sequence f holds LeftComp(f) misses L̃(f).

(44) For every non constant standard special circular sequence f holds i_{WN} f < i_{EN} f.

(45) Let f be a non constant standard special circular sequence. Then there exists i such that 1 i and i < lenthe Go-board of f and N-min L̃(f) = (the Go-board of f)_{i,width the Go-board of f}.

(46) Let f be a clockwise oriented non constant standard special circular sequence. Suppose i ∈ domthe Go-board of f and π_{1}f = (the Goboard of f)_{i,width the Go-board of f} and π_{1}f = N-min L̃(f). Then π_{2}f = (the Go-board of f)_{i+1,width the Go-board of f} and π_{len f− ′1}f = (the Go-board of f)_{i,width the Go-board of f− ′1}.

(47) Let f be a non constant standard special circular sequence. If 1 i and i < j and j lenf and π_{1}f ∈ L̃(mid(f, i, j)), then i = 1 or j = lenf.

(48) Let f be a clockwise oriented non constant standard special circular sequence. If π_{1}f = N-min L̃(f), then L(π_{1}f, π_{2}f) ⊆ L̃(SpStSeq L̃(f)).

6. Rectangular Sequences

We now state the proposition

(49) Let f be a rectangular finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f), then p_{1} = W-bound L̃(f) or p_{1} = E-bound L̃(f) or p_{2} = S-bound L̃(f) or p_{2} = N-bound L̃(f).

One can check that there exists a special circular sequence which is rectangular.

The following propositions are true:

(50) Let f be a rectangular special circular sequence and g be a S-sequence in ℝ^{2} . If π_{1}g ∈ LeftComp(f) and π_{len g}g ∈ RightComp(f), then L̃(f) meets L̃(g).

(51) For every rectangular special circular sequence f holds SpStSeq L̃(f) = f.

(52) Let f be a rectangular special circular sequence. Then L̃(f) = {p; p ranges over points of E_{T}^{2}: p_{1} = W-bound L̃(f) ∧ p_{2} N-bound L̃(f) ∧ p_{2} S-bound L̃(f) ∨ p_{1} E-bound L̃(f) ∧ p_{1} W-bound L̃(f) ∧ p_{2} = N-bound L̃(f) ∨ p_{1} E-bound L̃(f) ∧ p_{1} W-bound L̃(f) ∧ p_{2} = S-bound L̃(f) ∨ p_{1} = E-bound L̃(f) ∧ p_{2} N-bound L̃(f) ∧ p_{2} S-bound L̃(f)}.

(53) For every rectangular special circular sequence f holds the Go-board of

f = ( \begin{matrix}π_{4}f & π_{1}f \\ π_{3}f & π_{2}f \end{matrix} ) .

(54) Let f be a rectangular special circular sequence. Then LeftComp(f) = {p : W-bound L̃(f) p̸_{1} ∨ p_{1} E̸-bound L̃(f) ∨ S-bound L̃(f) p_{2} ∨ p_{2} N̸-bound L̃(f)} and RightComp(f) = {q : W-bound L̃(f) < q_{1} ∧ q_{1} < E-bound L̃(f) ∧ S-bound L̃(f) < q_{2} ∧ q_{2} < N-bound L̃(f)}.

One can check that there exists a rectangular special circular sequence which is clockwise oriented.

One can check that every rectangular special circular sequence is clockwise oriented.

Next we state four propositions:

(55) Let f be a rectangular special circular sequence and g be a S-sequence in ℝ^{2} . If π_{1}g ∈ LeftComp(f) and π_{len g}g ∈ RightComp(f), then LPoint(L̃(g), π_{1}g, π_{len g}g, L̃(f)) ≠ NW-corner L̃(f).

(56) Let f be a rectangular special circular sequence and g be a S-sequence in ℝ^{2} . If π_{1}g ∈ LeftComp(f) and π_{len g}g ∈ RightComp(f), then LPoint(L̃(g), π_{1}g, π_{len g}g, L̃(f)) ≠ SE-corner L̃(f).

(57) Let f be a rectangular special circular sequence and p be a point of E_{T}^{2}. If W-bound L̃(f) > p_{1} or p_{1} > E-bound L̃(f) or S-bound L̃(f) > p_{2} or p_{2} > N-bound L̃(f), then p ∈ LeftComp(f).

(58) For every clockwise oriented non constant standard special circular sequence f such that π_{1}f = N-min L̃(f) holds LeftComp(SpStSeq L̃(f)) ⊆ LeftComp(f).

7. In the Area

Next we state a number of propositions:

(59) Let f be a finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. Then ⟨p, q⟩ is in the area of f if and only if ⟨p⟩ is in the area of f and ⟨q⟩ is in the area of f.

(60) Let f be a rectangular finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. Suppose ⟨p⟩ is in the area of f but p_{1} = W-bound L̃(f) or p_{1} = E-bound L̃(f) or p_{2} = S-bound L̃(f) or p_{2} = N-bound L̃(f). Then p ∈ L̃(f).

(61) Let f be a finite sequence of elements of E_{T}^{2}, p, q be points of E_{T}^{2}, and r be a real number. Suppose 0 r and r 1 and ⟨p, q⟩ is in the area of f. Then ⟨(1 − r) · p + r · q⟩ is in the area of f.

(62) Let f, g be finite sequences of elements of E_{T}^{2}. If g is in the area of f and i ∈ domg, then ⟨π_{i}g⟩ is in the area of f.

(63) Let f, g be finite sequences of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If g is in the area of f and p ∈ L̃(g), then ⟨p⟩ is in the area of f.

(64) Let f be a rectangular finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. If q ∉ L̃(f) and ⟨p, q⟩ is in the area of f, then L(p, q) ∩ L̃(f) ⊆ {p}.

(65) Let f be a rectangular finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. If p ∈ L̃(f) and q ∉ L̃(f) and ⟨q⟩ is in the area of f, then L(p, q) ∩ L̃(f) = {p}.

some properties of special polygonal curves 271

(66) Let f be a non constant standard special circular sequence. Suppose 1 i and i lenthe Go-board of f and 1 j and j widththe Go-board of f. Then ⟨(the Go-board of f)_{i,j}⟩ is in the area of f.

(67) Let g be a finite sequence of elements of E_{T}^{2} and p, q be points of E_{T}^{2}. If ⟨p, q⟩ is in the area of g, then ⟨\frac{1}{2} · (p + q)⟩ is in the area of g.

(68) For all finite sequences f, g of elements of E_{T}^{2} such that g is in the area of f holds Rev(g) is in the area of f.

(69) Let f, g be finite sequences of elements of E_{T}^{2} and p be a point of E_{T}^{2}.

Suppose that

(i) g is in the area of f, (ii) ⟨p⟩ is in the area of f, (iii) g is a special sequence, and (iv) there exists a natural number i such that 1 i and i + 1 leng and

p ∈ L(g, i).

Then ⇂ g, p is in the area of f.

(70) Let f be a non constant standard special circular sequence and g be a finite sequence of elements of E_{T}^{2}. Then g is in the area of f if and only if g is in the area of SpStSeq L̃(f).

(71) Let f be a rectangular special circular sequence and g be a S-sequence in ℝ^{2} . If π_{1}g ∈ LeftComp(f) and π_{len g}g ∈ RightComp(f), then ⇃ LPoint(L̃(g), π_{1}g, π_{len g}g, L̃(f)), g is in the area of f.

(72) Let f be a non constant standard special circular sequence. Suppose 1 i and i < lenthe Go-board of f and 1 j and j < widththe Go-board of f. Then Intcell(the Go-board of f, i, j) misses L̃(SpStSeq L̃(f)).

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[9] Adam Grabowski and Yatsuka Nakamura. The ordering of points on a curve. Part II. Formalized Mathematics, 6(4):467–473, 1997.

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

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

272 andrzej trybulec and yatsuka nakamura

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

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

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

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

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

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

[18] Yatsuka Nakamura and Adam Grabowski. Bounding boxes for special sequences in E^{2}. Formalized Mathematics, 7(1):115–121, 1998.

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

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

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

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

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

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

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

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

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

[28] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

[29] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Formalized Mathematics, 6(4):531–539, 1997.

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

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

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

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

Real Linear-Metric Space and Isometric Functions

Robert Milewski University of Białystok

MML Identifier: VECTMETR.

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

1. Convex and Internal Metric Spaces

Let V be a non empty metric structure. We say that V is convex if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let x, y be elements of the carrier of V and r be a real number. Suppose 0 r and r 1. Then there exists an element z of the carrier of V such that ρ(x, z) = r · ρ(x, y) and ρ(z, y) = (1 − r) · ρ(x, y).

Let V be a non empty metric structure. We say that V is internal if and only if the condition (Def. 2) is satisfied.

(Def. 2) Let x, y be elements of the carrier of V and p, q be real numbers. Suppose p > 0 and q > 0. Then there exists a finite sequence f of elements of the carrier of V such that

(i) π_{1}f = x, (ii) π_{len f}f = y, (iii) for every natural number i such that 1 i and i lenf − 1 holds

ρ(π_{i}f, π_{i+1}f) < p, and (iv) for every finite sequence F of elements of ℝ such that lenF = lenf − 1

and for every natural number i such that 1 i and i lenF holds π_{i}F = ρ(π_{i}f, π_{i+1}f) holds |ρ(x, y) − ∑ F | < q.

One can prove the following proposition (1) Let V be a non empty metric space. Suppose V is convex. Let x, y be

elements of the carrier of V and p be a real number. Suppose p > 0. Then there exists a finite sequence f of elements of the carrier of V such that

(i) π_{1}f = x, (ii) π_{len f}f = y, (iii) for every natural number i such that 1 i and i lenf − 1 holds

ρ(π_{i}f, π_{i+1}f) < p, and (iv) for every finite sequence F of elements of ℝ such that lenF = lenf − 1

and for every natural number i such that 1 i and i lenF holds π_{i}F = ρ(π_{i}f, π_{i+1}f) holds ρ(x, y) = ∑ F.

Let us observe that every non empty metric space which is convex is also internal.

One can verify that there exists a non empty metric space which is convex. A Geometry is a Reflexive discernible symmetric triangle internal non empty

metric structure.

2. Isometric Functions

Let V be a non empty metric structure and let f be a map from V into V. We say that f is isometric if and only if:

(Def. 3) rngf = the carrier of V and for all elements x, y of the carrier of V

holds ρ(x, y) = ρ(f(x), f(y)). Let V be a non empty metric structure. The functor ISOMV yields a set

and is defined as follows: (Def. 4) For every set x holds x ∈ ISOMV iff there exists a map f from V into

V such that f = x and f is isometric. Let V be a non empty metric structure. Then ISOMV is a subset of (the

carrier of V)^{the carrier of V}.

One can prove the following proposition

(2) Let V be a discernible Reflexive non empty metric structure and f be a map from V into V. If f is isometric, then f is one-to-one.

Let V be a discernible Reflexive non empty metric structure. One can check that every map from V into V which is isometric is also one-to-one.

Let V be a non empty metric structure. Observe that there exists a map from V into V which is isometric.

The following three propositions are true:

(3) Let V be a discernible Reflexive non empty metric structure and f be an isometric map from V into V. Then f^{−1} is isometric.

(4) For every non empty metric structure V and for all isometric maps f, g from V into V holds f · g is isometric.

(5) For every non empty metric structure V holds id_{V} is isometric.

Let V be a non empty metric structure. Note that ISOMV is non empty.

3. Real Linear-Metric Spaces

We introduce RLSMetrStruct which are extensions of RLS structure and metric structure and are systems

⟨ a carrier, a distance, a zero, an addition, an external multiplication ⟩, where the carrier is a set, the distance is a function from [: the carrier, the carrier :] into ℝ, the zero is an element of the carrier, the addition is a binary operation on the carrier, and the external multiplication is a function from [: ℝ, the carrier :] into the carrier.

One can verify that there exists a RLSMetrStruct which is non empty and strict.

Let X be a non empty set, let F be a function from [: X, X:] into ℝ, let O be an element of X, let B be a binary operation on X, and let G be a function from [: ℝ, X:] into X. One can verify that ⟨X, F, O, B, G⟩ is non empty.

