FORMALIZED MATHEMATICS Volume 9, Number 4, 2001 University of Białystok

On the Instructions of SCM^{1}

Artur Korniłowicz University of Białystok

MML Identifier: AMI 6.

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

For simplicity, we adopt the following rules: a, b are data-locations, i_{1}, i_{2}, i_{3} are instruction-locations of SCM , s_{1}, s_{2} are states of SCM , T is an instruction type of SCM , and k is a natural number.

We now state a number of propositions:

(1) a ∉ the instruction locations of SCM .

(2) Data-Loc_{SCM} ≠ the instruction locations of SCM .

(3) For every object o of SCM holds o = IC_{SCM} or o ∈ the instruction locations of SCM or o is a data-location.

(4) If i_{2} ≠ i_{3}, then Next(i_{2}) ≠ Next(i_{3}).

(5) If s_{1} and s_{2} are equal outside the instruction locations of SCM , then s_{1}(a) = s_{2}(a).

(6) Let N be a set with non empty elements, S be a realistic IC-Ins-separated definite non empty non void AMI over N, t, u be states of S, i_{1} be an instruction-location of S, e be an element of ObjectKind(IC_{S}), and I be an element of ObjectKind(i_{1}). If e = i_{1} and u = t+·[IC_{S} ⟼→ e, i_{1} ⟼→ I], then u(i_{1}) = I and IC_{u} = i_{1} and IC_{Following (u)} = (Exec(u(IC_{u}), u))(IC_{S}).

(7) AddressPart(halt_{SCM} ) = ∅.

(8) AddressPart(a:=b) = ⟨a, b⟩.

(9) AddressPart(AddTo(a, b)) = ⟨a, b⟩.

(10) AddressPart(SubFrom(a, b)) = ⟨a, b⟩.

(11) AddressPart(MultBy(a, b)) = ⟨a, b⟩.

1This work has been partially supported by TYPES grant IST-1999-29001.

(12) AddressPart(Divide(a, b)) = ⟨a, b⟩.

(13) AddressPart(goto i_{2}) = ⟨i_{2}⟩.

(14) AddressPart(if a = 0 goto i_{2}) = ⟨i_{2}, a⟩.

(15) AddressPart(if a > 0 goto i_{2}) = ⟨i_{2}, a⟩.

(16) If T = 0, then AddressPartsT = {0}.

Let us consider T. One can check that AddressPartsT is non empty.

The following propositions are true:

(17) If T = 1, then dom∏_{AddressParts T} = {1, 2}.

(18) If T = 2, then dom∏_{AddressParts T} = {1, 2}.

(19) If T = 3, then dom∏_{AddressParts T} = {1, 2}.

(20) If T = 4, then dom∏_{AddressParts T} = {1, 2}.

(21) If T = 5, then dom∏_{AddressParts T} = {1, 2}.

(22) If T = 6, then dom∏_{AddressParts T} = {1}.

(23) If T = 7, then dom∏_{AddressParts T} = {1, 2}.

(24) If T = 8, then dom∏_{AddressParts T} = {1, 2}.

(25) ∏_{AddressParts InsCode (a:=b)}(1) = Data-Loc_{SCM}.

(26) ∏_{AddressParts InsCode (a:=b)}(2) = Data-Loc_{SCM}.

(27) ∏_{AddressParts InsCode (AddTo(a,b))}(1) = Data-Loc_{SCM}.

(28) ∏_{AddressParts InsCode (AddTo(a,b))}(2) = Data-Loc_{SCM}.

(29) ∏_{AddressParts InsCode (SubFrom (a,b))}(1) = Data-Loc_{SCM}.

(30) ∏_{AddressParts InsCode (SubFrom (a,b))}(2) = Data-Loc_{SCM}.

(31) ∏_{AddressParts InsCode (MultBy (a,b))}(1) = Data-Loc_{SCM}.

(32) ∏_{AddressParts InsCode (MultBy (a,b))}(2) = Data-Loc_{SCM}.

(33) ∏_{AddressParts InsCode (Divide(a,b))}(1) = Data-Loc_{SCM}.

(34) ∏_{AddressParts InsCode (Divide(a,b))}(2) = Data-Loc_{SCM}.

(35) ∏_{AddressParts InsCode (goto i_{2})}(1) = the instruction locations of SCM .

(36) ∏_{AddressParts InsCode (if a=0 goto i_{2})}(1) = the instruction locations of SCM .

(37) ∏_{AddressParts InsCode (if a=0 goto i_{2})}(2) = Data-Loc_{SCM}.

(38) ∏_{AddressParts InsCode (if a>0 goto i_{2})}(1) = the instruction locations of SCM .

(39) ∏_{AddressParts InsCode (if a>0 goto i_{2})}(2) = Data-Loc_{SCM}.

(40) NIC(halt_{SCM} , i_{1}) = {i_{1}}.

Let us note that JUMP(halt_{SCM} ) is empty.

One can prove the following proposition

(41) NIC(a:=b, i_{1}) = {Next(i_{1})}.

Let us consider a, b. One can verify that JUMP(a:=b) is empty. Next we state the proposition

(42) NIC(AddTo(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. Note that JUMP(AddTo(a, b)) is empty. The following proposition is true

(43) NIC(SubFrom(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. One can check that JUMP(SubFrom(a, b)) is empty. Next we state the proposition

(44) NIC(MultBy(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. Observe that JUMP(MultBy(a, b)) is empty. The following proposition is true

(45) NIC(Divide(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. Note that JUMP(Divide(a, b)) is empty. We now state two propositions:

(46) NIC(goto i_{2}, i_{1}) = {i_{2}}.

(47) JUMP(goto i_{2}) = {i_{2}}.

Let us consider i_{2}. One can check that JUMP(goto i_{2}) is non empty and trivial.

The following two propositions are true:

(48) i_{2} ∈ NIC(if a = 0 goto i_{2}, i_{1}) and NIC(if a = 0 goto i_{2}, i_{1}) ⊆ {i_{2}, Next(i_{1})}.

(49) JUMP(if a = 0 goto i_{2}) = {i_{2}}.

Let us consider a, i_{2}. Note that JUMP(if a = 0 goto i_{2}) is non empty and trivial.

One can prove the following propositions:

(50) i_{2} ∈ NIC(if a > 0 goto i_{2}, i_{1}) and NIC(if a > 0 goto i_{2}, i_{1}) ⊆ {i_{2}, Next(i_{1})}.

(51) JUMP(if a > 0 goto i_{2}) = {i_{2}}.

Let us consider a, i_{2}. One can check that JUMP(if a > 0 goto i_{2}) is non empty and trivial.

Next we state two propositions:

(52) SUCC(i_{1}) = {i_{1}, Next(i_{1})}.

(53) Let f be a function from ℕ into the instruction locations of SCM . Sup-

pose that for every natural number k holds f(k) = i_{k}. Then

(i) f is bijective, and (ii) for every natural number k holds f(k + 1) ∈ SUCC(f(k)) and for every

natural number j such that f(j) ∈ SUCC(f(k)) holds k j. Let us note that SCM is standard. One can prove the following three propositions:

(54) il_{SCM} (k) = i_{k}.

(55) Next(il_{SCM} (k)) = il_{SCM} (k + 1).

(56) Next(i_{1}) = NextLoci_{1}.

Let us observe that InsCode(halt_{SCM} ) is jump-only. Let us observe that halt_{SCM} is jump-only. Let us consider i_{2}. Observe that InsCode(goto i_{2}) is jump-only. Let us consider i_{2}. Note that goto i_{2} is jump-only non sequential and non

instruction location free.

Let us consider a, i_{2}. One can verify that InsCode(if a = 0 goto i_{2}) is jumponly and InsCode(if a > 0 goto i_{2}) is jump-only.

Let us consider a, i_{2}. One can verify that if a = 0 goto i_{2} is jump-only non sequential and non instruction location free and if a > 0 goto i_{2} is jump-only non sequential and non instruction location free.

Let us consider a, b. One can verify the following observations:

∗ InsCode(a:=b) is non jump-only,

∗ InsCode(AddTo(a, b)) is non jump-only,

∗ InsCode(SubFrom(a, b)) is non jump-only,

∗ InsCode(MultBy(a, b)) is non jump-only, and

∗ InsCode(Divide(a, b)) is non jump-only. Let us consider a, b. One can check the following observations:

∗ a:=b is non jump-only and sequential,

∗ AddTo(a, b) is non jump-only and sequential,

∗ SubFrom(a, b) is non jump-only and sequential,

∗ MultBy(a, b) is non jump-only and sequential, and

∗ Divide(a, b) is non jump-only and sequential. Let us note that SCM is homogeneous and has explicit jumps and no implicit

jumps.

Let us observe that SCM is regular. We now state three propositions:

(57) IncAddr(goto i_{2}, k) = goto il_{SCM} (locnum(i_{2}) + k).

(58) IncAddr(if a = 0 goto i_{2}, k) = if a = 0 goto il_{SCM} (locnum(i_{2}) + k).

(59) IncAddr(if a > 0 goto i_{2}, k) = if a > 0 goto il_{SCM} (locnum(i_{2}) + k). Let us note that SCM is IC-good and Exec-preserving.

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. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

on the instructions of SCM 663

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

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

[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] Artur Korniłowicz. On the composition of macro instructions of standard computers. Formalized Mathematics, 9(2):303–316, 2001.

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

[10] Yatsuka Nakamura and Andrzej Trybulec. On a mathematical model of programs. Formalized Mathematics, 3(2):241–250, 1992.

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

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

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

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

[15] Andrzej Trybulec, Piotr Rudnicki, and Artur Korniłowicz. Standard ordering of instruction locations. Formalized Mathematics, 9(2):291–301, 2001.

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

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

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

[19] Edmund Woronowicz. Relations 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.

664 artur korniłowicz

Input and Output of Instructions^{1}

Artur Korniłowicz University of Białystok

MML Identifier: AMI 7.

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

1. Preliminaries

In this paper N is a set with non empty elements. One can prove the following propositions:

(1) For all sets x, y, z such that x ≠ y and x ≠ z holds {x, y, z}\{x} = {y, z}.

(2) For every non empty non void AMI A over N and for every state s of A and for every object o of A holds s(o) ∈ ObjectKind(o).

(3) Let A be a realistic IC-Ins-separated definite non empty non void AMI over N, s be a state of A, f be an instruction-location of A, and w be an element of ObjectKind(IC_{A}). Then (s +·(IC_{A}, w))(f) = s(f).

Let N be a set with non empty elements, let A be an IC-Ins-separated definite non empty non void AMI over N, let s be a state of A, let o be an object of A, and let a be an element of ObjectKind(o). Then s+·(o, a) is a state of A.

We now state several propositions:

(4) Let A be a steady-programmed IC-Ins-separated definite non empty non void AMI over N, s be a state of A, o be an object of A, f be an instruction-location of A, I be an instruction of A, and w be an element of ObjectKind(o). If f ≠ o, then (Exec(I, s))(f) = (Exec(I, s +·(o, w)))(f).

1This work has been partially supported by TYPES grant IST-1999-29001.

(5) Let A be an IC-Ins-separated definite non empty non void AMI over N, s be a state of A, o be an object of A, and w be an element of ObjectKind(o). If o ≠ IC_{A}, then IC_{s} = IC_{s+·(o,w)}.

(6) Let A be a standard IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, s be a state of A, o be an object of A, and w be an element of ObjectKind(o). If I is sequential and o ≠ IC_{A}, then IC_{Exec(I,s)} = IC_{Exec(I,s+·(o,w))}.

(7) Let A be a standard IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, s be a state of A, o be an object of A, and w be an element of ObjectKind(o). If I is sequential and o ≠ IC_{A}, then IC_{Exec(I,s+·(o,w))} = IC_{Exec(I,s)+·(o,w)}.

(8) Let A be a standard steady-programmed IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, s be a state of A, o be an object of A, w be an element of ObjectKind(o), and i be an instruction-location of A. Then (Exec(I, s +·(o, w)))(i) = (Exec(I, s) +· (o, w))(i).

2. Input and Output of Instructions

Let N be a set and let A be an AMI over N. We say that A has non trivial instruction set if and only if:

(Def. 1) The instructions of A are non trivial.

Let N be a set and let A be a non empty AMI over N. We say that A has non trivial ObjectKinds if and only if:

(Def. 2) For every object o of A holds ObjectKind(o) is non trivial.

Let N be a set with non empty elements. One can verify that STC(N) has non trivial ObjectKinds.

Let N be a set with non empty elements. Observe that there exists a regular standard IC-Ins-separated definite non empty non void AMI over N which is halting, realistic, steady-programmed, programmable, IC-good, and Execpreserving and has explicit jumps, no implicit jumps, non trivial ObjectKinds, and non trivial instruction set.

Let N be a set with non empty elements. Note that every definite non empty non void AMI over N which has non trivial ObjectKinds has also non trivial instruction set.

Let N be a set with non empty elements. One can check that every IC-Ins-separated non empty AMI over N which has non trivial ObjectKinds has also non trivial instruction locations.

Let N be a set with non empty elements, let A be a non empty AMI over N with non trivial ObjectKinds, and let o be an object of A. Observe that ObjectKind(o) is non trivial.

Let N be a set with non empty elements and let A be an AMI over N with non trivial instruction set. Note that the instructions of A is non trivial.

Let N be a set with non empty elements and let A be an IC-Ins-separated non empty AMI over N with non trivial instruction locations. Note that ObjectKind(IC_{A}) is non trivial.

Let N be a set with non empty elements, let A be a non empty non void AMI over N, and let I be an instruction of A. The functor OutputI yielding a subset of the carrier of A is defined as follows:

(Def. 3) For every object o of A holds o ∈ OutputI iff there exists a state s of A

such that s(o) ≠ (Exec(I, s))(o). Let N be a set with non empty elements, let A be an IC-Ins-separated

definite non empty non void AMI over N, and let I be an instruction of A. The functor IODiff I yielding a subset of the carrier of A is defined by the condition (Def. 4).

(Def. 4) Let o be an object of A. Then o ∈ IODiff I if and only if for every state s of A and for every element a of ObjectKind(o) holds Exec(I, s) = Exec(I, s +·(o, a)).

The functor IOSumI yielding a subset of the carrier of A is defined by the condition (Def. 5).

(Def. 5) Let o be an object of A. Then o ∈ IOSumI if and only if there exists a state s of A and there exists an element a of ObjectKind(o) such that Exec(I, s +·(o, a)) ≠ Exec(I, s) +·(o, a).

Let N be a set with non empty elements, let A be an IC-Ins-separated definite non empty non void AMI over N, and let I be an instruction of A. The functor InputI yielding a subset of the carrier of A is defined as follows:

(Def. 6) InputI = IOSumI \ IODiff I.

The following propositions are true:

(9) Let A be an IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. Then IODiff I misses InputI.

(10) Let A be an IC-Ins-separated definite non empty non void AMI over N with non trivial ObjectKinds and I be an instruction of A. Then IODiff I ⊆ OutputI.

(11) For every IC-Ins-separated definite non empty non void AMI A over N and for every instruction I of A holds OutputI ⊆ IOSumI.

(12) For every IC-Ins-separated definite non empty non void AMI A over N and for every instruction I of A holds InputI ⊆ IOSumI.

(13) Let A be an IC-Ins-separated definite non empty non void AMI over N with non trivial ObjectKinds and I be an instruction of A. Then IODiff I = OutputI \ InputI.

(14) Let A be an IC-Ins-separated definite non empty non void AMI over N with non trivial ObjectKinds and I be an instruction of A. Then IOSumI = OutputI ∪ InputI.

(15) Let A be an IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, and o be an object of A. If ObjectKind(o) is trivial, then o ∉ IOSumI.

(16) Let A be an IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, and o be an object of A. If ObjectKind(o) is trivial, then o ∉ InputI.

(17) Let A be an IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, and o be an object of A. If ObjectKind(o) is trivial, then o ∉ OutputI.

(18) Let A be an IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. Then I is halting if and only if OutputI is empty.

(19) Let A be an IC-Ins-separated definite non empty non void AMI over N with non trivial ObjectKinds and I be an instruction of A. If I is halting, then IODiff I is empty.

(20) Let A be an IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. If I is halting, then IOSumI is empty.

(21) Let A be an IC-Ins-separated definite non empty non void AMI over N

and I be an instruction of A. If I is halting, then InputI is empty. Let N be a set with non empty elements, let A be a halting IC-Ins-separated

definite non empty non void AMI over N, and let I be a halting instruction of A. One can verify the following observations:

∗ InputI is empty,

∗ OutputI is empty, and

∗ IOSumI is empty. Let N be a set with non empty elements, let A be a halting IC-Ins-separated

definite non empty non void AMI over N with non trivial ObjectKinds, and let I be a halting instruction of A. Note that IODiff I is empty.

The following propositions are true:

(22) Let A be a steady-programmed IC-Ins-separated definite non empty non void AMI over N with non trivial instruction set, f be an instructionlocation of A, and I be an instruction of A. Then f ∉ IODiff I.

(23) Let A be a standard IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. If I is sequential, then IC_{A} ∉ IODiff I.

(24) Let A be an IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. If there exists a state s of A such that (Exec(I, s))(IC_{A}) ≠ IC_{s}, then IC_{A} ∈ OutputI.

(25) Let A be a standard IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. If I is sequential, then IC_{A} ∈ OutputI.

(26) Let A be an IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. If there exists a state s of A such that (Exec(I, s))(IC_{A}) ≠ IC_{s}, then IC_{A} ∈ IOSumI.

(27) Let A be a standard IC-Ins-separated definite non empty non void AMI over N and I be an instruction of A. If I is sequential, then IC_{A} ∈ IOSumI.

(28) Let A be an IC-Ins-separated definite non empty non void AMI over N, f be an instruction-location of A, and I be an instruction of A. Suppose that for every state s of A and for every programmed finite partial state p of A holds Exec(I, s+·p) = Exec(I, s)+·p. Then f ∉ IOSumI.

(29) Let A be an IC-Ins-separated definite non empty non void AMI over N, I be an instruction of A, and o be an object of A. If I is jump-only, then if o ∈ OutputI, then o = IC_{A}.

3. Input and Output of the Instructions of SCM

In the sequel a, b are data-locations, f is an instruction-location of SCM , and I is an instruction of SCM .

We now state two propositions:

(30) For every state s of SCM and for every element w of ObjectKind(IC_{SCM} ) holds (s +·(IC_{SCM} , w))(a) = s(a).

(31) f ≠ Next(f).

Let s be a state of SCM , let d_{1} be a data-location, and let k be an integer. Then s +·(d_{1}, k) is a state of SCM .

Let us observe that SCM has non trivial ObjectKinds.

Next we state a number of propositions:

(32) IODiff(a:=a) = ∅.

(33) If a ≠ b, then IODiff(a:=b) = {a}.

(34) IODiff AddTo(a, b) = ∅.

(35) IODiff SubFrom(a, a) = {a}.

(36) If a ≠ b, then IODiff SubFrom(a, b) = ∅.

(37) IODiff MultBy(a, b) = ∅.

(38) IODiff Divide(a, a) = {a}.

(39) If a ≠ b, then IODiff Divide(a, b) = ∅.

(40) IODiff goto f = {IC_{SCM} }.

(41) IODiff(if a = 0 goto f) = ∅.

(42) IODiff(if a > 0 goto f) = ∅.

(43) Output(a:=a) = {IC_{SCM} }.

(44) If a ≠ b, then Output(a:=b) = {a, IC_{SCM} }.

(45) OutputAddTo(a, b) = {a, IC_{SCM} }.

(46) OutputSubFrom(a, b) = {a, IC_{SCM} }.

(47) OutputMultBy(a, b) = {a, IC_{SCM} }.

(48) OutputDivide(a, b) = {a, b, IC_{SCM} }.

(49) Outputgoto f = {IC_{SCM} }.

(50) Output(if a = 0 goto f) = {IC_{SCM} }.

(51) Output(if a > 0 goto f) = {IC_{SCM} }.

(52) f ∉ IOSumI.

(53) IOSum(a:=a) = {IC_{SCM} }.

(54) If a ≠ b, then IOSum(a:=b) = {a, b, IC_{SCM} }.

(55) IOSumAddTo(a, b) = {a, b, IC_{SCM} }.

(56) IOSumSubFrom(a, b) = {a, b, IC_{SCM} }.

(57) IOSumMultBy(a, b) = {a, b, IC_{SCM} }.

(58) IOSumDivide(a, b) = {a, b, IC_{SCM} }.

(59) IOSumgoto f = {IC_{SCM} }.

(60) IOSum(if a = 0 goto f) = {a, IC_{SCM} }.

(61) IOSum(if a > 0 goto f) = {a, IC_{SCM} }.

(62) Input(a:=a) = {IC_{SCM} }.

(63) If a ≠ b, then Input(a:=b) = {b, IC_{SCM} }.

(64) InputAddTo(a, b) = {a, b, IC_{SCM} }.

(65) InputSubFrom(a, a) = {IC_{SCM} }.

(66) If a ≠ b, then InputSubFrom(a, b) = {a, b, IC_{SCM} }.

(67) InputMultBy(a, b) = {a, b, IC_{SCM} }.

(68) InputDivide(a, a) = {IC_{SCM} }.

(69) If a ≠ b, then InputDivide(a, b) = {a, b, IC_{SCM} }.

(70) Inputgoto f = ∅.

(71) Input(if a = 0 goto f) = {a, IC_{SCM} }.

(72) Input(if a > 0 goto f) = {a, IC_{SCM} }. input and output of instructions 671

References

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

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

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

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

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

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

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

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

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

[10] Andrzej Trybulec, Piotr Rudnicki, and Artur Korniłowicz. Standard ordering of instruction locations. Formalized Mathematics, 9(2):291–301, 2001.

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

672 artur korniłowicz

On the Instructions of SCM _{FSA}^{1}

Artur Korniłowicz University of Białystok

MML Identifier: SCMFSA10.

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

For simplicity, we use the following convention: a, b are integer locations, f is a finite sequence location, i_{1}, i_{2}, i_{3} are instruction-locations of SCM _{FSA}, T is an instruction type of SCM _{FSA}, and k is a natural number.

Next we state two propositions:

(1) For every function f and for all sets a, A, b, B, c, C such that a ≠ b and a ≠ c holds (f+·(a⟼→ . A)+·(b⟼→ . B)+·(c⟼→ . C))(a) = A.

(2) For all sets a, b holds ⟨a⟩ +·(1, b) = ⟨b⟩. Let l_{1}, l_{2} be integer locations and let a, b be integers. Then [l_{1} ⟼→ a, l_{2} ⟼→ b]

is a finite partial state of SCM _{FSA}.

One can prove the following propositions:

(3) a ∉ the instruction locations of SCM _{FSA}.

(4) f ∉ the instruction locations of SCM _{FSA}.

(5) Data-Loc_{SCM_{FSA}} ≠ the instruction locations of SCM _{FSA}.

(6) Data^{∗}-Loc_{SCM_{FSA}} ≠ the instruction locations of SCM _{FSA}.

(7) Let o be an object of SCM _{FSA}. Then

(i) o = IC_{SCM _{FSA}} , or (ii) o ∈ the instruction locations of SCM _{FSA}, or (iii) o is an integer location or a finite sequence location.

(8) If i_{2} ≠ i_{3}, then Next(i_{2}) ≠ Next(i_{3}).

(9) a:=b = ⟨⟨1, ⟨a, b⟩⟩⟩.

(10) AddTo(a, b) = ⟨⟨2, ⟨a, b⟩⟩⟩.

1This work has been partially supported by TYPES grant IST-1999-29001.

(11) SubFrom(a, b) = ⟨⟨3, ⟨a, b⟩⟩⟩.

(12) MultBy(a, b) = ⟨⟨4, ⟨a, b⟩⟩⟩.

(13) Divide(a, b) = ⟨⟨5, ⟨a, b⟩⟩⟩.

(14) goto i_{1} = ⟨⟨6, ⟨i_{1}⟩⟩⟩.

(15) if a = 0 goto i_{1} = ⟨⟨7, ⟨i_{1}, a⟩⟩⟩.

(16) if a > 0 goto i_{1} = ⟨⟨8, ⟨i_{1}, a⟩⟩⟩.

(17) AddressPart(halt_{SCM _{FSA}} ) = ∅.

(18) AddressPart(a:=b) = ⟨a, b⟩.

(19) AddressPart(AddTo(a, b)) = ⟨a, b⟩.

(20) AddressPart(SubFrom(a, b)) = ⟨a, b⟩.

(21) AddressPart(MultBy(a, b)) = ⟨a, b⟩.

(22) AddressPart(Divide(a, b)) = ⟨a, b⟩.

(23) AddressPart(goto i_{2}) = ⟨i_{2}⟩.

(24) AddressPart(if a = 0 goto i_{2}) = ⟨i_{2}, a⟩.

(25) AddressPart(if a > 0 goto i_{2}) = ⟨i_{2}, a⟩.

(26) AddressPart(b:=f_{a}) = ⟨b, f, a⟩.

(27) AddressPart(f_{a}:=b) = ⟨b, f, a⟩.

(28) AddressPart(a:=lenf) = ⟨a, f⟩.

(29) AddressPart(f:=⟨\underbrace{0, …, 0}_{a}⟩) = ⟨a, f⟩.

(30) If T = 0, then AddressPartsT = {0}.

Let us consider T. Observe that AddressPartsT is non empty. Next we state a number of propositions:

(31) If T = 1, then dom∏_{AddressParts T} = {1, 2}.

(32) If T = 2, then dom∏_{AddressParts T} = {1, 2}.

(33) If T = 3, then dom∏_{AddressParts T} = {1, 2}.

(34) If T = 4, then dom∏_{AddressParts T} = {1, 2}.

(35) If T = 5, then dom∏_{AddressParts T} = {1, 2}.

(36) If T = 6, then dom∏_{AddressParts T} = {1}.

(37) If T = 7, then dom∏_{AddressParts T} = {1, 2}.

(38) If T = 8, then dom∏_{AddressParts T} = {1, 2}.

(39) If T = 9, then dom∏_{AddressParts T} = {1, 2, 3}.

(40) If T = 10, then dom∏_{AddressParts T} = {1, 2, 3}.

(41) If T = 11, then dom∏_{AddressParts T} = {1, 2}.

(42) If T = 12, then dom∏_{AddressParts T} = {1, 2}.

(43) ∏_{AddressParts InsCode (a:=b)}(1) = Data-Loc_{SCM_{FSA}} .

(44) ∏_{AddressParts InsCode (a:=b)}(2) = Data-Loc_{SCM_{FSA}} .

(45) ∏_{AddressParts InsCode (AddTo(a,b))}(1) = Data-Loc_{SCM_{FSA}} .

(46) ∏_{AddressParts InsCode (AddTo(a,b))}(2) = Data-Loc_{SCM_{FSA}} .

(47) ∏_{AddressParts InsCode (SubFrom (a,b))}(1) = Data-Loc_{SCM_{FSA}} .

(48) ∏_{AddressParts InsCode (SubFrom (a,b))}(2) = Data-Loc_{SCM_{FSA}} .

(49) ∏_{AddressParts InsCode (MultBy (a,b))}(1) = Data-Loc_{SCM_{FSA}} .

(50) ∏_{AddressParts InsCode (MultBy (a,b))}(2) = Data-Loc_{SCM_{FSA}} .

(51) ∏_{AddressParts InsCode (Divide(a,b))}(1) = Data-Loc_{SCM_{FSA}} .

(52) ∏_{AddressParts InsCode (Divide(a,b))}(2) = Data-Loc_{SCM_{FSA}} .

(53) ∏_{AddressParts InsCode (goto i_{2})}(1) = the instruction locations of SCM _{FSA}.

(54) ∏_{AddressParts InsCode (if a=0 goto i_{2})}(1) = the instruction locations of SCM _{FSA}.

(55) ∏_{AddressParts InsCode (if a=0 goto i_{2})}(2) = Data-Loc_{SCM_{FSA}} .

(56) ∏_{AddressParts InsCode (if a>0 goto i_{2})}(1) = the instruction locations of SCM _{FSA}.

(57) ∏_{AddressParts InsCode (if a>0 goto i_{2})}(2) = Data-Loc_{SCM_{FSA}} .

(58) ∏_{AddressParts InsCode (b:=f_{a})}(1) = Data-Loc_{SCM_{FSA}} .

(59) ∏_{AddressParts InsCode (b:=f_{a})}(2) = Data^{∗}-Loc_{SCM_{FSA}} .

(60) ∏_{AddressParts InsCode (b:=f_{a})}(3) = Data-Loc_{SCM_{FSA}} .

(61) ∏_{AddressParts InsCode (f_{a}:=b)}(1) = Data-Loc_{SCM_{FSA}} .

(62) ∏_{AddressParts InsCode (f_{a}:=b)}(2) = Data^{∗}-Loc_{SCM_{FSA}} .

(63) ∏_{AddressParts InsCode (f_{a}:=b)}(3) = Data-Loc_{SCM_{FSA}} .

(64) ∏_{AddressParts InsCode (a:=lenf)}(1) = Data-Loc_{SCM_{FSA}} .

(65) ∏_{AddressParts InsCode (a:=lenf)}(2) = Data^{∗}-Loc_{SCM_{FSA}} .

(66) ∏_{AddressParts InsCode (f:=⟨\underbrace{0, …, 0}_{a}⟩)}(1) = Data-Loc_{SCM_{FSA}} .

(67) ∏_{AddressParts InsCode (f:=⟨\underbrace{0, …, 0}_{a}⟩)}(2) = Data^{∗}-Loc_{SCM_{FSA}} .

(68) NIC(halt_{SCM _{FSA}} , i_{1}) = {i_{1}}.

One can verify that JUMP(halt_{SCM _{FSA}} ) is empty.

We now state the proposition

(69) NIC(a:=b, i_{1}) = {Next(i_{1})}.

Let us consider a, b. Note that JUMP(a:=b) is empty.

One can prove the following proposition

(70) NIC(AddTo(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. Note that JUMP(AddTo(a, b)) is empty.

Next we state the proposition

(71) NIC(SubFrom(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. Note that JUMP(SubFrom(a, b)) is empty. One can prove the following proposition

(72) NIC(MultBy(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. Note that JUMP(MultBy(a, b)) is empty. Next we state the proposition

(73) NIC(Divide(a, b), i_{1}) = {Next(i_{1})}.

Let us consider a, b. One can verify that JUMP(Divide(a, b)) is empty. We now state two propositions:

(74) NIC(goto i_{2}, i_{1}) = {i_{2}}.

(75) JUMP(goto i_{2}) = {i_{2}}.

Let us consider i_{2}. One can verify that JUMP(goto i_{2}) is non empty and trivial.

We now state two propositions:

(76) i_{2} ∈ NIC(if a = 0 goto i_{2}, i_{1}) and NIC(if a = 0 goto i_{2}, i_{1}) ⊆ {i_{2}, Next(i_{1})}.

(77) JUMP(if a = 0 goto i_{2}) = {i_{2}}.

Let us consider a, i_{2}. One can check that JUMP(if a = 0 goto i_{2}) is non empty and trivial.

One can prove the following two propositions:

(78) i_{2} ∈ NIC(if a > 0 goto i_{2}, i_{1}) and NIC(if a > 0 goto i_{2}, i_{1}) ⊆ {i_{2}, Next(i_{1})}.

(79) JUMP(if a > 0 goto i_{2}) = {i_{2}}.

Let us consider a, i_{2}. Note that JUMP(if a > 0 goto i_{2}) is non empty and trivial.

The following proposition is true

(80) NIC(a:=f_{b}, i_{1}) = {Next(i_{1})}.

Let us consider a, b, f. Observe that JUMP(a:=f_{b}) is empty. Next we state the proposition

(81) NIC(f_{b}:=a, i_{1}) = {Next(i_{1})}.

Let us consider a, b, f. One can check that JUMP(f_{b}:=a) is empty. The following proposition is true

(82) NIC(a:=lenf, i_{1}) = {Next(i_{1})}.

Let us consider a, f. Observe that JUMP(a:=lenf) is empty. The following proposition is true

(83) NIC(f:=⟨\underbrace{0, …, 0}_{a}⟩, i_{1}) = \{Next(i_{1})\}.

Let us consider a, f. Note that JUMP(f:=⟨\underbrace{0, …, 0}_{a}⟩) is empty.

The following two propositions are true:

(84) SUCC(i_{1}) = {i_{1}, Next(i_{1})}.

(85) Let f be a function from ℕ into the instruction locations of SCM _{FSA}.

Suppose that for every natural number k holds f(k) = insloc(k). Then (i) f is bijective, and (ii) for every natural number k holds f(k + 1) ∈ SUCC(f(k)) and for every

natural number j such that f(j) ∈ SUCC(f(k)) holds k j. Let us observe that SCM _{FSA} is standard. The following propositions are true:

(86) il_{SCM _{FSA}} (k) = insloc(k).

(87) Next(il_{SCM _{FSA}} (k)) = il_{SCM _{FSA}} (k + 1).

(88) Next(i_{1}) = NextLoci_{1}.

Let us mention that InsCode(halt_{SCM _{FSA}} ) is jump-only. Let us mention that halt_{SCM _{FSA}} is jump-only. Let us consider i_{2}. One can verify that InsCode(goto i_{2}) is jump-only. Let us consider i_{2}. Observe that goto i_{2} is jump-only non sequential and non

instruction location free.

Let us consider a, i_{2}. One can check that InsCode(if a = 0 goto i_{2}) is jumponly and InsCode(if a > 0 goto i_{2}) is jump-only.

Let us consider a, i_{2}. Observe that if a = 0 goto i_{2} is jump-only non sequential and non instruction location free and if a > 0 goto i_{2} is jump-only non sequential and non instruction location free.

Let us consider a, b. One can verify the following observations:

∗ InsCode(a:=b) is non jump-only,

∗ InsCode(AddTo(a, b)) is non jump-only,

∗ InsCode(SubFrom(a, b)) is non jump-only,

∗ InsCode(MultBy(a, b)) is non jump-only, and

∗ InsCode(Divide(a, b)) is non jump-only. Let us consider a, b. One can verify the following observations:

∗ a:=b is non jump-only and sequential,

∗ AddTo(a, b) is non jump-only and sequential,

∗ SubFrom(a, b) is non jump-only and sequential,

∗ MultBy(a, b) is non jump-only and sequential, and

∗ Divide(a, b) is non jump-only and sequential. Let us consider a, b, f. One can check that InsCode(b:=f_{a}) is non jump-only

and InsCode(f_{a}:=b) is non jump-only.

Let us consider a, b, f. Observe that b:=f_{a} is non jump-only and sequential and f_{a}:=b is non jump-only and sequential.

678 artur korniłowicz

Let us consider a, f. One can check that InsCode(a:=lenf) is non jump-only and InsCode(f:=⟨\underbrace{0, …, 0}_{a}⟩) is non jump-only.

Let us consider a, f. Note that a:=lenf is non jump-only and sequential and f:=⟨\underbrace{0, …, 0}_{a}⟩ is non jump-only and sequential.

One can verify that SCM _{FSA} is homogeneous and has explicit jumps and no implicit jumps.

Let us note that SCM _{FSA} is regular. The following propositions are true:

(89) IncAddr(goto i_{2}, k) = goto il_{SCM _{FSA}} (locnum(i_{2}) + k).

(90) IncAddr(if a = 0 goto i_{2}, k) = if a = 0 goto il_{SCM _{FSA}} (locnum(i_{2}) + k).

(91) IncAddr(if a > 0 goto i_{2}, k) = if a > 0 goto il_{SCM _{FSA}} (locnum(i_{2}) + k).

Let us note that SCM _{FSA} is IC-good and Exec-preserving.

References

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

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

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

[4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 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. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[10] Artur Korniłowicz. On the composition of macro instructions of standard computers. Formalized Mathematics, 9(2):303–316, 2001.

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

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

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

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

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

[16] Andrzej Trybulec, Yatsuka Nakamura, and Piotr Rudnicki. The SCM _{FSA} computer. Formalized Mathematics, 5(4):519–528, 1996.

[17] Andrzej Trybulec, Yatsuka Nakamura, and Piotr Rudnicki. An extension of scm. Formalized Mathematics, 5(4):507–512, 1996.

on the instructions of SCM _{FSA} 679

[18] Andrzej Trybulec, Piotr Rudnicki, and Artur Korniłowicz. Standard ordering of instruction locations. Formalized Mathematics, 9(2):291–301, 2001.

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

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

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

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

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

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

680 artur korniłowicz

Robbins Algebras vs. Boolean Algebras^{1}

Adam Grabowski University of Białystok

Summary. In the early 1930s, Huntington proposed several axiom systems for Boolean algebras. Robbins slightly changed one of them and asked if the resulted system is still a basis for variety of Boolean algebras. The solution (afirmative answer) was given in 1996 by McCune with the help of automated theorem prover EQP/Otter. Some simplified and restucturized versions of this proof are known. In our version of proof that all Robbins algebras are Boolean we use the results of McCune [5], Huntington [2, 4, 3] and Dahn [1].

MML Identifier: ROBBINS1.

The papers [7] and [6] provide the terminology and notation for this paper.

1. Preliminaries

We introduce complemented lattice structures which are extensions of ⊔- semi lattice structure and are systems

⟨ a carrier, a join operation, a complement operation ⟩, where the carrier is a set, the join operation is a binary operation on the carrier, and the complement operation is a unary operation on the carrier.

We introduce ortholattice structures which are extensions of complemented lattice structure and lattice structure and are systems

⟨ a carrier, a join operation, a meet operation, a complement operation ⟩, where the carrier is a set, the join operation and the meet operation are binary operations on the carrier, and the complement operation is a unary operation on the carrier.

The strict complemented lattice structure TrivComplLat is defined as follows:

1This work has been partially supported by TYPES grant IST-1999-29001.

(Def. 1) TrivComplLat = ⟨{∅}, op_{2}, op_{1}⟩.

The strict ortholattice structure TrivOrtLat is defined by:

(Def. 2) TrivOrtLat = ⟨{∅}, op_{2}, op_{2}, op_{1}⟩.

Let us note that TrivComplLat is non empty and trivial and TrivOrtLat is non empty and trivial.

Let us mention that there exists an ortholattice structure which is strict, non empty, and trivial and there exists a complemented lattice structure which is strict, non empty, and trivial.

Let L be a non empty complemented lattice structure and let x be an element of the carrier of L. The functor x^{c} yielding an element of L is defined as follows:

(Def. 3) x^{c} = (the complement operation of L)(x).

Let L be a non empty complemented lattice structure and let x, y be elements of the carrier of L. We introduce x + y as a synonym of x ⊔ y.

Let L be a non empty complemented lattice structure and let x, y be elements of the carrier of L. The functor x ∗ y yields an element of L and is defined by:

(Def. 4) x ∗ y = (x^{c} ⊔ y^{c})^{c}.

Let L be a non empty complemented lattice structure. We say that L is Robbins if and only if:

(Def. 5) For all elements x, y of the carrier of L holds ((x + y)^{c} + (x + y^{c})^{c})^{c} = x. We say that L is Huntington if and only if:

(Def. 6) For all elements x, y of the carrier of L holds (x^{c} + y^{c})^{c} + (x^{c} + y)^{c} = x.

Let G be a non empty ⊔-semi lattice structure. We say that G is joinidempotent if and only if:

(Def. 7) For every element x of the carrier of G holds x ⊔ x = x.

Let us observe that TrivComplLat is join-commutative join-associative Robbins Huntington and join-idempotent and TrivOrtLat is join-commutative joinassociative Huntington and Robbins.

Let us mention that TrivOrtLat is meet-commutative meet-associative meetabsorbing and join-absorbing.

One can verify that there exists a non empty complemented lattice structure which is strict, join-associative, join-commutative, Robbins, join-idempotent, and Huntington.

Let us observe that there exists a non empty ortholattice structure which is strict, lattice-like, Robbins, and Huntington.

Let L be a join-commutative non empty complemented lattice structure and let x, y be elements of the carrier of L. Let us observe that the functor x + y is commutative.

Next we state several propositions:

(1) Let L be a Huntington join-commutative join-associative non empty complemented lattice structure and a, b be elements of the carrier of L. Then a ∗ b + a ∗ b^{c} = a.

(2) Let L be a Huntington join-commutative join-associative non empty complemented lattice structure and a be an element of the carrier of L. Then a + a^{c} = a^{c} + (a^{c})^{c}.

(3) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and x be an element of the carrier of L. Then (x^{c})^{c} = x.

(4) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. Then a + a^{c} = b + b^{c}.

(5) Let L be a join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure. Then there exists an element c of the carrier of L such that for every element a of the carrier of L holds c + a = c and a + a^{c} = c.

(6) Every join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure is upper-bounded.

One can verify that every non empty complemented lattice structure which is join-commutative, join-associative, join-idempotent, and Huntington is also upper-bounded.

Let L be a join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure. Then ⊤_{L} can be characterized by the condition:

(Def. 8) There exists an element a of the carrier of L such that ⊤_{L} = a + a^{c}. One can prove the following propositions:

(7) Let L be a join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure. Then there exists an element c of the carrier of L such that for every element a of the carrier of L holds c ∗ a = c and (a + a^{c})^{c} = c.

(8) Let L be a join-commutative join-associative non empty complemented lattice structure and a, b be elements of the carrier of L. Then a∗b = b∗a.

Let L be a join-commutative join-associative non empty complemented lattice structure and let x, y be elements of the carrier of L. Let us note that the functor x ∗ y is commutative.

Let L be a join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure. The functor ⊥_{L}^{C} yielding an element of L is defined as follows:

(Def. 9) For every element a of the carrier of L holds ⊥_{L}^{C} ∗ a = ⊥_{L}^{C}. One can prove the following propositions:

(9) Let L be a join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure and a be an element of the carrier of L. Then ⊥_{L}^{C} = (a + a^{c})^{c}.

(10) Let L be a join-commutative join-associative join-idempotent Huntington non empty complemented lattice structure. Then (⊤_{L})^{c} = ⊥_{L}^{C} and ⊤_{L} = (⊥_{L}^{C})^{c}.

(11) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. If a^{c} = b^{c}, then a = b.

(12) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. Then a + (b + b^{c})^{c} = a.

(13) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a be an element of the carrier of L. Then a + a = a.

Let us note that every non empty complemented lattice structure which is join-commutative, join-associative, and Huntington is also join-idempotent.

One can prove the following propositions:

(14) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a be an element of the carrier of L. Then a + ⊥_{L}^{C} = a.

(15) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a be an element of the carrier of L. Then a ∗ ⊤_{L} = a.

(16) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a be an element of the carrier of L. Then a ∗ a^{c} = ⊥_{L}^{C}.

(17) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then a ∗ (b ∗ c) = (a ∗ b) ∗ c.

(18) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. Then a + b = (a^{c} ∗ b^{c})^{c}.

(19) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a be an element of the carrier of L. Then a ∗ a = a.

(20) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a be an element of the carrier of L.

Then a + ⊤_{L} = ⊤_{L}.

(21) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. Then a + a ∗ b = a.

(22) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. Then a ∗ (a + b) = a.

(23) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. If a^{c} + b = ⊤_{L} and b^{c} + a = ⊤_{L}, then a = b.

(24) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of the carrier of L. If a + b = ⊤_{L} and a ∗ b = ⊥_{L}^{C}, then a^{c} = b.

(25) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then a ∗ b ∗ c + a ∗ b ∗ c^{c} + a ∗ b^{c} ∗ c + a ∗ b^{c} ∗ c^{c} + a^{c} ∗ b ∗ c + a^{c} ∗ b ∗ c^{c} + a^{c} ∗ b^{c} ∗ c + a^{c} ∗ b^{c} ∗ c^{c} = ⊤_{L}.

(26) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then

(i) a ∗ c ∗ (b ∗ c^{c}) = ⊥_{L}^{C}, (ii) a ∗ b ∗ c ∗ (a^{c} ∗ b ∗ c) = ⊥_{L}^{C}, (iii) a ∗ b^{c} ∗ c ∗ (a^{c} ∗ b ∗ c) = ⊥_{L}^{C}, (iv) a ∗ b ∗ c ∗ (a^{c} ∗ b^{c} ∗ c) = ⊥_{L}^{C},

(v) a ∗ b ∗ c^{c} ∗ (a^{c} ∗ b^{c} ∗ c^{c}) = ⊥_{L}^{C}, and (vi) a ∗ b^{c} ∗ c ∗ (a^{c} ∗ b ∗ c) = ⊥_{L}^{C}.

(27) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then a ∗ b + a ∗ c = a ∗ b ∗ c + a ∗ b ∗ c^{c} + a ∗ b^{c} ∗ c.

(28) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then (a ∗ (b + c))^{c} = a ∗ b^{c}∗ c^{c}+ a^{c}∗ b ∗ c + a^{c}∗ b ∗ c^{c}+ a^{c}∗ b^{c}∗ c + a^{c}∗ b^{c}∗ c^{c}.

(29) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then a ∗ b + a ∗ c + (a ∗ (b + c))^{c} = ⊤_{L}.

(30) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then (a ∗ b + a ∗ c) ∗ (a ∗ (b + c))^{c} = ⊥_{L}^{C}.

(31) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L.

Then a ∗ (b + c) = a ∗ b + a ∗ c.

(32) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b, c be elements of the carrier of L. Then a + b ∗ c = (a + b) ∗ (a + c).

2. Pre-Ortholattices

Let L be a non empty ortholattice structure. We say that L is well-comple-mented if and only if:

(Def. 10) For every element a of the carrier of L holds a^{c} is a complement of a. Let us observe that TrivOrtLat is Boolean and well-complemented. A pre-ortholattice is a lattice-like non empty ortholattice structure. Let us mention that there exists a pre-ortholattice which is strict, Boolean,

and well-complemented.

We now state two propositions:

(33) Let L be a distributive well-complemented pre-ortholattice and x be an element of the carrier of L. Then (x^{c})^{c} = x.

(34) Let L be a bounded distributive well-complemented pre-ortholattice and x, y be elements of the carrier of L. Then x ⊓ y = (x^{c} ⊔ y^{c})^{c}.

3. Correspondence between Boolean Pre-OrthoLattices and Boolean Lattices

Let L be a non empty complemented lattice structure. The functor CLattL yielding a strict ortholattice structure is defined by the conditions (Def. 11).

(Def. 11)(i) The carrier of CLattL = the carrier of L, (ii) the join operation of CLattL = the join operation of L, (iii) the complement operation of CLattL = the complement operation of

L, and (iv) for all elements a, b of the carrier of L holds (the meet operation of

CLattL)(a, b) = a ∗ b. Let L be a non empty complemented lattice structure. One can verify that

CLattL is non empty.

Let L be a join-commutative non empty complemented lattice structure. One can check that CLattL is join-commutative.

Let L be a join-associative non empty complemented lattice structure. One can check that CLattL is join-associative.

Let L be a join-commutative join-associative non empty complemented lattice structure. Observe that CLattL is meet-commutative.

The following proposition is true

(35) Let L be a non empty complemented lattice structure, a, b be elements of the carrier of L, and a′, b′be elements of the carrier of CLattL. If a = a′ and b = b′, then a ∗ b = a′⊓ b′ and a + b = a′⊔ b′ and a^{c} = a′^{c}.

Let L be a join-commutative join-associative Huntington non empty complemented lattice structure. Observe that CLattL is meet-associative join-absorbing and meet-absorbing.

Let L be a Huntington non empty complemented lattice structure. Note that CLattL is Huntington.

Let L be a join-commutative join-associative Huntington non empty complemented lattice structure. Note that CLattL is lower-bounded.

We now state the proposition

(36) For every join-commutative join-associative Huntington non empty complemented lattice structure L holds ⊥_{L}^{C} = ⊥_{CLatt L}.

Let L be a join-commutative join-associative Huntington non empty complemented lattice structure. One can check that CLattL is complemented distributive and bounded.

4. Proofs according to Bernd Ingo Dahn

Let G be a non empty complemented lattice structure and let x be an element of the carrier of G. We introduce − x as a synonym of x^{c}.

Let G be a join-commutative non empty complemented lattice structure. Let us observe that G is Huntington if and only if:

(Def. 12) For all elements x, y of the carrier of G holds − (− x + − y)+− (x + − y) =

y. Let G be a non empty complemented lattice structure. We say that G has

idempotent element if and only if:

(Def. 13) There exists an element x of the carrier of G such that x + x = x.

In the sequel G is a Robbins join-associative join-commutative non empty complemented lattice structure and x, y, z are elements of the carrier of G.

Let G be a non empty complemented lattice structure and let x, y be elements of the carrier of G. The functor δ(x, y) yielding an element of G is defined by:

(Def. 14) δ(x, y) = − (− x + y).

Let G be a non empty complemented lattice structure and let x, y be elements of the carrier of G. The functor Expand(x, y) yields an element of G and is defined by:

(Def. 15) Expand(x, y) = δ(x + y, δ(x, y)).

Let G be a non empty complemented lattice structure and let x be an element of the carrier of G. The functor x_{0} yielding an element of G is defined by:

(Def. 16) x_{0} = − (− x + x).

The functor 2x yielding an element of G is defined as follows: (Def. 17) 2x = x + x.

Let G be a non empty complemented lattice structure and let x be an element of the carrier of G. The functor x_{1} yielding an element of G is defined by:

(Def. 18) x_{1} = x_{0} + x.

The functor x_{2} yields an element of G and is defined as follows: (Def. 19) x_{2} = x_{0} + 2x.

The functor x_{3} yields an element of G and is defined by: (Def. 20) x_{3} = x_{0} + (2x + x).

The functor x_{4} yielding an element of G is defined as follows: (Def. 21) x_{4} = x_{0} + (2x + 2x).

We now state a number of propositions:

(37) δ(x + y, δ(x, y)) = y.

(38) Expand(x, y) = y.

(39) δ(− x + y, z) = − (δ(x, y) + z).

(40) δ(x, x) = x_{0}.

(41) δ(2x, x_{0}) = x.

(42) δ(x_{2}, x) = x_{0}.

(43) x_{2} + x = x_{3}.

(44) x_{4} + x_{0} = x_{3} + x_{1}.

(45) x_{3} + x_{0} = x_{2} + x_{1}.

(46) x_{3} + x = x_{4}.

(47) δ(x_{3}, x_{0}) = x.

(48) If − x = − y, then δ(x, z) = δ(y, z).

(49) δ(x, − y) = δ(y, − x).

(50) δ(x_{3}, x) = x_{0}.

(51) δ(x_{1} + x_{3}, x) = x_{0}.

(52) δ(x_{1} + x_{2}, x) = x_{0}.

(53) δ(x_{1} + x_{3}, x_{0}) = x.

Let us consider G, x. The functor β(x) yielding an element of G is defined as follows:

(Def. 22) β(x) = − (x_{1} + x_{3}) + x + − x_{3}.

We now state three propositions:

(54) δ(β(x), x) = − x_{3}.

(55) δ(β(x), x) = − (x_{1} + x_{3}).

(56) There exist y, z such that − (y + z) = − z.

5. Proofs according to William McCune

One can prove the following two propositions:

(57) If for every z holds − − z = z, then G is Huntington.

(58) If G has idempotent element, then G is Huntington.

Let us observe that TrivComplLat has idempotent element. One can check that every Robbins join-associative join-commutative non

empty complemented lattice structure which has idempotent element is Huntington.

One can prove the following two propositions:

(59) If there exist elements c, d of the carrier of G such that c + d = c, then G is Huntington.

(60) There exist y, z such that y + z = z.

One can verify that every join-associative join-commutative non empty complemented lattice structure which is Robbins is also Huntington.

Let L be a non empty ortholattice structure. We say that L is de Morgan if and only if:

(Def. 23) For all elements x, y of the carrier of L holds x ⊓ y = (x^{c} ⊔ y^{c})^{c}.

Let L be a non empty complemented lattice structure. One can verify that CLattL is de Morgan.

Next we state two propositions:

(61) Let L be a well-complemented join-commutative meet-commutative non empty ortholattice structure and x be an element of the carrier of L. Then x + x^{c} = ⊤_{L} and x ⊓ x^{c} = ⊥_{L}.

(62) For every bounded distributive well-complemented pre-ortholattice L

holds (⊤_{L})^{c} = ⊥_{L}. Let us observe that TrivOrtLat is de Morgan. One can verify that there exists a pre-ortholattice which is strict, de Morgan,

Boolean, Robbins, and Huntington.

690 adam grabowski

Let us note that every non empty ortholattice structure which is join-associative, join-commutative, and de Morgan is also meet-commutative.

One can prove the following proposition

(63) For every Huntington de Morgan pre-ortholattice L holds ⊥_{L}^{C} = ⊥_{L}. One can verify that every well-complemented pre-ortholattice which is Bo-

olean is also Huntington.

Let us note that every de Morgan pre-ortholattice which is Huntington is also Boolean.

One can verify that every pre-ortholattice which is Robbins and de Morgan is also Boolean and every well-complemented pre-ortholattice which is Boolean is also Robbins.

References

[1] B. I. Dahn. Robbins algebras are Boolean: A revision of McCune’s computer-generated solution of Robbins problem. Journal of Algebra, 208:526–532, 1998.

[2] E. V. Huntington. Sets of independent postulates for the algebra of logic. Trans. AMS, 5:288–309, 1904.

[3] E. V. Huntington. Boolean algebra. A correction. Trans. AMS, 35:557–558, 1933.

[4] E. V. Huntington. New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia Mathematica. Trans. AMS, 35:274–304, 1933.

[5] W. McCune. Solution of the Robbins problem. Journal of Automated Reasoning, 19:263– 276, 1997.

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

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

Received June 12, 2001

Properties of Fuzzy Relation

Noboru Endou Gifu National College of Technology Takashi Mitsuishi Miyagi University

Keiji Ohkubo Shinshu University Nagano

Summary. In this article, we introduce four fuzzy relations and the composition, and some useful properties are shown by them. In section 2, the definition of converse relation R^{−1} of fuzzy relation R and properties concerning it are described. In the next section, we define the composition of the fuzzy relation and show some properties. In the final section we describe the identity relation, the universe relation and the zero relation.

MML Identifier: FUZZY 4.

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

1. Basic Properties of the Membership Function

We follow the rules: x, y, z are sets and C_{1}, C_{2}, C_{3} are non empty sets. Let C_{1} be a non empty set and let F be a membership function of C_{1}. One

can check that rngF is non empty.

Next we state four propositions:

(1) Let F be a membership function of C_{1}. Then rngF is bounded and for every x such that x ∈ domF holds F (x) suprngF and for every x such that x ∈ domF holds F (x) inf rngF.

(2) For all membership functions F, G of C_{1} such that for every x such that x ∈ C_{1} holds F (x) G(x) holds suprngF suprngG.

(3) For every Membership function f of C_{1}, C_{2} and for every element c of [: C_{1}, C_{2} :] holds 0 f(c) and f(c) 1.

(4) For every Membership function f of C_{1}, C_{2} and for all x, y such that ⟨⟨x, y⟩⟩∈ [: C_{1}, C_{2} :] holds 0 f(⟨⟨x, y⟩⟩) and f(⟨⟨x, y⟩⟩) 1.

2. Definition of Converse Fuzzy Relation and some Properties

Let C_{1}, C_{2} be non empty sets and let h be a Membership function of C_{2}, C_{1}. The functor converseh yielding a Membership function of C_{1}, C_{2} is defined by:

(Def. 1) For all x, y such that ⟨⟨x, y⟩⟩ ∈ [: C_{1}, C_{2} :] holds (converseh)(⟨⟨x, y⟩⟩) =

h(⟨⟨y, x⟩⟩). Let C_{1}, C_{2} be non empty sets, let f be a Membership function of C_{2}, C_{1}, and

let R be a fuzzy relation of C_{2}, C_{1}, f. The functor R^{−1} yields a fuzzy relation of C_{1}, C_{2}, conversef and is defined by:

(Def. 2) R^{−1} = [: [: C_{1}, C_{2} :], (conversef)^{∘}[: C_{1}, C_{2} :] :].

The following propositions are true:

(5) For every Membership function f of C_{1}, C_{2} holds converseconversef = f.

(6) For every Membership function f of C_{1}, C_{2} and for every fuzzy relation R of C_{1}, C_{2}, f holds (R^{−1} )^{−1} = R.

(7) For every Membership function f of C_{1}, C_{2} holds 1-minusconversef = converse1-minusf.

(8) For every Membership function f of C_{1}, C_{2} and for every fuzzy relation R of C_{1}, C_{2}, f holds (R^{−1} )^{c} = (R^{c})^{−1} .

(9) For all Membership functions f, g of C_{1}, C_{2} holds conversemax(f, g) = max(conversef, converseg).

(10) Let f, g be Membership functions of C_{1}, C_{2}, R be a fuzzy relation of C_{1}, C_{2}, f, and S be a fuzzy relation of C_{1}, C_{2}, g. Then (R∪S)^{−1} = R^{−1} ∪S^{−1} .

(11) For all Membership functions f, g of C_{1}, C_{2} holds conversemin(f, g) = min(conversef, converseg).

(12) Let f, g be Membership functions of C_{1}, C_{2}, R be a fuzzy relation of C_{1}, C_{2}, f, and S be a fuzzy relation of C_{1}, C_{2}, g. Then (R∩S)^{−1} = R^{−1} ∩S^{−1} .

(13) Let f, g be Membership functions of C_{1}, C_{2} and given x, y. If x ∈ C_{1} and y ∈ C_{2}, then if f(⟨⟨x, y⟩⟩) g(⟨⟨x, y⟩⟩), then (conversef)(⟨⟨y, x⟩⟩) (converseg)(⟨⟨y, x⟩⟩).

(14) Let f, g be Membership functions of C_{1}, C_{2}, R be a fuzzy relation of C_{1}, C_{2}, f, and S be a fuzzy relation of C_{1}, C_{2}, g. If R ⊆ S, then R^{−1} ⊆ S^{−1} .

(15) For all Membership functions f, g of C_{1}, C_{2} holds conversemin(f, 1-minusg) = min(conversef, 1-minusconverseg).

(16) Let f, g be Membership functions of C_{1}, C_{2}, R be a fuzzy relation of C_{1}, C_{2}, f, and S be a fuzzy relation of C_{1}, C_{2}, g. Then (R \S)^{−1} = R^{−1} \S^{−1} .

(17) For all Membership functions f, g of C_{1}, C_{2} holds converse max(min(f, 1-minusg), min(1-minusf, g)) = max(min(conversef, 1-minusconverseg), min(1-minusconversef, converseg)).

(18) Let f, g be Membership functions of C_{1}, C_{2}, R be a fuzzy relation of C_{1}, C_{2}, f, and S be a fuzzy relation of C_{1}, C_{2}, g. Then (R−. S)^{−1} = R^{−1} −. S^{−1} .

3. Definition of the Composition and some Properties

Let C_{1}, C_{2}, C_{3} be non empty sets, let h be a Membership function of C_{1}, C_{2}, let g be a Membership function of C_{2}, C_{3}, and let x, z be sets. Let us assume that x ∈ C_{1} and z ∈ C_{3}. The functor min(h, g, x, z) yields a membership function of C_{2} and is defined by:

(Def. 3) For every element y of C_{2} holds (min(h, g, x, z))(y) = min(h(⟨⟨x,

y⟩⟩), g(⟨⟨y, z⟩⟩)). Let C_{1}, C_{2}, C_{3} be non empty sets, let h be a Membership function of C_{1},

C_{2}, and let g be a Membership function of C_{2}, C_{3}. The functor h g yields a Membership function of C_{1}, C_{3} and is defined by:

(Def. 4) For all x, z such that ⟨⟨x, z⟩⟩ ∈ [: C_{1}, C_{3} :] holds (h g)(⟨⟨x, z⟩⟩) =

suprngmin(h, g, x, z). Let C_{1}, C_{2}, C_{3} be non empty sets, let f be a Membership function of C_{1},

C_{2}, let g be a Membership function of C_{2}, C_{3}, let R be a fuzzy relation of C_{1}, C_{2}, f, and let S be a fuzzy relation of C_{2}, C_{3}, g. The functor R S yields a fuzzy relation of C_{1}, C_{3}, f g and is defined as follows:

(Def. 5) R S = [: [: C_{1}, C_{3} :], (f g)^{∘}[: C_{1}, C_{3} :] :].

Next we state a number of propositions:

(19) For every Membership function f of C_{1}, C_{2} and for all Membership functions g, h of C_{2}, C_{3} holds f max(g, h) = max(f g, f h).

(20) Let f be a Membership function of C_{1}, C_{2}, g, h be Membership functions of C_{2}, C_{3}, R be a fuzzy relation of C_{1}, C_{2}, f, S be a fuzzy relation of C_{2}, C_{3}, g, and T be a fuzzy relation of C_{2}, C_{3}, h. Then R (S ∪T) = R S ∪R T.

(21) For all Membership functions f, g of C_{1}, C_{2} and for every Membership function h of C_{2}, C_{3} holds max(f, g) h = max(f h, g h).

(22) Let f, g be Membership functions of C_{1}, C_{2}, h be a Membership function of C_{2}, C_{3}, R be a fuzzy relation of C_{1}, C_{2}, f, S be a fuzzy relation of C_{1}, C_{2}, g, and T be a fuzzy relation of C_{2}, C_{3}, h. Then (R ∪ S) T = R T ∪ S T.

(23) Let f be a Membership function of C_{1}, C_{2}, g, h be Membership functions of C_{2}, C_{3}, and x, z be sets. If x ∈ C_{1} and z ∈ C_{3}, then (f min(g, h))(⟨⟨x, z⟩⟩) (min(f g, f h))(⟨⟨x, z⟩⟩).

(24) Let f be a Membership function of C_{1}, C_{2}, g, h be Membership functions of C_{2}, C_{3}, R be a fuzzy relation of C_{1}, C_{2}, f, S be a fuzzy relation of C_{2}, C_{3}, g, and T be a fuzzy relation of C_{2}, C_{3}, h. Then R (S ∩ T) ⊆ (R S) ∩ (R T).

(25) Let f, g be Membership functions of C_{1}, C_{2}, h be a Membership function of C_{2}, C_{3}, and x, z be sets. If x ∈ C_{1} and z ∈ C_{3}, then (min(f, g) h)(⟨⟨x, z⟩⟩) (min(f h, g h))(⟨⟨x, z⟩⟩).

(26) Let f, g be Membership functions of C_{1}, C_{2}, h be a Membership function of C_{2}, C_{3}, R be a fuzzy relation of C_{1}, C_{2}, f, S be a fuzzy relation of C_{1}, C_{2}, g, and T be a fuzzy relation of C_{2}, C_{3}, h. Then (R ∩ S) T ⊆ (R T) ∩ (S T).

(27) For every Membership function f of C_{1}, C_{2} and for every Membership function g of C_{2}, C_{3} holds conversef g = converseg conversef.

(28) Let f be a Membership function of C_{1}, C_{2}, g be a Membership function of C_{2}, C_{3}, R be a fuzzy relation of C_{1}, C_{2}, f, and S be a fuzzy relation of C_{2}, C_{3}, g. Then (R S)^{−1} = S^{−1} R^{−1} .

(29) Let f, g be Membership functions of C_{1}, C_{2}, h, k be Membership functions of C_{2}, C_{3}, and x, z be sets. Suppose x ∈ C_{1} and z ∈ C_{3} and for every set y such that y ∈ C_{2} holds f(⟨⟨x, y⟩⟩) g(⟨⟨x, y⟩⟩) and h(⟨⟨y, z⟩⟩) k(⟨⟨y, z⟩⟩). Then (f h)(⟨⟨x, z⟩⟩) (g k)(⟨⟨x, z⟩⟩).

(30) Let f, g be Membership functions of C_{1}, C_{2}, h, k be Membership functions of C_{2}, C_{3}, R be a fuzzy relation of C_{1}, C_{2}, f, S be a fuzzy relation of C_{1}, C_{2}, g, T be a fuzzy relation of C_{2}, C_{3}, h, and W be a fuzzy relation of C_{2}, C_{3}, k. If R ⊆ S and T ⊆ W, then R T ⊆ S W.

4. Definition of Identity Relation and Properties of Universe and Zero Relation

Let C_{1}, C_{2} be non empty sets. The functor Imf(C_{1}, C_{2}) yields a Membership function of C_{1}, C_{2} and is defined as follows:

(Def. 6) For all x, y such that ⟨⟨x, y⟩⟩ ∈ [: C_{1}, C_{2} :] holds if x = y, then

(Imf(C_{1}, C_{2}))(⟨⟨x, y⟩⟩) = 1 and if x ≠ y, then (Imf(C_{1}, C_{2}))(⟨⟨x, y⟩⟩) = 0. One can prove the following propositions:

(31) For every element c of [: C_{1}, C_{2} :] holds (Zmf(C_{1}, C_{2}))(c) = 0 and (Umf(C_{1}, C_{2}))(c) = 1.

properties of fuzzy relation 695

(32) For all x, y such that ⟨⟨x, y⟩⟩∈ [: C_{1}, C_{2} :] holds (Zmf(C_{1}, C_{2}))(⟨⟨x, y⟩⟩) = 0 and (Umf(C_{1}, C_{2}))(⟨⟨x, y⟩⟩) = 1.

(33) Let f be a Membership function of C_{2}, C_{3}, O_{1} be a zero relation of C_{1}, C_{2}, O_{2} be a zero relation of C_{1}, C_{3}, and R be a fuzzy relation of C_{2}, C_{3}, f. Then O_{1} R = O_{2}.

(34) For every Membership function f of C_{1}, C_{2} holds f Zmf(C_{2}, C_{3}) = Zmf(C_{1}, C_{3}).

(35) Let f be a Membership function of C_{1}, C_{2}, O_{1} be a zero relation of C_{2}, C_{3}, O_{2} be a zero relation of C_{1}, C_{3}, and R be a fuzzy relation of C_{1}, C_{2}, f. Then R O_{1} = O_{2}.

(36) For every Membership function f of C_{1}, C_{1} holds f Zmf(C_{1}, C_{1}) = Zmf(C_{1}, C_{1}) f.

(37) Let f be a Membership function of C_{1}, C_{1}, O be a zero relation of C_{1}, C_{1}, and R be a fuzzy relation of C_{1}, C_{1}, f. Then R O = O R.

References

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

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

Received June 25, 2001 696 noboru endou et al.

On Outside Fashoda Meet Theorem

Yatsuka Nakamura Shinshu University Nagano

Summary. We have proven the “Fashoda Meet Theorem” in [12]. Here we prove the outside version of it. It says that if Britain and France intended to set the courses for ships to the opposite side of Africa, they must also meet.

MML Identifier: JGRAPH 2.

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

One can prove the following propositions:

(1) For all real numbers a, b such that a ≠ 0 and b ≠ 0 holds \frac{a}{b} · \frac{b}{a} = 1.

(2) For every real number a such that 1 a holds a a^{2}.

(3) For every real number a such that − 1 a holds − a a^{2}.

(4) For every real number a such that − 1 > a holds − a < a^{2}.

(5) For all real numbers a, b such that b^{2} a^{2} and a 0 holds − a b and b a.

(6) For all real numbers a, b such that b^{2} < a^{2} and a 0 holds − a < b and b < a.

(7) For all real numbers a, b such that − a b and b a holds b^{2} a^{2}.

(8) For all real numbers a, b such that − a < b and b < a holds b^{2} < a^{2}. In the sequel T, T_{1}, T_{2}, S denote non empty topological spaces. Next we state a number of propositions:

(9) Let f be a map from T_{1} into S, g be a map from T_{2} into S, and F_{1}, F_{2} be subsets of T. Suppose that T_{1} is a subspace of T and T_{2} is a subspace of T and F_{1} = Ω_{(T_{1})} and F_{2} = Ω_{(T_{2})} and Ω_{(T_{1})} ∪ Ω_{(T_{2})} = Ω_{T} and F_{1} is closed and F_{2} is closed and f is continuous and g is continuous and for every set p such that p ∈ Ω_{(T_{1})} ∩ Ω_{(T_{2})} holds f(p) = g(p). Then there exists a map h from T into S such that h = f+·g and h is continuous.

(10) Let n be a natural number, q_{2} be a point of E^{n}, q be a point of E_{T}^{n}, and r be a real number. If q = q_{2}, then Ball(q_{2}, r) = {q_{3}; q_{3} ranges over points of E_{T}^{n}: |q − q_{3}| < r}.

(11) (0_{E_{T}^{2}})_{1} = 0 and (0_{E_{T}^{2}})_{2} = 0.

(12) 1.REAL2 = ⟨(1 qua real number), (1 qua real number)⟩.

(13) (1.REAL2)_{1} = 1 and (1.REAL2)_{2} = 1.

(14) domproj1 = the carrier of E_{T}^{2} and domproj1 = R^{2}.

(15) domproj2 = the carrier of E_{T}^{2} and domproj2 = R^{2}.

(16) proj1 is a map from E_{T}^{2} into ℝ^{1}.

(17) proj2 is a map from E_{T}^{2} into ℝ^{1}.

(18) For every point p of E_{T}^{2} holds p = [proj1(p), proj2(p)].

(19) For every subset B of the carrier of E_{T}^{2} such that B = {0_{E_{T}^{2}}} holds B^{c} ≠ ∅ and (the carrier of E_{T}^{2}) \ B ≠ ∅.

(20) Let X, Y be non empty topological spaces and f be a map from X into Y. Then f is continuous if and only if for every point p of X and for every subset V of Y such that f(p) ∈ V and V is open there exists a subset W of X such that p ∈ W and W is open and f^{∘}W ⊆ V.

(21) Let p be a point of E_{T}^{2} and G be a subset of E_{T}^{2}. Suppose G is open and p ∈ G. Then there exists a real number r such that r > 0 and {q; q ranges over points of E_{T}^{2}: p_{1}−r < q_{1} ∧ q_{1} < p_{1}+r ∧ p_{2}−r < q_{2} ∧ q_{2} < p_{2}+r} ⊆ G.

(22) Let X, Y, Z be non empty topological spaces, B be a subset of Y, C be a subset of Z, f be a map from X into Y, and h be a map from Y ↾B into Z↾C. Suppose f is continuous and h is continuous and rngf ⊆ B and B ≠ ∅ and C ≠ ∅. Then there exists a map g from X into Z such that g is continuous and g = h · f.

In the sequel p, q are points of E_{T}^{2}.

The function OutInSq from (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} into (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} is defined by the condition (Def. 1).

(Def. 1) Let p be a point of E_{T}^{2} such that p ≠ 0_{E_{T}^{2}}. Then

(i) if p_{2} p_{1} and − p_{1} p_{2} or p_{2} p_{1} and p_{2} − p_{1}, then OutInSq(p) = [\frac{1}{p_{1}} , ^{\frac{\frac{p_{2}}{p_{1}}}{p_{1}}} ], and

(ii) if p_{2} p̸_{1} or − p_{1} p̸_{2} and if p_{2} p̸_{1} or p_{2} −̸ p_{1}, then OutInSq(p) =

[^{\frac{\frac{p_{1}}{p_{2}}}{p_{2}}} , \frac{1}{p_{2}} ]. Next we state a number of propositions:

(23) Let p be a point of E_{T}^{2}. Suppose p_{2} p̸_{1} or − p_{1} p̸_{2} but p_{2} p̸_{1} or p_{2} −̸ p_{1}. Then p_{1} p_{2} and − p_{2} p_{1} or p_{1} p_{2} and p_{1} − p_{2}.

(24) Let p be a point of E_{T}^{2} such that p ≠ 0_{E_{T}^{2}}. Then (i) if p_{1} p_{2} and − p_{2} p_{1} or p_{1} p_{2} and p_{1} − p_{2}, then OutInSq(p) =

[^{\frac{\frac{p_{1}}{p_{2}}}{p_{2}}} , \frac{1}{p_{2}} ], and (ii) if p_{1} p̸_{2} or − p_{2} p̸_{1} and if p_{1} p̸_{2} or p_{1} −̸ p_{2}, then OutInSq(p) =

[\frac{1}{p_{1}} , ^{\frac{\frac{p_{2}}{p_{1}}}{p_{1}}} ].

(25) Let D be a subset of E_{T}^{2} and K_{0} be a subset of (E_{T}^{2})↾D. Suppose K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then rng(OutInSq↾K_{0}) ⊆ the carrier of (E_{T}^{2})↾D↾K_{0}.

(26) Let D be a subset of E_{T}^{2} and K_{0} be a subset of (E_{T}^{2})↾D. Suppose K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then rng(OutInSq↾K_{0}) ⊆ the carrier of (E_{T}^{2})↾D↾K_{0}.

(27) Let K_{1} be a set and D be a non empty subset of E_{T}^{2}. Suppose K_{1} = {p; p ranges over points of E_{T}^{2}: (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}} and D^{c} = {0_{E_{T}^{2}}}. Then K_{1} is a non empty subset of (E_{T}^{2})↾D and a non empty subset of E_{T}^{2}.

(28) Let K_{1} be a set and D be a non empty subset of E_{T}^{2}. Suppose K_{1} = {p; p ranges over points of E_{T}^{2}: (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}} and D^{c} = {0_{E_{T}^{2}}}. Then K_{1} is a non empty subset of (E_{T}^{2})↾D and a non empty subset of E_{T}^{2}.

(29) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} + r_{2} and g is continuous.

(30) Let X be a non empty topological space and a be a real number. Then there exists a map g from X into ℝ^{1} such that for every point p of X holds g(p) = a and g is continuous.

(31) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} − r_{2} and g is continuous.

(32) Let X be a non empty topological space and f_{1} be a map from X into ℝ^{1}. Suppose f_{1} is continuous. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for every real number r_{1} such that f_{1}(p) = r_{1} holds g(p) = r_{1} · r_{1} and g is continuous.

(33) Let X be a non empty topological space, f_{1} be a map from X into ℝ^{1}, and a be a real number. Suppose f_{1} is continuous. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for every real number r_{1} such that f_{1}(p) = r_{1} holds g(p) = a · r_{1} and g is continuous.

(34) Let X be a non empty topological space, f_{1} be a map from X into ℝ^{1}, and a be a real number. Suppose f_{1} is continuous. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for every real number r_{1} such that f_{1}(p) = r_{1} holds g(p) = r_{1} + a and g is continuous.

(35) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} · r_{2} and g is continuous.

(36) Let X be a non empty topological space and f_{1} be a map from X into ℝ^{1}. Suppose f_{1} is continuous and for every point q of X holds f_{1}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for every real number r_{1} such that f_{1}(p) = r_{1} holds g(p) = \frac{1}{r_{1}} and g is continuous.

(37) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = \frac{r_{1}}{r_{2}} and g is continuous.

(38) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1}

and f_{2}(p) = r_{2} holds g(p) = ^{\frac{\frac{r_{1}}{r_{2}}}{r_{2}}} , and (ii) g is continuous.

(39) Let K_{0} be a subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into ℝ^{1}. If for every point p of (E_{T}^{2})↾K_{0} holds f(p) = proj1(p), then f is continuous.

(40) Let K_{0} be a subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into ℝ^{1}. If for every point p of (E_{T}^{2})↾K_{0} holds f(p) = proj2(p), then f is continuous.

(41) Let K_{2} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{2} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{2} holds f(p) = \frac{1}{p_{1}} , and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{2} holds q_{1} ≠ 0. Then f is continuous.

(42) Let K_{2} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{2} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{2} holds f(p) = \frac{1}{p_{2}} , and (ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{2} holds

q_{2} ≠ 0. Then f is continuous.

(43) Let K_{2} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{2} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{2} holds

f(p) = ^{\frac{\frac{p_{2}}{p_{1}}}{p_{1}}} , and (ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{2} holds

q_{1} ≠ 0. Then f is continuous.

(44) Let K_{2} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{2} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{2} holds

f(p) = ^{\frac{\frac{p_{1}}{p_{2}}}{p_{2}}} , and (ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{2} holds

q_{2} ≠ 0. Then f is continuous.

(45) Let K_{0}, B_{0} be subsets of E_{T}^{2}, f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0},

and f_{1}, f_{2} be maps from (E_{T}^{2})↾K_{0} into ℝ^{1}. Suppose that

(i) f_{1} is continuous, (ii) f_{2} is continuous, (iii) K_{0} ≠ ∅, (iv) B_{0} ≠ ∅, and

(v) for all real numbers x, y, r, s such that [x, y] ∈ K_{0} and r = f_{1}([x, y]) and s = f_{2}([x, y]) holds f([x, y]) = [r, s]. Then f is continuous.

(46) Let K_{0}, B_{0} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = OutInSq↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous.

(47) Let K_{0}, B_{0} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = OutInSq↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous.

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

{p; p ranges over points of E_{T}^{2}: P[p]} is a subset of E_{T}^{2} for all values of the parameters.

The scheme TopCompl deals with a subset A of E_{T}^{2} and a unary predicate P, and states that:

− A = {p; p ranges over points of E_{T}^{2}: not P[p]} provided the parameters meet the following requirement:

• A = {p; p ranges over points of E_{T}^{2}: P[p]}. The scheme ClosedSubset deals with two unary functors F and G yielding

real numbers, and states that:

{p; p ranges over points of E_{T}^{2}: F(p) G(p)} is a closed subset of E_{T}^{2}

provided the following conditions are met:

• For all points p, q of E_{T}^{2} holds F(p − q) = F(p) − F(q) and G(p − q) = G(p) − G(q), and

• For all points p, q of E_{T}^{2} holds |p − q|^{2} = |F(p − q)|^{2} + |G(p − q)|^{2}. One can prove the following propositions:

(48) Let B_{0} be a subset of E_{T}^{2}, K_{0} be a subset of (E_{T}^{2})↾B_{0}, and f be a map from (E_{T}^{2})↾B_{0}↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = OutInSq↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous and K_{0} is closed.

(49) Let B_{0} be a subset of E_{T}^{2}, K_{0} be a subset of (E_{T}^{2})↾B_{0}, and f be a map from (E_{T}^{2})↾B_{0}↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = OutInSq↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous and K_{0} is closed.

(50) Let D be a non empty subset of E_{T}^{2}. Suppose D^{c} = {0_{E_{T}^{2}}}. Then there exists a map h from (E_{T}^{2})↾D into (E_{T}^{2})↾D such that h = OutInSq and h is continuous.

(51) Let B, K_{0}, K_{3} be subsets of E_{T}^{2}. Suppose that (i) B = {0_{E_{T}^{2}}}, (ii) K_{0} = {p : − 1 < p_{1} ∧ p_{1} < 1 ∧ − 1 < p_{2} ∧ p_{2} < 1}, and (iii) K_{3} = {q : − 1 = q_{1} ∧ − 1 q_{2} ∧ q_{2} 1 ∨ q_{1} = 1 ∧ − 1 q_{2} ∧ q_{2}

1 ∨ − 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1 ∨ 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1}. Then there exists a map f from (E_{T}^{2})↾B^{c} into (E_{T}^{2})↾B^{c} such that

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

(v) for every point t of E_{T}^{2} such that t ∈ K_{0} and t ≠ 0_{E_{T}^{2}} holds f(t) ∉ K_{0} ∪ K_{3},

(vi) for every point r of E_{T}^{2} such that r ∉ K_{0} ∪ K_{3} holds f(r) ∈ K_{0}, and (vii) for every point s of E_{T}^{2} such that s ∈ K_{3} holds f(s) = s.

(52) Let f, g be maps from 𝕀 into E_{T}^{2}, K_{0} be a subset of E_{T}^{2}, and O, I be points of 𝕀. Suppose that O = 0 and I = 1 and f is continuous and oneto-one and g is continuous and one-to-one and K_{0} = {p : − 1 < p_{1} ∧ p_{1} < 1 ∧ − 1 < p_{2} ∧ p_{2} < 1} and f(O)_{1} = − 1 and f(I)_{1} = 1 and − 1 f(O)_{2} and f(O)_{2} 1 and − 1 f(I)_{2} and f(I)_{2} 1 and g(O)_{2} = − 1 and g(I)_{2} = 1 and − 1 g(O)_{1} and g(O)_{1} 1 and − 1 g(I)_{1} and g(I)_{1} 1 and rngf ∩ K_{0} = ∅ and rngg ∩ K_{0} = ∅. Then rngf ∩ rngg ≠ ∅.

on outside fashoda meet theorem 703

(53) Let A, B, C, D be real numbers and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose that for every point t of E_{T}^{2} holds f(t) = [A · t_{1} + B, C · t_{2} + D]. Then f is continuous.

(54) 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 O = 0 and I = 1 and f is continuous and one-to-one and g is continuous and one-to-one and f(O)_{1} = a and f(I)_{1} = b and c f(O)_{2} and f(O)_{2} d and c f(I)_{2} and f(I)_{2} d and g(O)_{2} = c and g(I)_{2} = d and a g(O)_{1} and g(O)_{1} b and a g(I)_{1} and g(I)_{1} b and a < b and c < d and it is not true that there exists a point r of 𝕀 such that a < f(r)_{1} and f(r)_{1} < b and c < f(r)_{2} and f(r)_{2} < d and it is not true that there exists a point r of 𝕀 such that a < g(r)_{1} and g(r)_{1} < b and c < g(r)_{2} and g(r)_{2} < d. Then rngf ∩ rngg ≠ ∅.

(55)(i) {p_{7}; p_{7} ranges over points of E_{T}^{2}: (p_{7})_{2} (p_{7})_{1}} is a closed subset of

E_{T}^{2}, and (ii) {p_{7}; p_{7} ranges over points of E_{T}^{2}: (p_{7})_{1} (p_{7})_{2}} is a closed subset of E_{T}^{2}. (56)(i) {p_{7}; p_{7} ranges over points of E_{T}^{2}: − (p_{7})_{1} (p_{7})_{2}} is a closed subset of

E_{T}^{2}, and (ii) {p_{7}; p_{7} ranges over points of E_{T}^{2}: (p_{7})_{2} − (p_{7})_{1}} is a closed subset of

E_{T}^{2}. (57)(i) {p_{7}; p_{7} ranges over points of E_{T}^{2}: − (p_{7})_{2} (p_{7})_{1}} is a closed subset of

E_{T}^{2}, and (ii) {p_{7}; p_{7} ranges over points of E_{T}^{2}: (p_{7})_{1} − (p_{7})_{2}} is a closed subset of

E_{T}^{2}.

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The Set of Primitive Recursive Functions^{1}

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Piotr Rudnicki University of Alberta Edmonton

Summary. We follow [23] in defining the set of primitive recursive functions. The important helper notion is the homogeneous function from finite sequences of natural numbers into natural numbers where homogeneous means that all the sequences in the domain are of the same length. The set of all such functions is then used to define the notion of a set closed under composition of functions and under primitive recursion. We call a set primitively recursively closed iff it contains the initial functions (nullary constant function returning 0, unary successor and projection functions for all arities) and is closed under composition and primitive recursion. The set of primitive recursive functions is then defined as the smallest set of functions which is primitive recursively closed. We show that this set can be obtained by primitive recursive approximation. We finish with showing that some simple and well known functions are primitive recursive.

MML Identifier: COMPUT 1.

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

1. Preliminaries

For simplicity, we adopt the following rules: i, j, k, c, m, n are natural numbers, a, x, y, z, X, Y are sets, D, E are non empty sets, R is a binary relation, f, g are functions, and p, q are finite sequences.

1This work has been supported by NSERC Grant OGP9207, NATO CRG 951368 and TYPES grant IST-1999-29001.

Let X be a non empty set, let n be a natural number, let p be an element of X^{n}, let i be a natural number, and let x be an element of X. Then p +·(i, x) is an element of X^{n}.

Let n be a natural number, let t be an element of ℕ^{n}, and let i be a natural number. Then t(i) is an element of ℕ.

The following propositions are true: (3)^{2} ⟨x, y⟩ +·(1, z) = ⟨z, y⟩ and ⟨x, y⟩ +·(2, z) = ⟨x, z⟩. (5)^{3} If f +·(a, x) = g +·(a, y), then f +·(a, z) = g +·(a, z).

(6) (p +·(i, x))_{↾i} = p_{↾i}.

(7) If p +·(i, a) = q +·(i, a), then p_{↾i} = q_{↾i}.

(8) X^{0} = {∅}.

(9) If n ≠ 0, then ∅^{n} = ∅.

(10) If ∅ ∈ rngf, then ∏^{∗} f = ∅.

(11) If rngf = D, then rng∏^{∗}⟨f⟩ = D^{1}.

(12) If 1 i and i n + 1, then for every element p of D^{n+1} holds p_{↾i} ∈ D^{n}.

(13) For every set X and for every set Y of finite sequences of X holds Y ⊆ X^{∗}.

2. Sets of Compatible Functions

Let X be a set. We say that X is compatible if and only if: (Def. 1) For all functions f, g such that f ∈ X and g ∈ X holds f ≈ g.

Let us observe that there exists a set which is non empty, functional, and compatible.

Let X be a functional compatible set. One can verify that ⋃ X is functionlike and relation-like.

The following proposition is true

(14) X is functional and compatible iff ⋃ X is a function.

Let X, Y be sets. One can verify that there exists a non empty set of partial functions from X to Y which is non empty and compatible.

The following propositions are true:

(15) For every non empty functional compatible set X holds dom⋃ X = ⋃{domf : f ranges over elements of X}.

(16) Let X be a functional compatible set and f be a function. If f ∈ X, then domf ⊆ dom⋃ X and for every set x such that x ∈ domf holds (⋃ X)(x) = f(x).

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

3The proposition (4) has been removed.

(17) For every non empty functional compatible set X holds rng⋃ X =

⋃{rngf : f ranges over elements of X}. Let us consider X, Y. Observe that every non empty set of partial functions

from X to Y is functional.

We now state the proposition

(18) Let P be a compatible non empty set of partial functions from X to Y. Then ⋃ P is a partial function from X to Y.

3. Homogeneous Relations

Let f be a binary relation. We introduce f is into ℕ as a synonym of f is natural-yielding.

Let f be a binary relation. We say that f is from tuples on ℕ if and only if: (Def. 2) domf ⊆ ℕ^{∗}.

One can check that there exists a function which is from tuples on ℕ and into ℕ.

Let f be a binary relation from tuples on ℕ. We say that f is length total if and only if:

(Def. 3) For all finite sequences x, y of elements of ℕ such that lenx = leny and

x ∈ domf holds y ∈ domf. Let f be a binary relation. We say that f is homogeneous if and only if:

(Def. 4) For all finite sequences x, y such that x ∈ domf and y ∈ domf holds

lenx = leny. One can prove the following proposition

(19) If domR ⊆ D^{n}, then R is homogeneous.

Let us observe that ∅ is homogeneous. Let p be a finite sequence and let x be a set. Observe that {p} ⟼→ x is non

empty and homogeneous.

Let us note that there exists a function which is non empty and homogeneous. Let f be a homogeneous function and let g be a function. Observe that g · f

is homogeneous.

Let X, Y be sets. Note that there exists a partial function from X^{∗} to Y which is homogeneous.

Let X, Y be non empty sets. Observe that there exists a partial function from X^{∗} to Y which is non empty and homogeneous.

Let X be a non empty set. Observe that there exists a partial function from X^{∗} to X which is non empty, homogeneous, and quasi total.

One can check that there exists a function from tuples on ℕ which is non empty, homogeneous, into ℕ, and length total.

One can check that every partial function from ℕ^{∗} to ℕ is into ℕ and from tuples on ℕ.

Let us observe that every partial function from ℕ^{∗} to ℕ which is quasi total is also length total.

The following proposition is true

(20) Every length total function from tuples on ℕ into ℕ is a quasi total

partial function from ℕ^{∗} to ℕ. Let f be a homogeneous binary relation. The functor arityf yielding a na-

tural number is defined by: (Def. 5)(i) For every finite sequence x such that x ∈ domf holds arityf = lenx

if there exists a finite sequence x such that x ∈ domf, (ii) arityf = 0, otherwise. The following propositions are true:

(21) arity∅ = 0.

(22) For every homogeneous binary relation f such that domf = {∅} holds arityf = 0.

(23) For every homogeneous partial function f from X^{∗} to Y holds domf ⊆ X^{arity f}.

(24) For every homogeneous function f from tuples on ℕ holds domf ⊆ ℕ^{arity f}.

(25) Let f be a homogeneous partial function from X^{∗} to X. Then f is quasi total and non empty if and only if domf = X^{arity f}.

(26) Let f be a homogeneous function into ℕ and from tuples on ℕ. Then f is length total and non empty if and only if domf = ℕ^{arity f}.

(27) For every non empty homogeneous partial function f from D^{∗} to D and for every n such that domf ⊆ D^{n} holds arityf = n.

(28) For every homogeneous partial function f from D^{∗} to D and for every

n such that domf = D^{n} holds arityf = n. Let R be a binary relation. We say that R has the same arity if and only if

the condition (Def. 6) is satisfied. (Def. 6) Let f, g be functions such that f ∈ rngR and g ∈ rngR. Then

(i) if f is empty, then g is empty or domg = {∅}, and (ii) if f is non empty and g is non empty, then there exists a natural

number n and there exists a non empty set X such that domf ⊆ X^{n} and domg ⊆ X^{n}.

Let us note that ∅ has the same arity. One can check that there exists a finite sequence which has the same arity.

Let X be a set. One can verify that there exists a finite sequence of elements of X which has the same arity and there exists an element of X^{∗} which has the same arity.

Let F be a binary relation with the same arity. The functor arityF yielding a natural number is defined as follows:

(Def. 7)(i) For every homogeneous function f such that f ∈ rngF holds arityF = arityf if there exists a homogeneous function f such that f ∈ rngF,

(ii) arityF = 0, otherwise. Next we state the proposition

(29) For every finite sequence F with the same arity such that lenF = 0

holds arityF = 0. Let X be a set. The functor HFuncsX yielding a non empty set of partial

functions from X^{∗} to X is defined by: (Def. 8) HFuncsX = {f; f ranges over elements of X^{∗} ˙→ X : f is homogeneous}.

Next we state the proposition

(30) ∅ ∈ HFuncsX.

Let X be a non empty set. Note that there exists an element of HFuncsX which is non empty, homogeneous, and quasi total.