Let V be a non empty RLSMetrStruct. We say that V is homogeneous if and only if:

(Def. 5) For every real number r and for all elements v, w of the carrier of V

holds ρ(r · v, r · w) = |r| · ρ(v, w). Let V be a non empty RLSMetrStruct. We say that V is translatible if and

only if: (Def. 6) For all elements u, w, v of the carrier of V holds ρ(v, w) = ρ(v+u, w+u).

Let V be a non empty RLSMetrStruct and let v be an element of the carrier of V. The functor Normv yielding a real number is defined as follows:

(Def. 7) Normv = ρ(0_{V} , v).

Let us note that there exists a non empty RLSMetrStruct which is strict, Abelian, add-associative, right zeroed, right complementable, real linear spacelike, Reflexive, discernible, symmetric, triangle, homogeneous, and translatible.

A RealLinearMetrSpace is an Abelian add-associative right zeroed right complementable real linear space-like Reflexive discernible symmetric triangle homogeneous translatible non empty RLSMetrStruct.

We now state three propositions:

(6) Let V be a homogeneous Abelian add-associative right zeroed right complementable real linear space-like non empty RLSMetrStruct, r be a real number, and v be an element of the carrier of V. Then Norm(r · v) = |r| · Normv.

(7) Let V be a translatible Abelian add-associative right zeroed right complementable triangle non empty RLSMetrStruct and v, w be elements of the carrier of V. Then Norm(v + w) Normv + Normw.

(8) Let V be a translatible add-associative right zeroed right complementable non empty RLSMetrStruct and v, w be elements of the carrier of V. Then ρ(v, w) = Norm(w − v).

Let n be a natural number. The functor RLMSpacen yielding a strict Real-LinearMetrSpace is defined by the conditions (Def. 8).

(Def. 8)(i) The carrier of RLMSpacen = R^{n}, (ii) the distance of RLMSpacen = ρ^{n}, (iii) the zero of RLMSpacen = ⟨\underbrace{0, …, 0}_{n}⟩,

(iv) for all elements x, y of R^{n} holds (the addition of RLMSpacen)(x,

y) = x + y, and

(v) for every element x of R^{n} and for every element r of ℝ holds (the external multiplication of RLMSpacen)(r, x) = r · x.

Next we state the proposition

(9) For every natural number n and for every isometric map f from RLMSpacen into RLMSpacen holds rngf = R^{n}.

4. Groups of Isometric Functions

Let n be a natural number. The functor IsomGroupn yielding a strict groupoid is defined by the conditions (Def. 9).

(Def. 9)(i) The carrier of IsomGroupn = ISOMRLMSpacen, and

(ii) for all functions f, g such that f ∈ ISOMRLMSpacen and g ∈

ISOMRLMSpacen holds (the multiplication of IsomGroupn)(f, g) = f ·g. Let n be a natural number. Note that IsomGroupn is non empty. Let n be a natural number. Note that IsomGroupn is associative and group-

like.

The following two propositions are true:

(10) For every natural number n holds 1_{IsomGroup n} = id_{RLMSpace n}.

(11) Let n be a natural number, f be an element of IsomGroupn, and g be

a map from RLMSpacen into RLMSpacen. If f = g, then f^{−1} = g^{−1} . Let n be a natural number and let G be a subgroup of IsomGroupn.

The functor SubIsomGroupRelG yielding a binary relation on the carrier of RLMSpacen is defined by the condition (Def. 10).

real linear-metric space and isometric … 277

(Def. 10) Let A, B be elements of RLMSpacen. Then ⟨⟨A, B⟩⟩∈ SubIsomGroupRelG if and only if there exists a function f such that f ∈ the carrier of G and f(A) = B.

Let n be a natural number and let G be a subgroup of IsomGroupn. Observe that SubIsomGroupRelG is equivalence relation-like.

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

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

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

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

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

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

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

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

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

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

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

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

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

278 robert milewski

Introduction to Meet-Continuous Topological Lattices^{1}

Artur Korniłowicz University of Białystok

MML Identifier: YELLOW13.

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

1. Preliminaries

Let S be a finite 1-sorted structure. One can verify that the carrier of S is finite.

Let S be a trivial 1-sorted structure. One can check that the carrier of S is trivial.

One can check that every set which is trivial is also finite. One can verify that every 1-sorted structure which is trivial is also finite. Let us mention that every 1-sorted structure which is non trivial is also non

empty.

One can check the following observations:

∗ there exists a 1-sorted structure which is strict, non empty, and trivial,

∗ there exists a relational structure which is strict, non empty, and trivial, and

∗ there exists a FR-structure which is strict, non empty, and trivial. We now state the proposition

(1) For every T_{1} non empty topological space T holds every finite subset of T is closed.

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

Let T be a compact topological structure. Observe that Ω_{T} is compact. Let us observe that there exists a topological space which is strict, non

empty, and trivial.

Let us mention that every non empty topological space which is finite and T_{1} is also discrete.

Let us observe that every topological space which is finite is also compact. One can prove the following propositions:

(2) Every discrete non empty topological space is a T_{4} space.

(3) Every discrete non empty topological space is a T_{3} space.

(4) Every discrete non empty topological space is a T_{2} space.

(5) Every discrete non empty topological space is a T_{1} space. One can check that every non empty topological space which is T_{4} and T_{1} is

also T_{3}.

Let us observe that every non empty topological space which is T_{3} and T_{1} is also T_{2}.

Let us note that every topological space which is T_{2} is also T_{1}. One can check that every topological space which is T_{1} is also T_{0}. Next we state three propositions:

(6) Let S be a reflexive relational structure, T be a reflexive transitive relational structure, f be a map from S into T, and X be a subset of S. Then ↓(f^{∘}X) ⊆ ↓(f^{∘}↓X).

(7) Let S be a reflexive relational structure, T be a reflexive transitive relational structure, f be a map from S into T, and X be a subset of S. If f is monotone, then ↓(f^{∘}X) = ↓(f^{∘}↓X).

(8) For every non empty poset N holds IdsMap(N) is one-to-one. One can prove the following proposition

(9) For every finite lattice N holds SupMap(N) is one-to-one. We now state three propositions:

(10) For every finite lattice N holds N and ⟨Ids(N), ⊆⟩ are isomorphic.

(11) Let N be a complete non empty poset, x be an element of N, and X be a non empty subset of N. Then x ⊓ □ preserves inf of X.

(12) For every complete non empty poset N and for every element x of N holds x ⊓ □ is meet-preserving.

2. On the Basis of Topological Spaces

Next we state several propositions:

(13) Let T be an anti-discrete non empty topological structure and p be a point of T. Then {the carrier of T } is a basis of p.

(14) Let T be an anti-discrete non empty topological structure, p be a point of T, and D be a basis of p. Then D = {the carrier of T }.

(15) Let T be a non empty topological space, P be a basis of T, and p be a point of T. Then {A; A ranges over subsets of T: A ∈ P ∧ p ∈ A} is a basis of p.

(16) Let T be a non empty topological structure, A be a subset of T, and p be a point of T. Then p ∈ \overline{A} if and only if for every basis K of p and for every subset Q of T such that Q ∈ K holds A ∩ Q ≠ ∅.

(17) Let T be a non empty topological structure, A be a subset of T, and p be a point of T. Then p ∈ \overline{A} if and only if there exists a basis K of p such that for every subset Q of T such that Q ∈ K holds A ∩ Q ≠ ∅.

Let T be a topological structure and let p be a point of T. A family of subsets of T is said to be a generalized basis of p if:

(Def. 1) For every subset A of T such that p ∈ IntA there exists a subset P of T

such that P ∈ it and p ∈ IntP and P ⊆ A. Let T be a non empty topological space and let p be a point of T. Let us note

that the generalized basis of p can be characterized by the following (equivalent) condition:

(Def. 2) For every neighbourhood A of p there exists a neighbourhood P of p

such that P ∈ it and P ⊆ A. The following propositions are true:

(18) Let T be a topological structure and p be a point of T. Then 2^{the carrier of T} is a generalized basis of p.

(19) For every non empty topological space T and for every point p of T holds

every generalized basis of p is non empty. Let T be a topological structure and let p be a point of T. Observe that

there exists a generalized basis of p which is non empty.

Let T be a topological structure, let p be a point of T, and let P be a generalized basis of p. We say that P is correct if and only if:

(Def. 3) For every subset A of T holds A ∈ P iff p ∈ IntA.

Let T be a topological structure and let p be a point of T. Note that there exists a generalized basis of p which is correct.

One can prove the following proposition (20) Let T be a topological structure and p be a point of T. Then {A; A

ranges over subsets of T: p ∈ IntA} is a correct generalized basis of p. Let T be a non empty topological space and let p be a point of T. Observe

that there exists a generalized basis of p which is non empty and correct.

One can prove the following three propositions:

(21) Let T be an anti-discrete non empty topological structure and p be a point of T. Then {the carrier of T } is a correct generalized basis of p.

(22) Let T be an anti-discrete non empty topological structure, p be a point of T, and D be a correct generalized basis of p. Then D = {the carrier of T }.

(23) For every non empty topological space T and for every point p of T holds

every basis of p is a generalized basis of p. Let T be a topological structure. A family of subsets of T is said to be a

generalized basis of T if: (Def. 4) For every point p of T holds it is a generalized basis of p.

Next we state two propositions:

(24) For every topological structure T holds 2^{the carrier of T} is a generalized basis of T.

(25) For every non empty topological space T holds every generalized basis

of T is non empty. Let T be a topological structure. Note that there exists a generalized basis

of T which is non empty.

Next we state two propositions:

(26) For every non empty topological space T and for every generalized basis P of T holds the topology of T ⊆ UniCl(IntP).

(27) For every topological space T holds every basis of T is a generalized basis

of T. Let T be a non empty topological space-like FR-structure. We say that T is

topological semilattice if and only if: (Def. 5) For every map f from [: T, (T qua topological space) :] into T such that

f = ⊓_{T} holds f is continuous. Let us note that every non empty topological space-like FR-structure which

is reflexive and trivial is also topological semilattice.

Let us mention that there exists a FR-structure which is reflexive, trivial, non empty, and topological space-like.

We now state the proposition

(28) Let T be a topological semilattice non empty topological space-like FRstructure and x be an element of T. Then x ⊓ □ is continuous. introduction to meet-continuous topological … 283

References

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

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

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

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

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

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

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

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

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

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

[11] Zbigniew Karno and Toshihiko Watanabe. Completeness of the lattices of domains of a topological space. Formalized Mathematics, 3(1):71–79, 1992.

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

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

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

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

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

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

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

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

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

284 artur korniłowicz

Bases of Continuous Lattices^{1}

Robert Milewski University of Białystok

Summary. The article is a Mizar formalization of [7, 168–169]. We show definition and fundamental theorems from theory of basis of continuous lattices.

MML Identifier: WAYBEL23.

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

1. Preliminaries

The following proposition is true

(1) For every non empty poset L and for every element x of L holds compactbelow(x) = ↓↓x ∩ the carrier of CompactSublatt(L).

Let L be a non empty reflexive transitive relational structure and let X be a subset of ⟨Ids(L), ⊆⟩. Then ⋃ X is a subset of L.

The following propositions are true:

(2) For every non empty relational structure L and for all subsets X, Y of the carrier of L such that X ⊆ Y holds finsups(X) ⊆ finsups(Y).

(3) Let L be a non empty transitive relational structure, S be a supsinheriting non empty full relational substructure of L, X be a subset of the carrier of L, and Y be a subset of the carrier of S. If X = Y, then finsups(X) ⊆ finsups(Y).

(4) Let L be a complete transitive antisymmetric non empty relational structure, S be a sups-inheriting non empty full relational substructure of L, X be a subset of the carrier of L, and Y be a subset of the carrier of S. If X = Y, then finsups(X) = finsups(Y).

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

(5) Let L be a complete sup-semilattice and S be a join-inheriting non empty full relational substructure of L. Suppose ⊥_{L} ∈ the carrier of S. Let X be a subset of L and Y be a subset of S. If X = Y, then finsups(Y) ⊆ finsups(X).

(6) For every lower-bounded sup-semilattice L and for every subset X of ⟨Ids(L), ⊆⟩ holds supX = ↓finsups(⋃ X).