Let X be a set. Observe that every element of HFuncsX is homogeneous. Let X be a non empty set and let S be a non empty subset of HFuncsX.

Note that every element of S is homogeneous.

The following propositions are true:

(31) Every homogeneous function into ℕ and from tuples on ℕ is an element of HFuncsℕ.

(32) Every length total homogeneous function from tuples on ℕ into ℕ is a quasi total element of HFuncsℕ.

(33) Let X be a non empty set and F be a binary relation such that rngF ⊆ HFuncsX and for all homogeneous functions f, g such that f ∈ rngF and g ∈ rngF holds arityf = arityg. Then F has the same arity.

Let n, m be natural numbers. The functor const_{n}(m) yields a homogeneous function into ℕ and from tuples on ℕ and is defined by:

(Def. 9) const_{n}(m) = ℕ^{n} ⟼→ m.

We now state the proposition

(34) const_{n}(m) ∈ HFuncsℕ.

Let n, m be natural numbers. One can check that const_{n}(m) is length total and non empty.

We now state two propositions:

(35) arityconst_{n}(m) = n.

(36) For every element t of ℕ^{n} holds (const_{n}(m))(t) = m.

Let n, i be natural numbers. The functor succ_{n}(i) yields a homogeneous function into ℕ and from tuples on ℕ and is defined by:

(Def. 10) domsucc_{n}(i) = ℕ^{n} and for every element p of ℕ^{n} holds (succ_{n}(i))(p) =

p_{i} + 1. We now state the proposition

(37) succ_{n}(i) ∈ HFuncsℕ.

Let n, i be natural numbers. One can check that succ_{n}(i) is length total and non empty.

Next we state the proposition

(38) aritysucc_{n}(i) = n.

Let n, i be natural numbers. The functor proj_{n}(i) yielding a homogeneous function into ℕ and from tuples on ℕ is defined by:

(Def. 11) proj_{n}(i) = proj(n ↦ ℕ, i).

The following two propositions are true:

(39) proj_{n}(i) ∈ HFuncsℕ.

(40) domproj_{n}(i) = ℕ^{n} and if 1 i and i n, then rngproj_{n}(i) = ℕ. Let n, i be natural numbers. One can verify that proj_{n}(i) is length total and

non empty.

We now state two propositions:

(41) arityproj_{n}(i) = n.

(42) For every element t of ℕ^{n} holds (proj_{n}(i))(t) = t(i).

Let X be a set. Observe that HFuncsX is functional. We now state three propositions:

(43) Let F be a function from D into HFuncsE. Suppose rngF is compatible and for every element x of D holds domF (x) ⊆ E^{n}. Then there exists an element f of HFuncsE such that f = ⋃ F and domf ⊆ E^{n}.

(44) For every function F from ℕ into HFuncsD such that for every i holds F (i) ⊆ F (i + 1) holds ⋃ F ∈ HFuncsD.

(45) For every finite sequence F of elements of HFuncsD with the same arity

holds dom∏^{∗} F ⊆ D^{arity F} . Let X be a non empty set and let F be a finite sequence of elements of

HFuncsX with the same arity. Observe that ∏^{∗} F is homogeneous.

The following proposition is true

(46) Let f be an element of HFuncsD and F be a finite sequence of elements of HFuncsD with the same arity. Then dom(f · ∏^{∗} F) ⊆ D^{arity F} and rng(f · ∏^{∗} F) ⊆ D and f · ∏^{∗} F ∈ HFuncsD.

Let X, Y be non empty sets, let P be a non empty set of partial functions from X to Y, and let S be a non empty subset of P. We see that the element of S is an element of P.

Let f be a homogeneous function from tuples on ℕ. One can check that ⟨f⟩ has the same arity.

Next we state several propositions:

(47) For every homogeneous function f into ℕ and from tuples on ℕ holds arity⟨f⟩ = arityf.

(48) Let f, g be non empty elements of HFuncsℕ and F be a finite sequence of elements of HFuncsℕ with the same arity. If g = f ·∏^{∗} F, then arityg = arityF.

(49) Let f be a non empty quasi total element of HFuncsD and F be a finite sequence of elements of HFuncsD with the same arity. Suppose arityf = lenF and F is non empty and for every element h of HFuncsD such that h ∈ rngF holds h is quasi total and non empty. Then f ·∏^{∗} F is a non empty quasi total element of HFuncsD and dom(f · ∏^{∗} F) = D^{arity F} .

(50) Let f be a quasi total element of HFuncsD and F be a finite sequence of elements of HFuncsD with the same arity. Suppose arityf = lenF and for every element h of HFuncsD such that h ∈ rngF holds h is quasi total. Then f · ∏^{∗} F is a quasi total element of HFuncsD.

(51) For all non empty quasi total elements f, g of HFuncsD such that arityf = 0 and arityg = 0 and f(∅) = g(∅) holds f = g.

(52) Let f, g be non empty length total homogeneous functions from tuples on ℕ into ℕ. If arityf = 0 and arityg = 0 and f(∅) = g(∅), then f = g.

4. Primitive Recursiveness

We adopt the following convention: f_{1}, f_{2} are non empty homogeneous functions into ℕ and from tuples on ℕ, e_{1}, e_{2} are homogeneous functions into ℕ and from tuples on ℕ, and p is an element of ℕ^{arity f_{1}+1}.

Let g, f_{1}, f_{2} be homogeneous functions into ℕ and from tuples on ℕ and let i be a natural number. We say that g is primitive recursively expressed by f_{1}, f_{2} and i if and only if the condition (Def. 12) is satisfied.

(Def. 12) There exists a natural number n such that (i) domg ⊆ ℕ^{n}, (ii) i 1, (iii) i n, (iv) arityf_{1} + 1 = n,

(v) n + 1 = arityf_{2}, and (vi) for every finite sequence p of elements of ℕ such that lenp = n holds

p +·(i, 0) ∈ domg iff p_{↾i} ∈ domf_{1} and if p +·(i, 0) ∈ domg, then g(p +· (i, 0)) = f_{1}(p_{↾i}) and for every natural number n holds p+·(i, n+1) ∈ domg iff p +·(i, n) ∈ domg and (p +·(i, n)) ^{⌢} ⟨g(p +·(i, n))⟩ ∈ domf_{2} and if p+·(i, n+1) ∈ domg, then g(p+·(i, n+1)) = f_{2}((p+·(i, n))^{⌢}⟨g(p+·(i, n))⟩).

Let f_{1}, f_{2} be homogeneous functions into ℕ and from tuples on ℕ, let i be a natural number, and let p be a finite sequence of elements of ℕ. The functor primrec(f_{1}, f_{2}, i, p) yielding an element of HFuncsℕ is defined by the condition (Def. 13).

(Def. 13) There exists a function F from ℕ into HFuncsℕ such that (i) primrec(f_{1}, f_{2}, i, p) = F (p_{i}), (ii) if i ∈ domp and p_{↾i} ∈ domf_{1}, then F (0) = {p +·(i, 0)} ⟼→ f_{1}(p_{↾i}), (iii) if i ∉ domp or p_{↾i} ∉ domf_{1}, then F (0) = ∅, and (iv) for every natural number m holds if i ∈ domp and p +·(i, m) ∈

domF (m) and (p+·(i, m))^{⌢}⟨F (m)(p+·(i, m))⟩ ∈ domf_{2}, then F (m+1) = F (m)+·({p +·(i, m + 1)} ⟼→ f_{2}((p +·(i, m))^{⌢} ⟨F (m)(p +·(i, m))⟩)) and if i ∉ domp or p+·(i, m) ∉ domF (m) or (p+·(i, m))^{⌢}⟨F (m)(p+·(i, m))⟩ ∉ domf_{2}, then F (m + 1) = F (m).

We now state several propositions:

(53) For all finite sequences p, q of elements of ℕ such that q ∈ domprimrec(e_{1}, e_{2}, i, p) there exists k such that q = p +·(i, k).

(54) For every finite sequence p of elements of ℕ such that i ∉ domp holds primrec(e_{1}, e_{2}, i, p) = ∅.

(55) For all finite sequences p, q of elements of ℕ holds primrec(e_{1}, e_{2}, i, p) ≈ primrec(e_{1}, e_{2}, i, q).

(56) For every finite sequence p of elements of ℕ holds domprimrec(e_{1}, e_{2}, i, p) ⊆ ℕ^{1+arity e_{1}}.

(57) For every finite sequence p of elements of ℕ such that e_{1} is empty holds primrec(e_{1}, e_{2}, i, p) is empty.

(58) If f_{1} is length total and f_{2} is length total and arityf_{1} + 2 = arityf_{2} and

1 i and i 1 + arityf_{1}, then p ∈ domprimrec(f_{1}, f_{2}, i, p). Let f_{1}, f_{2} be homogeneous functions into ℕ and from tuples on ℕ and let

i be a natural number. The functor primrec(f_{1}, f_{2}, i) yielding an element of HFuncsℕ is defined as follows:

(Def. 14) There exists a function G from ℕ^{arity f_{1}+1} into HFuncsℕ such that primrec(f_{1}, f_{2}, i) = ⋃ G and for every element p of ℕ^{arity f_{1}+1} holds G(p) = primrec(f_{1}, f_{2}, i, p).

One can prove the following propositions:

(59) If e_{1} is empty, then primrec(e_{1}, e_{2}, i) is empty.

(60) domprimrec(f_{1}, f_{2}, i) ⊆ ℕ^{arity f_{1}+1}.

(61) If f_{1} is length total and f_{2} is length total and arityf_{1} + 2 = arityf_{2} and 1 i and i 1 + arityf_{1}, then domprimrec(f_{1}, f_{2}, i) = ℕ^{arity f_{1}+1} and arityprimrec(f_{1}, f_{2}, i) = arityf_{1} + 1.

(62) If i ∈ domp, then p +·(i, 0) ∈ domprimrec(f_{1}, f_{2}, i) iff p_{↾i} ∈ domf_{1}.

(63) If i ∈ domp and p +· (i, 0) ∈ domprimrec(f_{1}, f_{2}, i), then (primrec(f_{1}, f_{2}, i))(p +·(i, 0)) = f_{1}(p_{↾i}).

(64) If i ∈ domp and f_{1} is length total, then (primrec(f_{1}, f_{2}, i))(p +·(i, 0)) = f_{1}(p_{↾i}).

(65) If i ∈ domp, then p +·(i, m + 1) ∈ domprimrec(f_{1}, f_{2}, i) iff p +·(i, m) ∈ domprimrec(f_{1}, f_{2}, i) and (p+·(i, m))^{⌢}⟨(primrec(f_{1}, f_{2}, i))(p+·(i, m))⟩ ∈ domf_{2}.

(66) If i ∈ domp and p +· (i, m + 1) ∈ domprimrec(f_{1}, f_{2}, i), then (primrec(f_{1}, f_{2}, i))(p +· (i, m + 1)) = f_{2}((p +· (i, m)) ^{⌢} ⟨(primrec(f_{1}, f_{2}, i))(p +·(i, m))⟩).

(67) Suppose f_{1} is length total and f_{2} is length total and arityf_{1}+2 = arityf_{2} and 1 i and i 1 + arityf_{1}. Then (primrec(f_{1}, f_{2}, i))(p +·(i, m + 1)) = f_{2}((p +·(i, m))^{⌢} ⟨(primrec(f_{1}, f_{2}, i))(p +·(i, m))⟩).

(68) If arityf_{1} + 2 = arityf_{2} and 1 i and i arityf_{1} + 1, then primrec(f_{1}, f_{2}, i) is primitive recursively expressed by f_{1}, f_{2} and i.

(69) Suppose 1 i and i arityf_{1}+1. Let g be an element of HFuncsℕ. If g is primitive recursively expressed by f_{1}, f_{2} and i, then g = primrec(f_{1}, f_{2}, i).

5. The Set of Primitive Recursive Functions

Let X be a set. We say that X is composition closed if and only if the condition (Def. 15) is satisfied.

(Def. 15) Let f be an element of HFuncsℕ and F be a finite sequence of elements of HFuncsℕ with the same arity. If f ∈ X and arityf = lenF and rngF ⊆ X, then f · ∏^{∗} F ∈ X.

We say that X is primitive recursion closed if and only if the condition (Def. 16) is satisfied.

(Def. 16) Let g, f_{1}, f_{2} be elements of HFuncsℕ and i be a natural number. Suppose g is primitive recursively expressed by f_{1}, f_{2} and i and f_{1} ∈ X and f_{2} ∈ X. Then g ∈ X.

Let X be a set. We say that X is primitive recursively closed if and only if the conditions (Def. 17) are satisfied.

(Def. 17)(i) const_{0}(0) ∈ X, (ii) succ_{1}(1) ∈ X, (iii) for all natural numbers n, i such that 1 i and i n holds proj_{n}(i) ∈

X, and (iv) X is composition closed and primitive recursion closed. We now state the proposition (70) HFuncsℕ is primitive recursively closed.

One can check that there exists a subset of HFuncsℕ which is primitive recursively closed and non empty.

In the sequel P is a primitive recursively closed non empty subset of HFuncsℕ. We now state several propositions:

(71) For every element g of HFuncsℕ such that e_{1} = ∅ and g is primitive recursively expressed by e_{1}, e_{2} and i holds g = ∅.

(72) Let g be an element of HFuncsℕ, f_{1}, f_{2} be quasi total elements of HFuncsℕ, and i be a natural number. Suppose g is primitive recursively expressed by f_{1}, f_{2} and i. Then g is quasi total and if f_{1} is non empty, then g is non empty.

(73) const_{n}(c) ∈ P.

(74) If 1 i and i n, then succ_{n}(i) ∈ P.

(75) ∅ ∈ P.

(76) Let f be an element of P and F be a finite sequence of elements of P with the same arity. If arityf = lenF, then f · ∏^{∗} F ∈ P.

(77) Let f_{1}, f_{2} be elements of P. Suppose arityf_{1} + 2 = arityf_{2}. Let i be a

natural number. If 1 i and i arityf_{1} + 1, then primrec(f_{1}, f_{2}, i) ∈ P. The subset PrimRec of HFuncsℕ is defined as follows:

(Def. 18) PrimRec = ⋂{R; R ranges over elements of 2^{HFuncs ℕ}: R is primitive

recursively closed}. The following proposition is true

(78) For every subset X of HFuncsℕ such that X is primitive recursively

closed holds PrimRec ⊆ X. Let us observe that PrimRec is non empty and primitive recursively closed. One can check that every element of PrimRec is homogeneous. Let x be a set. We say that x is primitive recursive if and only if:

(Def. 19) x ∈ PrimRec.

Let us note that every set which is primitive recursive is also relation-like and function-like.

Let us observe that every binary relation which is primitive recursive is also homogeneous, into ℕ, and from tuples on ℕ.

Let us observe that every element of PrimRec is primitive recursive. Let us note that there exists a function which is primitive recursive and there

exists an element of HFuncsℕ which is primitive recursive.

The initial functions constitute a subset of HFuncsℕ defined as follows: (Def. 20) The initial functions = {const_{0}(0), succ_{1}(1)} ∪ {proj_{n}(i); n ranges over

natural numbers, i ranges over natural numbers: 1 i ∧ i n}.

Let Q be a subset of HFuncsℕ. The primitive recursion closure of Q is a subset of HFuncsℕ and is defined by the condition (Def. 21).

(Def. 21) The primitive recursion closure of Q = Q ∪ {g; g ranges over elements of HFuncsℕ : ⋁_{f_{1},f_{2} : element of HFuncs ℕ} ⋁_{i: natural number} (f_{1} ∈ Q ∧ f_{2} ∈ Q ∧ g is primitive recursively expressed by f_{1}, f_{2} and i)}.

The composition closure of Q is a subset of HFuncsℕ and is defined by the condition (Def. 22).

(Def. 22) The composition closure of Q = Q ∪ {f · ∏^{∗} F; f ranges over elements of HFuncsℕ, F ranges over elements of (HFuncsℕ)^{∗} with the same arity: f ∈ Q ∧ arityf = lenF ∧ rngF ⊆ Q}.

The function PrimRec^{≈} from ℕ into 2^{HFuncs ℕ} is defined by the conditions (Def. 23).

(Def. 23)(i) PrimRec^{≈} (0) = the initial functions, and

(ii) for every natural number m holds PrimRec^{≈} (m + 1) = (the primitive recursion closure of PrimRec^{≈} (m)) ∪ (the composition closure of PrimRec^{≈} (m)).

One can prove the following propositions:

(79) If m n, then PrimRec^{≈} (m) ⊆ PrimRec^{≈} (n).

(80) ⋃(PrimRec^{≈} ) is primitive recursively closed.

(81) PrimRec = ⋃(PrimRec^{≈} ).

(82) For every element f of HFuncsℕ such that f ∈ PrimRec^{≈} (m) holds f is

quasi total. Let us note that every element of PrimRec is quasi total and homogeneous. Let us observe that every element of HFuncsℕ which is primitive recursive

is also quasi total.

Let us observe that every function from tuples on ℕ which is primitive recursive is also length total and there exists an element of PrimRec which is non empty.

6. Examples

Let f be a homogeneous binary relation. We say that f is nullary if and only if:

(Def. 24) arityf = 0.

We say that f is unary if and only if: (Def. 25) arityf = 1.

We say that f is binary if and only if: (Def. 26) arityf = 2.

We say that f is ternary if and only if: (Def. 27) arityf = 3.

One can check the following observations:

∗ every homogeneous function which is unary is also non empty,

∗ every homogeneous function which is binary is also non empty, and ∗ every homogeneous function which is ternary is also non empty. One can check the following observations:

∗ proj_{1}(1) is primitive recursive,

∗ proj_{2}(1) is primitive recursive,

∗ proj_{2}(2) is primitive recursive,

∗ succ_{1}(1) is primitive recursive, and

∗ succ_{3}(3) is primitive recursive. Let i be a natural number. One can check the following observations:

∗ const_{0}(i) is nullary,

∗ const_{1}(i) is unary,

∗ const_{2}(i) is binary,

∗ const_{3}(i) is ternary,

∗ proj_{1}(i) is unary,

∗ proj_{2}(i) is binary,

∗ proj_{3}(i) is ternary,

∗ succ_{1}(i) is unary,

∗ succ_{2}(i) is binary, and

∗ succ_{3}(i) is ternary. Let j be a natural number. One can check that const_{i}(j) is primitive recursive.

One can verify the following observations:

∗ there exists a homogeneous function which is nullary, primitive recursive, and non empty,

∗ there exists a homogeneous function which is unary and primitive recursive,

∗ there exists a homogeneous function which is binary and primitive recursive, and

∗ there exists a homogeneous function which is ternary and primitive recursive.

One can verify the following observations:

∗ there exists a homogeneous function from tuples on ℕ which is non empty, nullary, length total, and into ℕ, ∗ there exists a homogeneous function from tuples on ℕ which is non empty, unary, length total, and into ℕ,

∗ there exists a homogeneous function from tuples on ℕ which is non empty, binary, length total, and into ℕ, and

∗ there exists a homogeneous function from tuples on ℕ which is non empty, ternary, length total, and into ℕ.

Let f be a nullary non empty primitive recursive function and let g be a binary primitive recursive function. One can check that primrec(f, g, 1) is primitive recursive and unary.

Let f be a unary primitive recursive function and let g be a ternary primitive recursive function. One can verify that primrec(f, g, 1) is primitive recursive and binary and primrec(f, g, 2) is primitive recursive and binary.

The following propositions are true:

(83) Let f_{1} be a unary length total homogeneous function from tuples on ℕ into ℕ and f_{2} be a non empty homogeneous function into ℕ and from tuples on ℕ. Then (primrec(f_{1}, f_{2}, 2))(⟨i, 0⟩) = f_{1}(⟨i⟩).

(84) If f_{1} is length total and arityf_{1} = 0, then (primrec(f_{1}, f_{2}, 1))(⟨0⟩) = f_{1}(∅).

(85) Let f_{1} be a unary length total homogeneous function from tuples on ℕ into ℕ and f_{2} be a ternary length total homogeneous function from tuples on ℕ into ℕ. Then (primrec(f_{1}, f_{2}, 2))(⟨i, j + 1⟩) = f_{2}(⟨i, j, (primrec(f_{1}, f_{2}, 2))(⟨i, j⟩)⟩).

(86) If f_{1} is length total and f_{2} is length total and arityf_{1} = 0 and arityf_{2} =

2, then (primrec(f_{1}, f_{2}, 1))(⟨i + 1⟩) = f_{2}(⟨i, (primrec(f_{1}, f_{2}, 1))(⟨i⟩)⟩). Let g be a function. The functor ^{⟨1,?,2⟩}g yielding a function is defined by:

(Def. 28) ^{⟨1,?,2⟩}g = g · ∏^{∗}⟨proj_{3}(1), proj_{3}(3)⟩.

Let g be a function into ℕ and from tuples on ℕ. Observe that ^{⟨1,?,2⟩}g is into ℕ and from tuples on ℕ.

Let g be a homogeneous function. Note that ^{⟨1,?,2⟩}g is homogeneous. Let g be a binary length total homogeneous function from tuples on ℕ into

ℕ. Observe that ^{⟨1,?,2⟩}g is non empty ternary and length total.

The following propositions are true:

(87) Let f be a binary length total homogeneous function from tuples on ℕ into ℕ. Then (^{⟨1,?,2⟩}f)(⟨i, j, k⟩) = f(⟨i, k⟩).

(88) For every binary primitive recursive function g holds ^{⟨1,?,2⟩}g ∈ PrimRec. Let f be a binary primitive recursive homogeneous function. Observe that

⟨1,?,2⟩f is primitive recursive and ternary.

The binary primitive recursive function [+] is defined by: (Def. 29) [+] = primrec(proj_{1}(1), succ_{3}(3), 2).

We now state the proposition

(89) [+](⟨i, j⟩) = i + j.

The binary primitive recursive function [∗] is defined by: (Def. 30) [∗] = primrec(const_{1}(0),^{⟨1,?,2⟩}[+], 2).

Next we state the proposition

(90) For all natural numbers i, j holds [∗](⟨i, j⟩) = i · j. Let g, h be binary primitive recursive homogeneous functions. Note that ⟨g,

h⟩ has the same arity.

Let f, g, h be binary primitive recursive functions. Observe that f · ∏^{∗}⟨g, h⟩ is primitive recursive.

Let f, g, h be binary primitive recursive functions. Observe that f · ∏^{∗}⟨g, h⟩ is binary.

Let f be a unary primitive recursive function and let g be a primitive recursive function. Note that f · ∏^{∗}⟨g⟩ is primitive recursive.

Let f be a unary primitive recursive function and let g be a binary primitive recursive function. One can verify that f · ∏^{∗}⟨g⟩ is binary.

The unary primitive recursive function [!] is defined by: (Def. 31) [!] = primrec(const_{0}(1), [∗] · ∏^{∗}⟨succ_{1}(1) · ∏^{∗}⟨proj_{2}(1)⟩, proj_{2}(2)⟩, 1).

In this article we present several logical schemes. The scheme Primrec1 deals with a unary length total homogeneous function A from tuples on ℕ into ℕ, a binary length total homogeneous function B from tuples on ℕ into ℕ, a unary functor F yielding a natural number, and a binary functor G yielding a natural number, and states that:

For all natural numbers i, j holds (A · ∏^{∗}⟨B⟩)(⟨i, j⟩) = F(G(i, j)) provided the parameters meet the following requirements:

• For every natural number i holds A(⟨i⟩) = F(i), and

• For all natural numbers i, j holds B(⟨i, j⟩) = G(i, j).

The scheme Primrec2 deals with binary length total homogeneous functions A, B, C from tuples on ℕ into ℕ and three binary functors F, G, and H yielding natural numbers, and states that:

For all natural numbers i, j holds (A·∏^{∗}⟨B, C⟩)(⟨i, j⟩) = F(G(i, j), H(i, j)) provided the parameters meet the following conditions:

• For all natural numbers i, j holds A(⟨i, j⟩) = F(i, j),

• For all natural numbers i, j holds B(⟨i, j⟩) = G(i, j), and

• For all natural numbers i, j holds C(⟨i, j⟩) = H(i, j).

The following proposition is true

(91) [!](⟨i⟩) = i!.

The binary primitive recursive function [^{∧}] is defined by: (Def. 32) [^{∧}] = primrec(const_{1}(1),^{⟨1,?,2⟩}[∗], 2).

One can prove the following proposition

the set of primitive recursive functions 719

(92) [^{∧}](⟨i, j⟩) = i^{j}.

The unary primitive recursive function [pred] is defined as follows: (Def. 33) [pred] = primrec(const_{0}(0), proj_{2}(1), 1).

The following proposition is true

(93) [pred](⟨0⟩) = 0 and [pred](⟨i + 1⟩) = i. The binary primitive recursive function [−] is defined as follows:

(Def. 34) [−] = primrec(proj_{1}(1),^{⟨1,?,2⟩}([pred] · ∏^{∗}⟨proj_{2}(2)⟩), 2). The following proposition is true

(94) [−](⟨i, j⟩) = i − ′j.

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Introduction to Turing Machines

Jing-Chao Chen Bell Labs Research China Lucent Technologies Bejing Yatsuka Nakamura Shinshu University Nagano

Summary. A Turing machine can be viewed as a simple kind of computer, whose operations are constrainted to reading and writing symbols on a tape, or moving along the tape to the left or right. In theory, one has proven that the computability of Turing machines is equivalent to recursive functions. This article defines and verifies the Turing machines of summation and three primitive functions which are successor, zero and project functions. It is difficult to compute sophisticated functions by simple Turing machines. Therefore, we define the combination of two Turing machines.

MML Identifier: TURING 1.

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

1. Preliminaries

In this paper n, i, j, k denote natural numbers.

Let A, B be non empty sets, let f be a function from A into B, and let g be a partial function from A to B. Then f+·g is a function from A into B.

Let X, Y be non empty sets, let a be an element of X, and let b be an element of Y. Then a⟼→ . b is a partial function from X to Y.

Let n be a natural number. The functor Seg_{M} n yielding a subset of ℕ is defined as follows:

(Def. 1) Seg_{M} n = {k : k n}.

Let n be a natural number. One can verify that Seg_{M} n is finite and non empty.

One can prove the following propositions:

(1) k ∈ Seg_{M} n iff k n.

(2) For every function f and for all sets x, y, z, u, v such that u ≠ x holds (f+·(⟨⟨x, y⟩⟩⟼→ . z))(⟨⟨u, v⟩⟩) = f(⟨⟨u, v⟩⟩).

(3) For every function f and for all sets x, y, z, u, v such that v ≠ y holds (f+·(⟨⟨x, y⟩⟩⟼→ . z))(⟨⟨u, v⟩⟩) = f(⟨⟨u, v⟩⟩).

In the sequel i_{1}, i_{2}, i_{3}, i_{4} denote elements of ℤ.

We now state three propositions:

(4) ∑⟨i_{1}, i_{2}⟩ = i_{1} + i_{2}.

(5) ∑⟨i_{1}, i_{2}, i_{3}⟩ = i_{1} + i_{2} + i_{3}.

(6) ∑⟨i_{1}, i_{2}, i_{3}, i_{4}⟩ = i_{1} + i_{2} + i_{3} + i_{4}.

Let f be a finite sequence of elements of ℕ and let i be a natural number. The functor Prefix(f, i) yields a finite sequence of elements of ℤ and is defined by:

(Def. 2) Prefix(f, i) = f↾ Segi. Next we state two propositions:

(7) For all natural numbers x_{1}, x_{2} holds ∑ Prefix(⟨x_{1}, x_{2}⟩, 1) = x_{1} and ∑ Prefix(⟨x_{1}, x_{2}⟩, 2) = x_{1} + x_{2}.

(8) For all natural numbers x_{1}, x_{2}, x_{3} holds ∑ Prefix(⟨x_{1}, x_{2}, x_{3}⟩, 1) = x_{1} and ∑ Prefix(⟨x_{1}, x_{2}, x_{3}⟩, 2) = x_{1} +x_{2} and ∑ Prefix(⟨x_{1}, x_{2}, x_{3}⟩, 3) = x_{1} + x_{2} + x_{3}.

2. Definitions and Terminology for Turing Machine

We consider Turing machine structures as systems

⟨ symbols, control states, a transition, an initial state, an accepting state ⟩, where the symbols and the control states constitute finite non empty sets, the transition is a function from [: the control states, the symbols :] into [: the control states, the symbols, {− 1, 0, 1}:], and the initial state and the accepting state are elements of the control states.

Let T be a Turing machine structure. A state of T is an element of the control states of T. A tape of T is an element of (the symbols of T)^{ℤ}. A symbol of T is an element of the symbols of T.

Let T be a Turing machine structure, let t be a tape of T, let h be an integer, and let s be a symbol of T. The functor Tape-Chg(t, h, s) yields a tape of T and is defined as follows:

(Def. 3) Tape-Chg(t, h, s) = t+·(h⟼→ . s).

Let T be a Turing machine structure. A State of T is an element of [: the control states of T, ℤ, (the symbols of T)^{ℤ}:]. A transition-source of T is an element of [: the control states of T, the symbols of T :]. A transition-target of T is an element of [: the control states of T, the symbols of T, {− 1, 0, 1}:].

Let T be a Turing machine structure and let g be a transition-target of T. The functor offset(g) yields an integer and is defined as follows:

(Def. 4) offset(g) = g_{3}.

Let T be a Turing machine structure and let s be a State of T. The functor Head(s) yielding an integer is defined by:

(Def. 5) Head(s) = s_{2}.

Let T be a Turing machine structure and let s be a State of T. The functor s -target yielding a transition-target of T is defined by:

(Def. 6) s -target = (the transition of T)(⟨⟨s_{1}, (s_{3} qua tape of T)(Head(s))⟩⟩).

Let T be a Turing machine structure and let s be a State of T. The functor Following(s) yields a State of T and is defined as follows:

(Def. 7) Following(s) = \{ \begin{matrix}⟨⟨s -target_{1}, Head(s) + offset(s -target), \\ Tape-Chg(s_{3}, Head(s), s -target_{2})⟩⟩, \\ if s_{1} ≠ the accepting state of T, \\ s, otherwise. \end{matrix}

Let T be a Turing machine structure and let s be a State of T. The functor Computation(s) yielding a function from ℕ into [: the control states of T, ℤ, (the symbols of T)^{ℤ}:] is defined as follows:

(Def. 8) (Computation(s))(0) = s and for every i holds (Computation(s))(i+1) =

Following((Computation(s))(i)). In the sequel T is a Turing machine structure and s is a State of T. The following propositions are true:

(9) Let T be a Turing machine structure and s be a State of T. If s_{1} = the accepting state of T, then s = Following(s).

(10) (Computation(s))(0) = s.

(11) (Computation(s))(k + 1) = Following((Computation(s))(k)).

(12) (Computation(s))(1) = Following(s).

(13) (Computation(s))(i + k) = (Computation((Computation(s))(i)))(k).

(14) If i j and Following((Computation(s))(i)) = (Computation(s))(i), then (Computation(s))(j) = (Computation(s))(i).

(15) If i j and (Computation(s))(i)_{1} = the accepting state of T, then

(Computation(s))(j) = (Computation(s))(i). Let T be a Turing machine structure and let s be a State of T. We say that

s is accepting if and only if:

(Def. 9) There exists k such that (Computation(s))(k)_{1} = the accepting state of

T. Let T be a Turing machine structure and let s be a State of T. Let us assume

that s is accepting. The functor Result(s) yielding a State of T is defined by: (Def. 10) There exists k such that Result(s) = (Computation(s))(k) and

(Computation(s))(k)_{1} = the accepting state of T. We now state the proposition

(16) Let T be a Turing machine structure and s be a State of T. Suppose s

is accepting. Then there exists a natural number k such that

(i) (Computation(s))(k)_{1} = the accepting state of T, (ii) Result(s) = (Computation(s))(k), and (iii) for every natural number i such that i < k holds (Computation(s))(i)_{1} ≠

the accepting state of T. Let A, B be non empty sets and let y be a set. Let us assume that y ∈ B.

The functor id(A, B, y) yields a function from A into [: A, B:] and is defined as follows:

(Def. 11) For every element x of A holds (id(A, B, y))(x) = ⟨⟨x, y⟩⟩.

The function SumTran from [: Seg_{M} 5, {0, 1}:] into [: Seg_{M} 5, {0, 1}, {− 1, 0, 1}:] is defined as follows:

(Def. 12) SumTran = id([: Seg_{M} 5, {0, 1}:], {− 1, 0, 1}, 0)+·(⟨⟨0, 0⟩⟩⟼→ . ⟨⟨0, 0, 1⟩⟩)+·(⟨⟨0, 1⟩⟩⟼→ . ⟨⟨1, 0, 1⟩⟩)+·(⟨⟨1, 1⟩⟩⟼→ . ⟨⟨1, 1, 1⟩⟩)+·(⟨⟨1, 0⟩⟩⟼→ . ⟨⟨2, 1, 1⟩⟩)+·(⟨⟨2, 1⟩⟩⟼→ . ⟨⟨2, 1, 1⟩⟩)+·(⟨⟨2, 0⟩⟩⟼→ . ⟨⟨3, 0, − 1⟩⟩)+·(⟨⟨3, 1⟩⟩⟼→ . ⟨⟨4, 0, − 1⟩⟩)+·(⟨⟨4, 1⟩⟩⟼→ . ⟨⟨4, 1, − 1⟩⟩)+·(⟨⟨4, 0⟩⟩⟼→ . ⟨⟨5, 0, 0⟩⟩).

Next we state the proposition

(17) SumTran(⟨⟨0, 0⟩⟩) = ⟨⟨0, 0, 1⟩⟩ and SumTran(⟨⟨0, 1⟩⟩) = ⟨⟨1, 0, 1⟩⟩ and SumTran(⟨⟨1, 1⟩⟩) = ⟨⟨1, 1, 1⟩⟩ and SumTran(⟨⟨1, 0⟩⟩) = ⟨⟨2, 1, 1⟩⟩ and SumTran(⟨⟨2, 1⟩⟩) = ⟨⟨2, 1, 1⟩⟩ and SumTran(⟨⟨2, 0⟩⟩) = ⟨⟨3, 0, − 1⟩⟩ and SumTran(⟨⟨3, 1⟩⟩) = ⟨⟨4, 0, − 1⟩⟩ and SumTran(⟨⟨4, 1⟩⟩) = ⟨⟨4, 1, − 1⟩⟩ and SumTran(⟨⟨4, 0⟩⟩) = ⟨⟨5, 0, 0⟩⟩.

Let T be a Turing machine structure, let t be a tape of T, and let i, j be integers. We say that t is 1 between i, j if and only if:

(Def. 13) t(i) = 0 and t(j) = 0 and for every integer k such that i < k and k < j

holds t(k) = 1. Let f be a finite sequence of elements of ℕ, let T be a Turing machine

structure, and let t be a tape of T. We say that t stores data f if and only if: (Def. 14) For every natural number i such that 1 i and i < lenf holds t is 1

between ∑ Prefix(f, i) + 2 · (i − 1), ∑ Prefix(f, i + 1) + 2 · i. We now state several propositions:

(18) Let T be a Turing machine structure, t be a tape of T, and s, n be natural numbers. If t stores data ⟨s, n⟩, then t is 1 between s, s + n + 2.

(19) Let T be a Turing machine structure, t be a tape of T, and s, n be natural numbers. If t is 1 between s, s + n + 2, then t stores data ⟨s, n⟩.

(20) Let T be a Turing machine structure, t be a tape of T, and s, n be natural numbers. Suppose t stores data ⟨s, n⟩. Then t(s) = 0 and t(s + n + 2) = 0 and for every integer i such that s < i and i < s + n + 2 holds t(i) = 1.

(21) Let T be a Turing machine structure, t be a tape of T, and s, n_{1}, n_{2} be natural numbers. Suppose t stores data ⟨s, n_{1}, n_{2}⟩. Then t is 1 between s, s + n_{1} + 2 and 1 between s + n_{1} + 2, s + n_{1} + n_{2} + 4.

(22) Let T be a Turing machine structure, t be a tape of T, and s, n_{1}, n_{2} be

natural numbers. Suppose t stores data ⟨s, n_{1}, n_{2}⟩. Then

(i) t(s) = 0, (ii) t(s + n_{1} + 2) = 0, (iii) t(s + n_{1} + n_{2} + 4) = 0, (iv) for every integer i such that s < i and i < s + n_{1} + 2 holds t(i) = 1,

and

(v) for every integer i such that s + n_{1} + 2 < i and i < s + n_{1} + n_{2} + 4 holds t(i) = 1.

(23) Let f be a finite sequence of elements of ℕ and s be a natural number. If lenf 1, then ∑ Prefix(⟨s⟩^{⌢} f, 1) = s and ∑ Prefix(⟨s⟩^{⌢} f, 2) = s + f_{1}.

(24) Let f be a finite sequence of elements of ℕ and s be a natural number. Suppose lenf 3. Then ∑ Prefix(⟨s⟩^{⌢}f, 1) = s and ∑ Prefix(⟨s⟩^{⌢}f, 2) = s + f_{1} and ∑ Prefix(⟨s⟩^{⌢} f, 3) = s + f_{1} + f_{2} and ∑ Prefix(⟨s⟩^{⌢} f, 4) = s + f_{1} + f_{2} + f_{3}.

(25) Let T be a Turing machine structure, t be a tape of T, s be a natural number, and f be a finite sequence of elements of ℕ. If lenf 1 and t stores data ⟨s⟩^{⌢} f, then t is 1 between s, s + f_{1} + 2.

(26) Let T be a Turing machine structure, t be a tape of T, s be a natural number, and f be a finite sequence of elements of ℕ. Suppose lenf 3 and t stores data ⟨s⟩ ^{⌢} f. Then t is 1 between s, s + f_{1} + 2, 1 between s+f_{1} +2, s+f_{1} +f_{2} +4, and 1 between s+f_{1} +f_{2} +4, s+f_{1} +f_{2} +f_{3} +6.

3. Summation of Two Natural Numbers

The strict Turing machine structure SumTuring is defined by the conditions (Def. 15).

(Def. 15)(i) The symbols of SumTuring = {0, 1}, (ii) the control states of SumTuring = Seg_{M} 5, (iii) the transition of SumTuring = SumTran, (iv) the initial state of SumTuring = 0, and

(v) the accepting state of SumTuring = 5. Next we state several propositions:

(27) Let T be a Turing machine structure, s be a State of T, and p, h, t be sets. If s = ⟨⟨p, h, t⟩⟩, then Head(s) = h.

(28) Let T be a Turing machine structure, t be a tape of T, h be an integer, and s be a symbol of T. If t(h) = s, then Tape-Chg(t, h, s) = t.

(29) Let T be a Turing machine structure, s be a State of T, and p, h, t be sets. Suppose s = ⟨⟨p, h, t⟩⟩ and p ≠ the accepting state of T. Then Following(s) = ⟨⟨s -target_{1}, Head(s) + offset(s -target), Tape-Chg(s_{3}, Head(s), s -target_{2})⟩⟩.

(30) Let T be a Turing machine structure, t be a tape of T, h be an integer, s be a symbol of T, and i be a set. Then (Tape-Chg(t, h, s))(h) = s and if i ≠ h, then (Tape-Chg(t, h, s))(i) = t(i).

(31) Let s be a State of SumTuring, t be a tape of SumTuring, and h_{1}, n_{1}, n_{2} be natural numbers. Suppose s = ⟨⟨0, h_{1}, t⟩⟩ and t stores data ⟨h_{1}, n_{1}, n_{2}⟩. Then s is accepting and (Result(s))_{2} = 1+h_{1} and (Result(s))_{3} stores data ⟨1 + h_{1}, n_{1} + n_{2}⟩.

Let T be a Turing machine structure and let F be a function. We say that T computes F if and only if the condition (Def. 16) is satisfied.

(Def. 16) Let s be a State of T, t be a tape of T, a be a natural number, and x be a finite sequence of elements of ℕ. Suppose x ∈ domF and s = ⟨⟨the initial state of T, a, t⟩⟩ and t stores data ⟨a⟩^{⌢} x. Then s is accepting and there exist natural numbers b, y such that (Result(s))_{2} = b and y = F (x) and (Result(s))_{3} stores data ⟨b⟩^{⌢} ⟨y⟩.

Next we state two propositions:

(32) dom[+] ⊆ ℕ^{2}.

(33) SumTuring computes [+].

4. Computing Successor Function

The function SuccTran from [: Seg_{M} 4, {0, 1}:] into [: Seg_{M} 4, {0, 1}, {− 1, 0, 1}:] is defined as follows:

(Def. 17) SuccTran = id([: Seg_{M} 4, {0, 1}:], {− 1, 0, 1}, 0)+·(⟨⟨0, 0⟩⟩⟼→ . ⟨⟨1, 0, 1⟩⟩)+·(⟨⟨1, 1⟩⟩⟼→ . ⟨⟨1, 1, 1⟩⟩)+·(⟨⟨1, 0⟩⟩⟼→ . ⟨⟨2, 1, 1⟩⟩)+·(⟨⟨2, 0⟩⟩⟼→ . ⟨⟨3, 0, − 1⟩⟩)+·(⟨⟨2, 1⟩⟩⟼→ . ⟨⟨3, 0, − 1⟩⟩)+·(⟨⟨3, 1⟩⟩⟼→ . ⟨⟨3, 1, − 1⟩⟩)+·(⟨⟨3, 0⟩⟩⟼→ . ⟨⟨4, 0, 0⟩⟩).

We now state the proposition (34) SuccTran(⟨⟨0, 0⟩⟩) = ⟨⟨1, 0, 1⟩⟩ and SuccTran(⟨⟨1, 1⟩⟩) = ⟨⟨1, 1, 1⟩⟩ and

SuccTran(⟨⟨1, 0⟩⟩) = ⟨⟨2, 1, 1⟩⟩ and SuccTran(⟨⟨2, 0⟩⟩) = ⟨⟨3, 0, − 1⟩⟩ and SuccTran(⟨⟨2, 1⟩⟩) = ⟨⟨3, 0, − 1⟩⟩ and SuccTran(⟨⟨3, 1⟩⟩) = ⟨⟨3, 1, − 1⟩⟩ and SuccTran(⟨⟨3, 0⟩⟩) = ⟨⟨4, 0, 0⟩⟩.

The strict Turing machine structure SuccTuring is defined by the conditions (Def. 18).

(Def. 18)(i) The symbols of SuccTuring = {0, 1}, (ii) the control states of SuccTuring = Seg_{M} 4, (iii) the transition of SuccTuring = SuccTran, (iv) the initial state of SuccTuring = 0, and

(v) the accepting state of SuccTuring = 4. The following propositions are true:

(36)^{1} Let s be a State of SuccTuring, t be a tape of SuccTuring, and h_{1}, n be natural numbers. Suppose s = ⟨⟨0, h_{1}, t⟩⟩ and t stores data ⟨h_{1}, n⟩. Then s is accepting and (Result(s))_{2} = h_{1} and (Result(s))_{3} stores data ⟨h_{1}, n+1⟩.

(37) SuccTuring computes succ_{1}(1).

5. Computing Zero Function

The function ZeroTran from [: Seg_{M} 4, {0, 1}:] into [: Seg_{M} 4, {0, 1}, {− 1, 0, 1}:] is defined as follows:

(Def. 19) ZeroTran = id([: Seg_{M} 4, {0, 1}:], {− 1, 0, 1}, 1)+·(⟨⟨0, 0⟩⟩⟼→ . ⟨⟨1, 0, 1⟩⟩)+·(⟨⟨1, 1⟩⟩⟼→ . ⟨⟨2, 1, 1⟩⟩)+·(⟨⟨2, 0⟩⟩⟼→ . ⟨⟨3, 0, − 1⟩⟩)+·(⟨⟨2, 1⟩⟩⟼→ . ⟨⟨3, 0, − 1⟩⟩)+· (⟨⟨3, 1⟩⟩⟼→ . ⟨⟨4, 1, − 1⟩⟩).

Next we state the proposition

(38) ZeroTran(⟨⟨0, 0⟩⟩) = ⟨⟨1, 0, 1⟩⟩ and ZeroTran(⟨⟨1, 1⟩⟩) = ⟨⟨2, 1, 1⟩⟩ and ZeroTran(⟨⟨2, 0⟩⟩) = ⟨⟨3, 0, − 1⟩⟩ and ZeroTran(⟨⟨2, 1⟩⟩) = ⟨⟨3, 0, − 1⟩⟩ and ZeroTran(⟨⟨3, 1⟩⟩) = ⟨⟨4, 1, − 1⟩⟩.

The strict Turing machine structure ZeroTuring is defined by the conditions (Def. 20).

(Def. 20)(i) The symbols of ZeroTuring = {0, 1}, (ii) the control states of ZeroTuring = Seg_{M} 4, (iii) the transition of ZeroTuring = ZeroTran, (iv) the initial state of ZeroTuring = 0, and

(v) the accepting state of ZeroTuring = 4. We now state two propositions:

1The proposition (35) has been removed.