(7) For every reflexive transitive relational structure L and for every subset X of L holds ↓↓X = ↓X.

(8) For every reflexive transitive relational structure L and for every subset X of L holds ↑↑X = ↑X.

(9) For every non empty reflexive transitive relational structure L and for every element x of L holds ↓↓x = ↓x.

(10) For every non empty reflexive transitive relational structure L and for every element x of L holds ↑↑x = ↑x.

(11) Let L be a non empty relational structure, S be a non empty relational substructure of L, X be a subset of L, and Y be a subset of S. If X = Y, then ↓Y ⊆ ↓X.

(12) Let L be a non empty relational structure, S be a non empty relational substructure of L, X be a subset of L, and Y be a subset of S. If X = Y, then ↑Y ⊆ ↑X.

(13) Let L be a non empty relational structure, S be a non empty relational substructure of L, x be an element of L, and y be an element of S. If x = y, then ↓y ⊆ ↓x.

(14) Let L be a non empty relational structure, S be a non empty relational substructure of L, x be an element of L, and y be an element of S. If x = y, then ↑y ⊆ ↑x.

2. Relational Subsets

Let L be a non empty relational structure and let S be a subset of L. We say that S is meet-closed if and only if:

(Def. 1) sub(S) is meet-inheriting.

Let L be a non empty relational structure and let S be a subset of L. We say that S is join-closed if and only if:

(Def. 2) sub(S) is join-inheriting.

Let L be a non empty relational structure and let S be a subset of L. We say that S is infs-closed if and only if:

(Def. 3) sub(S) is infs-inheriting.

Let L be a non empty relational structure and let S be a subset of L. We say that S is sups-closed if and only if:

(Def. 4) sub(S) is sups-inheriting.

Let L be a non empty relational structure. Observe that every subset of L which is infs-closed is also meet-closed and every subset of L which is sups-closed is also join-closed.

Let L be a non empty relational structure. One can verify that there exists a subset of L which is infs-closed, sups-closed, and non empty.

One can prove the following propositions:

(15) Let L be a non empty relational structure and S be a subset of L. Then S is meet-closed if and only if for all elements x, y of L such that x ∈ S and y ∈ S and inf {x, y} exists in L holds inf{x, y} ∈ S.

(16) Let L be a non empty relational structure and S be a subset of L. Then S is join-closed if and only if for all elements x, y of L such that x ∈ S and y ∈ S and sup {x, y} exists in L holds sup{x, y} ∈ S.

(17) Let L be an antisymmetric relational structure with g.l.b.’s and S be a subset of L. Then S is meet-closed if and only if for all elements x, y of L such that x ∈ S and y ∈ S holds inf{x, y} ∈ S.

(18) Let L be an antisymmetric relational structure with l.u.b.’s and S be a subset of L. Then S is join-closed if and only if for all elements x, y of L such that x ∈ S and y ∈ S holds sup{x, y} ∈ S.

(19) Let L be a non empty relational structure and S be a subset of L. Then S is infs-closed if and only if for every subset X of S such that inf X exists in L holds ⌈−⌉_{L}X ∈ S.

(20) Let L be a non empty relational structure and S be a subset of L. Then S is sups-closed if and only if for every subset X of S such that sup X exists in L holds ⨆_{L} X ∈ S.

(21) Let L be a non empty transitive relational structure, S be an infs-closed non empty subset of L, and X be a subset of S. If inf X exists in L, then inf X exists in sub(S) and ⌈−⌉_{sub(S)}X = ⌈−⌉_{L}X.

(22) Let L be a non empty transitive relational structure, S be a sups-closed non empty subset of L, and X be a subset of S. If sup X exists in L, then sup X exists in sub(S) and ⨆_{sub(S)} X = ⨆_{L} X.

(23) Let L be a non empty transitive relational structure, S be a meet-closed non empty subset of L, and x, y be elements of S. Suppose inf {x, y} exists in L. Then inf {x, y} exists in sub(S) and ⌈−⌉_{sub(S)}{x, y} = ⌈−⌉_{L}{x, y}.

(24) Let L be a non empty transitive relational structure, S be a join-closed non empty subset of L, and x, y be elements of S. Suppose sup {x, y} exists in L. Then sup {x, y} exists in sub(S) and ⨆_{sub(S)}{x, y} = ⨆_{L}{x, y}.

(25) Let L be an antisymmetric transitive relational structure with g.l.b.’s and S be a non empty meet-closed subset of L. Then sub(S) has g.l.b.’s.

(26) Let L be an antisymmetric transitive relational structure with l.u.b.’s and S be a non empty join-closed subset of L. Then sub(S) has l.u.b.’s.

Let L be an antisymmetric transitive relational structure with g.l.b.’s and let S be a non empty meet-closed subset of L. Observe that sub(S) has g.l.b.’s.

Let L be an antisymmetric transitive relational structure with l.u.b.’s and let S be a non empty join-closed subset of L. Observe that sub(S) has l.u.b.’s.

The following four propositions are true:

(27) Let L be a complete transitive antisymmetric non empty relational structure, S be an infs-closed non empty subset of L, and X be a subset of S. Then ⌈−⌉_{sub(S)}X = ⌈−⌉_{L}X.

(28) Let L be a complete transitive antisymmetric non empty relational structure, S be a sups-closed non empty subset of L, and X be a subset of S. Then ⨆_{sub(S)} X = ⨆_{L} X.

(29) For every semilattice L holds every meet-closed subset of L is filtered.

(30) For every sup-semilattice L holds every join-closed subset of L is directed.

Let L be a semilattice. Observe that every subset of L which is meet-closed is also filtered.

Let L be a sup-semilattice. One can check that every subset of L which is join-closed is also directed.

The following propositions are true:

(31) Let L be a semilattice and S be an upper non empty subset of L. Then S is a filter of L if and only if S is meet-closed.

(32) Let L be a sup-semilattice and S be a lower non empty subset of L. Then S is an ideal of L if and only if S is join-closed.

(33) For every non empty relational structure L and for all join-closed subsets S_{1}, S_{2} of L holds S_{1} ∩ S_{2} is join-closed.

(34) For every non empty relational structure L and for all meet-closed subsets S_{1}, S_{2} of L holds S_{1} ∩ S_{2} is meet-closed.

(35) For every sup-semilattice L and for every element x of the carrier of L holds ↓x is join-closed.

(36) For every semilattice L and for every element x of the carrier of L holds ↓x is meet-closed.

(37) For every sup-semilattice L and for every element x of the carrier of L holds ↑x is join-closed.

(38) For every semilattice L and for every element x of the carrier of L holds ↑x is meet-closed.

Let L be a sup-semilattice and let x be an element of L. Observe that ↓x is join-closed and ↑x is join-closed.

Let L be a semilattice and let x be an element of L. Note that ↓x is meetclosed and ↑x is meet-closed.

Next we state three propositions:

(39) For every sup-semilattice L and for every element x of L holds ↓↓x is join-closed.

(40) For every semilattice L and for every element x of L holds ↓↓x is meetclosed.

(41) For every sup-semilattice L and for every element x of L holds ↑↑x is

join-closed. Let L be a sup-semilattice and let x be an element of L. Note that ↓↓x is

join-closed and ↑↑x is join-closed.

Let L be a semilattice and let x be an element of L. Observe that ↓↓x is meet-closed.

3. About Bases of Continuous Lattices

Let T be a topological structure. The functor weightT yields a cardinal number and is defined as follows:

(Def. 5) weightT = ⋂\{\overline{\overline{B}} : B ranges over bases of T \}.

Let T be a topological structure. We say that T is second-countable if and only if:

(Def. 6) weightT ⊆ ω.

Let L be a continuous sup-semilattice. A subset of L is called a CLbasis of L if:

(Def. 7) It is join-closed and for every element x of L holds x = sup(↓↓x ∩ it).

Let L be a non empty relational structure and let S be a subset of L. We say that S has bottom if and only if:

(Def. 8) ⊥_{L} ∈ S.

Let L be a non empty relational structure and let S be a subset of L. We say that S has top if and only if:

(Def. 9) ⊤_{L} ∈ S.

Let L be a non empty relational structure. Note that every subset of L which has bottom is non empty.

Let L be a non empty relational structure. Observe that every subset of L which has top is non empty.

Let L be a non empty relational structure. Note that there exists a subset of L which has bottom and there exists a subset of L which has top.

Let L be a continuous sup-semilattice. One can verify that there exists a CLbasis of L which has bottom and there exists a CLbasis of L which has top.

One can prove the following proposition

(42) Let L be a lower-bounded antisymmetric non empty relational structure

and S be a subset of L with bottom. Then sub(S) is lower-bounded. Let L be a lower-bounded antisymmetric non empty relational structure and

let S be a subset of L with bottom. One can verify that sub(S) is lower-bounded.

Let L be a continuous sup-semilattice. Observe that every CLbasis of L is join-closed.

One can check that there exists a continuous lattice which is bounded and non trivial.

Let L be a lower-bounded non trivial continuous sup-semilattice. One can verify that every CLbasis of L is non empty.

One can prove the following propositions:

(43) For every sup-semilattice L holds the carrier of CompactSublatt(L) is a join-closed subset of L.

(44) For every algebraic lower-bounded lattice L holds the carrier of CompactSublatt(L) is a CLbasis of L with bottom.

(45) Let L be a continuous lower-bounded sup-semilattice. If the carrier of CompactSublatt(L) is a CLbasis of L, then L is algebraic.

(46) Let L be a continuous lower-bounded lattice and B be a join-closed subset of L. Then B is a CLbasis of L if and only if for all elements x, y of L such that y x̸ there exists an element b of L such that b ∈ B and b x̸ and b ≪ y.

(47) Let L be a continuous lower-bounded lattice and B be a join-closed subset of L. Suppose ⊥_{L} ∈ B. Then B is a CLbasis of L if and only if for all elements x, y of L such that x ≪ y there exists an element b of L such that b ∈ B and x b and b ≪ y.

(48) Let L be a continuous lower-bounded lattice and B be a join-closed subset of L. Suppose ⊥_{L} ∈ B. Then B is a CLbasis of L if and only if the following conditions are satisfied:

(i) the carrier of CompactSublatt(L) ⊆ B, and (ii) for all elements x, y of L such that y x̸ there exists an element b of

L such that b ∈ B and b x̸ and b y.

(49) Let L be a continuous lower-bounded lattice and B be a join-closed subset of L. Suppose ⊥_{L} ∈ B. Then B is a CLbasis of L if and only if for all elements x, y of L such that y x̸ there exists an element b of L such that b ∈ B and b x̸ and b y.

(50) Let L be a lower-bounded sup-semilattice and S be a non empty full relational substructure of L. Suppose ⊥_{L} ∈ the carrier of S and the carrier of S is a join-closed subset of L. Let x be an element of L. Then ↓↓x ∩ the carrier of S is an ideal of S.

Let L be a non empty reflexive transitive relational structure and let S be a non empty full relational substructure of L. The functor supMapS yielding a map from ⟨Ids(S), ⊆⟩ into L is defined by:

(Def. 10) For every ideal I of S holds (supMapS)(I) = ⨆_{L} I.

Let L be a non empty reflexive transitive relational structure and let S be a non empty full relational substructure of L. The functor idsMapS yields a map from ⟨Ids(S), ⊆⟩ into ⟨Ids(L), ⊆⟩ and is defined by:

(Def. 11) For every ideal I of S there exists a subset J of L such that I = J and

(idsMapS)(I) = ↓J. Let L be a non empty relational structure and let B be a non empty subset

of the carrier of L. Observe that sub(B) is non empty.

Let L be a reflexive relational structure and let B be a subset of the carrier of L. Note that sub(B) is reflexive.

Let L be a transitive relational structure and let B be a subset of the carrier of L. Note that sub(B) is transitive.

Let L be an antisymmetric relational structure and let B be a subset of the carrier of L. Observe that sub(B) is antisymmetric.

Let L be a lower-bounded continuous sup-semilattice and let B be a CLbasis of L with bottom. The functor baseMapB yielding a map from L into ⟨Ids(sub(B)), ⊆⟩ is defined as follows:

(Def. 12) For every element x of L holds (baseMapB)(x) = ↓↓x ∩ B.

We now state a number of propositions:

(51) Let L be a non empty reflexive transitive relational structure and S be a non empty full relational substructure of L. Then domsupMapS = Ids(S) and rngsupMapS is a subset of L.

(52) Let L be a non empty reflexive transitive relational structure, S be a non empty full relational substructure of L, and x be a set. Then x ∈ domsupMapS if and only if x is an ideal of S.

(53) Let L be a non empty reflexive transitive relational structure and S be a non empty full relational substructure of L. Then domidsMapS = Ids(S) and rngidsMapS is a subset of Ids(L).

(54) Let L be a non empty reflexive transitive relational structure, S be a non empty full relational substructure of L, and x be a set. Then x ∈ domidsMapS if and only if x is an ideal of S.

(55) Let L be a non empty reflexive transitive relational structure, S be a non empty full relational substructure of L, and x be a set. If x ∈ rngidsMapS, then x is an ideal of L.

(56) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis of L with bottom. Then dombaseMapB = the carrier of L and rngbaseMapB is a subset of Ids(sub(B)).

(57) Let L be a lower-bounded continuous sup-semilattice, B be a CLbasis of L with bottom, and x be a set. If x ∈ rngbaseMapB, then x is an ideal of sub(B).

(58) For every up-complete non empty poset L and for every non empty full relational substructure S of L holds supMapS is monotone.

(59) Let L be a non empty reflexive transitive relational structure and S be a non empty full relational substructure of L. Then idsMapS is monotone.

(60) For every lower-bounded continuous sup-semilattice L and for every

CLbasis B of L with bottom holds baseMapB is monotone. Let L be an up-complete non empty poset and let S be a non empty full

relational substructure of L. Observe that supMapS is monotone.

Let L be a non empty reflexive transitive relational structure and let S be a non empty full relational substructure of L. One can check that idsMapS is monotone.

Let L be a lower-bounded continuous sup-semilattice and let B be a CLbasis of L with bottom. One can check that baseMapB is monotone.

The following propositions are true:

(61) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis of L with bottom. Then idsMapsub(B) is sups-preserving.

(62) For every up-complete non empty poset L and for every non empty full relational substructure S of L holds supMapS = SupMap(L) · idsMapS.

(63) For every lower-bounded continuous sup-semilattice L and for every CLbasis B of L with bottom holds ⟨⟨supMapsub(B), baseMapB⟩⟩ is Galois.

(64) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis of L with bottom. Then supMapsub(B) is upper adjoint and baseMapB is lower adjoint.

(65) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis of L with bottom. Then rngsupMapsub(B) = the carrier of L.

(66) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis of L with bottom. Then supMapsub(B) is infs-preserving and supspreserving.

(67) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis

of L with bottom. Then baseMapB is sups-preserving. Let L be a lower-bounded continuous sup-semilattice and let B be a CLbasis

of L with bottom. One can verify that supMapsub(B) is infs-preserving and sups-preserving and baseMapB is sups-preserving.

bases of continuous lattices 293

One can prove the following propositions: (69)^{2} Let L be a lower-bounded continuous sup-semilattice and B be a CLba-

sis of L with bottom. Then the carrier of CompactSublatt(⟨Ids(sub(B)), ⊆ ⟩) = {↓b : b ranges over elements of sub(B)}.

(70) Let L be a lower-bounded continuous sup-semilattice and B be a CLbasis of L with bottom. Then CompactSublatt(⟨Ids(sub(B)), ⊆⟩) and sub(B) are isomorphic.

(71) Let L be a continuous lower-bounded lattice and B be a CLbasis of L with bottom. Suppose that for every CLbasis B_{1} of L with bottom holds B ⊆ B_{1}. Let J be an element of ⟨Ids(sub(B)), ⊆⟩. Then J = ↓↓⨆_{L} J ∩ B.

(72) Let L be a continuous lower-bounded lattice. Then L is algebraic if and

only if the following conditions are satisfied:

(i) the carrier of CompactSublatt(L) is a CLbasis of L with bottom, and (ii) for every CLbasis B of L with bottom holds the carrier of

CompactSublatt(L) ⊆ B.

(73) Let L be a continuous lower-bounded lattice. Then L is algebraic if and only if there exists a CLbasis B of L with bottom such that for every CLbasis B_{1} of L with bottom holds B ⊆ B_{1}.

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 from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

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

[7] 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.

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

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

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

[11] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 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 and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

2The proposition (68) has been removed.

294 robert milewski

[15] Mariusz Żynel and Czesław Byliński. Properties of relational structures, posets, lattices

and maps. Formalized Mathematics, 6(1):123–130, 1997.

Received November 28, 1998

The Construction of SCM over Ring

Artur Korniłowicz University of Białystok

MML Identifier: SCMRING1.

The terminology and notation used in this paper have been introduced in the following articles: [6], [11], [2], [3], [9], [4], [5], [7], [1], [10], and [8].

For simplicity, we follow the rules: i, k are natural numbers, I is an element of ℤ_{8}, i_{1} is an element of Instr-Loc_{SCM}, d_{1} is an element of Data-Loc_{SCM}, and S is a non empty 1-sorted structure.

Let us observe that every non empty loop structure which is trivial is also Abelian, add-associative, right zeroed, and right complementable and every non empty double loop structure which is trivial is also right unital and rightdistributive.

Let us note that every element of Data-Loc_{SCM} is natural.

One can check the following observations:

∗ Data-Loc_{SCM} is non trivial,

∗ Instr_{SCM} is non trivial, and

∗ Instr-Loc_{SCM} is non trivial.

Let S be a non empty 1-sorted structure. The functor Instr_{SCM}(S) yields a subset of [: ℤ_{8}, (⋃{the carrier of S} ∪ ℕ)^{∗} :] and is defined by the condition (Def. 1).

(Def. 1) Instr_{SCM}(S) = {⟨⟨0, ε⟩⟩}∪{⟨⟨I, ⟨a, b⟩⟩⟩; I ranges over elements of ℤ_{8}, a ranges over elements of Data-Loc_{SCM}, b ranges over elements of Data-Loc_{SCM}: I ∈ {1, 2, 3, 4}} ∪ {⟨⟨6, ⟨i⟩⟩⟩: i ranges over elements of Instr-Loc_{SCM}} ∪ {⟨⟨7, ⟨i, a⟩⟩⟩: i ranges over elements of Instr-Loc_{SCM}, a ranges over elements of Data-Loc_{SCM}} ∪ {⟨⟨5, ⟨a, r⟩⟩⟩ : a ranges over elements of Data-Loc_{SCM}, r ranges over elements of the carrier of S}.

Let S be a non empty 1-sorted structure. Note that Instr_{SCM}(S) is non trivial.

Let S be a non empty 1-sorted structure. We say that S is good if and only if:

(Def. 2) The carrier of S ≠ Instr-Loc_{SCM} and the carrier of S ≠ Instr_{SCM}(S).

One can verify that every non empty 1-sorted structure which is trivial is also good.

Let us observe that there exists a 1-sorted structure which is strict, trivial, and non empty.

Let us observe that there exists a double loop structure which is strict, trivial, and non empty.

One can check that there exists a ring which is strict and trivial. In the sequel G denotes a good non empty 1-sorted structure. Let S be a non empty 1-sorted structure. The functor OK_{SCM}(S) yielding a

function from ℕ into {the carrier of S}∪{Instr_{SCM}(S), Instr-Loc_{SCM}} is defined as follows:

(Def. 3) (OK_{SCM}(S))(0) = Instr-Loc_{SCM} and for every natural number k holds (OK_{SCM}(S))(2 · k + 1) = the carrier of S and (OK_{SCM}(S))(2 · k + 2) = Instr_{SCM}(S).

Let S be a non empty 1-sorted structure. An SCM-state over S is an element of ∏ OK_{SCM}(S).

Next we state several propositions:

(1) Instr-Loc_{SCM} ≠ Instr_{SCM}(S).

(2) (OK_{SCM}(G))(i) = Instr-Loc_{SCM} iff i = 0.

(3) (OK_{SCM}(G))(i) = the carrier of G iff there exists k such that i = 2·k+1.

(4) (OK_{SCM}(G))(i) = Instr_{SCM}(G) iff there exists k such that i = 2 · k + 2.

(5) (OK_{SCM}(G))(d_{1}) = the carrier of G.

(6) (OK_{SCM}(G))(i_{1}) = Instr_{SCM}(G).

(7) π_{0} ∏ OK_{SCM}(S) = Instr-Loc_{SCM}.

(8) π_{d_{1}} ∏ OK_{SCM}(G) = the carrier of G.

(9) π_{i_{1}} ∏ OK_{SCM}(G) = Instr_{SCM}(G). Let S be a non empty 1-sorted structure and let s be an SCM-state over S.

The functor IC_{s} yielding an element of Instr-Loc_{SCM} is defined by: (Def. 4) IC_{s} = s(0).

Let R be a good non empty 1-sorted structure, let s be an SCM-state over R, and let u be an element of Instr-Loc_{SCM}. The functor Chg_{SCM}(s, u) yielding an SCM-state over R is defined by:

(Def. 5) Chg_{SCM}(s, u) = s+·(0⟼→ . u).

The following three propositions are true:

(10) For every SCM-state s over G and for every element u of Instr-Loc_{SCM} holds (Chg_{SCM}(s, u))(0) = u.

(11) For every SCM-state s over G and for every element u of Instr-Loc_{SCM} and for every element m_{1} of Data-Loc_{SCM} holds (Chg_{SCM}(s, u))(m_{1}) = s(m_{1}).

(12) For every SCM-state s over G and for all elements u, v of Instr-Loc_{SCM}

holds (Chg_{SCM}(s, u))(v) = s(v). Let R be a good non empty 1-sorted structure, let s be an SCM-state over

R, let t be an element of Data-Loc_{SCM}, and let u be an element of the carrier of R. The functor Chg_{SCM}(s, t, u) yielding an SCM-state over R is defined as follows:

(Def. 6) Chg_{SCM}(s, t, u) = s+·(t⟼→ . u).

One can prove the following propositions:

(13) Let s be an SCM-state over G, t be an element of Data-Loc_{SCM}, and u be an element of the carrier of G. Then (Chg_{SCM}(s, t, u))(0) = s(0).

(14) Let s be an SCM-state over G, t be an element of Data-Loc_{SCM}, and u be an element of the carrier of G. Then (Chg_{SCM}(s, t, u))(t) = u.

(15) Let s be an SCM-state over G, t be an element of Data-Loc_{SCM}, u be an element of the carrier of G, and m_{1} be an element of Data-Loc_{SCM}. If m_{1} ≠ t, then (Chg_{SCM}(s, t, u))(m_{1}) = s(m_{1}).

(16) Let s be an SCM-state over G, t be an element of Data-Loc_{SCM}, u be an element of the carrier of G, and v be an element of Instr-Loc_{SCM}. Then (Chg_{SCM}(s, t, u))(v) = s(v).

Let R be a good non empty 1-sorted structure, let s be an SCM-state over R, and let a be an element of Data-Loc_{SCM}. Then s(a) is an element of R.

Let S be a non empty 1-sorted structure and let x be an element of Instr_{SCM}(S). Let us assume that there exist elements m_{1}, m_{2} of Data-Loc_{SCM} and I such that x = ⟨⟨I, ⟨m_{1}, m_{2}⟩⟩⟩. The functor x address_{1} yielding an element of Data-Loc_{SCM} is defined by:

(Def. 7) There exists a finite sequence f of elements of Data-Loc_{SCM} such that

f = x_{2} and x address_{1} = π_{1}f. The functor x address_{2} yields an element of Data-Loc_{SCM} and is defined by:

(Def. 8) There exists a finite sequence f of elements of Data-Loc_{SCM} such that

f = x_{2} and x address_{2} = π_{2}f. One can prove the following proposition

(17) For every element x of Instr_{SCM}(S) and for all elements m_{1}, m_{2} of Data-Loc_{SCM} such that x = ⟨⟨I, ⟨m_{1}, m_{2}⟩⟩⟩ holds x address_{1} = m_{1} and x address_{2} = m_{2}.