(39) Let s be a State of ZeroTuring, t be a tape of ZeroTuring, h_{1} be a natural number, and f be a finite sequence of elements of ℕ. Suppose lenf 1 and s = ⟨⟨0, h_{1}, t⟩⟩ and t stores data ⟨h_{1}⟩ ^{⌢} f. Then s is accepting and (Result(s))_{2} = h_{1} and (Result(s))_{3} stores data ⟨h_{1}, 0⟩.

(40) If n 1, then ZeroTuring computes const_{n}(0).

6. Computing n-ary Project Function

The function n -proj3Tran from [: Seg_{M} 3, {0, 1}:] into [: Seg_{M} 3, {0, 1}, {− 1, 0, 1}:] is defined by:

(Def. 21) n -proj3Tran = id([: Seg_{M} 3, {0, 1}:], {− 1, 0, 1}, 0)+·(⟨⟨0, 0⟩⟩⟼→ . ⟨⟨1, 0, 1⟩⟩)+·(⟨⟨1, 1⟩⟩⟼→ . ⟨⟨1, 0, 1⟩⟩)+·(⟨⟨1, 0⟩⟩⟼→ . ⟨⟨2, 0, 1⟩⟩)+·(⟨⟨2, 1⟩⟩⟼→ . ⟨⟨2, 0, 1⟩⟩)+·(⟨⟨2, 0⟩⟩⟼→ . ⟨⟨3, 0, 0⟩⟩).

The following proposition is true

(41) n -proj3Tran(⟨⟨0, 0⟩⟩) = ⟨⟨1, 0, 1⟩⟩and n -proj3Tran(⟨⟨1, 1⟩⟩) = ⟨⟨1, 0, 1⟩⟩and n -proj3Tran(⟨⟨1, 0⟩⟩) = ⟨⟨2, 0, 1⟩⟩ and n -proj3Tran(⟨⟨2, 1⟩⟩) = ⟨⟨2, 0, 1⟩⟩ and n -proj3Tran(⟨⟨2, 0⟩⟩) = ⟨⟨3, 0, 0⟩⟩.

The strict Turing machine structure n -proj3Turing is defined by the conditions (Def. 22).

(Def. 22)(i) The symbols of n -proj3Turing = {0, 1}, (ii) the control states of n -proj3Turing = Seg_{M} 3, (iii) the transition of n -proj3Turing = n -proj3Tran, (iv) the initial state of n -proj3Turing = 0, and

(v) the accepting state of n -proj3Turing = 3. Next we state two propositions:

(42) Let s be a State of n -proj3Turing, t be a tape of n -proj3Turing, h_{1} be a natural number, and f be a finite sequence of elements of ℕ. Suppose lenf 3 and s = ⟨⟨0, h_{1}, t⟩⟩and t stores data ⟨h_{1}⟩^{⌢} f. Then s is accepting and (Result(s))_{2} = h_{1} + f_{1} + f_{2} + 4 and (Result(s))_{3} stores data ⟨h_{1} + f_{1} + f_{2} + 4, f_{3}⟩.

(43) If n 3, then n -proj3Turing computes proj_{n}(3).

7. Combining Two Turing Machines into One

Let t_{1}, t_{2} be Turing machine structures. The functor SeqStates(t_{1}, t_{2}) yielding a finite non empty set is defined by the condition (Def. 23).

(Def. 23) SeqStates(t_{1}, t_{2}) = [: the control states of t_{1}, {the initial state of t_{2}}:] ∪ [: {the accepting state of t_{1}}, the control states of t_{2} :].

One can prove the following four propositions:

(44) Let t_{1}, t_{2} be Turing machine structures. Then (i) ⟨⟨the initial state of t_{1}, the initial state of t_{2}⟩⟩∈ SeqStates(t_{1}, t_{2}), and (ii) ⟨⟨the accepting state of t_{1}, the accepting state of t_{2}⟩⟩∈ SeqStates(t_{1}, t_{2}).

(45) For all Turing machine structures s, t and for every state x of s holds ⟨⟨x, the initial state of t⟩⟩∈ SeqStates(s, t).

(46) For all Turing machine structures s, t and for every state x of t holds ⟨⟨the accepting state of s, x⟩⟩∈ SeqStates(s, t).

(47) Let s, t be Turing machine structures and x be an element of SeqStates(s, t). Then there exists a state x_{1} of s and there exists a state x_{2} of t such that x = ⟨⟨x_{1}, x_{2}⟩⟩.

Let s, t be Turing machine structures and let x be a transition-target of s. The functor 1^{st}SeqTran(s, t, x) yielding an element of [: SeqStates(s, t), (the symbols of s) ∪ (the symbols of t), {− 1, 0, 1}:] is defined as follows:

(Def. 24) 1^{st}SeqTran(s, t, x) = ⟨⟨⟨⟨x_{1}, the initial state of t⟩⟩, x_{2}, x_{3} ⟩⟩.

Let s, t be Turing machine structures and let x be a transition-target of t. The functor 2^{nd}SeqTran(s, t, x) yielding an element of [: SeqStates(s, t), (the symbols of s) ∪ (the symbols of t), {− 1, 0, 1}:] is defined as follows:

(Def. 25) 2^{nd}SeqTran(s, t, x) = ⟨⟨⟨⟨the accepting state of s, x_{1} ⟩⟩, x_{2}, x_{3} ⟩⟩.

Let s, t be Turing machine structures and let x be an element of SeqStates(s, t). Then x_{1} is a state of s. Then x_{2} is a state of t.

Let s, t be Turing machine structures and let x be an element of [: SeqStates(s, t), (the symbols of s)∪(the symbols of t) :]. The functor 1^{st}SeqStatex yields a state of s and is defined by:

(Def. 26) 1^{st}SeqStatex = (x_{1})_{1}.

The functor 2^{nd}SeqStatex yielding a state of t is defined as follows: (Def. 27) 2^{nd}SeqStatex = (x_{1})_{2}.

Let X, Y, Z be non empty sets and let x be an element of [: X, Y ∪ Z:]. Let us assume that there exist a set u and an element y of Y such that x = ⟨⟨u, y⟩⟩. The functor 1^{st}SeqSymbolx yielding an element of Y is defined as follows:

(Def. 28) 1^{st}SeqSymbolx = x_{2}.

Let X, Y, Z be non empty sets and let x be an element of [: X, Y ∪ Z:]. Let us assume that there exist a set u and an element z of Z such that x = ⟨⟨u, z⟩⟩. The functor 2^{nd}SeqSymbolx yielding an element of Z is defined by:

(Def. 29) 2^{nd}SeqSymbolx = x_{2}.

Let s, t be Turing machine structures and let x be an element of [: SeqStates(s, t), (the symbols of s)∪(the symbols of t) :]. The functor SeqTran(s, t, x) yielding an element of [: SeqStates(s, t), (the symbols of s) ∪ (the symbols of t), {− 1, 0, 1}:] is defined by:

(Def. 30) SeqTran(s, t, x) = \{ \begin{matrix}1^{st}SeqTran(s, t, (the transition of s)(⟨⟨1^{st}SeqStatex, \\ 1^{st}SeqSymbolx⟩⟩)), if there exists a state p of s \\ and there exists a symbol y of s such that x = \\ ⟨⟨⟨⟨p, the initial state of t⟩⟩, y⟩⟩ and p ≠ the accepting \\ state of s, \\ 2^{nd}SeqTran(s, t, (the transition of t)(⟨⟨2^{nd}SeqStatex, \\ 2^{nd}SeqSymbolx⟩⟩)), if there exists a state q of t \\ and there exists a symbol y of t such that x = \\ ⟨⟨⟨⟨the accepting state of s, q⟩⟩, y⟩⟩, \\ ⟨⟨x_{1}, x_{2}, − 1⟩⟩, otherwise. \end{matrix}

Let s, t be Turing machine structures. The functor SeqTran(s, t) yielding a function from [: SeqStates(s, t), (the symbols of s) ∪ (the symbols of t) :] into [: SeqStates(s, t), (the symbols of s) ∪ (the symbols of t), {− 1, 0, 1}:] is defined by:

(Def. 31) For every element x of [: SeqStates(s, t), (the symbols of s)∪(the symbols

of t) :] holds (SeqTran(s, t))(x) = SeqTran(s, t, x). Let T_{1}, T_{2} be Turing machine structures. The functor T_{1}; T_{2} yielding a strict

Turing machine structure is defined by the conditions (Def. 32). (Def. 32)(i) The symbols of T_{1}; T_{2} = (the symbols of T_{1}) ∪ (the symbols of T_{2}),

(ii) the control states of T_{1}; T_{2} = SeqStates(T_{1}, T_{2}), (iii) the transition of T_{1}; T_{2} = SeqTran(T_{1}, T_{2}), (iv) the initial state of T_{1}; T_{2} = ⟨⟨the initial state of T_{1}, the initial state of

T_{2}⟩⟩, and

(v) the accepting state of T_{1}; T_{2} = ⟨⟨the accepting state of T_{1}, the accepting state of T_{2}⟩⟩.

We now state several propositions:

(48) Let T_{1}, T_{2} be Turing machine structures, g be a transition-target of T_{1}, p be a state of T_{1}, and y be a symbol of T_{1}. Suppose p ≠ the accepting state of T_{1} and g = (the transition of T_{1})(⟨⟨p, y⟩⟩). Then (the transition of T_{1}; T_{2})(⟨⟨⟨⟨p, the initial state of T_{2}⟩⟩, y⟩⟩) = ⟨⟨⟨⟨g_{1}, the initial state of T_{2}⟩⟩, g_{2}, g_{3} ⟩⟩.

(49) Let T_{1}, T_{2} be Turing machine structures, g be a transition-target of T_{2}, q be a state of T_{2}, and y be a symbol of T_{2}. Suppose g = (the transition of T_{2})(⟨⟨q, y⟩⟩). Then (the transition of T_{1}; T_{2})(⟨⟨⟨⟨the accepting state of T_{1}, q⟩⟩, y⟩⟩) = ⟨⟨⟨⟨the accepting state of T_{1}, g_{1} ⟩⟩, g_{2}, g_{3} ⟩⟩.

(50) Let T_{1}, T_{2} be Turing machine structures, s_{1} be a State of T_{1}, h be a natural number, t be a tape of T_{1}, s_{2} be a State of T_{2}, and s_{3} be a State of T_{1}; T_{2}. Suppose that

(i) s_{1} is accepting, (ii) s_{1} = ⟨⟨the initial state of T_{1}, h, t⟩⟩, (iii) s_{2} is accepting,

introduction to turing machines 731

(iv) s_{2} = ⟨⟨the initial state of T_{2}, (Result(s_{1}))_{2}, (Result(s_{1}))_{3} ⟩⟩, and

(v) s_{3} = ⟨⟨the initial state of T_{1}; T_{2}, h, t⟩⟩. Then s_{3} is accepting and (Result(s_{3}))_{2} = (Result(s_{2}))_{2} and (Result(s_{3}))_{3} = (Result(s_{2}))_{3}.

(51) Let t_{3}, t_{4} be Turing machine structures and t be a tape of t_{3}. If the symbols of t_{3} = the symbols of t_{4}, then t is a tape of t_{3}; t_{4}.

(52) Let t_{3}, t_{4} be Turing machine structures and t be a tape of t_{3}; t_{4}. Suppose the symbols of t_{3} = the symbols of t_{4}. Then t is a tape of t_{3} and a tape of t_{4}.

(53) Let f be a finite sequence of elements of ℕ, t_{3}, t_{4} be Turing machine structures, t_{1} be a tape of t_{3}, and t_{2} be a tape of t_{4}. If t_{1} = t_{2} and t_{1} stores data f, then t_{2} stores data f.

(54) Let s be a State of ZeroTuring; SuccTuring, t be a tape of ZeroTuring, and h_{1}, n be natural numbers. Suppose s = ⟨⟨⟨⟨0, 0⟩⟩, h_{1}, t⟩⟩ and t stores data ⟨h_{1}, n⟩. Then s is accepting and (Result(s))_{2} = h_{1} and (Result(s))_{3} stores data ⟨h_{1}, 1⟩.

References

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On the Characterizations of Compactness

Grzegorz Bancerek University of Białystok

Noboru Endou Gifu National College of Technology

Yuji Sakai Shinshu University Nagano

Summary. In the paper we show equivalence of the convergence of filters on a topological space and the convergence of nets in the space. We also give, five characterizations of compactness. Namely, for any topological space T we proved that following condition are equivalent:

• T is compact,

• every ultrafilter on T is convergent,

• every proper filter on T has cluster point,

• every net in T has cluster point,

• every net in T has convergent subnet,

• every Cauchy net in T is convergent.

MML Identifier: YELLOW19.

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

In this paper X is a set. The following propositions are true:

(1) The carrier of 2_{⊆}^{X} = 2^{X}.

(2) For every non empty set X and for every proper filter F of 2_{⊆}^{X} and for every set A such that A ∈ F holds A is not empty.

Let T be a non empty topological space and let x be a point of T. The neighborhood system of x is a subset of 2_{⊆}^{Ω_{T}} and is defined by:

(Def. 1) The neighborhood system of x = {A : A ranges over neighbourhoods of

x}. The following proposition is true (3) Let T be a non empty topological space, x be a point of T, and A

be a set. Then A ∈ the neighborhood system of x if and only if A is a neighbourhood of x.

Let T be a non empty topological space and let x be a point of T. Observe that the neighborhood system of x is non empty proper upper and filtered.

One can prove the following propositions:

(4) Let T be a non empty topological space, x be a point of T, and F be an upper subset of 2_{⊆}^{Ω_{T}}. Then x is a convergence point of F, T if and only if the neighborhood system of x ⊆ F.

(5) For every non empty topological space T holds every point x of T is a convergence point of the neighborhood system of x, T.

(6) Let T be a non empty topological space and A be a subset of T. Then A is open if and only if for every point x of T such that x ∈ A and for every filter F of 2_{⊆}^{Ω_{T}} such that x is a convergence point of F, T holds A ∈ F.

Let S be a non empty 1-sorted structure and let N be a non empty net structure over S. A subset of S is called a subset of S reachable by N if:

(Def. 2) There exists an element i of N such that it = rng(the mapping of N↾i). The following proposition is true

(7) Let S be a non empty 1-sorted structure, N be a non empty net structure over S, and i be an element of N. Then rng(the mapping of N↾i) is a subset of S reachable by N.

Let S be a non empty 1-sorted structure and let N be a reflexive non empty net structure over S. Note that every subset of S reachable by N is non empty.

We now state three propositions:

(8) Let S be a non empty 1-sorted structure, N be a net in S, i be an element of N, and x be a set. Then x ∈ rng(the mapping of N↾i) if and only if there exists an element j of N such that i j and x = N(j).

(9) Let S be a non empty 1-sorted structure, N be a net in S, and A be a subset of S reachable by N. Then N is eventually in A.

(10) Let S be a non empty 1-sorted structure, N be a net in S, and F be a finite non empty set. Suppose every element of F is a subset of S reachable by N. Then there exists a subset B of S reachable by N such that B ⊆ ⋂ F.

Let T be a non empty 1-sorted structure and let N be a non empty net structure over T. The filter of N is a subset of 2_{⊆}^{Ω_{T}} and is defined by:

(Def. 3) The filter of N = {A; A ranges over subsets of T: N is eventually in A}.

The following proposition is true

(11) Let T be a non empty 1-sorted structure, N be a non empty net structure over T, and A be a set. Then A ∈ the filter of N if and only if N is eventually in A and A is a subset of T.

Let T be a non empty 1-sorted structure and let N be a non empty net structure over T. Note that the filter of N is non empty and upper.

Let T be a non empty 1-sorted structure and let N be a net in T. One can verify that the filter of N is proper and filtered.

We now state two propositions:

(12) Let T be a non empty topological space, N be a net in T, and x be a point of T. Then x is a cluster point of N if and only if x is a cluster point of the filter of N, T.

(13) Let T be a non empty topological space, N be a net in T, and x be a point of T. Then x ∈ LimN if and only if x is a convergence point of the filter of N, T.

Let L be a non empty 1-sorted structure, let O be a non empty subset of L, and let F be a filter of 2_{⊆}^{O}. The net of F is a strict non empty net structure over L and is defined by the conditions (Def. 4).

(Def. 4)(i) The carrier of the net of F = {⟨⟨a, f⟩⟩; a ranges over elements of L, f

ranges over elements of F: a ∈ f}, (ii) for all elements i, j of the net of F holds i j iff j_{2} ⊆ i_{2}, and (iii) for every element i of the net of F holds (the net of F)(i) = i_{1}. Let L be a non empty 1-sorted structure, let O be a non empty subset of L,

and let F be a filter of 2_{⊆}^{O}. Note that the net of F is reflexive and transitive.

Let L be a non empty 1-sorted structure, let O be a non empty subset of L, and let F be a proper filter of 2_{⊆}^{O}. One can verify that the net of F is directed.

The following propositions are true:

(14) For every non empty 1-sorted structure T and for every filter F of 2_{⊆}^{Ω_{T}} holds F \ {∅} = the filter of the net of F.

(15) Let T be a non empty 1-sorted structure and F be a proper filter of 2_{⊆}^{Ω_{T}}. Then F = the filter of the net of F.

(16) Let T be a non empty 1-sorted structure, F be a filter of 2_{⊆}^{Ω_{T}}, and A be a non empty subset of T. Then A ∈ F if and only if the net of F is eventually in A.

(17) Let T be a non empty topological space, F be a proper filter of 2_{⊆}^{Ω_{T}}, and x be a point of T. Then x is a cluster point of the net of F if and only if x is a cluster point of F, T.

(18) Let T be a non empty topological space, F be a proper filter of 2_{⊆}^{Ω_{T}}, and x be a point of T. Then x ∈ Lim(the net of F) if and only if x is a convergence point of F, T.

(19) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ \overline{A} if and only if for every neighbourhood O of x holds O meets A.

(20) Let T be a non empty topological space, x be a point of T, and A be a subset of T. Suppose x ∈ \overline{A}. Let F be a proper filter of 2_{⊆}^{Ω_{T}}. If F = the neighborhood system of x, then the net of F is often in A.

(21) Let T be a non empty 1-sorted structure, A be a set, and N be a net in T. If N is eventually in A, then every subnet of N is eventually in A.

(22) Let T be a non empty topological space and F, G, x be sets. Suppose F ⊆ G and x is a convergence point of F, T. Then x is a convergence point of G, T.

(23) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ \overline{A} if and only if there exists a net N in T such that N is eventually in A and x is a cluster point of N.

(24) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ \overline{A} if and only if there exists a convergent net N in T such that N is eventually in A and x ∈ LimN.

(25) Let T be a non empty topological space and A be a subset of T. Then A is closed if and only if for every net N in T such that N is eventually in A and for every point x of T such that x is a cluster point of N holds x ∈ A.

(26) Let T be a non empty topological space and A be a subset of T. Then A is closed if and only if for every convergent net N in T such that N is eventually in A and for every point x of T such that x ∈ LimN holds x ∈ A.

(27) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ \overline{A} if and only if there exists a proper filter F of 2_{⊆}^{Ω_{T}} such that A ∈ F and x is a cluster point of F, T.

(28) Let T be a non empty topological space, A be a subset of T, and x be a point of T. Then x ∈ \overline{A} if and only if there exists an ultra filter F of 2_{⊆}^{Ω_{T}} such that A ∈ F and x is a convergence point of F, T.

(29) Let T be a non empty topological space and A be a subset of T. Then A is closed if and only if for every proper filter F of 2_{⊆}^{Ω_{T}} such that A ∈ F and for every point x of T such that x is a cluster point of F, T holds x ∈ A.

(30) Let T be a non empty topological space and A be a subset of T. Then A is closed if and only if for every ultra filter F of 2_{⊆}^{Ω_{T}} such that A ∈ F and for every point x of T such that x is a convergence point of F, T holds x ∈ A.

(31) Let T be a non empty topological space, N be a net in T, and s be a point of T. Then s is a cluster point of N if and only if for every subset A of T reachable by N holds s ∈ \overline{A}.

(32) Let T be a non empty topological space and F be a family of subsets of

the carrier of T. If F is closed, then FinMeetCl(F) is closed.

(33) Let T be a non empty topological space. Then T is compact if and only if for every ultra filter F of 2_{⊆}^{Ω_{T}} holds there exists a point of T which is a convergence point of F, T.

(34) Let T be a non empty topological space. Then T is compact if and only if for every proper filter F of 2_{⊆}^{Ω_{T}} holds there exists a point of T which is a cluster point of F, T.

(35) Let T be a non empty topological space. Then T is compact if and only if for every net N in T holds there exists a point of T which is a cluster point of N.

(36) Let T be a non empty topological space. Then T is compact if and only if for every net N in T such that N ∈ NetUniv(T) holds there exists a point of T which is a cluster point of N.

Let L be a non empty 1-sorted structure and let N be a transitive net structure over L. Note that every full structure of a subnet of N is transitive.

Let L be a non empty 1-sorted structure and let N be a non empty directed net structure over L. Note that there exists a structure of a subnet of N which is strict, non empty, directed, and full.

The following proposition is true

(37) For every non empty topological space T holds T is compact iff for every

net N in T holds there exists a subnet of N which is convergent. Let S be a non empty 1-sorted structure and let N be a non empty net

structure over S. We say that N is Cauchy if and only if: (Def. 5) For every subset A of S holds N is eventually in A or eventually in − A.

Let S be a non empty 1-sorted structure and let F be an ultra filter of 2_{⊆}^{Ω_{S}}. Observe that the net of F is Cauchy.

Next we state the proposition

(38) Let T be a non empty topological space. Then T is compact if and only if for every net N in T such that N is Cauchy holds N is convergent.

References

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

[2] Grzegorz Bancerek. Prime ideals and filters. Formalized Mathematics, 6(2):241–247, 1997.

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

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

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

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

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

[8] Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125–135, 1993.

738 grzegorz bancerek et al.

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

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

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

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

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

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

[15] Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213–225, 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] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

Compactness of Lim-inf Topology

Grzegorz Bancerek University of Białystok

Noboru Endou Shinshu University Nagano

Summary. Formalization of [10], chapter III, section 3 (3.4–3.6).

MML Identifier: WAYBEL33.

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

Let L be a non empty poset, let X be a non empty subset of L, and let F be a filter of 2_{⊆}^{X}. The functor lim inf F yielding an element of L is defined by:

(Def. 1) lim inf F = ⨆_{L}{inf B; B ranges over subsets of L: B ∈ F }. One can prove the following proposition

(1) Let L_{1}, L_{2} be complete lattices. Suppose the relational structure of L_{1} = the relational structure of L_{2}. Let X_{1} be a non empty subset of L_{1}, X_{2} be a non empty subset of L_{2}, F_{1} be a filter of 2_{⊆}^{X_{1}}, and F_{2} be a filter of 2_{⊆}^{X_{2}}. If F_{1} = F_{2}, then lim inf F_{1} = lim inf F_{2}.

Let L be a non empty FR-structure. We say that L is lim-inf if and only if: (Def. 2) The topology of L = ξ(L).

Let us note that every non empty FR-structure which is lim-inf is also topological space-like.

One can check that every top-lattice which is trivial is also lim-inf. One can check that there exists a top-lattice which is lim-inf, continuous,

and complete.

We now state several propositions:

(2) Let L_{1}, L_{2} be non empty 1-sorted structures. Suppose the carrier of L_{1} = the carrier of L_{2}. Let N_{1} be a net structure over L_{1}. Then there exists a strict net structure N_{2} over L_{2} such that (i) the relational structure of N_{1} = the relational structure of N_{2}, and (ii) the mapping of N_{1} = the mapping of N_{2}.

(3) Let L_{1}, L_{2} be non empty 1-sorted structures. Suppose the carrier of L_{1} = the carrier of L_{2}. Let N_{1} be a net structure over L_{1}. Suppose N_{1} ∈ NetUniv(L_{1}). Then there exists a strict net N_{2} in L_{2} such that

(i) N_{2} ∈ NetUniv(L_{2}), (ii) the relational structure of N_{1} = the relational structure of N_{2}, and (iii) the mapping of N_{1} = the mapping of N_{2}.

(4) Let L_{1}, L_{2} be inf-complete up-complete semilattices. Suppose the relational structure of L_{1} = the relational structure of L_{2}. Let N_{1} be a net in L_{1} and N_{2} be a net in L_{2}. Suppose that

(i) the relational structure of N_{1} = the relational structure of N_{2}, and (ii) the mapping of N_{1} = the mapping of N_{2}.

Then lim inf N_{1} = lim inf N_{2}.

(5) Let L_{1}, L_{2} be non empty 1-sorted structures. Suppose the carrier of L_{1} = the carrier of L_{2}. Let N_{1} be a net in L_{1} and N_{2} be a net in L_{2}. Suppose that

(i) the relational structure of N_{1} = the relational structure of N_{2}, and (ii) the mapping of N_{1} = the mapping of N_{2}.

Let S_{1} be a subnet of N_{1}. Then there exists a strict subnet S_{2} of N_{2} such that

(iii) the relational structure of S_{1} = the relational structure of S_{2}, and (iv) the mapping of S_{1} = the mapping of S_{2}.

(6) Let L_{1}, L_{2} be inf-complete up-complete semilattices. Suppose the relational structure of L_{1} = the relational structure of L_{2}. Let N_{1} be a net structure over L_{1} and a be a set. Suppose ⟨⟨N_{1}, a⟩⟩∈ the lim inf convergence of L_{1}. Then there exists a strict net N_{2} in L_{2} such that

(i) ⟨⟨N_{2}, a⟩⟩∈ the lim inf convergence of L_{2}, (ii) the relational structure of N_{1} = the relational structure of N_{2}, and (iii) the mapping of N_{1} = the mapping of N_{2}.

(7) Let L_{1}, L_{2} be non empty 1-sorted structures, N_{1} be a non empty net structure over L_{1}, and N_{2} be a non empty net structure over L_{2}. Suppose that

(i) the relational structure of N_{1} = the relational structure of N_{2}, and (ii) the mapping of N_{1} = the mapping of N_{2}.

Let X be a set. If N_{1} is eventually in X, then N_{2} is eventually in X.

(8) Let L_{1}, L_{2} be inf-complete up-complete semilattices. Suppose the relational structure of L_{1} = the relational structure of L_{2}. Then ConvergenceSpace(the lim inf convergence of L_{1}) = ConvergenceSpace(the lim inf convergence of L_{2}).

(9) Let L_{1}, L_{2} be inf-complete up-complete semilattices. Suppose the relational structure of L_{1} = the relational structure of L_{2}. Then ξ(L_{1}) = ξ(L_{2}).

Let R be an inf-complete non empty reflexive relational structure. Note that every topological augmentation of R is inf-complete.

Let R be a semilattice. One can verify that every topological augmentation of R has g.l.b.’s.

Let L be an inf-complete up-complete semilattice. One can check that there exists a topological augmentation of L which is strict and lim-inf.

The following proposition is true

(10) Let L be an inf-complete up-complete semilattice and X be a lim-inf

topological augmentation of L. Then ξ(L) = the topology of X. Let L be an inf-complete up-complete semilattice. The functor Ξ(L) yielding

a strict topological augmentation of L is defined by: (Def. 3) Ξ(L) is lim-inf.

Let L be an inf-complete up-complete semilattice. One can check that Ξ(L) is lim-inf.

Next we state a number of propositions:

(11) For every complete lattice L and for every net N in L holds lim inf N = ⨆_{L}{inf(N↾i) : i ranges over elements of N}.

(12) Let L be a complete lattice, F be a proper filter of 2_{⊆}^{Ω_{L}}, and f be a subset of L. Suppose f ∈ F. Let i be an element of the net of F. If i_{2} = f, then inf f = inf((the net of F)↾i).

(13) For every complete lattice L and for every proper filter F of 2_{⊆}^{Ω_{L}} holds lim inf F = lim inf (the net of F).

(14) For every complete lattice L and for every proper filter F of 2_{⊆}^{Ω_{L}} holds the net of F ∈ NetUniv(L).

(15) Let L be a complete lattice, F be an ultra filter of 2_{⊆}^{Ω_{L}}, and p be a greater or equal to id map from the net of F into the net of F. Then lim inf F inf((the net of F) · p).

(16) Let L be a complete lattice, F be an ultra filter of 2_{⊆}^{Ω_{L}}, and M be a subnet of the net of F. Then lim inf F = lim inf M.

(17) Let L be a non empty 1-sorted structure, N be a net in L, and A be a set. Suppose N is often in A. Then there exists a strict subnet N′ of N such that rng(the mapping of N′) ⊆ A and N′ is a structure of a subnet of N.

(18) Let L be a complete lim-inf top-lattice and A be a non empty subset of L. Then A is closed if and only if for every ultra filter F of 2_{⊆}^{Ω_{L}} such that A ∈ F holds lim inf F ∈ A.

(19) For every non empty reflexive relational structure L holds σ(L) ⊆ ξ(L).

742 grzegorz bancerek and noboru endou

(20) Let T_{1}, T_{2} be non empty topological spaces and B be a prebasis of T_{1}. Suppose B ⊆ the topology of T_{2} and the carrier of T_{1} ∈ the topology of T_{2}. Then the topology of T_{1} ⊆ the topology of T_{2}.

(21) For every complete lattice L holds ω(L) ⊆ ξ(L).

(22) Let T_{1}, T_{2} be topological spaces and T be a non empty topological space. Suppose T is a topological extension of T_{1} and a topological extension of T_{2}. Let R be a refinement of T_{1} and T_{2}. Then T is a topological extension of R.

(23) Let T_{1} be a topological space, T_{2} be a topological extension of T_{1}, and A be a subset of T_{1}. Then

(i) if A is open, then A is an open subset of T_{2}, and (ii) if A is closed, then A is a closed subset of T_{2}.

(24) For every complete lattice L holds λ(L) ⊆ ξ(L).

(25) Let L be a complete lattice, T be a lim-inf topological augmentation of L, and S be a Lawson correct topological augmentation of L. Then T is a topological extension of S.

(26) For every complete lim-inf top-lattice L and for every ultra filter F of 2_{⊆}^{Ω_{L}} holds lim inf F is a convergence point of F, L.

(27) Every complete lim-inf top-lattice is compact and T_{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. Prime ideals and filters. Formalized Mathematics, 6(2):241–247, 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] Grzegorz Bancerek, Noboru Endou, and Yuji Sakai. On the characterizations of compactness. Formalized Mathematics, 9(4):733–738, 2001.

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

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

[10] 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.

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

[12] Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125–135, 1993.

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

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

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

[16] Bartłomiej Skorulski. Lim-inf convergence. Formalized Mathematics, 9(2):237–240, 2001.

compactness of lim-inf topology 743

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

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

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

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

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

744 grzegorz bancerek and noboru endou

Miscellaneous Facts about Functors

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Summary. In the paper we show useful facts concerning reverse and inclusion functors and the restriction of functors. We also introduce a new notation for the intersection of categories and the isomorphism under arbitrary functors.

MML Identifier: YELLOW20.

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

1. Reverse Functors

The following propositions are true:

(1) Let A, B be transitive non empty category structures with units and F be a feasible reflexive functor structure from A to B. Suppose F is coreflexive and bijective. Let a be an object of A and b be an object of B. Then F (a) = b if and only if F^{−1} (b) = a.

(2) Let A, B be transitive non empty category structures with units, F be a precovariant feasible functor structure from A to B, and G be a precovariant feasible functor structure from B to A. Suppose F is bijective and G = F^{−1} . Let a_{1}, a_{2} be objects of A. Suppose ⟨a_{1}, a_{2}⟩ ≠ ∅. Let f be a morphism from a_{1} to a_{2} and g be a morphism from F (a_{1}) to F (a_{2}). Then F (f) = g if and only if G(g) = f.

(3) Let A, B be transitive non empty category structures with units, F be a precontravariant feasible functor structure from A to B, and G be a precontravariant feasible functor structure from B to A. Suppose F is bijective and G = F^{−1} . Let a_{1}, a_{2} be objects of A. Suppose ⟨a_{1}, a_{2}⟩ ≠ ∅. Let f be a morphism from a_{1} to a_{2} and g be a morphism from F (a_{2}) to F (a_{1}). Then F (f) = g if and only if G(g) = f.

(4) Let A, B be categories and F be a functor from A to B. Suppose F is bijective. Let G be a functor from B to A. If F · G = id_{B}, then the functor structure of G = F^{−1} .

(5) Let A, B be categories and F be a functor from A to B. Suppose F is bijective. Let G be a functor from B to A. If G · F = id_{A}, then the functor structure of G = F^{−1} .

(6) Let A, B be categories and F be a covariant functor from A to B. Suppose F is bijective. Let G be a covariant functor from B to A. Suppose that

(i) for every object b of B holds F (G(b)) = b, and (ii) for all objects a, b of B such that ⟨a, b⟩ ≠ ∅ and for every morphism f

from a to b holds F (G(f)) = f. Then the functor structure of G = F^{−1} .

(7) Let A, B be categories and F be a contravariant functor from A to B. Suppose F is bijective. Let G be a contravariant functor from B to A. Suppose that

(i) for every object b of B holds F (G(b)) = b, and (ii) for all objects a, b of B such that ⟨a, b⟩ ≠ ∅ and for every morphism f

from a to b holds F (G(f)) = f. Then the functor structure of G = F^{−1} .

(8) Let A, B be categories and F be a covariant functor from A to B. Suppose F is bijective. Let G be a covariant functor from B to A. Suppose that

(i) for every object a of A holds G(F (a)) = a, and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f

from a to b holds G(F (f)) = f. Then the functor structure of G = F^{−1} .

(9) Let A, B be categories and F be a contravariant functor from A to B. Suppose F is bijective. Let G be a contravariant functor from B to A. Suppose that

(i) for every object a of A holds G(F (a)) = a, and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f

from a to b holds G(F (f)) = f. Then the functor structure of G = F^{−1} .

2. Intersection of Categories

Let A, B be category structures. We say that A and B have the same composition if and only if:

(Def. 1) For all sets a_{1}, a_{2}, a_{3} holds (the composition of A)(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩) ≈ (the

composition of B)(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩). Let us note that the predicate A and B have the same composition is symmetric.

Next we state three propositions:

(10) Let A, B be category structures. Then A and B have the same composition if and only if for all sets a_{1}, a_{2}, a_{3}, x such that x ∈ dom(the composition of A)(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩) and x ∈ dom(the composition of B)(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩) holds (the composition of A)(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩)(x) = (the composition of B)(⟨⟨a_{1}, a_{2}, a_{3}⟩⟩)(x).

(11) Let A, B be transitive non empty category structures. Then A and B have the same composition if and only if for all objects a_{1}, a_{2}, a_{3} of A such that ⟨a_{1}, a_{2}⟩ ≠ ∅ and ⟨a_{2}, a_{3}⟩ ≠ ∅ and for all objects b_{1}, b_{2}, b_{3} of B such that ⟨b_{1}, b_{2}⟩ ≠ ∅ and ⟨b_{2}, b_{3}⟩ ≠ ∅ and b_{1} = a_{1} and b_{2} = a_{2} and b_{3} = a_{3} and for every morphism f_{1} from a_{1} to a_{2} and for every morphism g_{1} from b_{1} to b_{2} such that g_{1} = f_{1} and for every morphism f_{2} from a_{2} to a_{3} and for every morphism g_{2} from b_{2} to b_{3} such that g_{2} = f_{2} holds f_{2} · f_{1} = g_{2} · g_{1}.

(12) For all para-functional semi-functional categories A, B holds A and B

have the same composition. Let f, g be functions. The functor Intersect(f, g) yielding a function is defi-

ned as follows: (Def. 2) domIntersect(f, g) = domf ∩ domg and for every set x such that x ∈

domf ∩ domg holds (Intersect(f, g))(x) = f(x) ∩ g(x). Let us notice that the functor Intersect(f, g) is commutative.

One can prove the following propositions:

(13) For every set I and for all many sorted sets A, B indexed by I holds Intersect(A, B) = A ∩ B.

(14) Let I, J be sets, A be a many sorted set indexed by I, and B be a many sorted set indexed by J. Then Intersect(A, B) is a many sorted set indexed by I ∩ J.

(15) Let I, J be sets, A be a many sorted set indexed by I, B be a function, and C be a many sorted set indexed by J. If C = Intersect(A, B), then C ˙⊆ A.

(16) Let A_{1}, A_{2}, B_{1}, B_{2} be sets, f be a function from A_{1} into A_{2}, and g be a function from B_{1} into B_{2}. If f ≈ g, then f ∩ g is a function from A_{1} ∩ B_{1} into A_{2} ∩ B_{2}.

(17) Let I_{1}, I_{2} be sets, A_{1}, B_{1} be many sorted sets indexed by I_{1}, A_{2}, B_{2} be many sorted sets indexed by I_{2}, and A, B be many sorted sets indexed by I_{1} ∩ I_{2}. Suppose A = Intersect(A_{1}, A_{2}) and B = Intersect(B_{1}, B_{2}). Let F be a many sorted function from A_{1} into B_{1} and G be a many sorted function from A_{2} into B_{2}. Suppose that for every set x such that x ∈ domF and x ∈ domG holds F (x) ≈ G(x). Then Intersect(F, G) is a many sorted function from A into B.

(18) Let I, J be sets, F be a many sorted set indexed by [: I, I:], and G be a many sorted set indexed by [: J, J :]. Then there exists a many sorted set H indexed by [: I ∩ J, I ∩ J :] such that H = Intersect(F, G) and Intersect({|F |}, {|G|}) = {|H|}.

(19) Let I, J be sets, F_{1}, F_{2} be many sorted sets indexed by [: I, I:], and G_{1}, G_{2} be many sorted sets indexed by [: J, J :]. Then there exist many sorted sets H_{1}, H_{2} indexed by [: I ∩J, I ∩J :] such that H_{1} = Intersect(F_{1}, G_{1}) and H_{2} = Intersect(F_{2}, G_{2}) and Intersect({|F_{1}, F_{2}|}, {|G_{1}, G_{2}|}) = {|H_{1}, H_{2}|}.

Let A, B be category structures. Let us assume that A and B have the same composition. The functor Intersect(A, B) yields a strict category structure and is defined by the conditions (Def. 3).

(Def. 3)(i) The carrier of Intersect(A, B) = (the carrier of A) ∩ (the carrier of

B), (ii) the arrows of Intersect(A, B) = Intersect(the arrows of A, the arrows

of B), and (iii) the composition of Intersect(A, B) = Intersect(the composition of A,

the composition of B). The following propositions are true:

(20) For all category structures A, B such that A and B have the same composition holds Intersect(A, B) = Intersect(B, A).

(21) Let A, B be category structures. Suppose A and B have the same composition. Then Intersect(A, B) is a substructure of A.

(22) Let A, B be category structures. Suppose A and B have the same composition. Let a_{1}, a_{2} be objects of A, b_{1}, b_{2} be objects of B, and o_{1}, o_{2} be objects of Intersect(A, B). If o_{1} = a_{1} and o_{1} = b_{1} and o_{2} = a_{2} and o_{2} = b_{2}, then ⟨o_{1}, o_{2}⟩ = (⟨a_{1}, a_{2}⟩) ∩ (⟨b_{1}, b_{2}⟩).

(23) Let A, B be transitive category structures. If A and B have the same composition, then Intersect(A, B) is transitive.

(24) Let A, B be category structures. Suppose A and B have the same composition. Let a_{1}, a_{2} be objects of A, b_{1}, b_{2} be objects of B, and o_{1}, o_{2} be objects of Intersect(A, B). Suppose o_{1} = a_{1} and o_{1} = b_{1} and o_{2} = a_{2} and o_{2} = b_{2} and ⟨a_{1}, a_{2}⟩ ≠ ∅ and ⟨b_{1}, b_{2}⟩ ≠ ∅. Let f be a morphism from a_{1} to a_{2} and g be a morphism from b_{1} to b_{2}. If f = g, then f ∈ ⟨o_{1}, o_{2}⟩.

(25) Let A, B be non empty category structures with units. Suppose A and B have the same composition. Let a be an object of A, b be an object of B, and o be an object of Intersect(A, B). If o = a and o = b and id_{a} = id_{b}, then id_{a} ∈ ⟨o, o⟩.

(26) Let A, B be categories. Suppose that (i) A and B have the same composition, (ii) Intersect(A, B) is non empty, and (iii) for every object a of A and for every object b of B such that a = b

holds id_{a} = id_{b} . Then Intersect(A, B) is a subcategory of A.

3. Subcategories

The scheme SubcategoryUniq deals with a category A, non empty subcategories B, C of A, a unary predicate P, and a ternary predicate Q, and states that:

The category structure of B = the category structure of C provided the following requirements are met:

• For every object a of A holds a is an object of B iff P[a],

• Let a, b be objects of A and a′, b′be objects of B. Suppose a′= a and b′= b and ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then f ∈ ⟨a′, b′⟩ if and only if Q[a, b, f],

• For every object a of A holds a is an object of C iff P[a], and

• Let a, b be objects of A and a′, b′be objects of C. Suppose a′= a and b′= b and ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then f ∈ ⟨a′, b′⟩ if and only if Q[a, b, f].

The following proposition is true

(27) Let A be a non empty category structure and B be a non empty substructure of A. Then B is full if and only if for all objects a_{1}, a_{2} of A and for all objects b_{1}, b_{2} of B such that b_{1} = a_{1} and b_{2} = a_{2} holds ⟨b_{1}, b_{2}⟩ = ⟨a_{1}, a_{2}⟩.

Now we present two schemes. The scheme FullSubcategoryEx deals with a category A and a unary predicate P, and states that:

There exists a strict full non empty subcategory B of A such that for every object a of A holds a is an object of B if and only if P[a]

provided the parameters satisfy the following condition:

• There exists an object a of A such that P[a]. The scheme FullSubcategoryUniq deals with a category A, full non empty

subcategories B, C of A, and a unary predicate P, and states that:

The category structure of B = the category structure of C provided the parameters meet the following conditions:

• For every object a of A holds a is an object of B iff P[a], and

• For every object a of A holds a is an object of C iff P[a].

4. Inclusion Functors and Functor Restrictions

Let f be a function yielding function and let x, y be sets. Observe that f(x,

y) is relation-like and function-like.

One can prove the following proposition

(28) Let A be a category, C be a non empty subcategory of A, and a, b be objects of C. If ⟨a, b⟩ ≠ ∅, then for every morphism f from a to b holds ({C \atop ↪} )(f) = f.

Let A be a category and let C be a non empty subcategory of A. Note that {C \atop ↪} is id-preserving and comp-preserving.

Let A be a category and let C be a non empty subcategory of A. One can verify that {C \atop ↪} is precovariant.

Let A be a category and let C be a non empty subcategory of A. Then {C \atop ↪} is a strict covariant functor from C to A.

Let A, B be categories, let C be a non empty subcategory of A, and let F be a covariant functor from A to B. Then F ↾C is a strict covariant functor from C to B.

Let A, B be categories, let C be a non empty subcategory of A, and let F be a contravariant functor from A to B. Then F ↾C is a strict contravariant functor from C to B.

Next we state several propositions:

(29) Let A, B be categories, C be a non empty subcategory of A, F be a functor structure from A to B, a be an object of A, and c be an object of C. If c = a, then (F ↾C)(c) = F (a).

(30) Let A, B be categories, C be a non empty subcategory of A, F be a covariant functor from A to B, a, b be objects of A, and c, d be objects of C. Suppose c = a and d = b and ⟨c, d⟩ ≠ ∅. Let f be a morphism from a to b and g be a morphism from c to d. If g = f, then (F ↾C)(g) = F (f).

(31) Let A, B be categories, C be a non empty subcategory of A, F be a contravariant functor from A to B, a, b be objects of A, and c, d be objects of C. Suppose c = a and d = b and ⟨c, d⟩ ≠ ∅. Let f be a morphism from a to b and g be a morphism from c to d. If g = f, then (F ↾C)(g) = F (f).

(32) Let A, B be non empty graphs and F be a bimap structure from A into B. Suppose F is precovariant and one-to-one. Let a, b be objects of A. If F (a) = F (b), then a = b.

(33) Let A, B be non empty reflexive graphs and F be a feasible precovariant functor structure from A to B. Suppose F is faithful. Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f, g be morphisms from a to b. If F (f) = F (g), then f = g.

(34) Let A, B be non empty graphs and F be a precovariant functor structure from A to B. Suppose F is surjective. Let a, b be objects of B. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then there exist objects c, d of A and there exists a morphism g from c to d such that a = F (c) and b = F (d) and ⟨c, d⟩ ≠ ∅ and f = F (g).

(35) Let A, B be non empty graphs and F be a bimap structure from A into B. Suppose F is precontravariant and one-to-one. Let a, b be objects of A. If F (a) = F (b), then a = b.

(36) Let A, B be non empty reflexive graphs and F be a feasible precontravariant functor structure from A to B. Suppose F is faithful. Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f, g be morphisms from a to b. If F (f) = F (g), then f = g.

(37) Let A, B be non empty graphs and F be a precontravariant functor structure from A to B. Suppose F is surjective. Let a, b be objects of B. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then there exist objects c, d of A and there exists a morphism g from c to d such that b = F (c) and a = F (d) and ⟨c, d⟩ ≠ ∅ and f = F (g).

5. Isomorphisms under Arbitrary Functor

Let A, B be categories, let F be a functor structure from A to B, and let A′, B′be categories. We say that A′and B′are isomorphic under F if and only if the conditions (Def. 4) are satisfied.

(Def. 4)(i) A′ is a subcategory of A, (ii) B′ is a subcategory of B, and (iii) there exists a covariant functor G from A′to B′such that G is bijective

and for every object a′ of A′ and for every object a of A such that a′= a holds G(a′) = F (a) and for all objects b′, c′ of A′ and for all objects b, c of A such that ⟨b′, c′⟩ ≠ ∅ and b′= b and c′= c and for every morphism f′ from b′ to c′ and for every morphism f from b to c such that f′= f holds G(f′) = (Morph-Map_{F} (b, c))(f).

We say that A′and B′are anti-isomorphic under F if and only if the conditions (Def. 5) are satisfied.

(Def. 5)(i) A′ is a subcategory of A, (ii) B′ is a subcategory of B, and (iii) there exists a contravariant functor G from A′ to B′ such that G is

bijective and for every object a′ of A′ and for every object a of A such that a′ = a holds G(a′) = F (a) and for all objects b′, c′ of A′ and for all objects b, c of A such that ⟨b′, c′⟩ ≠ ∅ and b′= b and c′= c and for every morphism f′from b′to c′and for every morphism f from b to c such that f′= f holds G(f′) = (Morph-Map_{F} (b, c))(f).

We now state several propositions:

(38) Let A, B, A_{1}, B_{1} be categories and F be a functor structure from A to B. If A_{1} and B_{1} are isomorphic under F, then A_{1} and B_{1} are isomorphic.

(39) Let A, B, A_{1}, B_{1} be categories and F be a functor structure from A to B. Suppose A_{1} and B_{1} are anti-isomorphic under F. Then A_{1}, B_{1} are anti-isomorphic.

(40) Let A, B be categories and F be a covariant functor from A to B. If A and B are isomorphic under F, then F is bijective.

(41) Let A, B be categories and F be a contravariant functor from A to B. If A and B are anti-isomorphic under F, then F is bijective.

(42) Let A, B be categories and F be a covariant functor from A to B. If F is bijective, then A and B are isomorphic under F.

(43) Let A, B be categories and F be a contravariant functor from A to B.

If F is bijective, then A and B are anti-isomorphic under F. Now we present two schemes. The scheme CoBijectRestriction deals with

non empty categories A, B, a covariant functor C from A to B, a non empty subcategory D of A, and a non empty subcategory E of B, and states that:

D and E are isomorphic under C provided the parameters satisfy the following conditions:

• C is bijective,

• For every object a of A holds a is an object of D iff C(a) is an object of E, and

• Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let a_{1}, b_{1} be objects of D. Suppose a_{1} = a and b_{1} = b. Let a_{2}, b_{2} be objects of E. Suppose a_{2} = C(a) and b_{2} = C(b). Let f be a morphism from a to b. Then f ∈ ⟨a_{1}, b_{1}⟩ if and only if C(f) ∈ ⟨a_{2}, b_{2}⟩.

The scheme ContraBijectRestriction deals with non empty categories A, B, a contravariant functor C from A to B, a non empty subcategory D of A, and a non empty subcategory E of B, and states that:

D and E are anti-isomorphic under C provided the parameters meet the following conditions:

• C is bijective,

• For every object a of A holds a is an object of D iff C(a) is an object of E, and • Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let a_{1}, b_{1} be objects

of D. Suppose a_{1} = a and b_{1} = b. Let a_{2}, b_{2} be objects of E. Suppose a_{2} = C(a) and b_{2} = C(b). Let f be a morphism from a to b. Then f ∈ ⟨a_{1}, b_{1}⟩ if and only if C(f) ∈ ⟨b_{2}, a_{2}⟩.

The following propositions are true:

(44) For every category A and for every non empty subcategory B of A holds B and B are isomorphic under id_{A}.

(45) For all functions f, g such that f ⊆ g holds ↶f ⊆ ↶g.

(46) For all functions f, g such that domf is a binary relation and ↶f ⊆ ↶g holds f ⊆ g.

(47) Let I, J be sets, A be a many sorted set indexed by [: I, I:], and B be a many sorted set indexed by [: J, J :]. If A ˙⊆ B, then ↶A ˙⊆ ↶B.

(48) Let A be a transitive non empty category structure and B be a transitive non empty substructure of A. Then B^{op} is a substructure of A^{op}.

(49) For every category A and for every non empty subcategory B of A holds B^{op} is a subcategory of A^{op}.

(50) Let A be a category and B be a non empty subcategory of A. Then B and B^{op} are anti-isomorphic under the dualizing functor from A into A^{op}.

(51) Let A_{1}, A_{2} be categories and F be a covariant functor from A_{1} to A_{2}. Suppose F is bijective. Let B_{1} be a non empty subcategory of A_{1} and B_{2} be a non empty subcategory of A_{2}. Suppose B_{1} and B_{2} are isomorphic under F. Then B_{2} and B_{1} are isomorphic under F^{−1} .

(52) Let A_{1}, A_{2} be categories and F be a contravariant functor from A_{1} to A_{2}. Suppose F is bijective. Let B_{1} be a non empty subcategory of A_{1} and B_{2} be a non empty subcategory of A_{2}. Suppose B_{1} and B_{2} are anti-isomorphic under F. Then B_{2} and B_{1} are anti-isomorphic under F^{−1} .

(53) Let A_{1}, A_{2}, A_{3} be categories, F be a covariant functor from A_{1} to A_{2}, G be a covariant functor from A_{2} to A_{3}, B_{1} be a non empty subcategory of A_{1}, B_{2} be a non empty subcategory of A_{2}, and B_{3} be a non empty subcategory of A_{3}. Suppose B_{1} and B_{2} are isomorphic under F and B_{2} and B_{3} are isomorphic under G. Then B_{1} and B_{3} are isomorphic under G · F.

(54) Let A_{1}, A_{2}, A_{3} be categories, F be a contravariant functor from A_{1} to A_{2}, G be a covariant functor from A_{2} to A_{3}, B_{1} be a non empty subcategory of A_{1}, B_{2} be a non empty subcategory of A_{2}, and B_{3} be a non empty subcategory of A_{3}. Suppose B_{1} and B_{2} are anti-isomorphic under F and B_{2} and B_{3} are isomorphic under G. Then B_{1} and B_{3} are anti-isomorphic under G · F.

754 grzegorz bancerek

(55) Let A_{1}, A_{2}, A_{3} be categories, F be a covariant functor from A_{1} to A_{2}, G be a contravariant functor from A_{2} to A_{3}, B_{1} be a non empty subcategory of A_{1}, B_{2} be a non empty subcategory of A_{2}, and B_{3} be a non empty subcategory of A_{3}. Suppose B_{1} and B_{2} are isomorphic under F and B_{2} and B_{3} are anti-isomorphic under G. Then B_{1} and B_{3} are anti-isomorphic under G · F.

(56) Let A_{1}, A_{2}, A_{3} be categories, F be a contravariant functor from A_{1} to A_{2}, G be a contravariant functor from A_{2} to A_{3}, B_{1} be a non empty subcategory of A_{1}, B_{2} be a non empty subcategory of A_{2}, and B_{3} be a non empty subcategory of A_{3}. Suppose B_{1} and B_{2} are anti-isomorphic under F and B_{2} and B_{3} are anti-isomorphic under G. Then B_{1} and B_{3} are isomorphic under G · F.

References

[1] Grzegorz Bancerek. Concrete categories. Formalized Mathematics, 9(3):605–621, 2001.

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

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

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

[7] Artur Korniłowicz. The composition of functors and transformations in alternative categories. Formalized Mathematics, 7(1):1–7, 1998.

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

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

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

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

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

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

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

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

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

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

Categorial Background for Duality Theory

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Summary. In the paper, we develop the notation of lattice-wise categories as concrete categories (see [8]) of lattices. Namely, the categories based on [17] with lattices as objects and at least monotone maps between them as morphisms. As examples, we introduce the categories UPS, CONT, and ALG with complete, continuous, and algebraic lattices, respectively, as objects and directed suprema preserving maps as morphisms. Some useful schemes to construct categories of lattices and functors between them are also presented.

MML Identifier: YELLOW21.

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

1. Lattice-wise Categories

In this paper x, y are sets. Let a be a set. a as 1-sorted is a 1-sorted structure and is defined as follows:

(Def. 1) a as 1-sorted = \{ \begin{matrix}a, if a is a 1-sorted structure, \\ ⟨a⟩, otherwise. \end{matrix}

Let W be a set. The functor POSETS(W) is defined as follows: (Def. 2) x ∈ POSETS(W) iff x is a strict poset and the carrier of x as 1-sorted

∈ W. Let W be a non empty set. One can check that POSETS(W) is non empty. Let W be a set with non empty elements. Note that POSETS(W) is poset-

membered.

Let C be a category. We say that C is carrier-underlaid if and only if:

(Def. 3) For every object a of C there exists a 1-sorted structure S such that

a = S and the carrier of a = the carrier of S. Let C be a category. We say that C is lattice-wise if and only if the conditions

(Def. 4) are satisfied. (Def. 4)(i) C is semi-functional and set-id-inheriting,

(ii) every object of C is a lattice, and (iii) for all objects a, b of C and for all lattices A, B such that A = a and

B = b holds ⟨a, b⟩ ⊆ B_{}^{A}. Let C be a category. We say that C has complete lattices if and only if:

(Def. 5) C is lattice-wise and every object of C is a complete lattice.

One can check that every category which has complete lattices is lattice-wise and every category which is lattice-wise is also concrete and carrier-underlaid.

One can verify that there exists a category which is strict and has complete lattices.

We now state two propositions:

(1) Let C be a carrier-underlaid category and a be an object of C. Then the carrier of a = the carrier of a as 1-sorted.

(2) Let C be a set-id-inheriting carrier-underlaid category and a be an object of C. Then id_{a} = id_{a as 1-sorted} .

Let C be a lattice-wise category and let a be an object of C. Then a as 1-sorted is a lattice and it can be characterized by the condition:

(Def. 6) a as 1-sorted = a.

We introduce 𝕃_{a} as a synonym of a as 1-sorted.

Let C be a category with complete lattices and let a be an object of C. Then a as 1-sorted is a complete lattice. We introduce 𝕃_{a} as a synonym of a as 1-sorted.

Let C be a lattice-wise category and let a, b be objects of C. Let us assume that ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. The functor ^{@}f yielding a monotone map from 𝕃_{a} into 𝕃_{b} is defined as follows:

(Def. 7) ^{@}f = f.

The following proposition is true

(3) Let C be a lattice-wise category and a, b, c be objects of C. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅. Let f be a morphism from a to b and g be a morphism from b to c. Then g · f = (^{@}g) · (^{@}f).

In this article we present several logical schemes. The scheme CLCatEx1 deals with a non empty set A and a ternary predicate P, and states that:

There exists a lattice-wise strict category C such that

(i) the carrier of C = A, and (ii) for all objects a, b of C and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff P[𝕃_{a}, 𝕃_{b}, f] provided the following conditions are met:

• Every element of A is a lattice,

• Let a, b, c be lattices. Suppose a ∈ A and b ∈ A and c ∈ A. Let f be a map from a into b and g be a map from b into c. If P[a, b, f] and P[b, c, g], then P[a, c, g · f], and

• For every lattice a such that a ∈ A holds P[a, a, id_{a}]. The scheme CLCatEx2 deals with a non empty set A, a unary predicate P,

and a ternary predicate Q, and states that:

There exists a lattice-wise strict category C such that

(i) for every lattice x holds x is an object of C iff x is strict and P[x] and the carrier of x ∈ A, and (ii) for all objects a, b of C and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff Q[𝕃_{a}, 𝕃_{b}, f]

provided the parameters satisfy the following conditions:

• There exists a strict lattice x such that P[x] and the carrier of x ∈ A,

• Let a, b, c be lattices. Suppose P[a] and P[b] and P[c]. Let f be a map from a into b and g be a map from b into c. If Q[a, b, f] and Q[b, c, g], then Q[a, c, g · f], and

• For every lattice a such that P[a] holds Q[a, a, id_{a}]. The scheme CLCatUniq1 deals with a non empty set A and a ternary pre-

dicate P, and states that:

Let C_{1}, C_{2} be lattice-wise categories. Suppose that

(i) the carrier of C_{1} = A, (ii) for all objects a, b of C_{1} and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff P[a, b, f], (iii) the carrier of C_{2} = A, and (iv) for all objects a, b of C_{2} and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff P[a, b, f].

Then the category structure of C_{1} = the category structure of C_{2}

for all values of the parameters.

The scheme CLCatUniq2 deals with a non empty set A, a unary predicate P, and a ternary predicate Q, and states that:

Let C_{1}, C_{2} be lattice-wise categories. Suppose that

(i) for every lattice x holds x is an object of C_{1} iff x is strict and P[x] and the carrier of x ∈ A, (ii) for all objects a, b of C_{1} and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff Q[a, b, f], (iii) for every lattice x holds x is an object of C_{2} iff x is strict and P[x] and the carrier of x ∈ A, and (iv) for all objects a, b of C_{2} and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff Q[a, b, f].

Then the category structure of C_{1} = the category structure of C_{2}

for all values of the parameters.

The scheme CLCovariantFunctorEx deals with lattice-wise categories A, B, a unary functor F yielding a lattice, a ternary functor G yielding a function, and two ternary predicates P, Q, and states that:

There exists a covariant strict functor F from A to B such that

(i) for every object a of A holds F (a) = F(𝕃_{a}), and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds F (f) = G(𝕃_{a}, 𝕃_{b},^{@}f)

provided the parameters meet the following conditions:

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of A)(a, b) if and only if a ∈ the carrier of A and b ∈ the carrier of A and P[a, b, f],

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of B)(a, b) if and only if a ∈ the carrier of B and b ∈ the carrier of B and Q[a, b, f],

• For every lattice a such that a ∈ the carrier of A holds F(a) ∈ the carrier of B,

• Let a, b be lattices and f be a map from a into b. If P[a, b, f], then G(a, b, f) is a map from F(a) into F(b) and Q[F(a), F(b), G(a, b, f)],

• For every lattice a such that a ∈ the carrier of A holds G(a, a, id_{a}) = id_{F(a)}, and

• Let a, b, c be lattices, f be a map from a into b, and g be a map from b into c. If P[a, b, f] and P[b, c, g], then G(a, c, g · f) = G(b, c, g) · G(a, b, f).

The scheme CLContravariantFunctorEx deals with lattice-wise categories A, B, a unary functor F yielding a lattice, a ternary functor G yielding a function, and two ternary predicates P, Q, and states that:

There exists a contravariant strict functor F from A to B such that

(i) for every object a of A holds F (a) = F(𝕃_{a}), and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds F (f) = G(𝕃_{a}, 𝕃_{b},^{@}f)

provided the parameters satisfy the following conditions:

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of A)(a, b) if and only if a ∈ the carrier of A and b ∈ the carrier of A and P[a, b, f], • Let a, b be lattices and f be a map from a into b. Then f ∈ (the

arrows of B)(a, b) if and only if a ∈ the carrier of B and b ∈ the carrier of B and Q[a, b, f],

• For every lattice a such that a ∈ the carrier of A holds F(a) ∈ the carrier of B,

• Let a, b be lattices and f be a map from a into b. If P[a, b, f], then G(a, b, f) is a map from F(b) into F(a) and Q[F(b), F(a), G(a, b, f)],

• For every lattice a such that a ∈ the carrier of A holds G(a, a, id_{a}) = id_{F(a)}, and

• Let a, b, c be lattices, f be a map from a into b, and g be a map from b into c. If P[a, b, f] and P[b, c, g], then G(a, c, g · f) = G(a, b, f) · G(b, c, g).

The scheme CLCatIsomorphism deals with lattice-wise categories A, B, a unary functor F yielding a lattice, a ternary functor G yielding a function, and two ternary predicates P, Q, and states that:

A and B are isomorphic provided the parameters meet the following conditions:

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of A)(a, b) if and only if a ∈ the carrier of A and b ∈ the carrier of A and P[a, b, f],

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of B)(a, b) if and only if a ∈ the carrier of B and b ∈ the carrier of B and Q[a, b, f],

• There exists a covariant functor F from A to B such that (i) for every object a of A holds F (a) = F(a), and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds F (f) = G(a, b, f),

• For all lattices a, b such that a ∈ the carrier of A and b ∈ the carrier of A holds if F(a) = F(b), then a = b,

• For all lattices a, b and for all maps f, g from a into b such that P[a, b, f] and P[a, b, g] holds if G(a, b, f) = G(a, b, g), then f = g, and

• Let a, b be lattices and f be a map from a into b. Suppose Q[a, b, f]. Then there exist lattices c, d and there exists a map g from c into d such that c ∈ the carrier of A and d ∈ the carrier of A and P[c, d, g] and a = F(c) and b = F(d) and f = G(c, d, g).

The scheme CLCatAntiIsomorphism deals with lattice-wise categories A, B, a unary functor F yielding a lattice, a ternary functor G yielding a function, and two ternary predicates P, Q, and states that:

A, B are anti-isomorphic provided the following conditions are met:

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of A)(a, b) if and only if a ∈ the carrier of A and b ∈ the carrier of A and P[a, b, f],

• Let a, b be lattices and f be a map from a into b. Then f ∈ (the arrows of B)(a, b) if and only if a ∈ the carrier of B and b ∈ the carrier of B and Q[a, b, f],

• There exists a contravariant functor F from A to B such that (i) for every object a of A holds F (a) = F(a), and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds F (f) = G(a, b, f),

• For all lattices a, b such that a ∈ the carrier of A and b ∈ the carrier of A holds if F(a) = F(b), then a = b,

• For all lattices a, b and for all maps f, g from a into b such that G(a, b, f) = G(a, b, g) holds f = g, and

• Let a, b be lattices and f be a map from a into b. Suppose Q[a, b, f]. Then there exist lattices c, d and there exists a map g from c into d such that c ∈ the carrier of A and d ∈ the carrier of A and P[c, d, g] and b = F(c) and a = F(d) and f = G(c, d, g).

2. Equivalence of Lattice-wise Categories

Let C be a lattice-wise category. We say that C has all isomorphisms if and only if:

(Def. 8) For all objects a, b of C and for every map f from 𝕃_{a} into 𝕃_{b} such that

f is isomorphic holds f ∈ ⟨a, b⟩. One can verify that there exists a strict lattice-wise category which has all

isomorphisms.

The following propositions are true:

(4) Let C be a lattice-wise category with all isomorphisms, a, b be objects of C, and f be a morphism from a to b. If ^{@}f is isomorphic, then f is iso.

(5) Let C be a lattice-wise category and a, b be objects of C. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, a⟩ ≠ ∅. Let f be a morphism from a to b. If f is iso, then @f is isomorphic.

The scheme CLCatEquivalence deals with lattice-wise categories A, B, two unary functors F and G yielding lattices, two ternary functors H and I yielding functions, two unary functors A and B yielding functions, and two ternary predicates P, Q, and states that:

A and B are equivalent provided the parameters satisfy the following conditions:

• For all objects a, b of A and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff P[𝕃_{a}, 𝕃_{b}, f],

• For all objects a, b of B and for every monotone map f from 𝕃_{a} into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff Q[𝕃_{a}, 𝕃_{b}, f],

• There exists a covariant functor F from A to B such that (i) for every object a of A holds F (a) = F(a), and (ii) for all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds F (f) = H(a, b, f),

• There exists a covariant functor G from B to A such that (i) for every object a of B holds G(a) = G(a), and (ii) for all objects a, b of B such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds G(f) = I(a, b, f),

• Let a be a lattice. Suppose a ∈ the carrier of A. Then there exists a monotone map f from G(F(a)) into a such that f = A(a) and f is isomorphic and P[G(F(a)), a, f] and P[a, G(F(a)), f^{−1} ],

• Let a be a lattice. Suppose a ∈ the carrier of B. Then there exists a monotone map f from a into F(G(a)) such that f = B(a) and f is isomorphic and Q[a, F(G(a)), f] and Q[F(G(a)), a, f^{−1} ],

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds A(b) · I(F(a), F(b), H(a, b, f)) = (^{@}f) · A(a), and

• For all objects a, b of B such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds H(G(a), G(b), I(a, b, f)) · B(a) = B(b) · (^{@}f).

3. UPS Category

Let R be a binary relation. We say that R is upper-bounded if and only if: (Def. 9) There exists x such that for every y such that y ∈ fieldR holds ⟨⟨y,

x⟩⟩∈ R. Let us note that every binary relation which is well-ordering is also reflexive,

transitive, antisymmetric, connected, and well founded. Let us mention that there exists a binary relation which is well-ordering. Next we state the proposition

(6) Let f be an one-to-one function and R be a binary relation. Then ⟨⟨x, y⟩⟩ ∈ f · R · f^{−1} if and only if x ∈ domf and y ∈ domf and ⟨⟨f(x), f(y)⟩⟩∈ R.

Let f be an one-to-one function and let R be a reflexive binary relation. Note that f · R · f^{−1} is reflexive.

Let f be an one-to-one function and let R be an antisymmetric binary relation. Note that f · R · f^{−1} is antisymmetric.

Let f be an one-to-one function and let R be a transitive binary relation. Note that f · R · f^{−1} is transitive.

Next we state the proposition

(7) Let X be a set and A be an ordinal number. If X ≈ A, then there exists an order R in X such that R well orders X and \overline{R} = A.

Let X be a non empty set. Observe that there exists an order in X which is upper-bounded and well-ordering.

Next we state four propositions:

(8) Let P be a reflexive non empty relational structure. Then P is upperbounded if and only if the internal relation of P is upper-bounded.

(9) Let P be an upper-bounded non empty poset. Suppose the internal relation of P is well-ordering. Then P is connected, complete, and continuous.

(10) Let P be an upper-bounded non empty poset. Suppose the internal relation of P is well-ordering. Let x, y be elements of P. If y < x, then there exists an element z of P such that z is compact and y z and z x.

(11) Let P be an upper-bounded non empty poset. If the internal relation of

P is well-ordering, then P is algebraic. Let X be a non empty set and let R be an upper-bounded well-ordering order

in X. Observe that ⟨X, R⟩ is complete connected continuous and algebraic. Let us observe that every set which is non trivial has a non-empty element. Let W be a non empty set. Let us assume that there exists an element w of

W such that w is non empty. The functor UPS_{W} yielding a lattice-wise strict category is defined by the conditions (Def. 10).

(Def. 10)(i) For every lattice x holds x is an object of UPS_{W} iff x is strict and

complete and the carrier of x ∈ W, and (ii) for all objects a, b of UPS_{W} and for every monotone map f from 𝕃_{a}

into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff f is directed-sups-preserving. Let W be a set with a non-empty element. Observe that UPS_{W} has complete

lattices and all isomorphisms.

One can prove the following four propositions:

(12) For every set W with a non-empty element holds the carrier of UPS_{W} ⊆ POSETS(W).

(13) Let W be a set with a non-empty element and given x. Then x is an object of UPS_{W} if and only if x is a complete lattice and x ∈ POSETS(W).

(14) Let W be a set with a non-empty element and L be a lattice. Suppose the carrier of L ∈ W. Then L is an object of UPS_{W} if and only if L is strict and complete.

(15) Let W be a set with a non-empty element, a, b be objects of UPS_{W}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is a directed-sups-preserving map from 𝕃_{a} into 𝕃_{b}.

Let W be a set with a non-empty element and let a, b be objects of UPS_{W}. Observe that ⟨a, b⟩ is non empty.

4. Lattice-wise Subcategories

Next we state the proposition

(16) Let A be a category, B be a non empty subcategory of A, a be an object of A, and b be an object of B. If b = a, then the carrier of b = the carrier of a.

Let A be a set-id-inheriting category. Observe that every non empty subcategory of A is set-id-inheriting.

Let A be a para-functional category. One can verify that every non empty subcategory of A is para-functional.

Let A be a semi-functional category. Note that every non empty transitive substructure of A is semi-functional.

Let A be a carrier-underlaid category. Note that every non empty subcategory of A is carrier-underlaid.

Let A be a lattice-wise category. Observe that every non empty subcategory of A is lattice-wise.

Let A be a lattice-wise category with all isomorphisms. Observe that every non empty subcategory of A which is full has all isomorphisms.

Let A be a category with complete lattices. One can check that every non empty subcategory of A has complete lattices.

Let W be a set with a non-empty element. The functor CONT_{W} yielding a strict full non empty subcategory of UPS_{W} is defined by:

(Def. 11) For every object a of UPS_{W} holds a is an object of CONT_{W} iff 𝕃_{a} is

continuous. Let W be a set with a non-empty element. The functor ALG_{W} yielding a

strict full non empty subcategory of CONT_{W} is defined by: (Def. 12) For every object a of CONT_{W} holds a is an object of ALG_{W} iff 𝕃_{a} is

algebraic. The following four propositions are true:

(17) Let W be a set with a non-empty element and L be a lattice. Suppose the carrier of L ∈ W. Then L is an object of CONT_{W} if and only if L is strict, complete, and continuous.

(18) Let W be a set with a non-empty element and L be a lattice. Suppose the carrier of L ∈ W. Then L is an object of ALG_{W} if and only if L is strict, complete, and algebraic.

(19) Let W be a set with a non-empty element, a, b be objects of CONT_{W}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is a directed-sups-preserving map from 𝕃_{a} into 𝕃_{b}.

(20) Let W be a set with a non-empty element, a, b be objects of ALG_{W}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is a directed-sups-preserving map from 𝕃_{a} into 𝕃_{b}.

Let W be a set with a non-empty element and let a, b be objects of CONT_{W}. One can check that ⟨a, b⟩ is non empty.

Let W be a set with a non-empty element and let a, b be objects of ALG_{W}. One can check that ⟨a, b⟩ is non empty.

References

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

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

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

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

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

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

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

[8] Grzegorz Bancerek. Concrete categories. Formalized Mathematics, 9(3):605–621, 2001.

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

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

[11] Beata Madras. On the concept of the triangulation. Formalized Mathematics, 5(3):457– 462, 1996.

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

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

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

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

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

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

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

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

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

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

categorial background for duality theory 765

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

1990.

Received August 1, 2001 766 grzegorz bancerek

Duality Based on the Galois Connection. Part I

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Summary. In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [12, p. 179–183; 1.1–1.12]. The duality is based on the concept of the Galois connection.

MML Identifier: WAYBEL34.

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

1. Infs-preserving and Sups-preserving Maps

Let S, T be complete lattices. One can check that there exists a connection between S and T which is Galois.

Next we state the proposition

(1) Let S, T, S′, T′ be non empty relational structures. Suppose that

(i) the relational structure of S = the relational structure of S′, and (ii) the relational structure of T = the relational structure of T′.

Let c be a connection between S and T and c′be a connection between S′ and T′. If c = c′, then if c is Galois, then c′ is Galois.

Let S, T be lattices and let g be a map from S into T. Let us assume that S is complete and T is complete and g is infs-preserving. The lower adjoint of g is a map from T into S and is defined as follows:

(Def. 1) ⟨⟨g, the lower adjoint of g⟩⟩ is Galois.

Let S, T be lattices and let d be a map from T into S. Let us assume that S is complete and T is complete and d is sups-preserving. The upper adjoint of d is a map from S into T and is defined as follows:

(Def. 2) ⟨⟨the upper adjoint of d, d⟩⟩ is Galois.

Let S, T be complete lattices and let g be an infs-preserving map from S into T. One can verify that the lower adjoint of g is lower adjoint.

Let S, T be complete lattices and let d be a sups-preserving map from T into S. One can check that the upper adjoint of d is upper adjoint.

The following two propositions are true:

(2) Let S, T be complete lattices, g be an infs-preserving map from S into T, and t be an element of T. Then (the lower adjoint of g)(t) = inf(g^{−1} (↑t)).

(3) Let S, T be complete lattices, d be a sups-preserving map from T into S, and s be an element of S. Then (the upper adjoint of d)(s) = sup(d^{−1} (↓s)).

Let S, T be relational structures and let f be a function from the carrier of S into the carrier of T. The functor f^{op} yielding a map from S^{op} into T^{op} is defined as follows:

(Def. 3) f^{op} = f.

Let S, T be complete lattices and let g be an infs-preserving map from S into T. One can verify that g^{op} is lower adjoint.

Let S, T be complete lattices and let d be a sups-preserving map from S into T. Observe that d^{op} is upper adjoint.

We now state several propositions:

(4) Let S, T be complete lattices and g be an infs-preserving map from S into T. Then the lower adjoint of g = the upper adjoint of g^{op}.

(5) Let S, T be complete lattices and d be a sups-preserving map from S into T. Then the lower adjoint of d^{op} = the upper adjoint of d.

(6) For every non empty relational structure L holds ⟨⟨id_{L}, id_{L}⟩⟩ is Galois.

(7) For every complete lattice L holds the lower adjoint of id_{L} = id_{L} and the upper adjoint of id_{L} = id_{L}.

(8) Let L_{1}, L_{2}, L_{3} be complete lattices, g_{1} be an infs-preserving map from L_{1} into L_{2}, and g_{2} be an infs-preserving map from L_{2} into L_{3}. Then the lower adjoint of g_{2} · g_{1} = (the lower adjoint of g_{1}) · (the lower adjoint of g_{2}).

(9) Let L_{1}, L_{2}, L_{3} be complete lattices, d_{1} be a sups-preserving map from L_{1} into L_{2}, and d_{2} be a sups-preserving map from L_{2} into L_{3}. Then the upper adjoint of d_{2} · d_{1} = (the upper adjoint of d_{1}) · (the upper adjoint of d_{2}).

(10) Let S, T be complete lattices and g be an infs-preserving map from S into T. Then the upper adjoint of the lower adjoint of g = g.

(11) Let S, T be complete lattices and d be a sups-preserving map from S into T. Then the lower adjoint of the upper adjoint of d = d.

(12) Let C be a non empty category structure and a, b, f be sets. Suppose f ∈ (the arrows of C)(a, b). Then there exist objects o_{1}, o_{2} of C such that o_{1} = a and o_{2} = b and f ∈ ⟨o_{1}, o_{2}⟩ and f is a morphism from o_{1} to o_{2}.

Let W be a non empty set. Let us assume that there exists an element w of W such that w is non empty. The functor INF_{W} yields a lattice-wise strict category and is defined by the conditions (Def. 4).

(Def. 4)(i) For every lattice x holds x is an object of INF_{W} iff x is strict and

complete and the carrier of x ∈ W, and (ii) for all objects a, b of INF_{W} and for every monotone map f from 𝕃_{a}

into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff f is infs-preserving. Let W be a non empty set. Let us assume that there exists an element w

of W such that w is non empty. The functor SUP_{W} yields a lattice-wise strict category and is defined by the conditions (Def. 5).

(Def. 5)(i) For every lattice x holds x is an object of SUP_{W} iff x is strict and

complete and the carrier of x ∈ W, and (ii) for all objects a, b of SUP_{W} and for every monotone map f from 𝕃_{a}

into 𝕃_{b} holds f ∈ ⟨a, b⟩ iff f is sups-preserving. Let W be a set with a non-empty element. Observe that INF_{W} has complete

lattices and SUP_{W} has complete lattices.

One can prove the following propositions:

(13) Let W be a set with a non-empty element and L be a lattice. Then L is an object of INF_{W} if and only if L is strict and complete and the carrier of L ∈ W.

(14) Let W be a set with a non-empty element, a, b be objects of INF_{W}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is an infs-preserving map from 𝕃_{a} into 𝕃_{b}.

(15) Let W be a set with a non-empty element and L be a lattice. Then L is an object of SUP_{W} if and only if L is strict and complete and the carrier of L ∈ W.

(16) Let W be a set with a non-empty element, a, b be objects of SUP_{W}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is a sups-preserving map from 𝕃_{a} into 𝕃_{b}.

(17) For every set W with a non-empty element holds the carrier of INF_{W} =

the carrier of SUP_{W}. Let W be a set with a non-empty element. The functor LowerAdj_{W} yields

a contravariant strict functor from INF_{W} to SUP_{W} and is defined by the conditions (Def. 6).

(Def. 6)(i) For every object a of INF_{W} holds LowerAdj_{W} (a) = 𝕃_{a}, and (ii) for all objects a, b of INF_{W} such that ⟨a, b⟩ ≠ ∅ and for every morphism

f from a to b holds LowerAdj_{W} (f) = the lower adjoint of ^{@}f. The functor UpperAdj_{W} yields a contravariant strict functor from SUP_{W} to INF_{W} and is defined by the conditions (Def. 7).

(Def. 7)(i) For every object a of SUP_{W} holds UpperAdj_{W} (a) = 𝕃_{a}, and

(ii) for all objects a, b of SUP_{W} such that ⟨a, b⟩ ≠ ∅ and for every morphism

f from a to b holds UpperAdj_{W} (f) = the upper adjoint of ^{@}f. Let W be a set with a non-empty element. Observe that LowerAdj_{W} is

bijective and UpperAdj_{W} is bijective.

We now state several propositions:

(18) For every set W with a non-empty element holds (LowerAdj_{W})^{−1} = UpperAdj_{W} and (UpperAdj_{W})^{−1} = LowerAdj_{W} .

(19) For every set W with a non-empty element holds LowerAdj_{W} · UpperAdj_{W} = id_{SUP_{W}} and UpperAdj_{W} · LowerAdj_{W} = id_{INF _{W}} .

(20) For every set W with a non-empty element holds INF_{W}, SUP_{W} are anti-isomorphic.

(21) For every set W with a non-empty element holds INF_{W} and SUP_{W} are anti-isomorphic under LowerAdj_{W}.

(22) For every set W with a non-empty element holds SUP_{W} and INF_{W} are anti-isomorphic under UpperAdj_{W}.

2. Scott Continuous Maps and Continuous Lattices

Next we state the proposition

(23) Let S, T be complete lattices and g be an infs-preserving map from S into T. Then g is directed-sups-preserving if and only if for every Scott topological augmentation X of T and for every Scott topological augmentation Y of S and for every open subset V of X holds ↑((the lower adjoint of g)^{∘}V) is an open subset of Y.

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

(Def. 8) For all elements x, y of S such that x ≪ y holds f(x) ≪ f(y).

We now state two propositions:

(24) Let S, T be complete lattices and g be an infs-preserving map from S into T. Suppose g is directed-sups-preserving. Then the lower adjoint of g is waybelow-preserving.

(25) Let S be a complete lattice, T be a complete continuous lattice, and g be an infs-preserving map from S into T. Suppose the lower adjoint of g is waybelow-preserving. Then g is directed-sups-preserving.

Let S, T be topological spaces and let f be a map from S into T. We say that f is relatively open if and only if:

(Def. 9) For every open subset V of S holds f^{∘}V is an open subset of T ↾ rngf.

One can prove the following propositions:

(26) Let X, Y be non empty topological spaces and d be a map from X into Y. Then d is relatively open if and only if d^{∘} is open.

(27) Let S, T be complete lattices, g be an infs-preserving map from S into T, X be a Scott topological augmentation of T, Y be a Scott topological augmentation of S, and V be an open subset of X. Then (the lower adjoint of g)^{∘}V = rng(the lower adjoint of g) ∩ ↑((the lower adjoint of g)^{∘}V).

(28) Let S, T be complete lattices, g be an infs-preserving map from S into T, X be a Scott topological augmentation of T, and Y be a Scott topological augmentation of S. Suppose that for every open subset V of X holds ↑((the lower adjoint of g)^{∘}V) is an open subset of Y. Let d be a map from X into Y. If d = the lower adjoint of g, then d is relatively open.

Let X, Y be complete lattices and let f be a sups-preserving map from X into Y. One can check that Imf is complete.

Next we state four propositions:

(29) Let S, T be complete lattices, g be an infs-preserving map from S into T, X be a Scott topological augmentation of T, Y be a Scott topological augmentation of S, Z be a Scott topological augmentation of Im(the lower adjoint of g), d be a map from X into Y, and d′ be a map from X into Z. Suppose d = the lower adjoint of g and d′ = d. If d is relatively open, then d′ is open.

(30) Let T_{1}, T_{2}, S_{1}, S_{2} be topological structures. Suppose that (i) the topological structure of T_{1} = the topological structure of T_{2}, and (ii) the topological structure of S_{1} = the topological structure of S_{2}.

If S_{1} is a subspace of T_{1}, then S_{2} is a subspace of T_{2}.

(31) For every topological structure T holds T ↾Ω_{T} = the topological structure of T.

(32) Let S, T be complete lattices and g be an infs-preserving map from S into T. Suppose g is one-to-one. Let X be a Scott topological augmentation of T, Y be a Scott topological augmentation of S, and d be a map from X into Y. Suppose d = the lower adjoint of g. Then g is directed-sups-preserving if and only if d is open.

Let X be a complete lattice and let f be a projection map from X into X. One can verify that Imf is complete.

We now state a number of propositions:

(33) Let L be a complete lattice and k be a kernel map from L into L. Then (i) k^{∘} is infs-preserving, (ii) k_{∘} is sups-preserving, (iii) the lower adjoint of k^{∘} = k_{∘}, and (iv) the upper adjoint of k_{∘} = k^{∘}.

(34) Let L be a complete lattice and k be a kernel map from L into L. Then k is directed-sups-preserving if and only if k^{∘} is directed-sups-preserving.

(35) Let L be a complete lattice and k be a kernel map from L into L. Then k is directed-sups-preserving if and only if for every Scott topological augmentation X of Imk and for every Scott topological augmentation Y of L and for every subset V of L such that V is an open subset of X holds ↑V is an open subset of Y.

(36) Let L be a complete lattice, S be a sups-inheriting non empty full relational substructure of L, x, y be elements of L, and a, b be elements of S. If a = x and b = y, then if x ≪ y, then a ≪ b.

(37) Let L be a complete lattice and k be a kernel map from L into L. Suppose k is directed-sups-preserving. Let x, y be elements of L and a, b be elements of Imk. If a = x and b = y, then x ≪ y iff a ≪ b.

(38) Let L be a complete lattice and k be a kernel map from L into L. Suppose

that

(i) Imk is continuous, and (ii) for all elements x, y of L and for all elements a, b of Imk such that

a = x and b = y holds x ≪ y iff a ≪ b. Then k is directed-sups-preserving.

(39) Let L be a complete lattice and c be a closure map from L into L. Then (i) c^{∘} is sups-preserving, (ii) c_{∘} is infs-preserving, (iii) the upper adjoint of c^{∘} = c_{∘}, and (iv) the lower adjoint of c_{∘} = c^{∘}.

(40) Let L be a complete lattice and c be a closure map from L into L. Then Imc is directed-sups-inheriting if and only if c_{∘} is directed-sups-preserving.

(41) Let L be a complete lattice and c be a closure map from L into L. Then Imc is directed-sups-inheriting if and only if for every Scott topological augmentation X of Imc and for every Scott topological augmentation Y of L and for every map f from Y into X such that f = c holds f is open.

(42) Let L be a complete lattice and c be a closure map from L into L. If Imc is directed-sups-inheriting, then c^{∘} is waybelow-preserving.

(43) Let L be a continuous complete lattice and c be a closure map from L into L. If c^{∘} is waybelow-preserving, then Imc is directed-sups-inheriting.

3. Duality of Subcategories of INF and SUP

Let W be a non empty set. The functor INF _{W}^{↑} yielding a strict non empty subcategory of INF_{W} is defined by the conditions (Def. 10).

(Def. 10)(i) Every object of INF_{W} is an object of INF _{W}^{↑}, and

(ii) for all objects a, b of INF_{W} and for all objects a′, b′ of INF _{W}^{↑} such that a′= a and b′= b and ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds f ∈ ⟨a′, b′⟩ iff ^{@}f is directed-sups-preserving.

Let W be a set with a non-empty element. The functor SUP_{W}^{0} yields a strict non empty subcategory of SUP_{W} and is defined by the conditions (Def. 11).

(Def. 11)(i) Every object of SUP_{W} is an object of SUP_{W}^{0} , and

(ii) for all objects a, b of SUP_{W} and for all objects a′, b′ of SUP_{W}^{0} such that a′= a and b′= b and ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds f ∈ ⟨a′, b′⟩ iff the upper adjoint of ^{@}f is directed-sups-preserving.

The following propositions are true:

(44) Let S be a non empty relational structure, T be a non empty reflexive antisymmetric relational structure, t be an element of T, and X be a non empty subset of S. Then S ⟼→ t preserves sup of X and S ⟼→ t preserves inf of X.

(45) Let S be a non empty relational structure and T be a lower-bounded non empty reflexive antisymmetric relational structure. Then S ⟼→ ⊥_{T} is sups-preserving.

(46) Let S be a non empty relational structure and T be an upper-bounded non empty reflexive antisymmetric relational structure. Then S ⟼→ ⊤_{T} is infs-preserving.

Let S be a non empty relational structure and let T be an upper-bounded non empty reflexive antisymmetric relational structure. Observe that S ⟼→ ⊤_{T} is directed-sups-preserving and infs-preserving.

Let S be a non empty relational structure and let T be a lower-bounded non empty reflexive antisymmetric relational structure. Observe that S ⟼→ ⊥_{T} is filtered-infs-preserving and sups-preserving.

Let S be a non empty relational structure and let T be an upper-bounded non empty reflexive antisymmetric relational structure. Note that there exists a map from S into T which is directed-sups-preserving and infs-preserving.

Let S be a non empty relational structure and let T be a lower-bounded non empty reflexive antisymmetric relational structure. One can check that there exists a map from S into T which is filtered-infs-preserving and sups-preserving.

Next we state several propositions:

(47) Let W be a set with a non-empty element and L be a lattice. Then L is an object of INF _{W}^{↑} if and only if L is strict and complete and the carrier of L ∈ W.

(48) Let W be a set with a non-empty element, a, b be objects of INF _{W}^{↑}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is a directed-sups-preserving infs-preserving map from 𝕃_{a} into 𝕃_{b}.

(49) Let W be a set with a non-empty element and L be a lattice. Then L is an object of SUP_{W}^{0} if and only if L is strict and complete and the carrier of L ∈ W.

(50) Let W be a set with a non-empty element, a, b be objects of SUP_{W}^{0} , and f be a set. Then f ∈ ⟨a, b⟩ if and only if there exists a sups-preserving map g from 𝕃_{a} into 𝕃_{b} such that g = f and the upper adjoint of g is directed-sups-preserving.

(51) For every set W with a non-empty element holds INF _{W}^{↑} =

Intersect(INF_{W} , UPS_{W}). Let W be a set with a non-empty element. The functor CL_{W} yielding a strict

full non empty subcategory of INF _{W}^{↑} is defined as follows: (Def. 12) For every object a of INF _{W}^{↑} holds a is an object of CL_{W} iff 𝕃_{a} is conti-

nuous. Let W be a set with a non-empty element. Observe that CL_{W} has complete

lattices.

One can prove the following two propositions:

(52) Let W be a set with a non-empty element and L be a lattice. Suppose the carrier of L ∈ W. Then L is an object of CL_{W} if and only if L is strict, complete, and continuous.

(53) Let W be a set with a non-empty element, a, b be objects of CL_{W}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if f is an infs-preserving directedsups-preserving map from 𝕃_{a} into 𝕃_{b}.

Let W be a set with a non-empty element. The functor CL_{W}^{op} yields a strict full non empty subcategory of SUP_{W}^{0} and is defined by:

(Def. 13) For every object a of SUP_{W}^{0} holds a is an object of CL_{W}^{op} iff 𝕃_{a} is conti-

nuous. Next we state several propositions:

(54) Let W be a set with a non-empty element and L be a lattice. Suppose the carrier of L ∈ W. Then L is an object of CL_{W}^{op} if and only if L is strict, complete, and continuous.

(55) Let W be a set with a non-empty element, a, b be objects of CL_{W}^{op}, and f be a set. Then f ∈ ⟨a, b⟩ if and only if there exists a sups-preserving map g from 𝕃_{a} into 𝕃_{b} such that g = f and the upper adjoint of g is directed-sups-preserving.

(56) For every set W with a non-empty element holds INF _{W}^{↑} and SUP_{W}^{0} are anti-isomorphic under LowerAdj_{W}.

(57) For every set W with a non-empty element holds SUP_{W}^{0} and INF _{W}^{↑} are anti-isomorphic under UpperAdj_{W}.

(58) For every set W with a non-empty element holds CL_{W} and CL_{W}^{op} are anti-isomorphic under LowerAdj_{W}.

(59) For every set W with a non-empty element holds CL_{W}^{op} and CL_{W} are anti-isomorphic under UpperAdj_{W}.

4. Compact Preserving Maps and Sup-semilattices Morphisms

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

(Def. 14) For every element s of S such that s is compact holds f(s) is compact.

One can prove the following propositions:

(60) Let S, T be complete lattices and d be a sups-preserving map from T into S. If d is waybelow-preserving, then d is compact-preserving.

(61) Let S, T be complete lattices and d be a sups-preserving map from T into S. Suppose T is algebraic and d is compact-preserving. Then d is waybelow-preserving.