Let R be a non empty 1-sorted structure and let x be an element of Instr_{SCM}(R). Let us assume that there exist an element m_{1} of Instr-Loc_{SCM} and I such that x = ⟨⟨I, ⟨m_{1}⟩⟩⟩. The functor x address_{j} yielding an element of Instr-Loc_{SCM} is defined as follows:

(Def. 9) There exists a finite sequence f of elements of Instr-Loc_{SCM} such that

f = x_{2} and x address_{j} = π_{1}f. Next we state the proposition

(18) For every element x of Instr_{SCM}(S) and for every element m_{1} of

Instr-Loc_{SCM} such that x = ⟨⟨I, ⟨m_{1}⟩⟩⟩ holds x address_{j} = m_{1}. Let S be a non empty 1-sorted structure and let x be an element of

Instr_{SCM}(S). Let us assume that there exist an element m_{1} of Instr-Loc_{SCM}, an element m_{2} of Data-Loc_{SCM}, and I such that x = ⟨⟨I, ⟨m_{1}, m_{2}⟩⟩⟩. The functor x address_{j} yields an element of Instr-Loc_{SCM} and is defined as follows:

(Def. 10) There exists an element m_{1} of Instr-Loc_{SCM} and there exists an element m_{2} of Data-Loc_{SCM} such that ⟨m_{1}, m_{2}⟩ = x_{2} and x address_{j} = π_{1}⟨m_{1},

m_{2}⟩.

The functor x address_{c} yields an element of Data-Loc_{SCM} and is defined as follows:

(Def. 11) There exists an element m_{1} of Instr-Loc_{SCM} and there exists an element m_{2} of Data-Loc_{SCM} such that ⟨m_{1}, m_{2}⟩ = x_{2} and x address_{c} = π_{2}⟨m_{1}, m_{2}⟩.

We now state the proposition

(19) Let x be an element of Instr_{SCM}(S), m_{1} be an element of Instr-Loc_{SCM}, and m_{2} be an element of Data-Loc_{SCM}. If x = ⟨⟨I, ⟨m_{1}, m_{2}⟩⟩⟩, then x address_{j} = m_{1} and x address_{c} = m_{2}.

Let S be a non empty 1-sorted structure, let d be an element of Data-Loc_{SCM}, and let s be an element of the carrier of S. Then ⟨d, s⟩ is a finite sequence of elements of Data-Loc_{SCM} ∪ the carrier of S.

Let S be a non empty 1-sorted structure and let x be an element of Instr_{SCM}(S). Let us assume that there exist an element m_{1} of Data-Loc_{SCM}, an element r of the carrier of S, and I such that x = ⟨⟨I, ⟨m_{1}, r⟩⟩⟩. The functor x const address yields an element of Data-Loc_{SCM} and is defined as follows:

(Def. 12) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪the carrier

of S such that f = x_{2} and x const address = π_{1}f. The functor x const value yields an element of the carrier of S and is defined by:

(Def. 13) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪the carrier

of S such that f = x_{2} and x const value = π_{2}f. We now state the proposition

(20) Let x be an element of Instr_{SCM}(S), m_{1} be an element of Data-Loc_{SCM}, and r be an element of the carrier of S. If x = ⟨⟨I, ⟨m_{1}, r⟩⟩⟩, then x const address = m_{1} and x const value = r.

Let R be a good ring, let x be an element of Instr_{SCM}(R), and let s be an SCM-state over R. The functor Exec-Res_{SCM}(x, s) yields an SCM-state over

R and is defined by: (Def. 14) Exec-Res_{SCM}(x, s) =

Chg_{SCM}(Chg_{SCM}(s, x address_{1}, s(x address_{2})), Next(IC_{s})), if there

exist elements m_{1}, m_{2} of Data-Loc_{SCM} such that x = ⟨⟨1, ⟨m_{1}, m_{2}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, x address_{1}, s(x address_{1}) + s(x address_{2})), Next(IC_{s})),

if there exist elements m_{1}, m_{2} of Data-Loc_{SCM} such that x = ⟨⟨2, ⟨m_{1}, m_{2}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, x address_{1}, s(x address_{1}) − s(x address_{2})), Next(IC_{s})),

if there exist elements m_{1}, m_{2} of Data-Loc_{SCM} such that x = ⟨⟨3, ⟨m_{1}, m_{2}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, x address_{1}, s(x address_{1}) · s(x address_{2})), Next(IC_{s})),

if there exist elements m_{1}, m_{2} of Data-Loc_{SCM} such that x = ⟨⟨4, ⟨m_{1}, m_{2}⟩⟩⟩, Chg_{SCM}(s, x address_{j}), if there exists an element m_{1} of Instr-Loc_{SCM}

such that x = ⟨⟨6, ⟨m_{1}⟩⟩⟩, Chg_{SCM}(s, (s(x address_{c}) = 0_{R} → x address_{j}, Next(IC_{s}))), if there exists

an element m_{1} of Instr-Loc_{SCM} and there exists an element m_{2} of Data-Loc_{SCM} such that x = ⟨⟨7, ⟨m_{1}, m_{2}⟩⟩⟩,

Chg_{SCM}(Chg_{SCM}(s, x const address, x const value), Next(IC_{s})), if there exists an element m_{1} of Data-Loc_{SCM} and there exists an element r of the carrier of R such that x = ⟨⟨5, ⟨m_{1}, r⟩⟩⟩,

s, otherwise. Let S be a non empty 1-sorted structure, let f be a function from

Instr_{SCM}(S) into (∏ OK_{SCM}(S))^{∏ OK_{SCM} (S)}, and let x be an element of Instr_{SCM}(S). One can check that f(x) is function-like and relation-like.

Let R be a good ring. The functor Exec_{SCM}(R) yielding a function from Instr_{SCM}(R) into (∏ OK_{SCM}(R))^{∏ OK_{SCM} (R)} is defined as follows:

(Def. 15) For every element x of Instr_{SCM}(R) and for every SCM-state y over R holds (Exec_{SCM}(R))(x)(y) = Exec-Res_{SCM}(x, y).

References

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

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

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

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

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

[6] Yatsuka Nakamura and Andrzej Trybulec. On a mathematical model of programs. Formalized Mathematics, 3(2):241–250, 1992.

[7] Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623–627, 1991.

[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.

300 artur korniłowicz

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

thematics, 1(1):17–23, 1990.

Received November 29, 1998

The Basic Properties of SCM over Ring

Artur Korniłowicz University of Białystok

MML Identifier: SCMRING2.

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

1. SCM over Ring

In this paper I is an element of ℤ_{8}, S is a non empty 1-sorted structure, t is an element of the carrier of S, and x is a set.

Let R be a good ring. The functor SCM (R) yields a strict AMI over {the carrier of R} and is defined by the conditions (Def. 1).

(Def. 1)(i) The objects of SCM (R) = ℕ, (ii) the instruction counter of SCM (R) = 0, (iii) the instruction locations of SCM (R) = Instr-Loc_{SCM}, (iv) the instruction codes of SCM (R) = ℤ_{8},

(v) the instructions of SCM (R) = Instr_{SCM}(R), (vi) the object kind of SCM (R) = OK_{SCM}(R), and (vii) the execution of SCM (R) = Exec_{SCM}(R).

Let R be a good ring, let s be a state of SCM (R), and let a be an element of Data-Loc_{SCM}. Then s(a) is an element of R.

Let R be a good ring. An object of SCM (R) is called a Data-Location of R if:

(Def. 2) It ∈ (the objects of SCM (R)) \ (Instr-Loc_{SCM} ∪ {0}).

For simplicity, we use the following convention: R is a good ring, r is an element of the carrier of R, a, b, c, d_{1}, d_{2} are Data-Location of R, and i_{1} is an instruction-location of SCM (R).

Next we state the proposition (1) x is a Data-Location of R iff x ∈ Data-Loc_{SCM}. Let R be a good ring, let s be a state of SCM (R), and let a be a Data-

Location of R. Then s(a) is an element of R.

We now state several propositions:

(2) ⟨⟨0, ε⟩⟩∈ Instr_{SCM}(S).

(3) ⟨⟨0, ε⟩⟩ is an instruction of SCM (R).

(4) If x ∈ {1, 2, 3, 4}, then ⟨⟨x, ⟨d_{1}, d_{2}⟩⟩⟩∈ Instr_{SCM}(S).

(5) ⟨⟨5, ⟨d_{1}, t⟩⟩⟩∈ Instr_{SCM}(S).

(6) ⟨⟨6, ⟨i_{1}⟩⟩⟩∈ Instr_{SCM}(S).

(7) ⟨⟨7, ⟨i_{1}, d_{1}⟩⟩⟩∈ Instr_{SCM}(S).

Let R be a good ring and let a, b be Data-Location of R. The functor a:=b yielding an instruction of SCM (R) is defined by:

(Def. 3) a:=b = ⟨⟨1, ⟨a, b⟩⟩⟩.

The functor AddTo(a, b) yielding an instruction of SCM (R) is defined by: (Def. 4) AddTo(a, b) = ⟨⟨2, ⟨a, b⟩⟩⟩.

The functor SubFrom(a, b) yielding an instruction of SCM (R) is defined by: (Def. 5) SubFrom(a, b) = ⟨⟨3, ⟨a, b⟩⟩⟩.

The functor MultBy(a, b) yielding an instruction of SCM (R) is defined as follows:

(Def. 6) MultBy(a, b) = ⟨⟨4, ⟨a, b⟩⟩⟩.

Let R be a good ring, let a be a Data-Location of R, and let r be an element of the carrier of R. The functor a:=r yields an instruction of SCM (R) and is defined by:

(Def. 7) a:=r = ⟨⟨5, ⟨a, r⟩⟩⟩.

Let R be a good ring and let l be an instruction-location of SCM (R). The functor goto l yielding an instruction of SCM (R) is defined by:

(Def. 8) goto l = ⟨⟨6, ⟨l⟩⟩⟩.

Let R be a good ring, let l be an instruction-location of SCM (R), and let a be a Data-Location of R. The functor if a = 0 goto l yielding an instruction of SCM (R) is defined as follows:

(Def. 9) if a = 0 goto l = ⟨⟨7, ⟨l, a⟩⟩⟩. One can prove the following proposition

(8) Let I be a set. Then I is an instruction of SCM (R) if and only if one of the following conditions is satisfied:

(i) I = ⟨⟨0, ε⟩⟩, or (ii) there exist a, b such that I = a:=b, or (iii) there exist a, b such that I = AddTo(a, b), or (iv) there exist a, b such that I = SubFrom(a, b), or (v) there exist a, b such that I = MultBy(a, b), or (vi) there exists i_{1} such that I = goto i_{1}, or (vii) there exist a, i_{1} such that I = if a = 0 goto i_{1}, or (viii) there exist a, r such that I = a:=r.

In the sequel s denotes a state of SCM (R). Let us consider R. Observe that SCM (R) is von Neumann. The following two propositions are true:

(9) IC_{SCM (R)} = 0.

(10) For every SCM-state S over R such that S = s holds IC_{s} = IC_{S}. Let R be a good ring and let i_{1} be an instruction-location of SCM (R). The

functor Next(i_{1}) yields an instruction-location of SCM (R) and is defined by: (Def. 10) There exists an element m_{1} of Instr-Loc_{SCM} such that m_{1} = i_{1} and

Next(i_{1}) = Next(m_{1}). The following propositions are true:

(11) For every instruction-location i_{1} of SCM (R) and for every element m_{1} of Instr-Loc_{SCM} such that m_{1} = i_{1} holds Next(m_{1}) = Next(i_{1}).

(12) Let I be an instruction of SCM (R) and i be an element of Instr_{SCM}(R). If i = I, then for every SCM-state S over R such that S = s holds Exec(I, s) = Exec-Res_{SCM}(i, S).

2. Users Guide

Next we state several propositions:

(13) (Exec(a:=b, s))(IC_{SCM (R)}) = Next(IC_{s}) and (Exec(a:=b, s))(a) = s(b) and for every c such that c ≠ a holds (Exec(a:=b, s))(c) = s(c).

(14) (Exec(AddTo(a, b), s))(IC_{SCM (R)}) = Next(IC_{s}) and (Exec(AddTo(a, b), s))(a) = s(a) + s(b) and for every c such that c ≠ a holds (Exec(AddTo(a, b), s))(c) = s(c).

(15) (Exec(SubFrom(a, b), s))(IC_{SCM (R)}) = Next(IC_{s}) and (Exec(SubFrom(a, b), s))(a) = s(a) − s(b) and for every c such that c ≠ a holds (Exec(SubFrom(a, b), s))(c) = s(c).

(16) (Exec(MultBy(a, b), s))(IC_{SCM (R)}) = Next(IC_{s}) and (Exec(MultBy(a, b), s))(a) = s(a) · s(b) and for every c such that c ≠ a holds (Exec(MultBy(a, b), s))(c) = s(c).

(17) (Exec(goto i_{1}, s))(IC_{SCM (R)}) = i_{1} and (Exec(goto i_{1}, s))(c) = s(c).

(18) If s(a) = 0_{R}, then (Exec(if a = 0 goto i_{1}, s))(IC_{SCM (R)}) = i_{1} and if s(a) ≠ 0_{R}, then (Exec(if a = 0 goto i_{1}, s))(IC_{SCM (R)}) = Next(IC_{s}) and (Exec(if a = 0 goto i_{1}, s))(c) = s(c).

(19) (Exec(a:=r, s))(IC_{SCM (R)}) = Next(IC_{s}) and (Exec(a:=r, s))(a) = r and for every c such that c ≠ a holds (Exec(a:=r, s))(c) = s(c).

3. Halt Instruction

The following two propositions are true:

(20) For every instruction I of SCM (R) such that there exists s such that (Exec(I, s))(IC_{SCM (R)}) = Next(IC_{s}) holds I is non halting.

(21) For every instruction I of SCM (R) such that I = ⟨⟨0, ε⟩⟩ holds I is

halting. Let us consider R, a, b. One can check the following observations:

∗ a:=b is non halting,

∗ AddTo(a, b) is non halting,

∗ SubFrom(a, b) is non halting, and

∗ MultBy(a, b) is non halting. Let us consider R, i_{1}. Observe that goto i_{1} is non halting. Let us consider R, a, i_{1}. Observe that if a = 0 goto i_{1} is non halting. Let us consider R, a, r. Note that a:=r is non halting. Let us consider R. One can check that SCM (R) is halting definite data-

oriented steady-programmed and realistic.

One can prove the following propositions: (29)^{1} For every instruction I of SCM (R) such that I is halting holds I =

halt_{SCM (R)}.

(30) halt_{SCM (R)} = ⟨⟨0, ε⟩⟩.

References

[1] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 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] Artur Korniłowicz. The construction of SCM over ring. Formalized Mathematics, 7(2):295–300, 1998.

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

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

[7] Yatsuka Nakamura and Andrzej Trybulec. On a mathematical model of programs. Formalized Mathematics, 3(2):241–250, 1992.

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

1The propositions (22)–(28) have been removed.

the basic properties of SCM over ring 305

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

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

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

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

306 artur korniłowicz

A Theory of Boolean Valued Functions and Quantifiers with Respect to Partitions

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we define the coordinate of partitions. We also introduce the universal quantifier and the existential quantifier of Boolean valued functions with respect to partitions. Some predicate calculus formulae containing such quantifiers are proved. Such a theory gives a discussion of semantics to usual predicate logic.

MML Identifier: BVFUNC 2.

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

1. Preliminaries

In this paper Y denotes a non empty set and G denotes a subset of PARTITIONS(Y).

Let X be a set. Then PARTITIONS(X) is a partition family of X.

Let X be a set and let F be a non empty partition family of X. We see that the element of F is a partition of X.

The following proposition is true

(1) Let y be an element of Y. Then there exists a subset X of Y such that

(i) y ∈ X, and (ii) there exists a function h and there exists a family F of subsets of Y

such that domh = G and rngh = F and for every set d such that d ∈ G holds h(d) ∈ d and X = Intersect(F) and X ≠ ∅.

Let us consider Y and let G be a subset of PARTITIONS(Y). The functor ⋀ G yielding a partition of Y is defined by the condition (Def. 1).

(Def. 1) Let x be a set. Then x ∈ ⋀ G if and only if there exists a function h and there exists a family F of subsets of Y such that domh = G and rngh = F and for every set d such that d ∈ G holds h(d) ∈ d and x = Intersect(F) and x ≠ ∅.

Let us consider Y, let G be a subset of PARTITIONS(Y), and let b be a set. We say that b is upper min depend of G if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) For every partition d of Y such that d ∈ G holds b is a dependent set

of d, and (ii) for every set e such that e ⊆ b and for every partition d of Y such that

d ∈ G holds e is a dependent set of d holds e = b. One can prove the following proposition

(2) For every element y of Y such that G ≠ ∅ there exists a subset X of Y such that y ∈ X and X is upper min depend of G.

Let us consider Y and let G be a subset of PARTITIONS(Y). The functor ⋁ G yielding a partition of Y is defined by:

(Def. 3)(i) For every set x holds x ∈ ⋁ G iff x is upper min depend of G if G ≠ ∅, (ii) ⋁ G = I(Y), otherwise. The following propositions are true:

(3) For every subset G of PARTITIONS(Y) and for every partition P_{1} of Y such that P_{1} ∈ G holds P_{1} ⋑ ⋀ G.

(4) For every subset G of PARTITIONS(Y) and for every partition P_{1} of Y such that P_{1} ∈ G holds P_{1} ⋐ ⋁ G.

2. Coordinate and Quantifiers

Let us consider Y and let G be a subset of PARTITIONS(Y). We say that G is a generating family of partitions if and only if:

(Def. 4) ⋀ G = I(Y).

Let us consider Y and let G be a subset of PARTITIONS(Y). We say that G is an independent family of partitions if and only if the condition (Def. 5) is satisfied.

(Def. 5) Let h be a function and F be a family of subsets of Y. Suppose domh = G and rngh = F and for every set d such that d ∈ G holds h(d) ∈ d. Then Intersect(F) ≠ ∅.

Let us consider Y and let G be a subset of PARTITIONS(Y). We say that G is a coordinate if and only if the conditions (Def. 6) are satisfied.

(Def. 6)(i) G is an independent family of partitions , (ii) G is a generating family of partitions, and (iii) for all partitions d_{1}, d_{2} of Y such that d_{1} ∈ G and d_{2} ∈ G and d_{1} ≠ d_{2}

holds d_{1} ∨ d_{2} = O(Y). Let us consider Y and let P_{1} be a partition of Y. Then {P_{1}} is a subset of

PARTITIONS(Y).

Let us consider Y, let P_{1} be a partition of Y, and let G be a subset of PARTITIONS(Y). The functor CompF(P_{1}, G) yielding a partition of Y is defined by:

(Def. 7) CompF(P_{1}, G) = ⋀ G \ {P_{1}}.

Let us consider Y, let a be an element of BVF(Y), let G be a subset of PARTITIONS(Y), and let P_{1} be a partition of Y. We say that a is independent of P_{1}, G if and only if:

(Def. 8) a is dependent of CompF(P_{1}, G).

Let us consider Y, let a be an element of BVF(Y), let G be a subset of PARTITIONS(Y), and let P_{1} be a partition of Y. The functor ∀_{a,P_{1}}G yielding an element of BVF(Y) is defined by:

(Def. 9) ∀_{a,P_{1}}G = INF(a, CompF(P_{1}, G)).

Let us consider Y, let a be an element of BVF(Y), let G be a subset of PARTITIONS(Y), and let P_{1} be a partition of Y. The functor ∃_{a,P_{1}}G yielding an element of BVF(Y) is defined as follows:

(Def. 10) ∃_{a,P_{1}}G = SUP(a, CompF(P_{1}, G)). One can prove the following propositions:

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a,P_{1}}G is dependent of CompF(P_{1}, G).

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{a,P_{1}}G is dependent of CompF(P_{1}, G).

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{true(Y),P_{1}}G = true(Y).

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{true(Y),P_{1}}G = true(Y).

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{false (Y),P_{1}}G = false(Y).

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{false (Y),P_{1}}G =

false(Y).

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a,P_{1}}G ⋐ a.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then a ⋐ ∃_{a,P_{1}}G.

(13) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a∧b,P_{1}} G = ∀_{a,P_{1}}G ∧ ∀_{b,P_{1}}G.

(14) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a,P_{1}}G∨ ∀_{b,P_{1}}G ⋐ ∀_{a∨b,P_{1}} G.

(15) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a⇒b,P_{1}} G ⋐ ∀_{a,P_{1}}G ⇒ ∀_{b,P_{1}}G.

(16) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{a∨b,P_{1}} G = ∃_{a,P_{1}}G ∨ ∃_{b,P_{1}}G.

(17) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{a∧b,P_{1}} G ⋐ ∃_{a,P_{1}}G ∧ ∃_{b,P_{1}}G.

(18) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{a,P_{1}}G⊕ ∃_{b,P_{1}}G ⋐ ∃_{a⊕b,P_{1}} G.

(19) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∃_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G ⋐ ∃_{a⇒b,P_{1}} G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ¬∀_{a,P_{1}}G = ∃_{¬a,P_{1}} G.

(21) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ¬∃_{a,P_{1}}G = ∀_{¬a,P_{1}} G.

(22) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{u⇒a,P_{1}} G = u ⇒ ∀_{a,P_{1}}G.

(23) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a⇒u,P_{1}} G = ∃_{a,P_{1}}G ⇒ u.

(24) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u

is independent of P_{1}, G. Then ∀_{u∨a,P_{1}} G = u ∨ ∀_{a,P_{1}}G.

(25) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a∨u,P_{1}} G = ∀_{a,P_{1}}G ∨ u.

(26) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a∨u,P_{1}} G ⋐ ∃_{a,P_{1}}G ∨ u.

(27) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{u∧a,P_{1}} G = u ∧ ∀_{a,P_{1}}G.

(28) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a∧u,P_{1}} G = ∀_{a,P_{1}}G ∧ u.

(29) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a∧u,P_{1}} G ⋐ ∃_{a,P_{1}}G ∧ u.

(30) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{u⊕a,P_{1}} G ⋐ u ⊕ ∀_{a,P_{1}}G.

(31) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a⊕u,P_{1}} G ⋐ ∀_{a,P_{1}}G ⊕ u.

(32) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{u⇔a,P_{1}} G ⋐ u ⇔ ∀_{a,P_{1}}G.

(33) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a⇔u,P_{1}} G ⋐ ∀_{a,P_{1}}G ⇔ u.

(34) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{u∨a,P_{1}} G = u ∨ ∃_{a,P_{1}}G.

(35) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{a∨u,P_{1}} G = ∃_{a,P_{1}}G ∨ u.

(36) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{u∧a,P_{1}} G = u ∧ ∃_{a,P_{1}}G.

(37) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{a∧u,P_{1}} G = ∃_{a,P_{1}}G ∧ u.

312 shunichi kobayashi and yatsuka nakamura

(38) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then u ⇒ ∃_{a,P_{1}}G ⋐ ∃_{u⇒a,P_{1}} G.

(39) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{a,P_{1}}G ⇒ u ⋐ ∃_{a⇒u,P_{1}} G.

(40) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then u ⊕ ∃_{a,P_{1}}G ⋐ ∃_{u⊕a,P_{1}} G.

(41) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{a,P_{1}}G ⊕ u ⋐ ∃_{a⊕u,P_{1}} G.

References

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

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

[3] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 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] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233–236, 1996.

[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.

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

Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we have 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: BVFUNC 3.

The terminology and notation used here are introduced in the following articles: [1], [2], [3], and [4].

For simplicity, we adopt the following convention: Y denotes a non empty set, G denotes a subset of PARTITIONS(Y), a, b, c, u denote elements of BVF(Y), and P_{1} denotes a partition of Y.

The following propositions are true:

(1) a ⇒ b ⋐ ∀_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G.

(2) ∀_{a,P_{1}}G ∧ ∀_{b,P_{1}}G ⋐ a ∧ b.

(3) a ∧ b ⋐ ∃_{a,P_{1}}G ∧ ∃_{b,P_{1}}G.

(4) ¬(∀_{a,P_{1}}G ∧ ∀_{b,P_{1}}G) = ∃_{¬a,P_{1}} G ∨ ∃_{¬b,P_{1}} G.