(62) Let R, S, T be non empty relational structures, X be a subset of R, f be a map from R into S, and g be a map from S into T. Suppose f preserves sup of X and g preserves sup of f^{∘}X. Then g · f preserves sup of X.

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

(Def. 15) For every finite subset X of S holds f preserves sup of X.

We say that f is bottom-preserving if and only if: (Def. 16) f preserves sup of ∅_{S}.

Next we state the proposition

(63) Let R, S, T be non empty relational structures, f be a map from R into S, and g be a map from S into T. Suppose f is finite-sups-preserving and g is finite-sups-preserving. Then g · f is finite-sups-preserving.

Let S, T be non empty antisymmetric lower-bounded relational structures and let f be a map from S into T. Let us observe that f is bottom-preserving if and only if:

(Def. 17) f(⊥_{S}) = ⊥_{T} .

Let L be a non empty relational structure and let S be a relational substructure of L. We say that S is finite-sups-inheriting if and only if:

(Def. 18) For every finite subset X of S such that sup X exists in L holds ⨆_{L} X ∈

the carrier of S. We say that S is bottom-inheriting if and only if:

(Def. 19) ⊥_{L} ∈ the carrier of S.

Let S, T be non empty relational structures. Observe that every map from S into T which is sups-preserving is also bottom-preserving.

Let L be a lower-bounded antisymmetric non empty relational structure. Note that every relational substructure of L which is finite-sups-inheriting is also bottom-inheriting and join-inheriting.

Let L be a non empty relational structure. One can check that every relational substructure of L which is sups-inheriting is also finite-sups-inheriting.

Let S, T be lower-bounded non empty posets. One can verify that there exists a map from S into T which is sups-preserving.

Let L be a lower-bounded antisymmetric non empty relational structure. Observe that every full relational substructure of L which is bottom-inheriting is also non empty and lower-bounded.

Let L be a lower-bounded antisymmetric non empty relational structure. Note that there exists a relational substructure of L which is non empty, supsinheriting, finite-sups-inheriting, bottom-inheriting, and full.

Next we state the proposition

(64) Let L be a lower-bounded antisymmetric non empty relational structure and S be a non empty bottom-inheriting full relational substructure of L. Then ⊥_{S} = ⊥_{L}.

Let L be a lower-bounded non empty poset with l.u.b.’s. Note that every full relational substructure of L which is bottom-inheriting and join-inheriting is also finite-sups-inheriting.

Next we state two propositions:

(65) Let S, T be non empty relational structures and f be a map from S into T. Suppose f is finite-sups-preserving. Then f is join-preserving and bottom-preserving.

(66) Let S, T be lower-bounded posets with l.u.b.’s and f be a map from S into T. Suppose f is join-preserving and bottom-preserving. Then f is finite-sups-preserving.

Let S, T be non empty relational structures. One can check that every map from S into T which is sups-preserving is also finite-sups-preserving and every map from S into T which is finite-sups-preserving is also join-preserving and bottom-preserving.

Let S be a non empty relational structure and let T be a lower-bounded non empty reflexive antisymmetric relational structure. Observe that there exists a map from S into T which is sups-preserving and finite-sups-preserving.

Let L be a lower-bounded non empty poset. One can check that CompactSublatt(L) is lower-bounded.

One can prove the following propositions:

(67) Let S be a relational structure, T be a non empty relational structure, f be a map from S into T, S′be a relational substructure of S, and T′be a relational substructure of T. Suppose f^{∘}(the carrier of S′) ⊆ the carrier of T′. Then f↾the carrier of S′ is a map from S′ into T′.

(68) Let S, T be lattices, f be a join-preserving map from S into T, S′ be a non empty join-inheriting full relational substructure of S, T′ be a non empty join-inheriting full relational substructure of T, and g be a map from S′ into T′. If g = f↾the carrier of S′, then g is join-preserving.

(69) Let S, T be lower-bounded lattices, f be a finite-sups-preserving map from S into T, S′ be a non empty finite-sups-inheriting full relational substructure of S, T′ be a non empty finite-sups-inheriting full relational substructure of T, and g be a map from S′ into T′. If g = f↾the carrier of S′, then g is finite-sups-preserving.

Let L be a complete lattice. One can verify that CompactSublatt(L) is finitesups-inheriting.

Next we state two propositions:

(70) Let S, T be complete lattices and d be a sups-preserving map from T into S. Then d is compact-preserving if and only if d↾the carrier of CompactSublatt(T) is a finite-sups-preserving map from CompactSublatt(T) into CompactSublatt(S).

(71) Let S, T be complete lattices. Suppose T is algebraic. Let g be an infspreserving map from S into T. Then g is directed-sups-preserving if and only if (the lower adjoint of g)↾the carrier of CompactSublatt(T) is a finitesups-preserving map from CompactSublatt(T) into CompactSublatt(S).

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Yet Another Construction of Free Algebra

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Artur Korniłowicz^{1} University of Białystok

MML Identifier: MSAFREE3.

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

In this paper X, x, z are sets.

Let S be a non empty non void many sorted signature and let A be a non empty algebra over S. Observe that ⋃ (the sorts of A) is non empty.

Let S be a non empty non void many sorted signature and let A be a non empty algebra over S.

(Def. 1) An element of ⋃ (the sorts of A) is said to be an element of A. We now state two propositions:

(1) For every function f such that X ⊆ domf and f is one-to-one holds f^{−1} (f^{∘}X) = X.

(2) Let I be a set, A be a many sorted set indexed by I, and F be a many sorted function indexed by I. If F is “1-1” and A ⊆ dom_{κ} F (κ), then F^{−1} (F ^{∘} A) = A.

Let S be a non void signature and let X be a many sorted set indexed by the carrier of S. The functor Free_{S}(X) yields a strict algebra over S and is defined by:

(Def. 2) There exists a subset A of Free(X ∪ ((the carrier of S) ⟼→ {0})) such that Free_{S}(X) = Gen(A) and A = (Reverse(X ∪ ((the carrier of S) ⟼→ {0})))^{−1} (X).

We now state four propositions:

1This work has been partially supported by TYPES grant IST-1999-29001.

(3) Let S be a non void signature, X be a non-empty many sorted set indexed by the carrier of S, and s be a sort symbol of S. Then ⟨⟨x, s⟩⟩∈ the carrier of DTConMSA(X) if and only if x ∈ X(s).

(4) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, X be a many sorted set indexed by the carrier of S, and s be a sort symbol of S. Then x ∈ X(s) and x ∈ Y (s) if and only if the root tree of ⟨⟨x, s⟩⟩∈ ((Reverse(Y))^{−1} (X))(s).

(5) Let S be a non void signature, X be a many sorted set indexed by the carrier of S, and s be a sort symbol of S. If x ∈ X(s), then the root tree of ⟨⟨x, s⟩⟩∈ (the sorts of Free_{S}(X))(s).

(6) Let S be a non void signature, X be a many sorted set indexed by the carrier of S, and o be an operation symbol of S. Suppose Arity(o) = ∅. Then the root tree of ⟨⟨o, the carrier of S⟩⟩ ∈ (the sorts of Free_{S}(X))(the result sort of o).

Let S be a non void signature and let X be a non empty yielding many sorted set indexed by the carrier of S. Observe that Free_{S}(X) is non empty.

One can prove the following three propositions:

(7) Let S be a non void signature and X be a non-empty many sorted set indexed by the carrier of S. Then x is an element of Free(X) if and only if x is a term of S over X.

(8) Let S be a non void signature, X be a non-empty many sorted set indexed by the carrier of S, s be a sort symbol of S, and x be a term of S over X. Then x ∈ (the sorts of Free(X))(s) if and only if the sort of x = s.

(9) Let S be a non void signature and X be a non empty yielding many sorted set indexed by the carrier of S. Then every element of Free_{S}(X) is a term of S over X ∪ ((the carrier of S) ⟼→ {0}).

Let S be a non empty non void many sorted signature and let X be a non empty yielding many sorted set indexed by the carrier of S. Note that every element of Free_{S}(X) is relation-like and function-like.

Let S be a non empty non void many sorted signature and let X be a non empty yielding many sorted set indexed by the carrier of S. Note that every element of Free_{S}(X) is finite and decorated tree-like.

Let S be a non empty non void many sorted signature and let X be a non empty yielding many sorted set indexed by the carrier of S. Observe that every element of Free_{S}(X) is finite-branching.

One can check that every decorated tree is non empty. Let S be a many sorted signature and let t be a non empty binary relation.

The functor Var_{S} t yields a many sorted set indexed by the carrier of S and is defined as follows:

(Def. 3) For every set s such that s ∈ the carrier of S holds (Var_{S} t)(s) = {a_{1}; a

ranges over elements of rngt : a_{2} = s}. Let S be a many sorted signature, let X be a many sorted set indexed by

the carrier of S, and let t be a non empty binary relation. The functor Var_{X} t yielding a many sorted subset indexed by X is defined by:

(Def. 4) Var_{X} t = X ∩ Var_{S} t.

We now state several propositions:

(10) Let S be a many sorted signature, X be a many sorted set indexed by the carrier of S, t be a non empty binary relation, and V be a many sorted subset indexed by X. Then V = Var_{X} t if and only if for every set s such that s ∈ the carrier of S holds V (s) = X(s) ∩ {a_{1}; a ranges over elements of rngt : a_{2} = s}.

(11) Let S be a many sorted signature and s, x be sets. Then

(i) if s ∈ the carrier of S, then (Var_{S} (the root tree of ⟨⟨x, s⟩⟩))(s) = {x}, and

(ii) for every set s′such that s′ ≠ s or s ∉ the carrier of S holds (Var_{S} (the

root tree of ⟨⟨x, s⟩⟩))(s′) = ∅.

(12) Let S be a many sorted signature and s be a set. Suppose s ∈ the carrier of S. Let p be a decorated tree yielding finite sequence. Then x ∈ (Var_{S}(⟨⟨z, the carrier of S⟩⟩-tree(p)))(s) if and only if there exists a decorated tree t such that t ∈ rngp and x ∈ (Var_{S} t)(s).

(13) Let S be a many sorted signature, X be a many sorted set indexed by

the carrier of S, and s, x be sets. Then

(i) if x ∈ X(s), then (Var_{X} (the root tree of ⟨⟨x, s⟩⟩))(s) = {x}, and (ii) for every set s′ such that s′ ≠ s or x ∉ X(s) holds (Var_{X} (the root tree

of ⟨⟨x, s⟩⟩))(s′) = ∅.

(14) Let S be a many sorted signature, X be a many sorted set indexed by the carrier of S, and s be a set. Suppose s ∈ the carrier of S. Let p be a decorated tree yielding finite sequence. Then x ∈ (Var_{X}(⟨⟨z, the carrier of S⟩⟩-tree(p)))(s) if and only if there exists a decorated tree t such that t ∈ rngp and x ∈ (Var_{X} t)(s).

(15) Let S be a non void signature, X be a non-empty many sorted set indexed

by the carrier of S, and t be a term of S over X. Then Var_{S} t ⊆ X. Let S be a non void signature, let X be a non-empty many sorted set indexed

by the carrier of S, and let t be a term of S over X. The functor Var_{t} yielding a many sorted subset indexed by X is defined by:

(Def. 5) Var_{t} = Var_{S} t.

The following proposition is true

(16) Let S be a non void signature, X be a non-empty many sorted set indexed by the carrier of S, and t be a term of S over X. Then Var_{t} = Var_{X} t.

Let S be a non void signature, let Y be a non-empty many sorted set indexed by the carrier of S, and let X be a many sorted set indexed by the carrier of S. The functor S -Terms^{Y} (X) yielding a subset of Free(Y) is defined as follows:

(Def. 6) For every sort symbol s of S holds (S -Terms^{Y} (X))(s) = {t; t ranges over

terms of S over Y: the sort of t = s ∧ Var_{t} ⊆ X}. One can prove the following propositions:

(17) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, X be a many sorted set indexed by the carrier of S, and s be a sort symbol of S. If x ∈ (S -Terms^{Y} (X))(s), then x is a term of S over Y.

(18) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, X be a many sorted set indexed by the carrier of S, t be a term of S over Y, and s be a sort symbol of S. If t ∈ (S -Terms^{Y} (X))(s), then the sort of t = s and Var_{t} ⊆ X.

(19) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, X be a many sorted set indexed by the carrier of S, and s be a sort symbol of S. Then the root tree of ⟨⟨x, s⟩⟩∈ (S -Terms^{Y} (X))(s) if and only if x ∈ X(s) and x ∈ Y (s).

(20) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, X be a many sorted set indexed by the carrier of S, o be an operation symbol of S, and p be an argument sequence of Sym(o, Y). Then Sym(o, Y)-tree(p) ∈ (S -Terms^{Y} (X))(the result sort of o) if and only if rngp ⊆ ⋃(S -Terms^{Y} (X)).

(21) Let S be a non void signature, X be a non-empty many sorted set indexed by the carrier of S, and A be a subset of Free(X). Then A is operations closed if and only if for every operation symbol o of S and for every argument sequence p of Sym(o, X) such that rngp ⊆ ⋃ A holds Sym(o, X)-tree(p) ∈ A(the result sort of o).

(22) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, and X be a many sorted set indexed by the carrier of S. Then S -Terms^{Y} (X) is operations closed.

(23) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, and X be a many sorted set indexed by the carrier of S. Then (Reverse(Y))^{−1} (X) ⊆ S -Terms^{Y} (X).

(24) Let S be a non void signature, X be a many sorted set indexed by the carrier of S, t be a term of S over X ∪ ((the carrier of S) ⟼→ {0}), and s be a sort symbol of S. If t ∈ (S -Terms^{X∪((the carrier of S)⟼→{0})} (X))(s), then t ∈ (the sorts of Free_{S}(X))(s).

(25) Let S be a non void signature and X be a many sorted set indexed by the carrier of S. Then the sorts of Free_{S}(X) =

S -Terms^{X∪((the carrier of S)⟼→{0})} (X).

(26) Let S be a non void signature and X be a many sorted set indexed by the carrier of S. Then Free(X ∪ ((the carrier of S) ⟼→ {0}))↾(S -Terms^{X∪((the carrier of S)⟼→{0})} (X)) = Free_{S}(X).

(27) Let S be a non void signature, X, Y be non-empty many sorted sets indexed by the carrier of S, A be a subalgebra of Free(X), and B be a subalgebra of Free(Y). Suppose the sorts of A = the sorts of B. Then the algebra of A = the algebra of B.

(28) Let S be a non void signature, X be a non empty yielding many sorted set indexed by the carrier of S, Y be a many sorted set indexed by the carrier of S, and t be an element of Free_{S}(X). Then Var_{S} t ⊆ X.

(29) Let S be a non void signature, X be a non-empty many sorted set indexed by the carrier of S, and t be a term of S over X. Then Var_{t} ⊆ X.

(30) Let S be a non void signature, X, Y be non-empty many sorted sets indexed by the carrier of S, t_{1} be a term of S over X, and t_{2} be a term of S over Y. If t_{1} = t_{2}, then the sort of t_{1} = the sort of t_{2}.

(31) Let S be a non void signature, X, Y be non-empty many sorted sets indexed by the carrier of S, and t be a term of S over Y. If Var_{t} ⊆ X, then t is a term of S over X.

(32) Let S be a non void signature and X be a non-empty many sorted set indexed by the carrier of S. Then Free_{S}(X) = Free(X).

(33) Let S be a non void signature, Y be a non-empty many sorted set indexed by the carrier of S, t be a term of S over Y, and p be an element of domt. Then Var_{t↾p} ⊆ Var_{t}.

(34) Let S be a non void signature, X be a non empty yielding many sorted set indexed by the carrier of S, t be an element of Free_{S}(X), and p be an element of domt. Then t↾p is an element of Free_{S}(X).

(35) Let S be a non void signature, X be a non-empty many sorted set indexed by the carrier of S, t be a term of S over X, and a be an element of rngt. Then a = ⟨⟨a_{1}, a_{2} ⟩⟩.

(36) Let S be a non void signature, X be a non empty yielding many sorted set indexed by the carrier of S, t be an element of Free_{S}(X), and s be a sort symbol of S. Then

(i) if x ∈ (Var_{S} t)(s), then ⟨⟨x, s⟩⟩∈ rngt, and (ii) if ⟨⟨x, s⟩⟩∈ rngt, then x ∈ (Var_{S} t)(s) and x ∈ X(s).

(37) Let S be a non void signature and X be a many sorted set indexed by the carrier of S. Suppose that for every sort symbol s of S such that X(s) = ∅ there exists an operation symbol o of S such that the result sort of o = s and Arity(o) = ∅. Then Free_{S}(X) is non-empty.

(38) Let S be a non void signature, A be an algebra over S, B be a subalgebra 784 grzegorz bancerek and artur korniłowicz

of A, and o be an operation symbol of S. Then Args(o, B) ⊆ Args(o, A).

(39) For every non void signature S and for every feasible algebra A over S holds every subalgebra of A is feasible.

The following proposition is true

(40) Let S be a non void signature and X be a many sorted set indexed by the carrier of S. Then Free_{S}(X) is feasible and free.

References

[1] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[2] Grzegorz Bancerek. The reflection theorem. Formalized Mathematics, 1(5):973–977, 1990.

[3] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

[4] Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195–204, 1992.

[5] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77–82, 1993.

[6] Grzegorz Bancerek. Terms over many sorted universal algebra. Formalized Mathematics, 5(2):191–198, 1996.

[7] Grzegorz Bancerek. Translations, endomorphisms, and stable equational theories. Formalized Mathematics, 5(4):553–564, 1996.

[8] Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279–287, 1997.

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

[10] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91–101, 1993.

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

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

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

[14] Artur Korniłowicz. Extensions of mappings on generator set. Formalized Mathematics, 5(2):269–272, 1996.

[15] Artur Korniłowicz. Equations in many sorted algebras. Formalized Mathematics, 6(3):363–369, 1997.

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

[17] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

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

[19] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

[20] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67– 74, 1996.

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

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

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

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

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

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

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

yet another construction of free algebra 785

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

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

786 grzegorz bancerek and artur korniłowicz

Upper and Lower Sequence of a Cage^{1}

Robert Milewski University of Białystok

MML Identifier: JORDAN1E.

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

1. Preliminaries

In this paper n is a natural number. One can prove the following propositions:

(1) For every non empty subset X of E_{T}^{2} and for every compact subset Y of E_{T}^{2} such that X ⊆ Y holds N-boundX N-boundY.

(2) For every non empty subset X of E_{T}^{2} and for every compact subset Y of E_{T}^{2} such that X ⊆ Y holds E-boundX E-boundY.

(3) For every non empty subset X of E_{T}^{2} and for every compact subset Y of E_{T}^{2} such that X ⊆ Y holds S-boundX S-boundY.

(4) For every non empty subset X of E_{T}^{2} and for every compact subset Y of E_{T}^{2} such that X ⊆ Y holds W-boundX W-boundY.

(5) Let f, g be finite sequences of elements of E_{T}^{2}. Suppose f is in the area of g. Let p be an element of the carrier of E_{T}^{2}. If p ∈ rngf, then f − : p is in the area of g.

(6) Let f, g be finite sequences of elements of E_{T}^{2}. Suppose f is in the area of g. Let p be an element of the carrier of E_{T}^{2}. If p ∈ rngf, then f :− p is in the area of g.

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(7) For every non empty finite sequence f of elements of E_{T}^{2} and for every point p of E_{T}^{2} such that p ∈ L̃(f) holds ⇃ p, f ≠ ∅.

(8) Let f be a non empty finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If p ∈ L̃(f) and len⇂ f, p 2, then f(1) ∈ L̃(⇂ f, p).

(9) Let f be a non empty finite sequence of elements of E_{T}^{2}. Suppose f is a special sequence. Let p be a point of E_{T}^{2}. If p ∈ L̃(f), then f(1) ∉ L̃(mid(f, Index(p, f) + 1, lenf)).

(10) For all natural numbers i, j, m, n such that i + j = m + n and i m and j n holds i = m.

(11) Let f be a non empty finite sequence of elements of E_{T}^{2}. Suppose f is a special sequence. Let p be a point of E_{T}^{2}. If p ∈ L̃(f) and f(1) ∈ L̃(⇃ p, f), then f(1) = p.

2. About Upper and Lower Sequence of a Cage

Let C be a compact non vertical non horizontal subset of E_{T}^{2} and let n be a natural number. The functor UpperSeq(C, n) yielding a finite sequence of elements of E_{T}^{2} is defined as follows:

(Def. 1) UpperSeq(C, n) = ((Cage(C, n))_{⟲}^{W-min L̃(Cage(C,n))})− :E-max L̃(Cage(C, n)).

The following proposition is true

(12) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds lenUpperSeq(C, n) = (E-max L̃(Cage(C, n))) ↫ ((Cage(C, n))_{⟲}^{W-min L̃(Cage(C,n))}).

Let C be a compact non vertical non horizontal subset of E_{T}^{2} and let n be a natural number. The functor LowerSeq(C, n) yields a finite sequence of elements of E_{T}^{2} and is defined as follows:

(Def. 2) LowerSeq(C, n) = ((Cage(C, n))_{⟲}^{W-min L̃(Cage(C,n))}):− E-max L̃(Cage(C, n)).

Next we state the proposition

(13) Let C be a compact non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Then lenLowerSeq(C, n) = (len((Cage(C, n))_{⟲}^{W-min L̃(Cage(C,n))}) − (E-max L̃(Cage(C, n))) ↫

((Cage(C, n))_{⟲}^{W-min L̃(Cage(C,n))})) + 1. Let C be a compact non vertical non horizontal subset of E_{T}^{2} and let n be a

natural number. Note that UpperSeq(C, n) is non empty and LowerSeq(C, n) is non empty.

Let C be a compact non vertical non horizontal subset of E_{T}^{2} and let n be a natural number. Observe that UpperSeq(C, n) is one-to-one special unfolded and s.n.c. and LowerSeq(C, n) is one-to-one special unfolded and s.n.c..

The following propositions are true:

(14) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds lenUpperSeq(C, n) + lenLowerSeq(C, n) = lenCage(C, n) + 1.

(15) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds (Cage(C, n))_{⟲}^{W-min L̃(Cage(C,n))} = UpperSeq(C, n) ⌢⌢ LowerSeq(C, n).

(16) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds L̃(Cage(C, n)) = L̃(UpperSeq(C, n) ⌢⌢ LowerSeq(C, n)).

(17) For every compact non vertical non horizontal non empty subset C of E_{T}^{2} and for every natural number n holds L̃(Cage(C, n)) = L̃(UpperSeq(C, n)) ∪ L̃(LowerSeq(C, n)).

(18) For every simple closed curve P holds W-minP ≠ E-minP.

(19) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds lenUpperSeq(C, n) 3 and lenLowerSeq(C, n) 3.

Let C be a compact non vertical non horizontal subset of E_{T}^{2} and let n be a natural number. Observe that UpperSeq(C, n) is special sequence and LowerSeq(C, n) is special sequence.

Next we state several propositions:

(20) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds L̃(UpperSeq(C, n)) ∩ L̃(LowerSeq(C, n)) = {W-min L̃(Cage(C, n)), E-max L̃(Cage(C, n))}.

(21) For every compact non vertical non horizontal subset C of E_{T}^{2} holds UpperSeq(C, n) is in the area of Cage(C, n).

(22) For every compact non vertical non horizontal subset C of E_{T}^{2} holds LowerSeq(C, n) is in the area of Cage(C, n).

(23) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds ((Cage(C, n))_{2})_{2} = N-bound L̃(Cage(C, n)).

(24) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and k be a natural number. If 1 k and k + 1 lenCage(C, n) and (Cage(C, n))_{k} = E-max L̃(Cage(C, n)), then ((Cage(C, n))_{k+1})_{1} = E-bound L̃(Cage(C, n)).

(25) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and k be a natural number. If 1 k and k + 1 lenCage(C, n) and (Cage(C, n))_{k} = S-max L̃(Cage(C, n)), then ((Cage(C, n))_{k+1})_{2} = S-bound L̃(Cage(C, n)).

(26) Let C be a compact connected non vertical non horizontal subset of 790 robert milewski

E_{T}^{2} and k be a natural number. If 1 k and k + 1 lenCage(C, n) and (Cage(C, n))_{k} = W-min L̃(Cage(C, n)), then ((Cage(C, n))_{k+1})_{1} = W-bound L̃(Cage(C, n)).

References

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

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

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

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

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

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

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

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

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

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

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

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

[13] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Formalized Mathematics, 6(2):255–263, 1997.

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

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

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

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

[18] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

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

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

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

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

On Polynomials with Coefficients in a Ring of Polynomials

Barbara Dzienis University of Białystok

Summary. The main result of the paper is, that the ring of polynomials with o_{1} variables and coefficients in the ring of polynomials with o_{2} variables and coefficient in a ring L is isomorphic with the ring with o_{1} + o_{2} variables, and coefficients in L.

MML Identifier: POLYNOM6.

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

1. Preliminaries

In this paper o_{1}, o_{2} are ordinal numbers.

Let L_{1}, L_{2} be non empty double loop structures. Let us note that the predicate L_{1} is ring isomorphic to L_{2} is reflexive. We introduce L_{1} and L_{2} are isomorphic as a synonym of L_{1} is ring isomorphic to L_{2}.

We now state the proposition

(1) Let B be a set. Suppose that for every set x holds x ∈ B iff there exists an ordinal number o such that x = o_{1}+o and o ∈ o_{2}. Then o_{1}+o_{2} = o_{1}∪B.

Let o_{1} be an ordinal number and let o_{2} be a non empty ordinal number. Note that o_{1} + o_{2} is non empty and o_{2} + o_{1} is non empty.

One can prove the following proposition

(2) Let n be an ordinal number and a, b be bags of n. Suppose a < b. Then there exists an ordinal number o such that o ∈ n and a(o) < b(o) and for every ordinal number l such that l ∈ o holds a(l) = b(l).

2. About Bags

Let o_{1}, o_{2} be ordinal numbers, let a be an element of Bagso_{1}, and let b be an element of Bagso_{2}. The functor a + b yielding an element of Bags(o_{1} + o_{2}) is defined as follows:

(Def. 1) For every ordinal number o holds if o ∈ o_{1}, then (a + b)(o) = a(o) and if

o ∈ (o_{1} + o_{2}) \ o_{1}, then (a + b)(o) = b(o − o_{1}). One can prove the following propositions:

(3) For every element a of Bagso_{1} and for every element b of Bagso_{2} such that o_{2} = ∅ holds a + b = a.

(4) For every element a of Bagso_{1} and for every element b of Bagso_{2} such that o_{1} = ∅ holds a + b = b.

(5) For every element b_{1} of Bagso_{1} and for every element b_{2} of Bagso_{2} holds b_{1} +b_{2} = EmptyBag(o_{1} +o_{2}) iff b_{1} = EmptyBago_{1} and b_{2} = EmptyBago_{2}.

(6) For every element c of Bags(o_{1} +o_{2}) there exists an element c_{1} of Bagso_{1} and there exists an element c_{2} of Bagso_{2} such that c = c_{1} + c_{2}.

(7) For all elements b_{1}, c_{1} of Bagso_{1} and for all elements b_{2}, c_{2} of Bagso_{2} such that b_{1} + b_{2} = c_{1} + c_{2} holds b_{1} = c_{1} and b_{2} = c_{2}.

(8) Let n be an ordinal number, L be an Abelian add-associative right zeroed right complementable distributive associative non empty double loop structure, and p, q, r be serieses of n, L. Then (p + q) ∗ r = p ∗ r + q ∗ r.

3. Main Results

Let n be an ordinal number and let L be a right zeroed Abelian addassociative right complementable unital distributive associative non trivial non empty double loop structure. Observe that Polynom-Ring(n, L) is non trivial and distributive.

Let o_{1}, o_{2} be non empty ordinal numbers, let L be a right zeroed addassociative right complementable unital distributive non trivial non empty double loop structure, and let P be a polynomial of o_{1}, Polynom-Ring(o_{2}, L). The functor CompressP yields a polynomial of o_{1} + o_{2}, L and is defined by the condition (Def. 2).

(Def. 2) Let b be an element of Bags(o_{1} + o_{2}). Then there exists an element b_{1} of Bagso_{1} and there exists an element b_{2} of Bagso_{2} and there exists a polynomial Q_{1} of o_{2}, L such that Q_{1} = P (b_{1}) and b = b_{1} + b_{2} and (CompressP)(b) = Q_{1}(b_{2}).

Next we state several propositions:

(9) For all elements b_{1}, c_{1} of Bagso_{1} and for all elements b_{2}, c_{2} of Bagso_{2} such that b_{1} | c_{1} and b_{2} | c_{2} holds b_{1} + b_{2} | c_{1} + c_{2}.

(10) Let b be a bag of o_{1} + o_{2}, b_{1} be an element of Bagso_{1}, and b_{2} be an element of Bagso_{2}. Suppose b | b_{1} + b_{2}. Then there exists an element c_{1} of Bagso_{1} and there exists an element c_{2} of Bagso_{2} such that c_{1} | b_{1} and c_{2} | b_{2} and b = c_{1} + c_{2}.

(11) For all elements a_{1}, b_{1} of Bagso_{1} and for all elements a_{2}, b_{2} of Bagso_{2} holds a_{1} + a_{2} < b_{1} + b_{2} iff a_{1} < b_{1} or a_{1} = b_{1} and a_{2} < b_{2}.

(12) Let b_{1} be an element of Bagso_{1}, b_{2} be an element of Bagso_{2}, and G be

a finite sequence of elements of (Bags(o_{1} + o_{2}))^{∗}. Suppose that

(i) domG = Seglendivisorsb_{1}, and (ii) for every natural number i such that i ∈ Seglendivisorsb_{1} there exists

an element a′_{1} of Bagso_{1} and there exists a finite sequence F_{1} of elements of Bags(o_{1} + o_{2}) such that F_{1} = G_{i} and π_{i} divisorsb_{1} = a′_{1} and lenF_{1} = lendivisorsb_{2} and for every natural number m such that m ∈ domF_{1} there exists an element a′′_{1} of Bagso_{2} such that π_{m} divisorsb_{2} = a′′_{1} and π_{m}F_{1} = a′_{1} + a′′_{1}. Then divisors(b_{1} + b_{2}) = Flat(G).

(13) For all elements a_{1}, b_{1}, c_{1} of Bagso_{1} and for all elements a_{2}, b_{2}, c_{2} of Bagso_{2} such that c_{1} = b_{1}− ′a_{1} and c_{2} = b_{2}− ′a_{2} holds (b_{1}+b_{2})− ′(a_{1}+a_{2}) = c_{1} + c_{2}.

(14) Let b_{1} be an element of Bagso_{1}, b_{2} be an element of Bagso_{2}, and G be

a finite sequence of elements of ((Bags(o_{1} + o_{2}))^{2})^{∗}. Suppose that

(i) domG = Seglendecompb_{1}, and (ii) for every natural number i such that i ∈ Seglendecompb_{1} there exist

elements a′_{1}, b′_{1} of Bagso_{1} and there exists a finite sequence F_{1} of elements of (Bags(o_{1} + o_{2}))^{2} such that F_{1} = G_{i} and π_{i} decompb_{1} = ⟨a′_{1}, b′_{1}⟩ and lenF_{1} = lendecompb_{2} and for every natural number m such that m ∈ domF_{1} there exist elements a′′_{1}, b′′_{1} of Bagso_{2} such that π_{m} decompb_{2} = ⟨a′′_{1}, b′′_{1}⟩ and π_{m}F_{1} = ⟨a′_{1} + a′′_{1}, b′_{1} + b′′_{1}⟩. Then decomp(b_{1} + b_{2}) = Flat(G).

(15) Let o_{1}, o_{2} be non empty ordinal numbers and L be an Abelian right zeroed add-associative right complementable unital distributive associative well unital non trivial non empty double loop structure. Then Polynom-Ring(o_{1}, Polynom-Ring(o_{2}, L)) and Polynom-Ring(o_{1}+o_{2}, L) are isomorphic.

References

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

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

794 barbara dzienis

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

[4] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91–101, 1993.

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

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

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

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

[9] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339– 345, 1996.

[10] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.

[11] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[12] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

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

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

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

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

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

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

On Cosets in Segre’s Product of Partial Linear Spaces

Adam Naumowicz University of Białystok

Summary. This paper is a continuation of [12]. We prove that the family of cosets in the Segre’s product of partial linear spaces remains invariant under automorphisms.

MML Identifier: PENCIL 2.

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

1. Preliminaries on Finite Sequences

Let D be a set, let p be a finite sequence of elements of D, and let i, j be natural numbers. The functor Del(p, i, j) yields a finite sequence of elements of D and is defined by:

(Def. 1) Del(p, i, j) = (p↾(i − ′1))^{⌢} (p_{⇂j}). We now state several propositions:

(1) For every set D and for every finite sequence p of elements of D and for all natural numbers i, j holds rngDel(p, i, j) ⊆ rngp.

(2) Let D be a set, p be a finite sequence of elements of D, and i, j be natural numbers. If i ∈ domp and j ∈ domp, then lenDel(p, i, j) = ((lenp − j) + i) − 1.

(3) Let D be a set, p be a finite sequence of elements of D, and i, j be natural numbers. If i ∈ domp and j ∈ domp, then if lenDel(p, i, j) = 0, then i = 1 and j = lenp.

(4) Let D be a set, p be a finite sequence of elements of D, and i, j, k be natural numbers. If i ∈ domp and 1 k and k i − 1, then (Del(p, i, j))(k) = p(k).

(5) For all finite sequences p, q and for every natural number k such that lenp + 1 k holds (p^{⌢} q)(k) = q(k − lenp).

(6) Let D be a set, p be a finite sequence of elements of D, and i, j, k be natural numbers. Suppose i ∈ domp and j ∈ domp and i j and i k and k ((lenp − j) + i) − 1. Then (Del(p, i, j))(k) = p((j − ′i) + k + 1).

The scheme FinSeqOneToOne deals with sets A, B, C, a finite sequence D of elements of C, and a binary predicate P, and states that:

There exists an one-to-one finite sequence g of elements of C such that A = g(1) and B = g(leng) and rngg ⊆ rngD and for every natural number j such that 1 j and j < leng holds P[g(j), g(j+ 1)]

provided the following requirements are met:

• A = D(1) and B = D(lenD), and

• For every natural number i and for all sets d_{1}, d_{2} such that 1 i and i < lenD and d_{1} = D(i) and d_{2} = D(i + 1) holds P[d_{1}, d_{2}].

2. Segre Cosets

Next we state the proposition

(7) Let I be a non empty set, A be a 1-sorted yielding many sorted set indexed by I, L be a many sorted subset indexed by the support of A, i be an element of I, and S be a subset of the carrier of A(i). Then L+·(i, S) is a many sorted subset indexed by the support of A.

Let I be a non empty set and let A be a non-Trivial-yielding TopStructyielding many sorted set indexed by I. A subset of Segre ProductA is called a Segre-Coset of A if it satisfies the condition (Def. 2).

(Def. 2) There exists a Segre-like non trivial-yielding many sorted subset L indexed by the support of A such that it = ∏ L and L(index(L)) = Ω_{A(index(L))}.

The following proposition is true

(8) Let I be a non empty set, A be a non-Trivial-yielding TopStruct-yielding many sorted set indexed by I, and B_{1}, B_{2} be Segre-Cosets of A. If 2 ⊆ \overline{\overline{B_{1} ∩ B_{2}}} , then B_{1} = B_{2}.

Let S be a topological structure and let X, Y be subsets of the carrier of S. We say that X and Y are joinable if and only if the condition (Def. 3) is satisfied.

(Def. 3) There exists a finite sequence f of elements of 2^{the carrier of S} such that (i) X = f(1), (ii) Y = f(lenf), (iii) for every subset W of the carrier of S such that W ∈ rngf holds W is

closed under lines and strong, and (iv) for every natural number i such that 1 i and i < lenf holds 2 ⊆

\overline{\overline{f(i) ∩ f(i + 1)}} . One can prove the following three propositions:

(9) Let S be a topological structure and X, Y be subsets of the carrier of S. Suppose X and Y are joinable. Then there exists an one-to-one finite sequence f of elements of 2^{the carrier of S} such that

(i) X = f(1), (ii) Y = f(lenf), (iii) for every subset W of the carrier of S such that W ∈ rngf holds W is

closed under lines and strong, and (iv) for every natural number i such that 1 i and i < lenf holds 2 ⊆

\overline{\overline{f(i) ∩ f(i + 1)}} .

(10) Let S be a topological structure and X be a subset of the carrier of S. If X is closed under lines and strong, then X and X are joinable.

(11) Let I be a non empty set, A be a PLS-yielding many sorted set indexed

by I, and X, Y be subsets of the carrier of Segre ProductA. Suppose that (i) X is non trivial, closed under lines, and strong, (ii) Y is non trivial, closed under lines, and strong, and (iii) X and Y are joinable.

Let X_{1}, Y_{1} be Segre-like non trivial-yielding many sorted subsets indexed by the support of A. Suppose X = ∏ X_{1} and Y = ∏ Y_{1}. Then index(X_{1}) = index(Y_{1}) and for every set i such that i ≠ index(X_{1}) holds X_{1}(i) = Y_{1}(i).

3. Collineations of Segre Product

One can prove the following proposition

(12) Let S be a 1-sorted structure, T be a non empty 1-sorted structure, and

f be a map from S into T. If f is bijective, then f^{−1} is bijective. Let S, T be topological structures and let f be a map from S into T. We

say that f is isomorphic if and only if: (Def. 4) f is bijective and open and f^{−1} is bijective and open.

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

Let S be a non empty topological structure. A collineation of S is an isomorphic map from S into S.

Let S be a non empty non void topological structure, let f be a collineation of S, and let l be a block of S. Then f^{∘}l is a block of S.

Let S be a non empty non void topological structure, let f be a collineation of S, and let l be a block of S. Then f^{−1} (l) is a block of S.

Next we state a number of propositions:

(13) For every non empty topological structure S and for every collineation f of S holds f^{−1} is a collineation of S.

(14) Let S be a non empty topological structure, f be a collineation of S, and X be a subset of the carrier of S. If X is non trivial, then f^{∘}X is non trivial.

(15) Let S be a non empty topological structure, f be a collineation of S, and X be a subset of the carrier of S. If X is non trivial, then f^{−1} (X) is non trivial.

(16) Let S be a non empty non void topological structure, f be a collineation of S, and X be a subset of the carrier of S. If X is strong, then f^{∘}X is strong.

(17) Let S be a non empty non void topological structure, f be a collineation of S, and X be a subset of the carrier of S. If X is strong, then f^{−1} (X) is strong.

(18) Let S be a non empty non void topological structure, f be a collineation of S, and X be a subset of the carrier of S. If X is closed under lines, then f^{∘}X is closed under lines.

(19) Let S be a non empty non void topological structure, f be a collineation of S, and X be a subset of the carrier of S. If X is closed under lines, then f^{−1} (X) is closed under lines.

(20) Let S be a non empty non void topological structure, f be a collineation of S, and X, Y be subsets of the carrier of S. Suppose X is non trivial and Y is non trivial and X and Y are joinable. Then f^{∘}X and f^{∘}Y are joinable.

(21) Let S be a non empty non void topological structure, f be a collineation of S, and X, Y be subsets of the carrier of S. Suppose X is non trivial and Y is non trivial and X and Y are joinable. Then f^{−1} (X) and f^{−1} (Y) are joinable.

(22) Let I be a non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let W be a subset of the carrier of Segre ProductA. Suppose W is non trivial, strong, and closed under lines. Then ⋃{Y; Y ranges over subsets of the carrier of Segre ProductA : Y is non trivial, strong, and

on cosets in segre’s product of partial … 799

closed under lines ∧ W and Y are joinable} is a Segre-Coset of A.

(23) Let I be a non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let B be a set. Then B is a Segre-Coset of A if and only if there exists a subset W of the carrier of Segre ProductA such that W is non trivial, strong, and closed under lines and B = ⋃{Y; Y ranges over subsets of the carrier of Segre ProductA : Y is non trivial, strong, and closed under lines ∧ W and Y are joinable}.

(24) Let I be a non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let B be a Segre-Coset of A and f be a collineation of Segre ProductA. Then f^{∘}B is a Segre-Coset of A.

(25) Let I be a non empty set and A be a PLS-yielding many sorted set indexed by I. Suppose that for every element i of I holds A(i) is strongly connected. Let B be a Segre-Coset of A and f be a collineation of Segre ProductA. Then f^{−1} (B) is a Segre-Coset of A.

References

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

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

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

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

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

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

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

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

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

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

[11] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

[12] Adam Naumowicz. On Segre’s product of partial line spaces. Formalized Mathematics, 9(2):383–390, 2001.

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

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

[15] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

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

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

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

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

800 adam naumowicz

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

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

On the Simple Closed Curve Property of the Circle and the Fashoda Meet Theorem

Yatsuka Nakamura Shinshu University Nagano

Summary. First, we prove the fact that the circle is the simple closed curve, which was defined as a curve homeomorphic to the square. For this proof, we introduce a mapping which is a homeomorphism from 2-dimensional plane to itself. This mapping maps the square to the circle. Secondly, we prove the Fashoda meet theorem for the circle using this homeomorphism.

MML Identifier: JGRAPH 3.

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

1. Preliminaries

In this paper x, y, z, u, a are real numbers.

We now state a number of propositions:

(1) If x^{2} = y^{2}, then x = y or x = − y.

(2) If x^{2} = 1, then x = 1 or x = − 1.

(3) If 0 x and x 1, then x^{2} x.

(4) If a 0 and (x − a) · (x + a) 0, then − a x and x a.

(5) If x^{2} − 1 0, then − 1 x and x 1.

(6) x < y and x < z iff x < min(y, z).

(7) If 0 < x, then \frac{x}{3} < x and \frac{x}{4} < x.

(8) If x 1, then \sqrt{x} 1 and if x > 1, then \sqrt{x} > 1.

(9) If x y and z u, then [y, z] ⊆ [x, u].

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

(11) For every function f and for all sets B, C holds (f↾B)^{∘}C = f^{∘}(C ∩ B).

(12) Let X be a topological structure, Y be a non empty topological structure, f be a map from X into Y, and P be a subset of X. Then f↾P is a map from X↾P into Y.

(13) Let X, Y be non empty topological spaces, p_{0} be a point of X, D be a non empty subset of X, E be a non empty subset of Y, and f be a map from X into Y. Suppose that D^{c} = {p_{0}} and E^{c} = {f(p_{0})} and X is a T_{2} space and Y is a T_{2} space and for every point p of X↾D holds f(p) ≠ f(p_{0}) and there exists a map h from X↾D into Y ↾E such that h = f↾D and h is continuous and for every subset V of Y such that f(p_{0}) ∈ V and V is open there exists a subset W of X such that p_{0} ∈ W and W is open and f^{∘}W ⊆ V. Then f is continuous.

2. The Circle is a Simple Closed Curve

In the sequel p, q denote points of E_{T}^{2}.

The function SqCirc from the carrier of E_{T}^{2} into the carrier of E_{T}^{2} is defined by the condition (Def. 1).

(Def. 1) Let p be a point of E_{T}^{2}. Then (i) if p = 0_{E_{T}^{2}}, then SqCirc(p) = p, (ii) if p_{2} p_{1} and − p_{1} p_{2} or p_{2} p_{1} and p_{2} − p_{1} and if p ≠ 0_{E_{T}^{2}}, then

SqCirc(p) = [_{\frac{p_{1}}{\sqrt{1+( \frac{p_{2}}{p_{1}} )^{2}}}} , _{\frac{p_{2}}{\sqrt{1+( \frac{p_{2}}{p_{1}} )^{2}}}} ], and

(iii) if p_{2} p̸_{1} or − p_{1} p̸_{2} but p_{2} p̸_{1} or p_{2} −̸ p_{1} and p ≠ 0_{E_{T}^{2}}, then SqCirc(p) = [_{\frac{p_{1}}{\sqrt{1+( \frac{p_{1}}{p_{2}} )^{2}}}} , _{\frac{p_{2}}{\sqrt{1+( \frac{p_{1}}{p_{2}} )^{2}}}} ].

We now state a number of propositions:

(14) Let p be a point of E_{T}^{2} such that p ≠ 0_{E_{T}^{2}}. Then

(i) if p_{1} p_{2} and − p_{2} p_{1} or p_{1} p_{2} and p_{1} − p_{2}, then SqCirc(p) = [_{\frac{p_{1}}{\sqrt{1+( \frac{p_{1}}{p_{2}} )^{2}}}} , _{\frac{p_{2}}{\sqrt{1+( \frac{p_{1}}{p_{2}} )^{2}}}} ], and

(ii) if p_{1} p̸_{2} or − p_{2} p̸_{1} and if p_{1} p̸_{2} or p_{1} −̸ p_{2}, then SqCirc(p) = [_{\frac{p_{1}}{\sqrt{1+( \frac{p_{2}}{p_{1}} )^{2}}}} , _{\frac{p_{2}}{\sqrt{1+( \frac{p_{2}}{p_{1}} )^{2}}}} ].

(15) Let X be a non empty topological space and f_{1} be a map from X into ℝ^{1}. Suppose f_{1} is continuous and for every point q of X there exists a real number r such that f_{1}(q) = r and r 0. Then there exists a map g from X into ℝ^{1} such that for every point p of X and for every real number r_{1} such that f_{1}(p) = r_{1} holds g(p) = \sqrt{r_{1}} and g is continuous.

(16) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = (\frac{r_{1}}{r_{2}} )^{2}, and

(ii) g is continuous.

(17) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = 1 + (\frac{r_{1}}{r_{2}} )^{2}, and

(ii) g is continuous.

(18) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = \sqrt{1 + (\frac{r_{1}}{r_{2}} )^{2}}, and

(ii) g is continuous.

(19) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = _{\frac{r_{1}}{\sqrt{1+( \frac{r_{1}}{r_{2}} )^{2}}}} , and

(ii) g is continuous.

(20) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = _{\frac{r_{2}}{\sqrt{1+( \frac{r_{1}}{r_{2}} )^{2}}}} , and

(ii) g is continuous.

(21) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = _{\frac{p_{1}}{\sqrt{1+( \frac{p_{2}}{p_{1}} )^{2}}}} , and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{1} ≠ 0. Then f is continuous.

(22) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into ℝ^{1}. Suppose that (i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds

f(p) = _{\frac{p_{2}}{\sqrt{1+( \frac{p_{2}}{p_{1}} )^{2}}}} , and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{1} ≠ 0. Then f is continuous.

(23) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = _{\frac{p_{2}}{\sqrt{1+( \frac{p_{1}}{p_{2}} )^{2}}}} , and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{2} ≠ 0. Then f is continuous.

(24) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = _{\frac{p_{1}}{\sqrt{1+( \frac{p_{1}}{p_{2}} )^{2}}}} , and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{2} ≠ 0. Then f is continuous.

(25) Let K_{0}, B_{0} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous.

(26) Let K_{0}, B_{0} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous.

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

{p : P[p] ∧ p ≠ 0_{E_{T}^{2}}} ⊆ (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} for all values of the parameters.

The scheme TopInter concerns a unary predicate P, and states that: {p : P[p] ∧ p ≠ 0_{E_{T}^{2}}} = {p_{7}; p_{7} ranges over points of E_{T}^{2}: P[p_{7}]} ∩ ((the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}})

for all values of the parameters.

Next we state several propositions:

(27) Let B_{0} be a subset of E_{T}^{2}, K_{0} be a subset of (E_{T}^{2})↾B_{0}, and f be a map from (E_{T}^{2})↾B_{0}↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous and K_{0} is closed.

(28) Let B_{0} be a subset of E_{T}^{2}, K_{0} be a subset of (E_{T}^{2})↾B_{0}, and f be a map from (E_{T}^{2})↾B_{0}↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous and K_{0} is closed.

(29) Let D be a non empty subset of E_{T}^{2}. Suppose D^{c} = {0_{E_{T}^{2}}}. Then there exists a map h from (E_{T}^{2})↾D into (E_{T}^{2})↾D such that h = SqCirc↾D and h is continuous.

(30) For every non empty subset D of E_{T}^{2} such that D = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} holds D^{c} = {0_{E_{T}^{2}}}.

(31) There exists a map h from E_{T}^{2} into E_{T}^{2} such that h = SqCirc and h is continuous.

(32) SqCirc is one-to-one.

Let us observe that SqCirc is one-to-one. One can prove the following propositions:

(33) Let K_{2}, C_{1} be subsets of E_{T}^{2}. Suppose that

(i) K_{2} = {q : − 1 = q_{1} ∧ − 1 q_{2} ∧ q_{2} 1 ∨ q_{1} = 1 ∧ − 1 q_{2} ∧ q_{2} 1 ∨ − 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1 ∨ 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1}, and

(ii) C_{1} = {p_{2}; p_{2} ranges over points of E_{T}^{2}: |p_{2}| = 1}.

Then SqCirc^{∘} K_{2} = C_{1}.

(34) Let P, K_{2} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{2} into (E_{T}^{2})↾P.

Suppose that

(i) K_{2} = {q : − 1 = q_{1} ∧ − 1 q_{2} ∧ q_{2} 1 ∨ q_{1} = 1 ∧ − 1 q_{2} ∧ q_{2} 1 ∨ − 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1 ∨ 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1}, and

(ii) f is a homeomorphism.

Then P is a simple closed curve.

(35) Let K_{2} be a subset of E_{T}^{2}. Suppose K_{2} = {q : − 1 = q_{1} ∧ − 1 q_{2} ∧ q_{2} 1 ∨ q_{1} = 1 ∧ − 1 q_{2} ∧ q_{2} 1 ∨ − 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1 ∨ 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1}. Then K_{2} is a simple closed curve and compact.

(36) For every subset C_{1} of E_{T}^{2} such that C_{1} = {p; p ranges over points of E_{T}^{2}: |p| = 1} holds C_{1} is a simple closed curve.

3. The Fashoda Meet Theorem for the Circle

Next we state a number of propositions:

(37) Let K_{0}, C_{0} be subsets of E_{T}^{2}. Suppose K_{0} = {p : − 1 p_{1} ∧ p_{1} 1 ∧ − 1 p_{2} ∧ p_{2} 1} and C_{0} = {p_{1}; p_{1} ranges over points of E_{T}^{2}: |p_{1}| 1}. Then SqCirc^{−1} (C_{0}) ⊆ K_{0}.

(38) Let given p. Then (i) if p = 0_{E_{T}^{2}}, then SqCirc^{−1} (p) = 0_{E_{T}^{2}}, (ii) if p_{2} p_{1} and − p_{1} p_{2} or p_{2} p_{1} and p_{2} − p_{1} and if p ≠ 0_{E_{T}^{2}}, then

SqCirc^{−1} (p) = [p_{1} · \sqrt{1 + (\frac{p_{2}}{p_{1}} )^{2}}, p_{2} · \sqrt{1 + (\frac{p_{2}}{p_{1}} )^{2}}], and (iii) if p_{2} p̸_{1} or − p_{1} p̸_{2} but p_{2} p̸_{1} or p_{2} −̸ p_{1} and p ≠ 0_{E_{T}^{2}}, then

SqCirc^{−1} (p) = [p_{1} · \sqrt{1 + (\frac{p_{1}}{p_{2}} )^{2}}, p_{2} · \sqrt{1 + (\frac{p_{1}}{p_{2}} )^{2}}].

(39) SqCirc^{−1} is a map from E_{T}^{2} into E_{T}^{2}.

(40) Let p be a point of E_{T}^{2} such that p ≠ 0_{E_{T}^{2}}. Then

(i) if p_{1} p_{2} and − p_{2} p_{1} or p_{1} p_{2} and p_{1} − p_{2}, then SqCirc^{−1} (p) = [p_{1} · \sqrt{1 + (\frac{p_{1}}{p_{2}} )^{2}}, p_{2} · \sqrt{1 + (\frac{p_{1}}{p_{2}} )^{2}}], and

(ii) if p_{1} p̸_{2} or − p_{2} p̸_{1} and if p_{1} p̸_{2} or p_{1} −̸ p_{2}, then SqCirc^{−1} (p) = [p_{1} · \sqrt{1 + (\frac{p_{2}}{p_{1}} )^{2}}, p_{2} · \sqrt{1 + (\frac{p_{2}}{p_{1}} )^{2}}].

(41) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} · \sqrt{1 + (\frac{r_{1}}{r_{2}} )^{2}}, and

(ii) g is continuous.

(42) Let X be a non empty topological space and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous and for every point q of X holds f_{2}(q) ≠ 0. Then there exists a map g from X into ℝ^{1} such that

(i) for every point p of X and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{2} · \sqrt{1 + (\frac{r_{1}}{r_{2}} )^{2}}, and

(ii) g is continuous.

(43) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = p_{1} · \sqrt{1 + (\frac{p_{2}}{p_{1}} )^{2}}, and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{1} ≠ 0. Then f is continuous.

(44) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = p_{2} · \sqrt{1 + (\frac{p_{2}}{p_{1}} )^{2}}, and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{1} ≠ 0.

Then f is continuous.

(45) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = p_{2} · \sqrt{1 + (\frac{p_{1}}{p_{2}} )^{2}}, and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{2} ≠ 0. Then f is continuous.

(46) Let K_{1} be a non empty subset of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{1} into

ℝ^{1}. Suppose that

(i) for every point p of E_{T}^{2} such that p ∈ the carrier of (E_{T}^{2})↾K_{1} holds f(p) = p_{1} · \sqrt{1 + (\frac{p_{1}}{p_{2}} )^{2}}, and

(ii) for every point q of E_{T}^{2} such that q ∈ the carrier of (E_{T}^{2})↾K_{1} holds q_{2} ≠ 0. Then f is continuous.

(47) Let K_{0}, B_{0} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc^{−1} ↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous.

(48) Let K_{0}, B_{0} be subsets of E_{T}^{2} and f be a map from (E_{T}^{2})↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc^{−1} ↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous.

(49) Let B_{0} be a subset of E_{T}^{2}, K_{0} be a subset of (E_{T}^{2})↾B_{0}, and f be a map from (E_{T}^{2})↾B_{0}↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc^{−1} ↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{2} p_{1} ∧ − p_{1} p_{2} ∨ p_{2} p_{1} ∧ p_{2} − p_{1}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous and K_{0} is closed.

(50) Let B_{0} be a subset of E_{T}^{2}, K_{0} be a subset of (E_{T}^{2})↾B_{0}, and f be a map from (E_{T}^{2})↾B_{0}↾K_{0} into (E_{T}^{2})↾B_{0}. Suppose f = SqCirc^{−1} ↾K_{0} and B_{0} = (the carrier of E_{T}^{2}) \ {0_{E_{T}^{2}}} and K_{0} = {p : (p_{1} p_{2} ∧ − p_{2} p_{1} ∨ p_{1} p_{2} ∧ p_{1} − p_{2}) ∧ p ≠ 0_{E_{T}^{2}}}. Then f is continuous and K_{0} is closed.

(51) Let D be a non empty subset of E_{T}^{2}. Suppose D^{c} = {0_{E_{T}^{2}}}. Then there exists a map h from (E_{T}^{2})↾D into (E_{T}^{2})↾D such that h = SqCirc^{−1} ↾D and h is continuous.

(52) There exists a map h from E_{T}^{2} into E_{T}^{2} such that h = SqCirc^{−1} and h is continuous.

(54)^{1}(i) SqCirc is a map from E_{T}^{2} into E_{T}^{2}, (ii) rngSqCirc = the carrier of E_{T}^{2}, and 808 yatsuka nakamura

1The proposition (53) has been removed.

(iii) for every map f from E_{T}^{2} into E_{T}^{2} such that f = SqCirc holds f is a homeomorphism.

(55) Let f, g be maps from 𝕀 into E_{T}^{2}, C_{0}, K_{3}, K_{4}, K_{5}, K_{6} be subsets of E_{T}^{2}, and O, I be points of 𝕀. Suppose that O = 0 and I = 1 and f is continuous and one-to-one and g is continuous and one-to-one and C_{0} = {p : |p| 1} and K_{3} = {q_{1}; q_{1} ranges over points of E_{T}^{2}: |q_{1}| = 1 ∧ (q_{1})_{2} (q_{1})_{1} ∧ (q_{1})_{2} − (q_{1})_{1}} and K_{4} = {q_{2}; q_{2} ranges over points of E_{T}^{2}: |q_{2}| = 1 ∧ (q_{2})_{2} (q_{2})_{1} ∧ (q_{2})_{2} − (q_{2})_{1}} and K_{5} = {q_{3}; q_{3} ranges over points of E_{T}^{2}: |q_{3}| = 1 ∧ (q_{3})_{2} (q_{3})_{1} ∧ (q_{3})_{2} − (q_{3})_{1}} and K_{6} = {q_{4}; q_{4} ranges over points of E_{T}^{2}: |q_{4}| = 1 ∧ (q_{4})_{2} (q_{4})_{1} ∧ (q_{4})_{2} − (q_{4})_{1}} and f(O) ∈ K_{4} and f(I) ∈ K_{3} and g(O) ∈ K_{6} and g(I) ∈ K_{5} and rngf ⊆ C_{0} and rngg ⊆ C_{0}. Then rngf ∩ rngg ≠ ∅.

References

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Pythagorean Triples

Freek Wiedijk University of Nijmegen

Summary. A Pythagorean triple is a set of positive integers {a, b, c} with a^{2} + b^{2} = c^{2}. We prove that every Pythagorean triple is of the form

a = n^{2} − m^{2}  b = 2mn  c = n^{2} + m^{2}

or is a multiple of such a triple. Using this characterization we show that for every n > 2 there exists a Pythagorean triple X with n ∈ X. Also we show that even the set of simplified Pythagorean triples is infinite.

MML Identifier: PYTHTRIP.

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

1. Relative Primeness

We follow the rules: a, b, c, k, m, n are natural numbers and i is an integer.

Let us consider m, n. Let us observe that m and n are relative prime if and only if:

(Def. 1) For every k such that k | m and k | n holds k = 1.

Let us consider m, n. Let us observe that m and n are relative prime if and only if:

(Def. 2) For every prime natural number p holds p ∤ m or p ∤ n.

2. Squares

Let n be a number. We say that n is square if and only if: (Def. 3) There exists m such that n = m^{2}.

Let us observe that every number which is square is also natural. Let n be a natural number. Observe that n^{2} is square. Let us observe that there exists a natural number which is even and square. One can check that there exists a natural number which is odd and square. One can check that there exists a number which is even and square. One can check that there exists a number which is odd and square. Let m, n be square numbers. Observe that m · n is square. We now state the proposition

(1) If m · n is square and m and n are relative prime, then m is square and n is square.

Let i be an even integer. Observe that i^{2} is even. Let i be an odd integer. Observe that i^{2} is odd. Next we state three propositions:

(2) i is even iff i^{2} is even.

(3) If i is even, then i^{2} mod 4 = 0.

(4) If i is odd, then i^{2} mod 4 = 1. Let m, n be odd square numbers. Note that m + n is non square. One can prove the following two propositions:

(5) If m^{2} = n^{2}, then m = n.

(6) m | n iff m^{2} | n^{2}.

3. Distributive Law for HCF

We now state two propositions:

(7) m | n or k = 0 iff k · m | k · n.

(8) gcd(k · m, k · n) = k · gcd(m, n).

4. Unbounded Sets are Infinite

We now state the proposition

(9) For every set X such that for every m there exists n such that n m and n ∈ X holds X is infinite.

5. Pythagorean Triples

We now state three propositions:

(10) If a and b are relative prime, then a is odd or b is odd.

(11) Suppose a^{2} + b^{2} = c^{2} and a and b are relative prime and a is odd. Then there exist m, n such that m n and a = n^{2} − m^{2} and b = 2 · m · n and c = n^{2} + m^{2}.

(12) If a = n^{2} − m^{2} and b = 2 · m · n and c = n^{2} + m^{2}, then a^{2} + b^{2} = c^{2}.

A subset of ℕ is called a Pythagorean triple if: (Def. 4) There exist a, b, c such that a^{2} + b^{2} = c^{2} and it = {a, b, c}.

In the sequel X is a Pythagorean triple. Let us note that every Pythagorean triple is finite. Let us note that the Pythagorean triple can be characterized by the following

(equivalent) condition: (Def. 5) There exist k, m, n such that m n and it = {k · (n^{2} − m^{2}), k · (2 · m ·

n), k · (n^{2} + m^{2})}. Let us consider X. We say that X is degenerate if and only if:

(Def. 6) 0 ∈ X.

We now state the proposition

(13) If n > 2, then there exists X such that X is non degenerate and n ∈ X.

Let us consider X. We say that X is simplified if and only if: (Def. 7) For every k such that for every n such that n ∈ X holds k | n holds

k = 1. Let us consider X. Let us observe that X is simplified if and only if:

(Def. 8) There exist m, n such that m ∈ X and n ∈ X and m and n are relative

prime. One can prove the following proposition

(14) If n > 0, then there exists X such that X is non degenerate and simplified

and 4 · n ∈ X. Let us note that there exists a Pythagorean triple which is non degenerate

and simplified.

The following propositions are true:

(15) {3, 4, 5} is a non degenerate simplified Pythagorean triple.

(16) {X : X is non degenerate ∧ X is simplified} is infinite.

812 freek wiedijk

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[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] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

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

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

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

Some Remarks on Finite Sequences on Go-boards^{1}

Adam Naumowicz University of Białystok

Summary. This paper shows some properties of finite sequences on Goboards. It also provides the partial correspondence between two ways of decomposition of curves induced by cages.

MML Identifier: JORDAN1F.

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

We follow the rules: i, j, k, n denote natural numbers, f denotes a finite sequence of elements of the carrier of E_{T}^{2}, and G denotes a Go-board.

We now state several propositions:

(1) Suppose that

(i) f is a sequence which elements belong to G, (ii) L(G ∘ (i, j), G ∘ (i, k)) meets L̃(f), (iii) ⟨⟨i, j⟩⟩∈ the indices of G, (iv) ⟨⟨i, k⟩⟩∈ the indices of G, and

(v) j k. Then there exists n such that j n and n k and (G ∘ (i, n))_{2} = inf(proj2^{∘}(L(G ∘ (i, j), G ∘ (i, k)) ∩ L̃(f))).

(2) Suppose that

(i) f is a sequence which elements belong to G, (ii) L(G ∘ (i, j), G ∘ (i, k)) meets L̃(f), (iii) ⟨⟨i, j⟩⟩∈ the indices of G, (iv) ⟨⟨i, k⟩⟩∈ the indices of G, and (v) j k.

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

Then there exists n such that j n and n k and (G ∘ (i, n))_{2} = sup(proj2^{∘}(L(G ∘ (i, j), G ∘ (i, k)) ∩ L̃(f))).

(3) Suppose that

(i) f is a sequence which elements belong to G, (ii) L(G ∘ (j, i), G ∘ (k, i)) meets L̃(f), (iii) ⟨⟨j, i⟩⟩∈ the indices of G, (iv) ⟨⟨k, i⟩⟩∈ the indices of G, and

(v) j k. Then there exists n such that j n and n k and (G ∘ (n, i))_{1} = inf(proj1^{∘}(L(G ∘ (j, i), G ∘ (k, i)) ∩ L̃(f))).

(4) Suppose that

(i) f is a sequence which elements belong to G, (ii) L(G ∘ (j, i), G ∘ (k, i)) meets L̃(f), (iii) ⟨⟨j, i⟩⟩∈ the indices of G, (iv) ⟨⟨k, i⟩⟩∈ the indices of G, and

(v) j k. Then there exists n such that j n and n k and (G ∘ (n, i))_{1} = sup(proj1^{∘}(L(G ∘ (j, i), G ∘ (k, i)) ∩ L̃(f))).

(5) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds (UpperSeq(C, n))_{1} = W-min L̃(Cage(C, n)).

(6) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds (LowerSeq(C, n))_{1} = E-max L̃(Cage(C, n)).

(7) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds (UpperSeq(C, n))_{len UpperSeq (C,n)} = E-max L̃(Cage(C, n)).

(8) For every compact non vertical non horizontal subset C of E_{T}^{2} and for every natural number n holds (LowerSeq(C, n))_{len LowerSeq (C,n)} = W-min L̃(Cage(C, n)).

(9) Let C be a compact non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Then L̃(UpperSeq(C, n)) = UpperArc L̃(Cage(C, n)) and L̃(LowerSeq(C, n)) = LowerArc L̃(Cage(C, n)) or L̃(UpperSeq(C, n)) = LowerArc L̃(Cage(C, n)) and L̃(LowerSeq(C, n)) = UpperArc L̃(Cage(C, n)).

We adopt the following convention: C is a compact non vertical non horizontal non empty subset of E_{T}^{2} satisfying conditions of simple closed curve, p is a point of E_{T}^{2}, and i_{1}, j_{1}, i_{2}, j_{2} are natural numbers.

Next we state four propositions:

(10) Let C be a connected compact non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Then UpperSeq(C, n) is a sequence which elements belong to Gauge(C, n).

some remarks on finite sequences on go-boards 815

(11) Let f be a finite sequence of elements of E_{T}^{2}. Suppose that (i) f is a sequence which elements belong to G, (ii) there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and p = G ∘ (i, j),

and (iii) for all i_{1}, j_{1}, i_{2}, j_{2} such that ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G and ⟨⟨i_{2}, j_{2}⟩⟩∈ the

indices of G and p = G∘(i_{1}, j_{1}) and f_{1} = G∘(i_{2}, j_{2}) holds |i_{2}−i_{1}|+|j_{2}−j_{1}| = 1. Then ⟨p⟩^{⌢} f is a sequence which elements belong to G.

(12) Let C be a connected compact non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Then LowerSeq(C, n) is a sequence which elements belong to Gauge(C, n).

(13) Suppose p_{1} = \frac{W-bound C+E-bound C}{2} and p_{2} = inf(proj2^{∘}(L(Gauge(C, 1) ∘ (CenterGauge(C, 1), 1), Gauge(C, 1) ∘ (CenterGauge(C, 1), widthGauge (C, 1))) ∩ UpperArc L̃(Cage(C, i + 1)))). Then there exists j such that 1 j and j widthGauge(C, i + 1) and p = Gauge(C, i + 1) ∘ (CenterGauge(C, i + 1), j).

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Upper and Lower Sequence on the Cage. Part II^{1}

Robert Milewski University of Białystok

MML Identifier: JORDAN1G.

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

In this paper n is a natural number. Let us note that there exists a finite sequence which is trivial. The following proposition is true

(1) For every trivial finite sequence f holds f is empty or there exists a set x such that f = ⟨x⟩.

Let p be a non trivial finite sequence. Observe that Rev(p) is non trivial. We now state four propositions:

(2) Let D be a non empty set, f be a finite sequence of elements of D, G be a matrix over D, and p be a set. Suppose f is a sequence which elements belong to G. Then f − : p is a sequence which elements belong to G.

(3) Let D be a non empty set, f be a finite sequence of elements of D, G be a matrix over D, and p be an element of D. Suppose p ∈ rngf. Suppose f is a sequence which elements belong to G. Then f:− p is a sequence which elements belong to G.

(4) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2}. Then UpperSeq(C, n) is a sequence which elements belong to Gauge(C, n).

(5) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2}. Then LowerSeq(C, n) is a sequence which elements belong to Gauge(C, n).

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and let n be a natural number. Note that UpperSeq(C, n) is standard and LowerSeq(C, n) is standard.

One can prove the following propositions:

(6) Let G be a column Y-constant line Y-increasing matrix over E_{T}^{2} and i_{1}, i_{2}, j_{1}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G. If (G ∘ (i_{1}, j_{1}))_{2} = (G ∘ (i_{2}, j_{2}))_{2}, then j_{1} = j_{2}.

(7) Let G be a line X-constant column X-increasing matrix over E_{T}^{2} and i_{1}, i_{2}, j_{1}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G. If (G ∘ (i_{1}, j_{1}))_{1} = (G ∘ (i_{2}, j_{2}))_{1}, then i_{1} = i_{2}.

(8) For every non trivial finite sequence f of elements of E_{T}^{2} holds N-min L̃(f) ∈ rngf.

(9) For every non trivial finite sequence f of elements of E_{T}^{2} holds N-max L̃(f) ∈ rngf.

(10) For every non trivial finite sequence f of elements of E_{T}^{2} holds E-min L̃(f) ∈ rngf.

(11) For every non trivial finite sequence f of elements of E_{T}^{2} holds E-max L̃(f) ∈ rngf.

(12) For every non trivial finite sequence f of elements of E_{T}^{2} holds S-min L̃(f) ∈ rngf.

(13) For every non trivial finite sequence f of elements of E_{T}^{2} holds S-max L̃(f) ∈ rngf.

(14) For every non trivial finite sequence f of elements of E_{T}^{2} holds W-min L̃(f) ∈ rngf.

(15) For every non trivial finite sequence f of elements of E_{T}^{2} holds W-max L̃(f) ∈ rngf.

(16) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ N-min L̃(f) and f_{len f} ≠ N-min L̃(f) or f_{1} ≠ N-max L̃(f) and f_{len f} ≠ N-max L̃(f), then (N-min L̃(f))_{1} < (N-max L̃(f))_{1}.

(17) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ N-min L̃(f) and f_{len f} ≠ N-min L̃(f) or f_{1} ≠ N-max L̃(f) and f_{len f} ≠ N-max L̃(f), then N-min L̃(f) ≠ N-max L̃(f).

(18) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ S-min L̃(f) and f_{len f} ≠ S-min L̃(f) or f_{1} ≠ S-max L̃(f) and f_{len f} ≠ S-max L̃(f), then (S-min L̃(f))_{1} < (S-max L̃(f))_{1}.

(19) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ S-min L̃(f) and f_{len f} ≠ S-min L̃(f) or f_{1} ≠

S-max L̃(f) and f_{len f} ≠ S-max L̃(f), then S-min L̃(f) ≠ S-max L̃(f).

(20) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ W-min L̃(f) and f_{len f} ≠ W-min L̃(f) or f_{1} ≠ W-max L̃(f) and f_{len f} ≠ W-max L̃(f), then (W-min L̃(f))_{2} < (W-max L̃(f))_{2}.

(21) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ W-min L̃(f) and f_{len f} ≠ W-min L̃(f) or f_{1} ≠ W-max L̃(f) and f_{len f} ≠ W-max L̃(f), then W-min L̃(f) ≠ W-max L̃(f).

(22) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ E-min L̃(f) and f_{len f} ≠ E-min L̃(f) or f_{1} ≠ E-max L̃(f) and f_{len f} ≠ E-max L̃(f), then (E-min L̃(f))_{2} < (E-max L̃(f))_{2}.

(23) Let f be a standard special unfolded non trivial finite sequence of elements of E_{T}^{2}. If f_{1} ≠ E-min L̃(f) and f_{len f} ≠ E-min L̃(f) or f_{1} ≠ E-max L̃(f) and f_{len f} ≠ E-max L̃(f), then E-min L̃(f) ≠ E-max L̃(f).

(24) Let D be a non empty set, f be a finite sequence of elements of D, and p, q be elements of D. If p ∈ rngf and q ∈ rngf and q ↫ f p ↫ f, then (f − : p) :− q = (f :− q) − : p.

(25) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Then L̃(Cage(C, n) − : W-min L̃(Cage(C, n))) ∩ L̃(Cage(C, n) :− W-min L̃(Cage(C, n))) = {N-min L̃(Cage(C, n)), W-min L̃(Cage(C, n))}.

(26) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds LowerSeq(C, n) = ((Cage(C, n))_{⟲}^{E-max L̃(Cage(C,n))}) − : W-min L̃(Cage(C, n)).

(27) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (W-min L̃(Cage(C, n))) ↫ UpperSeq(C, n) = 1.

(28) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (W-min L̃(Cage(C, n))) ↫ UpperSeq(C, n) < (W-max L̃(Cage(C, n))) ↫ UpperSeq(C, n).

(29) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (W-max L̃(Cage(C, n))) ↫ UpperSeq(C, n) (N-min L̃(Cage(C, n))) ↫ UpperSeq(C, n).

(30) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (N-min L̃(Cage(C, n))) ↫ UpperSeq(C, n) < (N-max L̃(Cage(C, n))) ↫ UpperSeq(C, n).

(31) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (N-max L̃(Cage(C, n))) ↫ UpperSeq(C, n) (E-max L̃(Cage(C, n))) ↫ UpperSeq(C, n).

(32) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (E-max L̃(Cage(C, n))) ↫ UpperSeq(C, n) = lenUpperSeq(C, n).

(33) For every compact non vertical non horizontal subset C of E_{T}^{2} holds (E-max L̃(Cage(C, n))) ↫ LowerSeq(C, n) = 1.

(34) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds (E-max L̃(Cage(C, n))) ↫ LowerSeq(C, n) < (E-min L̃(Cage(C, n))) ↫ LowerSeq(C, n).

(35) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds (E-min L̃(Cage(C, n))) ↫ LowerSeq(C, n) (S-max L̃(Cage(C, n))) ↫ LowerSeq(C, n).

(36) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds (S-max L̃(Cage(C, n))) ↫ LowerSeq(C, n) < (S-min L̃(Cage(C, n))) ↫ LowerSeq(C, n).

(37) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds (S-min L̃(Cage(C, n))) ↫ LowerSeq(C, n) (W-min L̃(Cage(C, n))) ↫ LowerSeq(C, n).

(38) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds (W-min L̃(Cage(C, n))) ↫ LowerSeq(C, n) = lenLowerSeq(C, n).

(39) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds ((UpperSeq(C, n))_{2})_{1} = W-bound L̃(Cage(C, n)).

(40) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds ((LowerSeq(C, n))_{2})_{1} = E-bound L̃(Cage(C, n)).

(41) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds W-bound L̃(Cage(C, n)) + E-bound L̃(Cage(C, n)) = W-boundC + E-boundC.

(42) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds S-bound L̃(Cage(C, n))+N-bound L̃(Cage(C, n)) = S-boundC + N-boundC.

(43) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n, i be natural numbers. If 1 i and i widthGauge(C, n) and n > 0, then (Gauge(C, n) ∘ (CenterGauge(C, n), i))_{1} = \frac{W-bound C+E-bound C}{2} .

(44) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n, i be natural numbers. If 1 i and i lenGauge(C, n) and n > 0, then (Gauge(C, n) ∘ (i, CenterGauge(C, n)))_{2} = \frac{S-bound C+N-bound C}{2} .

(45) Let f be a S-sequence in ℝ^{2} and k_{1}, k_{2} be natural numbers. If 1 k_{1} and k_{1} lenf and 1 k_{2} and k_{2} lenf and f_{1} ∈ L̃(mid(f, k_{1}, k_{2})), then k_{1} = 1 or k_{2} = 1.

(46) Let f be a S-sequence in ℝ^{2} and k_{1}, k_{2} be natural numbers. If 1 k_{1} and k_{1} lenf and 1 k_{2} and k_{2} lenf and f_{len f} ∈ L̃(mid(f, k_{1}, k_{2})), then k_{1} = lenf or k_{2} = lenf.

(47) Let C be a compact non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Then rngUpperSeq(C, n) ⊆ rngCage(C, n) and rngLowerSeq(C, n) ⊆ rngCage(C, n).

(48) For every compact non vertical non horizontal subset C of E_{T}^{2} holds UpperSeq(C, n) is a h.c. for Cage(C, n).

(49) For every compact non vertical non horizontal subset C of E_{T}^{2} holds Rev(LowerSeq(C, n)) is a h.c. for Cage(C, n).

(50) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and i be a natural number. If 1 < i and i lenGauge(C, n), then Gauge(C, n) ∘ (i, 1) ∉ rngUpperSeq(C, n).

(51) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and i be a natural number. If 1 i and i < lenGauge(C, n), then Gauge(C, n) ∘ (i, widthGauge(C, n)) ∉ rngLowerSeq(C, n).

(52) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and i be a natural number. If 1 < i and i lenGauge(C, n), then Gauge(C, n) ∘ (i, 1) ∉ L̃(UpperSeq(C, n)).

(53) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and i be a natural number. If 1 i and i < lenGauge(C, n), then Gauge(C, n) ∘ (i, widthGauge(C, n)) ∉ L̃(LowerSeq(C, n)).

(54) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and i, j be natural numbers. Suppose 1 i and i lenGauge(C, n) and 1 j and j widthGauge(C, n) and Gauge(C, n)∘(i, j) ∈ L̃(Cage(C, n)). Then L(Gauge(C, n)∘(i, 1), Gauge(C, n)∘(i, j)) meets L̃(LowerSeq(C, n)).

(55) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. If n > 0, then FPoint(L̃(UpperSeq(C, n)), W-min L̃(Cage(C, n)), E-max L̃(Cage(C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}) ∈ rngUpperSeq(C, n).

(56) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. If n > 0, then LPoint(L̃(LowerSeq(C, n)), E-max L̃(Cage(C, n)), W-min L̃(Cage(C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}) ∈ rngLowerSeq(C, n).

(57) For every S-sequence f in ℝ^{2} and for every point p of E_{T}^{2} such that p ∈ rngf holds ⇂ f, p = mid(f, 1, p ↫ f).

(58) Let f be a S-sequence in ℝ^{2} and Q be a closed subset of E_{T}^{2}. Suppose L̃(f) meets Q and f_{1} ∉ Q and FPoint(L̃(f), f_{1}, f_{len f}, Q) ∈ rngf. Then L̃(mid(f, 1, (FPoint(L̃(f), f_{1}, f_{len f}, Q)) ↫ f)) ∩ Q = {FPoint(L̃(f), f_{1}, f_{len f}, Q)}.

(59) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Suppose n > 0.

Let k be a natural number. Suppose 1 k and k < (FPoint(L̃(UpperSeq(C, n)), W-min L̃(Cage(C, n)), E-max L̃(Cage(C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2})) ↫ UpperSeq(C, n). Then ((UpperSeq(C, n))_{k})_{1} < \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}.

(60) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Suppose n > 0. Let k be a natural number. Suppose 1 k and k < (FPoint(L̃(Rev(LowerSeq(C, n))), W-min L̃(Cage(C, n)), E-max L̃(Cage (C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2})) ↫ Rev(LowerSeq(C, n)). Then ((Rev(LowerSeq(C, n)))_{k})_{1} < \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}.

(61) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Suppose n > 0. Let q be a point of E_{T}^{2}. Suppose q ∈ rngmid(UpperSeq(C, n), 2, (FPoint(L̃(UpperSeq(C, n)), W-min L̃(Cage (C, n)), E-max L̃(Cage(C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2})) ↫ UpperSeq(C, n)). Then q_{1} \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}.

(62) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Suppose n > 0. Then (FPoint(L̃(UpperSeq(C, n)), W-min L̃(Cage(C, n)), E-max L̃(Cage(C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}))_{2} > (LPoint(L̃ (LowerSeq(C, n)), E-max L̃(Cage(C, n)), W-min L̃(Cage(C, n)), VerticalLine \frac{W-bound L̃(Cage(C,n))+E-bound L̃(Cage(C,n))}{2}))_{2}.

(63) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. If n > 0, then L̃(UpperSeq(C, n)) = UpperArc L̃(Cage(C, n)).

(64) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. If n > 0, then L̃(LowerSeq(C, n)) = LowerArc L̃(Cage(C, n)).

(65) Let C be a compact connected non vertical non horizontal subset of E_{T}^{2} and n be a natural number. Suppose n > 0. Let i, j be natural numbers. Suppose 1 i and i lenGauge(C, n) and 1 j and j widthGauge(C, n) and Gauge(C, n) ∘ (i, j) ∈ L̃(Cage(C, n)). Then L(Gauge(C, n)∘(i, 1), Gauge(C, n)∘(i, j)) meets LowerArc L̃(Cage(C, n)).

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.

upper and lower sequence on the cage. part … 823

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

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

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

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

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

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

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

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

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

[12] Adam Grabowski and Yatsuka Nakamura. The ordering of points on a curve. Part II. Formalized Mathematics, 6(4):467–473, 1997.

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

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

[15] Artur Korniłowicz, Robert Milewski, Adam Naumowicz, and Andrzej Trybulec. Gauges and cages. Part I. Formalized Mathematics, 9(3):501–509, 2001.

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

[17] Robert Milewski. Upper and lower sequence of a cage. Formalized Mathematics, 9(4):787– 790, 2001.

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

[19] Yatsuka Nakamura and 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] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.

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

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

[26] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

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

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

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

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

Received September 28, 2001 824 robert milewski

Zero-Based Finite Sequences

Tetsuya Tsunetou Kyushu University Grzegorz Bancerek Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

MML Identifier: AFINSQ 1.

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

We follow the rules: k, n are natural numbers, x, y, z, y_{1}, y_{2}, X are sets, and f is a function.

One can prove the following propositions:

(1) n ∈ n + 1.

(2) If k n, then k = k ∩ n.

(3) If k = k ∩ n, then k n.

(4) n ∪ {n} = n + 1.

(5) Segn ⊆ n + 1.

(6) n + 1 = {0} ∪ Segn.

(7) For every function r holds r is finite and transfinite sequence-like iff there exists n such that domr = n.

Let us mention that there exists a function which is finite and transfinite sequence-like.

A finite 0-sequence is a finite transfinite sequence. In the sequel p, q, r denote finite 0-sequences. Observe that every set which is natural is also finite. Let us consider p. One

can verify that domp is natural.

Let us consider p. Then \overline{\overline{p}} is a natural number and it can be characterized by the condition:

(Def. 1) \overline{\overline{p}} = domp.

We introduce lenp as a synonym of \overline{\overline{p}}.

Let us consider p. Then domp is a subset of ℕ. Next we state the proposition (8) If there exists k such that domf ⊆ k, then there exists p such that f ⊆ p. In this article we present several logical schemes. The scheme XSeqEx deals

with a natural number A and a binary predicate P, and states that: There exists p such that domp = A and for every k such that k ∈ A holds P[k, p(k)]

provided the following conditions are satisfied:

• For all k, y_{1}, y_{2} such that k ∈ A and P[k, y_{1}] and P[k, y_{2}] holds y_{1} = y_{2}, and

• For every k such that k ∈ A there exists x such that P[k, x]. The scheme SeqLambda deals with a natural number A and a unary functor

F yielding a set, and states that:

There exists a finite 0-sequence p such that lenp = A and for every k such that k ∈ A holds p(k) = F(k)

for all values of the parameters.

Next we state several propositions:

(9) If z ∈ p, then there exists k such that k ∈ domp and z = ⟨⟨k, p(k)⟩⟩.

(10) If domp = domq and for every k such that k ∈ domp holds p(k) = q(k), then p = q.

(11) If lenp = lenq and for every k such that k < lenp holds p(k) = q(k), then p = q.

(12) p↾n is a finite 0-sequence.

(13) If rngp ⊆ domf, then f · p is a finite 0-sequence.

(14) If k < lenp and q = p↾k, then lenq = k and domq = k.

Let D be a set. Observe that there exists a transfinite sequence of elements of D which is finite.

Let D be a set. A finite 0-sequence of D is a finite transfinite sequence of elements of D.

We now state the proposition

(15) For every set D holds every finite 0-sequence of D is a partial function

from ℕ to D. One can verify that ∅ is transfinite sequence-like. Let D be a set. Observe that there exists a partial function from ℕ to D

which is finite and transfinite sequence-like.

In the sequel D is a set. Next we state two propositions:

(16) For every finite 0-sequence p of D holds p↾k is a finite 0-sequence of D.

(17) For every non empty set D there exists a finite 0-sequence p of D such

that lenp = k. One can verify that there exists a finite 0-sequence which is empty. One can prove the following propositions:

(18) lenp = 0 iff p = ∅.

(19) For every set D holds ∅ is a finite 0-sequence of D.

Let D be a set. One can verify that there exists a finite 0-sequence of D which is empty.

Let us consider x. The functor ⟨_{0}x⟩ yielding a set is defined as follows: (Def. 2) ⟨_{0}x⟩ = {⟨⟨0, x⟩⟩}.

Let D be a set. The functor ⟨⟩_{D} yields an empty finite 0-sequence of D and is defined by:

(Def. 3) ⟨⟩_{D} = ∅.

Let us consider p, q. Observe that p^{⌢} q is finite. Then p^{⌢} q can be characterized by the condition:

(Def. 4) dom(p ^{⌢} q) = lenp + lenq and for every k such that k ∈ domp holds (p^{⌢} q)(k) = p(k) and for every k such that k ∈ domq holds (p^{⌢} q)(lenp + k) = q(k).

The following propositions are true:

(20) len(p^{⌢} q) = lenp + lenq.

(21) If lenp k and k < lenp + lenq, then (p^{⌢} q)(k) = q(k − lenp).

(22) If lenp k and k < len(p^{⌢} q), then (p^{⌢} q)(k) = q(k − lenp).

(23) If k ∈ dom(p^{⌢} q), then k ∈ domp or there exists n such that n ∈ domq and k = lenp + n.

(24) For all transfinite sequences p, q holds domp ⊆ dom(p^{⌢} q).

(25) If x ∈ domq, then there exists k such that k = x and lenp + k ∈ dom(p^{⌢} q).

(26) If k ∈ domq, then lenp + k ∈ dom(p^{⌢} q).

(27) rngp ⊆ rng(p^{⌢} q).

(28) rngq ⊆ rng(p^{⌢} q).

(29) rng(p^{⌢} q) = rngp ∪ rngq.

(30) (p^{⌢} q)^{⌢} r = p^{⌢} (q^{⌢} r).

(31) If p^{⌢} r = q^{⌢} r or r^{⌢} p = r^{⌢} q, then p = q.

(32) p^{⌢} ∅ = p and ∅^{⌢} p = p.

(33) If p^{⌢} q = ∅, then p = ∅ and q = ∅.

Let D be a set and let p, q be finite 0-sequences of D. Then p ^{⌢} q is a transfinite sequence of elements of D.

Let us consider x. Then ⟨_{0}x⟩ is a function and it can be characterized by the condition:

(Def. 5) dom⟨_{0}x⟩ = 1 and ⟨_{0}x⟩(0) = x.

Let us consider x. One can verify that ⟨_{0}x⟩ is function-like and relation-like.

Let us consider x. One can check that ⟨_{0}x⟩ is finite and transfinite sequencelike.

One can prove the following proposition

(34) Suppose p^{⌢} q is a finite 0-sequence of D. Then p is a finite 0-sequence

of D and q is a finite 0-sequence of D. Let us consider x, y. The functor ⟨_{0}x, y⟩ yielding a set is defined by:

(Def. 6) ⟨_{0}x, y⟩ = ⟨_{0}x⟩^{⌢} ⟨_{0}y⟩.

Let us consider z. The functor ⟨_{0}x, y, z⟩ yields a set and is defined by: (Def. 7) ⟨_{0}x, y, z⟩ = ⟨_{0}x⟩^{⌢} ⟨_{0}y⟩^{⌢} ⟨_{0}z⟩.

Let us consider x, y. One can check that ⟨_{0}x, y⟩ is function-like and relationlike. Let us consider z. One can verify that ⟨_{0}x, y, z⟩ is function-like and relationlike.

Let us consider x, y. One can check that ⟨_{0}x, y⟩ is finite and transfinite sequence-like. Let us consider z. Observe that ⟨_{0}x, y, z⟩ is finite and transfinite sequence-like.

One can prove the following propositions:

(35) ⟨_{0}x⟩ = {⟨⟨0, x⟩⟩}.

(36) p = ⟨_{0}x⟩ iff domp = 1 and rngp = {x}.

(37) p = ⟨_{0}x⟩ iff lenp = 1 and rngp = {x}.

(38) p = ⟨_{0}x⟩ iff lenp = 1 and p(0) = x.

(39) (⟨_{0}x⟩^{⌢} p)(0) = x.

(40) (p^{⌢} ⟨_{0}x⟩)(lenp) = x.

(41) ⟨_{0}x, y, z⟩ = ⟨_{0}x⟩^{⌢} ⟨_{0}y, z⟩ and ⟨_{0}x, y, z⟩ = ⟨_{0}x, y⟩^{⌢} ⟨_{0}z⟩.

(42) p = ⟨_{0}x, y⟩ iff lenp = 2 and p(0) = x and p(1) = y.

(43) p = ⟨_{0}x, y, z⟩ iff lenp = 3 and p(0) = x and p(1) = y and p(2) = z.

(44) If p ≠ ∅, then there exist q, x such that p = q^{⌢} ⟨_{0}x⟩.

Let D be a non empty set and let x be an element of D. Then ⟨_{0}x⟩ is a finite 0-sequence of D.

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

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

• P[∅], and

• For all p, x such that P[p] holds P[p^{⌢} ⟨_{0}x⟩].

We now state the proposition

(45) For all finite 0-sequences p, q, r, s such that p^{⌢}q = r^{⌢}s and lenp lenr

there exists a finite 0-sequence t such that p^{⌢} t = r. Let D be a set. The functor D^{ω} yields a set and is defined as follows:

(Def. 8) x ∈ D^{ω} iff x is a finite 0-sequence of D.

zero-based finite sequences 829

Let D be a set. One can check that D^{ω} is non empty. One can prove the following propositions:

(46) x ∈ D^{ω} iff x is a finite 0-sequence of D.

(47) ∅ ∈ D^{ω}.

The scheme SepSeq deals with a non empty set A and a unary predicate P, and states that:

There exists X such that for every x holds x ∈ X iff there exists p such that p ∈ A^{ω} and P[p] and x = p

for all values of the parameters.

Let p be a finite 0-sequence and let i, x be sets. Note that p +·(i, x) is finite and transfinite sequence-like. We introduce Replace(p, i, x) as a synonym of p +·(i, x).

One can prove the following proposition

(48) Let p be a finite 0-sequence, i be a natural number, and x be a set. Then lenReplace(p, i, x) = lenp and if i < lenp, then (Replace(p, i, x))(i) = x and for every natural number j such that j ≠ i holds (Replace(p, i, x))(j) = p(j).