(5) ¬(∃_{a,P_{1}}G ∧ ∃_{b,P_{1}}G) = ∀_{¬a,P_{1}} G ∨ ∀_{¬b,P_{1}} G.

(6) ∀_{a,P_{1}}G ∨ ∀_{b,P_{1}}G ⋐ a ∨ b.

(7) a ∨ b ⋐ ∃_{a,P_{1}}G ∨ ∃_{b,P_{1}}G.

(8) a ⊕ b ⋐ ¬(∃_{¬a,P_{1}} G ⊕ ∃_{b,P_{1}}G) ∨ ¬(∃_{a,P_{1}}G ⊕ ∃_{¬b,P_{1}} G).

(9) ∀_{a∨b,P_{1}} G ⋐ ∀_{a,P_{1}}G ∨ ∃_{b,P_{1}}G.

(10) ∀_{a∨b,P_{1}} G ⋐ ∃_{a,P_{1}}G ∨ ∀_{b,P_{1}}G.

(11) ∀_{a∨b,P_{1}} G ⋐ ∃_{a,P_{1}}G ∨ ∃_{b,P_{1}}G.

(12) ∃_{a,P_{1}}G ∧ ∀_{b,P_{1}}G ⋐ ∃_{a∧b,P_{1}} G.

(13) ∀_{a,P_{1}}G ∧ ∃_{b,P_{1}}G ⋐ ∃_{a∧b,P_{1}} G.

(14) ∀_{a⇒b,P_{1}} G ⋐ ∀_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G.

(15) ∀_{a⇒b,P_{1}} G ⋐ ∃_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G.

(16) ∃_{a,P_{1}}G ⇒ ∀_{b,P_{1}}G ⋐ ∀_{a⇒b,P_{1}} G.

(17) a ⇒ b ⋐ a ⇒ ∃_{b,P_{1}}G.

(18) a ⇒ b ⋐ ∀_{a,P_{1}}G ⇒ b.

(19) ∃_{a⇒b,P_{1}} G ⋐ ∀_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G.

(20) ∀_{a,P_{1}}G ⋐ ∃_{b,P_{1}}G ⇒ ∃_{a∧b,P_{1}} G.

(21) If u is independent of P_{1}, G, then ∃_{u⇒a,P_{1}} G ⋐ u ⇒ ∃_{a,P_{1}}G.

(22) If u is independent of P_{1}, G, then ∃_{a⇒u,P_{1}} G ⋐ ∀_{a,P_{1}}G ⇒ u.

(23) ∀_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G = ∃_{a⇒b,P_{1}} G.

(24) ∀_{a,P_{1}}G ⇒ ∀_{b,P_{1}}G ⋐ ∀_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G.

(25) ∃_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G ⋐ ∀_{a,P_{1}}G ⇒ ∃_{b,P_{1}}G.

(26) ∀_{a⇒b,P_{1}} G = ∀_{¬a∨b,P_{1}} G.

(27) If G is a coordinate and P_{1} ∈ G, then ∀_{a⇒b,P_{1}} G = ¬∃_{a∧¬b,P_{1}} G.

(28) ∃_{a,P_{1}}G ⋐ ¬(∀_{a⇒b,P_{1}} G ∧ ∀_{a⇒¬b,P_{1}} G).

(29) ∃_{a,P_{1}}G ⋐ ¬(¬∃_{a∧b,P_{1}} G ∧ ¬∃_{a∧¬b,P_{1}} G).

(30) ∃_{a,P_{1}}G ∧ ∀_{a⇒b,P_{1}} G ⋐ ∃_{a∧b,P_{1}} G.

(31) ∃_{a,P_{1}}G ∧ ¬∃_{a∧b,P_{1}} G ⋐ ¬∀_{a⇒b,P_{1}} G.

(32) ∀_{a⇒c,P_{1}} G ∧ ∀_{c⇒b,P_{1}} G ⋐ ∀_{a⇒b,P_{1}} G.

(33) ∀_{c⇒b,P_{1}} G ∧ ∃_{a∧c,P_{1}} G ⋐ ∃_{a∧b,P_{1}} G.

(34) ∀_{b⇒¬c,P_{1}} G ∧ ∀_{a⇒c,P_{1}} G ⋐ ∀_{a⇒¬b,P_{1}} G.

(35) ∀_{b⇒c,P_{1}} G ∧ ∀_{a⇒¬c,P_{1}} G ⋐ ∀_{a⇒¬b,P_{1}} G.

(36) ∀_{b⇒¬c,P_{1}} G ∧ ∃_{a∧c,P_{1}} G ⋐ ∃_{a∧¬b,P_{1}} G.

(37) ∀_{b⇒c,P_{1}} G ∧ ∃_{a∧¬c,P_{1}} G ⋐ ∃_{a∧¬b,P_{1}} G.

(38) ∃_{c,P_{1}}G ∧ ∀_{c⇒b,P_{1}} G ∧ ∀_{c⇒a,P_{1}} G ⋐ ∃_{a∧b,P_{1}} G.

(39) ∀_{b⇒c,P_{1}} G ∧ ∀_{c⇒¬a,P_{1}} G ⋐ ∀_{a⇒¬b,P_{1}} G.

(40) ∃_{b,P_{1}}G ∧ ∀_{b⇒c,P_{1}} G ∧ ∀_{c⇒a,P_{1}} G ⋐ ∃_{a∧b,P_{1}} G.

(41) ∃_{c,P_{1}}G ∧ ∀_{b⇒¬c,P_{1}} G ∧ ∀_{c⇒a,P_{1}} G ⋐ ∃_{a∧¬b,P_{1}} 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 … 315

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

Received December 21, 1998 316 shunichi kobayashi and yatsuka nakamura

Public-Key Cryptography and Pepin’s Test for the Primality of Fermat Numbers

Yoshinori Fujisawa Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Hidetaka Shimizu Information Technology Research Institute of Nagano Prefecture

Summary. In this article, we have proved the correctness of the Public-Key Cryptography and the Pepin’s Test for the Primality of Fermat Numbers (F (n) = 2^{2^{n}} + 1). It is a very important result in the IDEA Cryptography that F(4) is a prime number. At first, we prepared some useful theorems. Then, we proved the correctness of the Public-Key Cryptography. Next, we defined the Order’s function and proved some properties. This function is very important in the proof of the Pepin’s Test. Next, we proved some theorems about the Fermat Number. And finally, we proved the Pepin’s Test using some properties of the Order’s Function. And using the obtained result we have proved that F(1), F(2), F(3) and F(4) are prime number.

MML Identifier: PEPIN.

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

1. Some Useful Theorems

We adopt the following convention: d, i, j, k, m, n, p, q, k_{1}, k_{2} are natural numbers and a, b, c, i_{1}, i_{2}, i_{3}, i_{4}, i_{5} are integers.

One can prove the following four propositions:

(1) For every i holds i and i + 1 are relative prime.

(2) For every p such that p is prime holds m and p are relative prime or gcd(m, p) = p.

(3) If k | n · m and n and k are relative prime, then k | m.

(4) If n | m and k | m and n and k are relative prime, then n · k | m.

Let n be a natural number. Then n^{2} is a natural number.

We now state a number of propositions:

(5) If c > 1, then 1 mod c = 1.

(6) For every i such that i ≠ 0 holds i | n iff n mod i = 0.

(7) If m ≠ 0 and m | n mod m, then m | n.

(8) If 0 < n and m mod n = k, then n | m − k.

(9) If i · p ≠ 0 and p is prime and k modi · p < p, then k modi · p = k modp.

(10) If p ≠ 0, then (a · p + 1) mod p = 1 mod p.

(11) If 1 < m and n · k mod m = k mod m and k and m are relative prime, then n mod m = 1.

(12) If m ≠ 0, then (p_{ℕ}^{k}) mod m = ((p mod m)_{ℕ}^{k}) mod m.

(13) If i ≠ 0, then i^{2} mod (i + 1) = 1.

(14) If j ≠ 0 and k^{2} < j and i mod j = k, then i^{2} mod j = k^{2}.

(15) If p is prime and i mod p = − 1, then i^{2} mod p = 1.

(16) If n is even, then n + 1 is odd.

(17) If p > 2 and p is prime, then p is odd.

(18) If n > 0, then the n-th power of 2 is even.

(19) If i is odd and j is odd, then i · j is odd.

(20) For every k such that i is odd holds i_{ℕ}^{k} is odd.

(21) If k > 0 and i is even, then i_{ℕ}^{k} is even.

(22) 2 | n iff n is even.

(23) If m · n is even, then m is even or n is even.

(24) n_{ℕ}^{2} = n^{2}.

(25) 2_{ℕ}^{k} = the k-th power of 2.

(26) If m > 1 and n > 0, then m_{ℕ}^{n} > 1.

(27) If n ≠ 0 and p ≠ 0, then n_{ℕ}^{p} = n · n_{ℕ}^{p− ′1}.

(28) For all n, m such that m mod 2 = 0 holds (n_{ℕ}^{m÷2} )^{2} = n_{ℕ}^{m}.

(29) If n ≠ 0 and 1 k, then (n_{ℕ}^{k}) ÷ n = n_{ℕ}^{k− ′1}.

(30) 2_{ℕ}^{n+1} = (2_{ℕ}^{n}) + 2_{ℕ}^{n}.

(31) If k > 1 and k_{ℕ}^{n} = k_{ℕ}^{m}, then n = m.

(32) m n iff 2_{ℕ}^{m} | 2_{ℕ}^{n}.

(33) If p is prime and i | p_{ℕ}^{n}, then i = 1 or there exists a natural number k such that i = p · k.

(34) For every n such that n ≠ 0 and p is prime and n < p_{ℕ}^{k+1} holds n | p_{ℕ}^{k+1} iff n | p_{ℕ}^{k}.

(35) For every k such that p is prime and d | p_{ℕ}^{k} and d ≠ 0 there exists a natural number t such that d = p_{ℕ}^{t} and t k.

(36) If p > 1 and i mod p = 1, then (i_{ℕ}^{n}) mod p = 1.

(37) If m > 0 and n > 0, then (n_{ℕ}^{m}) mod n = 0.

(38) If p is prime and n and p are relative prime, then (n_{ℕ}^{p− ′1}) mod p = 1.

(39) If p is prime and d > 1 and d | p_{ℕ}^{k} and d ∤ (p_{ℕ}^{k}) ÷ p, then d = p_{ℕ}^{k}. Let a be an integer. Then a^{2} is a natural number. We now state several propositions:

(40) For every n such that n > 1 holds m mod n = 1 iff m ≡ 1(modn).

(41) If a ≡ b(modc), then a^{2} ≡ b^{2}(modc).

(42) If i_{5} = i_{3} · i_{4} and i_{1} ≡ i_{2}(modi_{5}), then i_{1} ≡ i_{2}(modi_{3}) and i_{1} ≡ i_{2}(modi_{4}).

(43) If i_{1} ≡ i_{2}(modi_{5}) and i_{1} ≡ i_{3}(modi_{5}), then i_{2} ≡ i_{3}(modi_{5}).

(44) 3 is prime.

(45) If n ≠ 0, then Eulern ≠ 0.

(46) If n ≠ 0, then − n < n.

(47) For all m, n such that n > 0 and n > m holds m ÷ n = 0.

(48) If n ≠ 0, then n ÷ n = 1.

2. Public-Key Cryptography

Let us consider k, m, n. The functor Crypto(m, n, k) yielding a natural number is defined as follows:

(Def. 1) Crypto(m, n, k) = (m_{ℕ}^{k}) mod n.

One can prove the following proposition

(49) Suppose p is prime and q is prime and p ≠ q and n = p · q and k_{1} and Eulern are relative prime and k_{1} · k_{2} mod Eulern = 1. Let m be a natural number. If m < n, then Crypto(Crypto(m, n, k_{1}), n, k_{2}) = m.

3. Order’s Function

Let us consider i, p. Let us assume that p > 1 and i and p are relative prime. The functor order(i, p) yields a natural number and is defined as follows:

(Def. 2) order(i, p) > 0 and (i_{ℕ}^{order(i,p)}) modp = 1 and for every k such that k > 0

and (i_{ℕ}^{k}) mod p = 1 holds 0 < order(i, p) and order(i, p) k. One can prove the following propositions:

(50) If p > 1, then order(1, p) = 1.

(51) If p > 1 and i and p are relative prime, then order(i, p) ≠ 0.

(52) If p > 1 and n > 0 and (i_{ℕ}^{n}) mod p = 1 and i and p are relative prime, then order(i, p) | n.

(53) If p > 1 and i and p are relative prime and order(i, p) | n, then (i_{ℕ}^{n}) mod p = 1.

(54) If p is prime and i and p are relative prime, then order(i, p) | p − ′1.

4. Fermat Number

Let n be a natural number. The functor Fermatn yielding a natural number is defined as follows:

(Def. 3) Fermatn = (2_{ℕ}^{2_{ℕ}^{n}}) + 1.

Next we state several propositions:

(55) Fermat0 = 3.

(56) Fermat1 = 5.

(57) Fermat2 = 17.

(58) Fermat3 = 257.

(59) Fermat4 = 256 · 256 + 1.

(60) Fermatn > 2.

(61) If p is prime and p > 2 and p | Fermatn, then there exists a natural number k such that p = k · 2_{ℕ}^{n+1} + 1.

(62) If n ≠ 0, then 3 and Fermatn are relative prime. public-key cryptography and pepin’s test … 321

5. Pepin’s Test

We now state several propositions:

(63) If n > 0 and 3_{ℕ}^{(Fermat n− ′1)÷2} ≡ − 1(modFermatn), then Fermatn is prime.

(64) 5 is prime.

(65) 17 is prime.

(66) 257 is prime.

(67) 256 · 256 + 1 is prime.

References

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

[2] Yoshinori Fujisawa and Yasushi Fuwa. The Euler’s function. Formalized Mathematics, 6(4):549–551, 1997.

[3] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Euler’s Theorem and small Fermat’s Theorem. Formalized Mathematics, 7(1):123–126, 1998.

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

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

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

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

[8] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

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

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

322 yoshinori fujisawa et al.

Lattice of Substitutions is a Heyting Algebra

Adam Grabowski University of Białystok

MML Identifier: HEYTING2.

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

1. Preliminaries

We adopt the following convention: V, C, x are sets and A, B are elements of SubstitutionSet(V, C).

Let a, b be sets. Note that {⟨⟨a, b⟩⟩} is function-like and relation-like.

Let A, B be sets. Observe that A ˙→ B is functional.

Next we state several propositions:

(1) For all non empty sets V, C there exists an element f of V ˙→ C such that f ≠ ∅.

(2) For all sets a, b such that b ∈ SubstitutionSet(V, C) and a ∈ b holds a is a finite function.

(3) For every element f of V ˙→ C and for every set g such that g ⊆ f holds g ∈ V ˙→ C.

(4) V ˙→ C ⊆ 2^{[: V, C:]}.

(5) If V is finite and C is finite, then V ˙→ C is finite.

One can check that there exists a set which is functional, finite, and non empty.

2. Some Properties of Sets of Substitutions

One can prove the following four propositions:

(6) For every finite element a of V ˙→ C holds {a} ∈ SubstitutionSet(V, C).

(7) If A^{⌢} B = A, then for every set a such that a ∈ A there exists a set b such that b ∈ B and b ⊆ a.

(8) If μ(A^{⌢} B) = A, then for every set a such that a ∈ A there exists a set b such that b ∈ B and b ⊆ a.

(9) If for every set a such that a ∈ A there exists a set b such that b ∈ B and b ⊆ a, then μ(A^{⌢} B) = A.

Let V be a set, let C be a finite set, and let A be an element of Fin(V ˙→ C). The functor InvolvedA is defined by:

(Def. 1) x ∈ InvolvedA iff there exists a finite function f such that f ∈ A and

x ∈ domf. In the sequel C denotes a finite set. The following propositions are true:

(10) For every set V and for every finite set C and for every element A of Fin(V ˙→ C) holds InvolvedA ⊆ V.

(11) For every set V and for every finite set C and for every element A of Fin(V ˙→ C) such that A = ∅ holds InvolvedA = ∅.

(12) For every set V and for every finite set C and for every element A of Fin(V ˙→ C) holds InvolvedA is finite.

(13) For every finite set C and for every element A of Fin(∅ ˙→ C) holds

InvolvedA = ∅. Let V be a set, let C be a finite set, and let A be an element of Fin(V ˙→ C).

The functor − A yielding an element of Fin(V ˙→ C) is defined as follows: (Def. 2) − A = {f; f ranges over elements of InvolvedA ˙→ C : ⋀_{g: element of V ˙→ C} (g ∈

A ⇒ f ≉ g)}. One can prove the following propositions:

(14) A^{⌢} − A = ∅.

(15) If A = ∅, then − A = {∅}.

(16) If A = {∅}, then − A = ∅.

(17) For every set V and for every finite set C and for every element A of SubstitutionSet(V, C) holds μ(A^{⌢} − A) = ⊥_{SubstLatt (V,C)}.

(18) For every non empty set V and for every finite non empty set C and for every element A of SubstitutionSet(V, C) such that A = ∅ holds μ(− A) = ⊤_{SubstLatt (V,C)}.

(19) Let V be a set, C be a finite set, A be an element of SubstitutionSet(V, C), a be an element of V ˙→ C, and B be an element of SubstitutionSet(V, C). Suppose B = {a}. If A ^{⌢} B = ∅, then there exists a finite set b such that b ∈ − A and b ⊆ a.

Let V be a set, let C be a finite set, and let A, B be elements of Fin(V ˙→ C). The functor A ↣ B yielding an element of Fin(V ˙→ C) is defined as follows:

(Def. 3) A ↣ B = (V ˙→ C) ∩ {⋃{f(i) \ i; i ranges over elements of V ˙→ C :

i ∈ A}; f ranges over elements of A ˙→ B : domf = A}. Next we state two propositions:

(20) Let A, B be elements of Fin(V ˙→ C) and s be a set. Suppose s ∈ A ↣ B. Then there exists a partial function f from A to B such that s = ⋃{f(i) \ i; i ranges over elements of V ˙→ C : i ∈ A} and domf = A.

(21) For every set V and for every finite set C and for every element A of

Fin(V ˙→ C) such that A = ∅ holds A ↣ A = {∅}. We adopt the following convention: u, v are elements of the carrier

of SubstLatt(V, C), a is an element of V ˙→ C, and K, L are elements of SubstitutionSet(V, C).

The following proposition is true

(22) For every set X such that X ⊆ u holds X is an element of the carrier of SubstLatt(V, C).

3. Lattice of Substitutions is Implicative

Let us consider V, C. The functor pseudo compl(V, C) yielding a unary operation on the carrier of SubstLatt(V, C) is defined as follows:

(Def. 4) For every element u′ of SubstitutionSet(V, C) such that u′ = u holds

(pseudo compl(V, C))(u) = μ(− u′). The functor StrongImpl(V, C) yielding a binary operation on the carrier of SubstLatt(V, C) is defined by:

(Def. 5) For all elements u′, v′ of SubstitutionSet(V, C) such that u′ = u and

v′= v holds (StrongImpl(V, C))(u, v) = μ(u′↣ v′). Let us consider u. The functor 2^{u} yielding an element of Fin(the carrier of SubstLatt(V, C)) is defined by:

(Def. 6) 2^{u} = 2^{u}.

The functor □ \_{u} □ yielding a unary operation on the carrier of SubstLatt(V, C) is defined by:

(Def. 7) (□ \_{u} □)(v) = u \ v.

Let us consider V, C. The functor Atom(V, C) yielding a function from V ˙→ C into the carrier of SubstLatt(V, C) is defined as follows:

(Def. 8) For every element a of V ˙→ C holds (Atom(V, C))(a) = μ{a}.

Next we state a number of propositions:

(23) ⨆_{K}^{f} Atom(V, C) = FinUnion(K, singleton_{V ˙→ C}).

(24) For every element u of SubstitutionSet(V, C) holds u = ⨆_{u}^{f} Atom(V, C).

(25) (□ \_{u} □)(v) ⊑ u.

(26) For every element a of V ˙→ C such that a is finite and for every set c such that c ∈ (Atom(V, C))(a) holds c = a.

(27) For every element a of V ˙→ C such that K = {a} and L = u and L^{⌢}K = ∅ holds (Atom(V, C))(a) ⊑ (pseudo compl(V, C))(u).

(28) For every finite element a of V ˙→ C holds a ∈ (Atom(V, C))(a).

(29) Let u, v be elements of SubstitutionSet(V, C). Suppose that for every set c such that c ∈ u there exists a set b such that b ∈ v and b ⊆ c ∪ a. Then there exists a set b such that b ∈ u ↣ v and b ⊆ a.

(30) Let a be a finite element of V ˙→ C. Suppose for every element b of V ˙→ C such that b ∈ u holds b ≈ a and u ⊓ (Atom(V, C))(a) ⊑ v. Then (Atom(V, C))(a) ⊑ (StrongImpl(V, C))(u, v).

(31) u ⊓ (pseudo compl(V, C))(u) = ⊥_{SubstLatt (V,C)}.

(32) u ⊓ (StrongImpl(V, C))(u, v) ⊑ v.

Let us consider V, C. Observe that SubstLatt(V, C) is implicative.

One can prove the following proposition

(33) u ⇒ v = ⨆_{2^{u}}^{f}((the meet operation of SubstLatt(V, C))^{∘}(pseudo compl(V, C), (StrongImpl(V, C))^{∘}(□ \_{u} □, v))).

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Index of MML Identifiers

BVFUNC 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .249

BVFUNC 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307

BVFUNC 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313

CONLAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233

GRAPH 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

HEYTING2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

IDEA 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

JGRAPH 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193

MSSUBLAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

PARTIT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

PEPIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

SCMRING1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

SCMRING2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

SIN COS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

SPRECT 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265

TOPGRP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217

VECTMETR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

WAYBEL19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

WAYBEL20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

WAYBEL21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

WAYBEL22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

WAYBEL23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

YELLOW13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Contents Formaliz. Math. 7 (2)

The Lawson Topology

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Kernel Projections and Quotient Lattices

By Piotr Rudnicki  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Lawson Topology in Continuous Lattices

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Representation Theorem for Free Continuous Lattices

By Piotr Rudnicki  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Oriented Chains

By Yatsuka Nakamura and Piotr Rudnicki  . . . . . . . . . . . . . . . 189

Graph Theoretical Properties of Arcs in the Plane and Fashoda

Meet Theorem

By Yatsuka Nakamura  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Algebraic Group on Fixed-length Bit Integer and its Adaptation

to IDEA Cryptography

By Yasushi Fuwa and Yoshinori Fujisawa  . . . . . . . . . . . . . . . . . 203

The Definition and Basic Properties of Topological Groups

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

The Correspondence Between Lattices of Subalgebras of Universal

Algebras and Many Sorted Algebras

By Adam Naumowicz and Agnieszka Julia Marasik  . . . . . . 227

Introduction to Concept Lattices

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Continued on inside back cover

A Theory of Partitions. Part I

By Shunichi Kobayashi and Kui Jia  . . . . . . . . . . . . . . . . . . . . . . . . 243

A Theory of Boolean Valued Functions and Partitions

By Shunichi Kobayashi and Kui Jia  . . . . . . . . . . . . . . . . . . . . . . . . 249

Trigonometric Functions and Existence of Circle Ratio

By Yuguang Yang and Yasunari Shidama  . . . . . . . . . . . . . . . . . 255

Some Properties of Special Polygonal Curves

By Andrzej Trybulec and Yatsuka Nakamura  . . . . . . . . . . . 265

Real Linear-Metric Space and Isometric Functions

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Introduction to Meet-Continuous Topological Lattices

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Bases of Continuous Lattices

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

The Construction of SCM over Ring

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

The Basic Properties of SCM over Ring

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

A Theory of Boolean Valued Functions and Quantifiers with Re-

spect to Partitions

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 307

Predicate Calculus for Boolean Valued Functions. Part I

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 313

Public-Key Cryptography and Pepin’s Test for the Primality of

Fermat Numbers

By Yoshinori Fujisawa et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Lattice of Substitutions is a Heyting Algebra

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328