Let D be a non empty set, let p be a finite 0-sequence of D, let i be a natural number, and let a be an element of D. Then Replace(p, i, a) is a finite 0-sequence of D.

References

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

[2] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711–714, 1990.

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

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

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

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

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

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

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

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

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

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

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

Received September 28, 2001 830 tetsuya tsunetou et al.

More on the External Approximation of a Continuum^{1}

Andrzej Trybulec University of Białystok

Summary. The main goal was to prove two facts:

• the gauge is the Go-board of a corresponding cage,

• the left components of the complement of the curve determined by a cage

are monotonic w.r.t. the index of the approximation. Some auxiliary facts are proved, too. At the end new notions needed for internal approximation are defined and some useful lemmas are proved.

MML Identifier: JORDAN1H.

The terminology and notation used in this paper have been introduced in the following articles: [28], [40], [1], [3], [12], [29], [14], [4], [5], [37], [33], [13], [6], [20], [21], [26], [32], [9], [35], [24], [18], [27], [25], [8], [11], [17], [2], [36], [38], [30], [10], [16], [41], [43], [42], [19], [23], [34], [39], [31], [15], [44], [22], and [7].

1. Preliminaries

For simplicity, we follow the rules: m, k, j, j_{1}, i, i_{1}, i_{2}, n are natural numbers, r, s, r_{1}, t are real numbers, C, D are compact non vertical non horizontal non empty subsets of E_{T}^{2}, f is a finite sequence of elements of the carrier of E_{T}^{2}, G is a Go-board, and p is a point of E_{T}^{2}.

We now state three propositions:

(1) For all sets A, x, y such that A meets {x, y} holds x ∈ A or y ∈ A.

(2) If r < 0 and r_{1} r and 0 t, then \frac{t}{r} \frac{t}{r_{1}} .

(3) For every set X and for every binary relation R such that R is reflexive in X holds X ⊆ fieldR.

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

Let us observe that there exists a set which has a non-empty element. Let D be a non empty set with a non-empty element. Observe that there

exists a finite sequence of elements of D^{∗} which is non empty and non-empty.

Let D be a non empty set with non empty elements. One can check that there exists a finite sequence of elements of D^{∗} which is non empty and non-empty.

Let F be a non-empty function yielding function. Note that rng_{κ} F (κ) is non-empty.

Let us note that every finite sequence of elements of ℝ which is increasing is also one-to-one.

One can prove the following propositions:

(4) For all points p, q of E_{T}^{2} holds L(p, q) \ {p, q} is convex.

(5) For all points p, q of E_{T}^{2} holds L(p, q) \ {p, q} is connected.

(6) For all points p, q of E_{T}^{2} such that p ≠ q holds p ∈ \overline{L(p, q) \ \{p, q\}}.

(7) For all points p, q of E_{T}^{2} such that p ≠ q holds \overline{L(p, q) \ \{p, q\}} = L(p, q).

(8) Let S be a subset of the carrier of E_{T}^{2} and p, q be points of E_{T}^{2}. If p ≠ q and L(p, q) \ \{p, q\} ⊆ S, then L(p, q) ⊆ \overline{S}.

2. Transforming Finite Sets to Finite Sequences

The binary relation RealOrd on ℝ is defined by: (Def. 1) RealOrd = {⟨⟨r, s⟩⟩: r s}.

Next we state two propositions:

(9) If ⟨⟨r, s⟩⟩∈ RealOrd, then r s.

(10) fieldRealOrd = ℝ.

Let us note that RealOrd is ordering and linear-order. The following propositions are true:

(11) RealOrd linearly orders ℝ.

(12) For every finite subset A of ℝ holds SgmX(RealOrd, A) is increasing.

(13) For every finite sequence f of elements of ℝ and for every finite subset

A of ℝ such that A = rngf holds SgmX(RealOrd, A) = Inc(f). Let A be a finite subset of ℝ. One can verify that SgmX(RealOrd, A) is

increasing.

Next we state two propositions:

(14) Let X be a non empty set, A be a finite subset of X, and R be an order in X. If R linearly orders A, then lenSgmX(R, A) = cardA.

(15) For every non empty set X and for every finite subset A of X and for every linear-order order R in X holds lenSgmX(R, A) = cardA.

3. On the Construction of Go-boards

Next we state two propositions:

(16) For every finite sequence f of elements of E_{T}^{2} holds X-coordinate(f) = proj1·f.

(17) For every finite sequence f of elements of E_{T}^{2} holds Y-coordinate(f) =

proj2·f. Let D be a non empty set and let M be a finite sequence of elements of D^{∗}.

Then ValuesM is a subset of D.

Let D be a non empty set with non empty elements and let M be a non empty non-empty finite sequence of elements of D^{∗}. One can verify that ValuesM is non empty.

The following propositions are true:

(18) For every non empty set D and for every matrix M over D and for every i such that i ∈ SegwidthM holds rng(M_{□,i}) ⊆ ValuesM.

(19) For every non empty set D and for every matrix M over D and for every i such that i ∈ domM holds rngLine(M, i) ⊆ ValuesM.

(20) For every column X-increasing non empty yielding matrix G over E_{T}^{2} holds lenG card(proj1^{∘} ValuesG).

(21) For every line X-constant matrix G over E_{T}^{2} holds card(proj1^{∘} ValuesG) lenG.

(22) For every line X-constant column X-increasing non empty yielding matrix G over E_{T}^{2} holds lenG = card(proj1^{∘} ValuesG).

(23) For every line Y-increasing non empty yielding matrix G over E_{T}^{2} holds widthG card(proj2^{∘} ValuesG).

(24) For every column Y-constant non empty yielding matrix G over E_{T}^{2} holds card(proj2^{∘} ValuesG) widthG.

(25) For every column Y-constant line Y-increasing non empty yielding matrix G over E_{T}^{2} holds widthG = card(proj2^{∘} ValuesG).

4. More about Go-boards

Next we state several propositions:

(26) For every standard special circular sequence f such that 1 k and k + 1 lenf holds L(f, k) ⊆ leftcell(f, k).

(27) For every standard special circular sequence f such that 1 k and k + 1 lenf holds left cell(f, k, the Go-board of f) = leftcell(f, k).

(28) For every standard special circular sequence f such that 1 k and k + 1 lenf holds L(f, k) ⊆ rightcell(f, k).

(29) For every standard special circular sequence f such that 1 k and k + 1 lenf holds right cell(f, k, the Go-board of f) = rightcell(f, k).

(30) Let P be a subset of E_{T}^{2} and f be a non constant standard special circular sequence. If P is a component of (L̃(f))^{c}, then P = RightComp(f) or P = LeftComp(f).

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

(32) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers and G be a Go-board. Suppose ⟨⟨i_{1}, j_{1}⟩⟩ ∈ the indices of G and ⟨⟨i_{2}, j_{2}⟩⟩ ∈ the indices of G and G ∘ (i_{1}, j_{1}) = G ∘ (i_{2}, j_{2}). Then i_{1} = i_{2} and j_{1} = j_{2}.

(33) Let f be a finite sequence of elements of E_{T}^{2} and M be a Go-board. Suppose f is a sequence which elements belong to M. Then mid(f, i_{1}, i_{2}) is a sequence which elements belong to M.

Let us mention that every Go-board is non empty and non-empty. The following propositions are true:

(34) For every Go-board G such that 1 i and i lenG holds (SgmX(RealOrd, proj1^{∘} ValuesG))(i) = (G ∘ (i, 1))_{1}.

(35) For every Go-board G such that 1 j and j widthG holds (SgmX(RealOrd, proj2^{∘} ValuesG))(j) = (G ∘ (1, j))_{2}.

(36) Let f be a non empty finite sequence of elements of E_{T}^{2} and G be a

Go-board. Suppose that

(i) f is a sequence which elements belong to G, (ii) there exists i such that ⟨⟨1, i⟩⟩∈ the indices of G and G ∘ (1, i) ∈ rngf,

and (iii) there exists i such that ⟨⟨lenG, i⟩⟩∈ the indices of G and G∘(lenG, i) ∈

rngf. Then proj1^{∘} rngf = proj1^{∘} ValuesG.

(37) Let f be a non empty finite sequence of elements of E_{T}^{2} and G be a

Go-board. Suppose that

(i) f is a sequence which elements belong to G, (ii) there exists i such that ⟨⟨i, 1⟩⟩∈ the indices of G and G ∘ (i, 1) ∈ rngf,

and (iii) there exists i such that ⟨⟨i, widthG⟩⟩ ∈ the indices of G and G ∘

(i, widthG) ∈ rngf. Then proj2^{∘} rngf = proj2^{∘} ValuesG.

Let G be a Go-board. Observe that ValuesG is non empty. One can prove the following three propositions:

(38) For every Go-board G holds G = the Go-board of SgmX(RealOrd,

proj1^{∘} ValuesG), SgmX(RealOrd, proj2^{∘} ValuesG).

(39) Let f be a non empty finite sequence of elements of E_{T}^{2} and G be a Go-board. If proj1^{∘} rngf = proj1^{∘} ValuesG and proj2^{∘} rngf = proj2^{∘} ValuesG, then G = the Go-board of f.

(40) Let f be a non empty finite sequence of elements of E_{T}^{2} and G be a

Go-board. Suppose that

(i) f is a sequence which elements belong to G, (ii) there exists i such that ⟨⟨1, i⟩⟩∈ the indices of G and G ∘ (1, i) ∈ rngf, (iii) there exists i such that ⟨⟨i, 1⟩⟩∈ the indices of G and G ∘ (i, 1) ∈ rngf, (iv) there exists i such that ⟨⟨lenG, i⟩⟩∈ the indices of G and G∘(lenG, i) ∈

rngf, and

(v) there exists i such that ⟨⟨i, widthG⟩⟩ ∈ the indices of G and G ∘ (i, widthG) ∈ rngf. Then G = the Go-board of f.

5. More about Gauges

The following propositions are true:

(41) If m n and 1 i and i + 1 lenGauge(C, n), then ⌊\frac{i−2}{2^{n− ′m}} + 2⌋ is a natural number.

(42) If m n and 1 i and i + 1 lenGauge(C, n), then 1 ⌊\frac{i−2}{2^{n− ′m}} + 2⌋ and ⌊\frac{i−2}{2^{n− ′m}} + 2⌋+ 1 lenGauge(C, m).

(43) Suppose m n and 1 i and i+1 lenGauge(C, n) and 1 j and j + 1 widthGauge(C, n). Then there exist i_{1}, j_{1} such that i_{1} = ⌊\frac{i−2}{2^{n− ′m}} + 2⌋ and j_{1} = ⌊\frac{j−2}{2^{n− ′m}} +2⌋and cell(Gauge(C, n), i, j) ⊆ cell(Gauge(C, m), i_{1}, j_{1}).

(44) Suppose m n and 1 i and i + 1 lenGauge(C, n) and 1 j and j + 1 widthGauge(C, n). Then there exist i_{1}, j_{1} such that 1 i_{1} and i_{1} + 1 lenGauge(C, m) and 1 j_{1} and j_{1} + 1 widthGauge(C, m) and cell(Gauge(C, n), i, j) ⊆ cell(Gauge(C, m), i_{1}, j_{1}).

(45) If i lenGauge(C, n), then cell(Gauge(C, n), i, 0) ⊆ UBDC.

(46) If i lenGauge(C, n), then cell(Gauge(C, n), i, widthGauge(C, n)) ⊆ UBDC.

(47) For every subset P of E_{T}^{2} such that P is Bounded holds UBDP is not Bounded.

(48) Let f be a non constant standard special circular sequence. If f_{⟲}^{p} is clockwise oriented, then f is clockwise oriented.

(49) For every non constant standard special circular sequence f such that LeftComp(f) = UBD L̃(f) holds f is clockwise oriented.

6. More about Cages

The following propositions are true:

(50) \overline{LeftComp(Cage(C, i))}^{c} = RightComp(Cage(C, i)).

(51) If C is connected, then the Go-board of Cage(C, n) = Gauge(C, n).

(52) If C is connected, then N-minC ∈ rightcell(Cage(C, n), 1).

(53) If C is connected and i j, then L̃(Cage(C, j)) ⊆ \overline{RightComp(Cage(C, i))}.

(54) If C is connected and i j, then LeftComp(Cage(C, i)) ⊆ LeftComp(Cage(C, j)).

(55) If C is connected and i j, then RightComp(Cage(C, j)) ⊆ RightComp(Cage(C, i)).

7. Preparing the Internal Approximation

Let us consider C, n. The functor X-SpanStart(C, n) yielding a natural number is defined as follows:

(Def. 2) X-SpanStart(C, n) = 2^{n− ′1} + 2.

Next we state three propositions:

(56) X-SpanStart(C, n) = CenterGauge(C, n).

(57) 2 < X-SpanStart(C, n) and X-SpanStart(C, n) < lenGauge(C, n).

(58) 1 X-SpanStart(C, n) − ′ 1 and X-SpanStart(C, n) − ′ 1 <

lenGauge(C, n). Let us consider C, n. We say that n is sufficiently large for C if and only if:

(Def. 3) There exists j such that j < widthGauge(C, n) and cell(Gauge(C, n),

X-SpanStart(C, n) − ′1, j) ⊆ BDDC. One can prove the following propositions:

(59) If n is sufficiently large for C, then n 1.

(60) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{1} be natural numbers. Suppose that

(i) left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{1}), (iv) ⟨⟨i_{1}, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1}, j_{1} + 1). Then ⟨⟨i_{1} − ′1, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n).

(61) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{1} be natural numbers. Suppose that

(i) left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{1}), (iv) ⟨⟨i_{1} + 1, j_{1}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1} + 1, j_{1}). Then ⟨⟨i_{1} + 1, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n).

(62) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let j_{1}, i_{2} be natural numbers. Suppose that

(i) left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{2} + 1, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{2} + 1, j_{1}), (iv) ⟨⟨i_{2}, j_{1}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{2}, j_{1}). Then ⟨⟨i_{2}, j_{1} − ′1⟩⟩∈ the indices of Gauge(C, n).

(63) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{2} be natural numbers. Suppose that

(i) left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{2} + 1⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{2} + 1), (iv) ⟨⟨i_{1}, j_{2}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1}, j_{2}). Then ⟨⟨i_{1} + 1, j_{2}⟩⟩∈ the indices of Gauge(C, n).

(64) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{1} be natural numbers. Suppose that

(i) front left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{1}), (iv) ⟨⟨i_{1}, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1}, j_{1} + 1). Then ⟨⟨i_{1}, j_{1} + 2⟩⟩∈ the indices of Gauge(C, n).

(65) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{1} be natural numbers. Suppose that

(i) front left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{1}), (iv) ⟨⟨i_{1} + 1, j_{1}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1} + 1, j_{1}). Then ⟨⟨i_{1} + 2, j_{1}⟩⟩∈ the indices of Gauge(C, n).

(66) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let j_{1}, i_{2} be natural numbers. Suppose that

(i) front left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{2} + 1, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{2} + 1, j_{1}), (iv) ⟨⟨i_{2}, j_{1}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{2}, j_{1}). Then ⟨⟨i_{2} − ′1, j_{1}⟩⟩∈ the indices of Gauge(C, n).

(67) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{2} be natural numbers. Suppose that

(i) front left cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{2} + 1⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{2} + 1), (iv) ⟨⟨i_{1}, j_{2}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1}, j_{2}). Then ⟨⟨i_{1}, j_{2} − ′1⟩⟩∈ the indices of Gauge(C, n).

(68) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{1} be natural numbers. Suppose that

(i) front right cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{1}), (iv) ⟨⟨i_{1}, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1}, j_{1} + 1). Then ⟨⟨i_{1} + 1, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n).

(69) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{1} be natural numbers. Suppose that

(i) front right cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{1}), (iv) ⟨⟨i_{1} + 1, j_{1}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1} + 1, j_{1}). Then ⟨⟨i_{1} + 1, j_{1} − ′1⟩⟩∈ the indices of Gauge(C, n).

(70) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let j_{1}, i_{2} be natural numbers. Suppose that

(i) front right cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{2} + 1, j_{1}⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{2} + 1, j_{1}), (iv) ⟨⟨i_{2}, j_{1}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{2}, j_{1}). Then ⟨⟨i_{2}, j_{1} + 1⟩⟩∈ the indices of Gauge(C, n).

(71) Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2}, given n, and f be a finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to Gauge(C, n) and lenf > 1. Let i_{1}, j_{2} be natural numbers. Suppose that

(i) front right cell(f, lenf − ′1, Gauge(C, n)) meets C, (ii) ⟨⟨i_{1}, j_{2} + 1⟩⟩∈ the indices of Gauge(C, n), (iii) f_{len f− ′1} = Gauge(C, n) ∘ (i_{1}, j_{2} + 1), (iv) ⟨⟨i_{1}, j_{2}⟩⟩∈ the indices of Gauge(C, n), and

(v) f_{len f} = Gauge(C, n) ∘ (i_{1}, j_{2}). Then ⟨⟨i_{1} − ′1, j_{2}⟩⟩∈ the indices of Gauge(C, n).

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Received October 7, 2001 842 andrzej trybulec

More on the Finite Sequences on the Plane^{1}

Andrzej Trybulec University of Białystok

Summary. We continue proving lemmas needed for the proof of the Jordan curve theorem. The main goal was to prove the last theorem being a mutation of the first theorem in [13].

MML Identifier: TOPREAL8.

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

1. Preliminaries

The following proposition is true

(1) For all sets A, x, y such that A ⊆ {x, y} and x ∈ A and y ∉ A holds A = {x}.

Let us note that there exists a function which is trivial.

2. Finite Sequences

We adopt the following convention: G denotes a Go-board and i, j, k, m, n denote natural numbers.

Let us note that there exists a finite sequence which is non constant. Next we state a number of propositions:

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(2) For every non trivial finite sequence f holds 1 < lenf.

(3) For every non trivial set D and for every non constant circular finite sequence f of elements of D holds lenf > 2.

(4) For every finite sequence f and for every set x holds x ∈ rngf or x ↫ f = 0.

(5) Let p be a set, 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. If p ↫ f = lenf, then f ^{⌢} g → p = g.

(6) For every non empty set D and for every non empty one-to-one finite sequence f of elements of D holds f_{len f} ↫ f = lenf.

(7) For all finite sequences f, g holds lenf len(f ⌢⌢ g).

(8) For all finite sequences f, g and for every set x such that x ∈ rngf holds x ↫ f = x ↫ (f ⌢⌢ g).

(9) For every non empty finite sequence f and for every finite sequence g holds leng len(f ⌢⌢ g).

(10) For all finite sequences f, g holds rngf ⊆ rng(f ⌢⌢ g).

(11) Let D be a non empty set, f be a non empty finite sequence of elements of D, and g be a non trivial finite sequence of elements of D. If g_{len g} = f_{1}, then f ⌢⌢ g is circular.

(12) Let D be a non empty set, M be a matrix over D, f be a finite sequence of elements of D, and g be a non empty finite sequence of elements of D. Suppose f_{len f} = g_{1} and f is a sequence which elements belong to M and g is a sequence which elements belong to M. Then f ⌢⌢ g is a sequence which elements belong to M.

(13) For every set D and for every finite sequence f of elements of D such that 1 k holds ⟨f(k + 1), …, f(lenf)⟩ = f_{⇂k}.

(14) For every set D and for every finite sequence f of elements of D such that k lenf holds ⟨f(1), …, f(k)⟩ = f↾k.

(15) Let p be a set, 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. If p ↫ f = lenf, then f ^{⌢} g ← p = ⟨f(1), …, f(lenf − ′1)⟩.

(16) Let D be a non empty set and f, g be non empty finite sequences of elements of D. If g_{1} ↫ f = lenf, then (f ⌢⌢ g) :− g_{1} = g.

(17) Let D be a non empty set and f, g be non empty finite sequences of elements of D. If g_{1} ↫ f = lenf, then (f ⌢⌢ g) − : g_{1} = f.

(18) Let D be a non trivial set, f be a non empty finite sequence of elements of D, and g be a non trivial finite sequence of elements of D. Suppose g_{1} = f_{len f} and for every i such that 1 i and i < lenf holds f_{i} ≠ g_{1}. Then (f ⌢⌢ g)_{⟲}^{g_{1}} = g ⌢⌢ f.

3. On the Plane

We now state several propositions:

(19) For every non trivial finite sequence f of elements of E_{T}^{2} holds L(f, 1) = L̃(f↾2).

(20) For every s.c.c. finite sequence f of elements of E_{T}^{2} and for every n such that n < lenf holds f↾n is s.n.c..

(21) For every s.c.c. finite sequence f of elements of E_{T}^{2} and for every n such that 1 n holds f_{⇂n} is s.n.c..

(22) Let f be a circular s.c.c. finite sequence of elements of E_{T}^{2} and given n. If n < lenf and lenf > 4, then f↾n is one-to-one.

(23) Let f be a circular s.c.c. finite sequence of elements of E_{T}^{2}. Suppose lenf > 4. Let i, j be natural numbers. If 1 < i and i < j and j lenf, then f_{i} ≠ f_{j}.

(24) Let f be a circular s.c.c. finite sequence of elements of E_{T}^{2} and given n. If 1 n and lenf > 4, then f_{⇂n} is one-to-one.

(25) For every special non empty finite sequence f of elements of E_{T}^{2} holds ⟨f(m), …, f(n)⟩ is special.

(26) Let f be a special non empty finite sequence of elements of E_{T}^{2} and g be a special non trivial finite sequence of elements of E_{T}^{2}. If f_{len f} = g_{1}, then f ⌢⌢ g is special.

(27) For every circular unfolded s.c.c. finite sequence f of elements of E_{T}^{2} such

that lenf > 4 holds L(f, 1) ∩ L̃(f_{⇂1}) = {f_{1}, f_{2}}. Let us note that there exists a finite sequence of elements of E_{T}^{2} which is

one-to-one, special, unfolded, s.n.c., and non empty.

We now state several propositions:

(28) For all finite sequences f, g of elements of E_{T}^{2} such that j < lenf holds L(f ⌢⌢ g, j) = L(f, j).

(29) For all non empty finite sequences f, g of elements of E_{T}^{2} such that 1 j and j + 1 < leng holds L(f ⌢⌢ g, lenf + j) = L(g, j + 1).

(30) Let f be a non empty finite sequence of elements of E_{T}^{2} and g be a non trivial finite sequence of elements of E_{T}^{2}. If f_{len f} = g_{1}, then L(f ⌢⌢ g, lenf) = L(g, 1).

(31) Let f be a non empty finite sequence of elements of E_{T}^{2} and g be a non trivial finite sequence of elements of E_{T}^{2}. If j + 1 < leng and f_{len f} = g_{1}, then L(f ⌢⌢ g, lenf + j) = L(g, j + 1).

(32) Let f be a non empty s.n.c. unfolded finite sequence of elements of E_{T}^{2} and given i. If 1 i and i < lenf, then L(f, i) ∩ rngf = {f_{i}, f_{i+1}}.

846 andrzej trybulec

(33) Let f, g be non trivial s.n.c. one-to-one unfolded finite sequences of elements of E_{T}^{2}. If L̃(f) ∩ L̃(g) = {f_{1}, g_{1}} and f_{1} = g_{len g} and g_{1} = f_{len f}, then f ⌢⌢ g is s.c.c..

In the sequel f, g are finite sequences of elements of E_{T}^{2}. The following propositions are true:

(34) If f is unfolded and g is unfolded and f_{len f} = g_{1} and L(f, lenf − ′1) ∩ L(g, 1) = {f_{len f}}, then f ⌢⌢ g is unfolded.

(35) If f is non empty and g is non trivial and f_{len f} = g_{1}, then L̃(f ⌢⌢ g) = L̃(f) ∪ L̃(g).

(36) Suppose that

(i) for every n such that n ∈ domf there exist i, j such that ⟨⟨i, j⟩⟩∈ the indices of G and f_{n} = G ∘ (i, j),

(ii) f is non constant, circular, unfolded, s.c.c., and special, and (iii) lenf > 4.

Then there exists g such that (iv) g is a sequence which elements belong to G, unfolded, s.c.c., and special,

(v) L̃(f) = L̃(g), (vi) f_{1} = g_{1}, (vii) f_{len f} = g_{len g}, and (viii) lenf leng.

References

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

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

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

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

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

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

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

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

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

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

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

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

[13] Jarosław Kotowicz and Yatsuka Nakamura. Properties of Go-board - part III. Formalized Mathematics, 3(1):123–124, 1992.

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

more on the finite sequences on the plane 847

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

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

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

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

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

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

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

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

Received October 25, 2001 848 andrzej trybulec

More on Multivariate Polynomials:

Monomials and Constant Polynomials

Christoph Schwarzweller University of Tuebingen

Summary. In this article we give some technical concepts for multivariate polynomials with arbitrary number of variables. Monomials and constant polynomials are introduced and their properties with respect to the eval functor are shown. In addition, the multiplication of polynomials with coefficients is defined and investigated.

MML Identifier: POLYNOM7.

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

1. Preliminaries

Let us note that there exists a non empty zero structure which is non trivial. Let us observe that every zero structure which is non trivial is also non

empty.

Let us mention that there exists a non trivial double loop structure which is Abelian, left zeroed, right zeroed, add-associative, right complementable, unital, associative, commutative, distributive, and integral domain-like.

Let R be a non empty zero structure and let a be an element of R. We say that a is non-zero if and only if:

(Def. 1) a ≠ 0_{R}.

Let R be a non trivial zero structure. Note that there exists an element of R which is non-zero.

Let X be a set, let R be a non empty zero structure, and let p be a series of X, R. We say that p is non-zero if and only if:

(Def. 2) p ≠ 0 (X, R).

Let X be a set and let R be a non trivial zero structure. One can check that there exists a series of X, R which is non-zero.

Let n be an ordinal number and let R be a non trivial zero structure. Note that there exists a polynomial of n, R which is non-zero.

The following two propositions are true:

(1) Let X be a set, R be a non empty zero structure, and s be a series of X, R. Then s = 0 (X, R) if and only if Supports = ∅.

(2) Let X be a set and R be a non empty zero structure. Then R is non trivial if and only if there exists a series s of X, R such that Supports ≠ ∅.

Let X be a set and let b be a bag of X. We say that b is univariate if and only if:

(Def. 3) There exists an element u of X such that supportb = {u}.

Let X be a non empty set. Note that there exists a bag of X which is univariate.

Let X be a non empty set. Note that every bag of X which is univariate is also non empty.

2. Polynomials without Variables

We now state three propositions:

(3) For every bag b of ∅ holds b = EmptyBag∅.

(4) Let L be a right zeroed add-associative right complementable well unital distributive non trivial double loop structure, p be a polynomial of ∅, L, and x be a function from ∅ into L. Then eval(p, x) = p(EmptyBag∅).

(5) Let L be a right zeroed add-associative right complementable well unital distributive non trivial double loop structure. Then Polynom-Ring(∅, L) is ring isomorphic to L.

3. Monomials

Let X be a set, let L be a non empty zero structure, and let p be a series of X, L. We say that p is monomial-like if and only if:

(Def. 4) There exists a bag b of X such that for every bag b′of X such that b′ ≠ b

holds p(b′) = 0_{L}. Let X be a set and let L be a non empty zero structure. Note that there

exists a series of X, L which is monomial-like.

Let X be a set and let L be a non empty zero structure. A monomial of X, L is a monomial-like series of X, L.

Let X be a set and let L be a non empty zero structure. One can check that every series of X, L which is monomial-like is also finite-Support.

The following proposition is true

(6) Let X be a set, L be a non empty zero structure, and p be a series of X, L. Then p is a monomial of X, L if and only if Supportp = ∅ or there exists a bag b of X such that Supportp = {b}.

Let X be a set, let L be a non empty zero structure, let a be an element of L, and let b be a bag of X. The functor Monom(a, b) yields a monomial of X, L and is defined as follows:

(Def. 5) Monom(a, b) = 0 (X, L) +·(b, a).

Let X be a set, let L be a non empty zero structure, and let m be a monomial of X, L. The functor termm yielding a bag of X is defined by:

(Def. 6) m(termm) ≠ 0_{L} or Supportm = ∅ and termm = EmptyBagX.

Let X be a set, let L be a non empty zero structure, and let m be a monomial of X, L. The functor coefficientm yields an element of L and is defined by:

(Def. 7) coefficientm = m(termm).

One can prove the following propositions:

(7) For every set X and for every non empty zero structure L and for every monomial m of X, L holds Supportm = ∅ or Supportm = {termm}.

(8) For every set X and for every non empty zero structure L and for every bag b of X holds coefficientMonom(0_{L}, b) = 0_{L} and termMonom(0_{L}, b) = EmptyBagX.

(9) Let X be a set, L be a non empty zero structure, a be an element of L, and b be a bag of X. Then coefficientMonom(a, b) = a.

(10) Let X be a set, L be a non trivial zero structure, a be a non-zero element of L, and b be a bag of X. Then termMonom(a, b) = b.

(11) For every set X and for every non empty zero structure L and for every monomial m of X, L holds Monom(coefficientm, termm) = m.

(12) Let n be an ordinal number, L be a right zeroed add-associative right complementable unital distributive non trivial double loop structure, m be a monomial of n, L, and x be a function from n into L. Then eval(m, x) = coefficientm · eval(termm, x).

(13) Let n be an ordinal number, L be a right zeroed add-associative right complementable unital distributive non trivial double loop structure, a be an element of L, b be a bag of n, and x be a function from n into L. Then eval(Monom(a, b), x) = a · eval(b, x).

4. Constant Polynomials

Let X be a set, let L be a non empty zero structure, and let p be a series of X, L. We say that p is constant if and only if:

(Def. 8) For every bag b of X such that b ≠ EmptyBagX holds p(b) = 0_{L}.

Let X be a set and let L be a non empty zero structure. Observe that there exists a series of X, L which is constant.

Let X be a set and let L be a non empty zero structure. A constant polynomial of X, L is a constant series of X, L.

Let X be a set and let L be a non empty zero structure. One can check that every series of X, L which is constant is also monomial-like.

The following proposition is true

(14) Let X be a set, L be a non empty zero structure, and p be a series of X, L. Then p is a constant polynomial of X, L if and only if p = 0 (X, L) or Supportp = {EmptyBagX}.

Let X be a set and let L be a non empty zero structure. Observe that 0 (X, L) is constant.

Let X be a set and let L be a unital non empty double loop structure. One can check that 1 (X, L) is constant.

The following propositions are true:

(15) Let X be a set, L be a non empty zero structure, and c be a constant polynomial of X, L. Then Supportc = ∅ or Supportc = {EmptyBagX}.

(16) Let X be a set, L be a non empty zero structure, and c be a constant polynomial of X, L. Then termc = EmptyBagX and coefficientc = c(EmptyBagX).

Let X be a set, let L be a non empty zero structure, and let a be an element of L. The functor a (X, L) yielding a series of X, L is defined by:

(Def. 9) a (X, L) = 0 (X, L) +·(EmptyBagX, a).

Let X be a set, let L be a non empty zero structure, and let a be an element of L. Observe that a (X, L) is constant.

We now state several propositions:

(17) Let X be a set, L be a non empty zero structure, and p be a series of X, L. Then p is a constant polynomial of X, L if and only if there exists an element a of L such that p = a (X, L).

(18) Let X be a set, L be a non empty multiplicative loop with zero structure, and a be an element of L. Then (a (X, L))(EmptyBagX) = a and for every bag b of X such that b ≠ EmptyBagX holds (a (X, L))(b) = 0_{L}.

(19) For every set X and for every non empty zero structure L holds 0_{L} (X, L) = 0 (X, L).

(20) For every set X and for every unital non empty multiplicative loop with zero structure L holds 1_{L} (X, L) = 1 (X, L).

(21) Let X be a set, L be a non empty zero structure, and a, b be elements of L. Then a (X, L) = b (X, L) if and only if a = b.

(22) For every set X and for every non empty zero structure L and for every element a of L holds Supporta (X, L) = ∅ or Supporta (X, L) = {EmptyBagX}.

(23) For every set X and for every non empty zero structure L and for every element a of L holds terma (X, L) = EmptyBagX and coefficienta (X, L) = a.

(24) Let n be an ordinal number, L be a right zeroed add-associative right complementable unital distributive non trivial double loop structure, c be a constant polynomial of n, L, and x be a function from n into L. Then eval(c, x) = coefficientc.

(25) Let n be an ordinal number, L be a right zeroed add-associative right complementable unital distributive non trivial double loop structure, a be an element of L, and x be a function from n into L. Then eval(a (n, L), x) = a.

5. Multiplication with Coefficients

Let X be a set, let L be a non empty multiplicative loop with zero structure, let p be a series of X, L, and let a be an element of L. The functor a · p yields a series of X, L and is defined by:

(Def. 10) For every bag b of X holds (a · p)(b) = a · p(b).

The functor p · a yields a series of X, L and is defined by: (Def. 11) For every bag b of X holds (p · a)(b) = p(b) · a.

Let X be a set, let L be a left zeroed right zeroed add-cancelable distributive non empty double loop structure, let p be a finite-Support series of X, L, and let a be an element of L. Note that a · p is finite-Support and p · a is finite-Support.

One can prove the following propositions:

(26) Let X be a set, L be a commutative non empty multiplicative loop with zero structure, p be a series of X, L, and a be an element of L. Then a · p = p · a.

(27) Let n be an ordinal number, L be an add-associative right complementable right zeroed left distributive non empty double loop structure, p be a series of n, L, and a be an element of L. Then a · p = (a (n, L)) ∗ p.

(28) Let n be an ordinal number, L be an add-associative right complementable right zeroed right distributive non empty double loop structure, p be a series of n, L, and a be an element of L. Then p · a = p ∗ (a (n, L)).

(29) Let n be an ordinal number, L be an Abelian left zeroed right zeroed add-associative right complementable unital associative commutative distributive non trivial double loop structure, p be a polynomial of n, L, a be an element of L, and x be a function from n into L. Then eval(a · p, x) = a · eval(p, x).

(30) Let n be an ordinal number, L be a left zeroed right zeroed add-left-cancelable add-associative right complementable unital associative integral domain-like distributive non trivial double loop structure, p be a polynomial of n, L, a be an element of L, and x be a function from n into L. Then eval(a · p, x) = a · eval(p, x).

(31) Let n be an ordinal number, L be an Abelian left zeroed right zeroed add-associative right complementable unital associative commutative distributive non trivial double loop structure, p be a polynomial of n, L, a be an element of L, and x be a function from n into L. Then eval(p · a, x) = eval(p, x) · a.

(32) Let n be an ordinal number, L be a left zeroed right zeroed add-left-cancelable add-associative right complementable unital associative commutative distributive integral domain-like non trivial double loop structure, p be a polynomial of n, L, a be an element of L, and x be a function from n into L. Then eval(p · a, x) = eval(p, x) · a.

(33) Let n be an ordinal number, L be an Abelian left zeroed right zeroed add-associative right complementable unital associative commutative distributive non trivial double loop structure, p be a polynomial of n, L, a be an element of L, and x be a function from n into L. Then eval((a (n, L)) ∗ p, x) = a · eval(p, x).

(34) Let n be an ordinal number, L be an Abelian left zeroed right zeroed add-associative right complementable unital associative commutative distributive non trivial double loop structure, p be a polynomial of n, L, a be an element of L, and x be a function from n into L. Then eval(p ∗ (a (n, L)), x) = eval(p, x) · a.

References

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

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

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

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

[5] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

more on multivariate polynomials: monomials … 855

[6] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

[7] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[8] Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69–79, 1998.

[9] Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Formalized Mathematics, 9(2):331–338, 2001.

[10] Wojciech Skaba and Michał Muzalewski. From double loops to fields. Formalized Mathematics, 2(1):185–191, 1991.

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

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

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

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

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

Received November 28, 2001 856 christoph schwarzweller

On State Machines of Calculating Type

Hisayoshi Kunimune Shinshu University Nagano

Grzegorz Bancerek Białystok Technical University

Yatsuka Nakamura Shinshu University Nagano

Summary. In this article, we show properties of calculating type state machines. In the first section, we have defined calculating type state machines of which the state transition only depends on the first input. We have also proved theorems of the state machines. In the second section, we defined Moore machines with final states. We also introduced the concept of result of the Moore machines. In the last section, we proved the correctness of several calculating type of Moore machines.

MML Identifier: FSM 2.

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

1. Calculating Type

For simplicity, we use the following convention: m denotes a natural number, x, y denote real numbers, i, j denote non empty natural numbers, I, O denote non empty sets, s, s_{1}, s_{2}, s_{3} denote elements of I, w, w_{1}, w_{2} denote finite sequences of elements of I, t denotes an element of O, S denotes a non empty FSM over I, and q, q_{1} denote states of S.

Let us consider I, S, q, w. We introduce GEN(w, q) as a synonym of (q, w)-admissible.

Let us consider I, S, q, w. Note that GEN(w, q) is non empty. The following propositions are true:

(1) GEN(⟨s⟩, q) = ⟨q, (the transition of S)(⟨⟨q, s⟩⟩)⟩.

(2) GEN(⟨s_{1}, s_{2}⟩, q) = ⟨q, (the transition of S)(⟨⟨q, s_{1}⟩⟩), (the transition of S)(⟨⟨(the transition of S)(⟨⟨q, s_{1}⟩⟩), s_{2}⟩⟩)⟩.

(3) GEN(⟨s_{1}, s_{2}, s_{3}⟩, q) = ⟨q, (the transition of S)(⟨⟨q, s_{1}⟩⟩), (the transition of S)(⟨⟨(the transition of S)(⟨⟨q, s_{1}⟩⟩), s_{2}⟩⟩), (the transition of S)(⟨⟨(the transition of S)(⟨⟨(the transition of S)(⟨⟨q, s_{1}⟩⟩), s_{2}⟩⟩), s_{3}⟩⟩)⟩.

Let us consider I, S. We say that S is calculating type if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let given j and given w_{1}, w_{2}. Suppose w_{1}(1) = w_{2}(1) and j lenw_{1} + 1 and j lenw_{2} + 1. Then (GEN(w_{1}, the initial state of S))(j) = (GEN(w_{2}, the initial state of S))(j).

The following propositions are true:

(4) Suppose S is calculating type. Let given w_{1}, w_{2}. Suppose w_{1}(1) = w_{2}(1). Then GEN(w_{1}, the initial state of S) and GEN(w_{2}, the initial state of S) are c=-comparable.

(5) Suppose that for all w_{1}, w_{2} such that w_{1}(1) = w_{2}(1) holds GEN(w_{1}, the initial state of S) and GEN(w_{2}, the initial state of S) are c=-comparable. Then S is calculating type.

(6) Suppose S is calculating type. Let given w_{1}, w_{2}. Suppose w_{1}(1) = w_{2}(1) and lenw_{1} = lenw_{2}. Then GEN(w_{1}, the initial state of S) = GEN(w_{2}, the initial state of S).

(7) Suppose that for all w_{1}, w_{2} such that w_{1}(1) = w_{2}(1) and lenw_{1} = lenw_{2} holds GEN(w_{1}, the initial state of S) = GEN(w_{2}, the initial state of S). Then S is calculating type.

Let us consider I, S, q, s. We say that q is accessible via s if and only if: (Def. 2) There exists a finite sequence w of elements of I such that the initial

state of S −→^{⟨s⟩^{⌢}w}q.

Let us consider I, S, q. We say that q is accessible if and only if: (Def. 3) There exists a finite sequence w of elements of I such that the initial

state of S −→^{w} q.

We now state four propositions:

(8) If q is accessible via s, then q is accessible.

(9) If q is accessible and q ≠ the initial state of S, then there exists s such that q is accessible via s.

(10) The initial state of S is accessible.

(11) Suppose S is calculating type and q is accessible via s. Then there exists a non empty natural number m such that for every w if lenw + 1 m and w(1) = s, then q = (GEN(w, the initial state of S))(m) and for every i such that i < m holds (GEN(w, the initial state of S))(i) ≠ q.

Let us consider I, S. We say that S is regular if and only if: (Def. 4) Every state of S is accessible.

We now state several propositions:

(12) If for all s_{1}, s_{2}, q holds (the transition of S)(⟨⟨q, s_{1}⟩⟩) = (the transition of S)(⟨⟨q, s_{2}⟩⟩), then S is calculating type.

(13) Let given S. Suppose that

(i) for all s_{1}, s_{2}, q such that q ≠ the initial state of S holds (the transition of S)(⟨⟨q, s_{1}⟩⟩) = (the transition of S)(⟨⟨q, s_{2}⟩⟩), and

(ii) for all s, q_{1} holds (the transition of S)(⟨⟨q_{1}, s⟩⟩) ≠ the initial state of S.

Then S is calculating type.

(14) Suppose S is regular and calculating type. Let given s_{1}, s_{2}, q. If q ≠ the initial state of S, then (GEN(⟨s_{1}⟩, q))(2) = (GEN(⟨s_{2}⟩, q))(2).

(15) Suppose S is regular and calculating type. Let given s_{1}, s_{2}, q. Suppose q ≠ the initial state of S. Then (the transition of S)(⟨⟨q, s_{1}⟩⟩) = (the transition of S)(⟨⟨q, s_{2}⟩⟩).

(16) Suppose S is regular and calculating type. Let given s, s_{1}, s_{2}, q. Suppose (the transition of S)(⟨⟨the initial state of S, s_{1}⟩⟩) ≠ (the transition of S)(⟨⟨the initial state of S, s_{2}⟩⟩). Then (the transition of S)(⟨⟨q, s⟩⟩) ≠ the initial state of S.

2. State Machine with Final States

Let I be a set. We introduce state machines over I with final states which are extensions of FSM over I and are systems

⟨ a carrier, a transition, an initial state, final states ⟩, where the carrier is a set, the transition is a function from [: the carrier, I:] into the carrier, the initial state is an element of the carrier, and the final states constitute a subset of the carrier.

Let I be a set. One can check that there exists a state machine over I with final states which is non empty.

Let us consider I, s and let S be a non empty state machine over I with final states. We say that s leads to final state of S if and only if:

(Def. 5) There exists a state q of S such that q is accessible via s and q ∈ the

final states of S. Let us consider I and let S be a non empty state machine over I with final

states. We say that S is halting if and only if: (Def. 6) s leads to final state of S.

Let I be a set and let O be a non empty set. We consider Moore state machines over I and O with final states as extensions of state machine over I with final states and Moore-FSM over I, O as systems

⟨ a carrier, a transition, an output function, an initial state, final states ⟩, where the carrier is a set, the transition is a function from [: the carrier, I:] into the carrier, the output function is a function from the carrier into O, the initial state is an element of the carrier, and the final states constitute a subset of the carrier.

Let I be a set and let O be a non empty set. Observe that there exists a Moore state machine over I and O with final states which is non empty and strict.

Let us consider I, O, let i, f be sets, and let o be a function from {i, f} into O. The functor I -TwoStatesMooreSM(i, f, o) yielding a non empty strict Moore state machine over I and O with final states is defined by the conditions (Def. 7).

(Def. 7)(i) The carrier of I -TwoStatesMooreSM(i, f, o) = {i, f}, (ii) the transition of I -TwoStatesMooreSM(i, f, o) = [: {i, f}, I:] ⟼→ f, (iii) the output function of I -TwoStatesMooreSM(i, f, o) = o, (iv) the initial state of I -TwoStatesMooreSM(i, f, o) = i, and

(v) the final states of I -TwoStatesMooreSM(i, f, o) = {f}.

One can prove the following proposition

(17) Let i, f be sets, o be a function from {i, f} into O, and given j. If 1 < j and j lenw + 1, then (GEN(w, the initial state of I -TwoStatesMooreSM(i, f, o)))(j) = f.

Let us consider I, O, let i, f be sets, and let o be a function from {i, f} into O. Observe that I -TwoStatesMooreSM(i, f, o) is calculating type.

Let us consider I, O, let i, f be sets, and let o be a function from {i, f} into O. One can check that I -TwoStatesMooreSM(i, f, o) is halting.

In the sequel n, m are non empty natural numbers and M is a non empty Moore state machine over I and O with final states.

Next we state the proposition

(18) Suppose that (i) M is calculating type, (ii) s leads to final state of M, and (iii) the initial state of M ∉ the final states of M.

Then there exists a non empty natural number m such that (iv) for every w such that lenw + 1 m and w(1) = s holds (GEN(w, the

initial state of M))(m) ∈ the final states of M, and

(v) for all w, j such that j lenw + 1 and w(1) = s and j < m holds (GEN(w, the initial state of M))(j) ∉ the final states of M.

3. Correctness of a Result of State Machine

Let us consider I, O, M, s and let t be a set. We say that t is a result of s in M if and only if the condition (Def. 8) is satisfied.

(Def. 8) There exists m such that for every w if w(1) = s, then if m lenw + 1, then t = (the output function of M)((GEN(w, the initial state of M))(m)) and (GEN(w, the initial state of M))(m) ∈ the final states of M and for every n such that n < m and n lenw +1 holds (GEN(w, the initial state of M))(n) ∉ the final states of M.

We now state several propositions:

(19) Suppose the initial state of M ∈ the final states of M. Then (the output function of M)(the initial state of M) is a result of s in M.

(20) Suppose that (i) M is calculating type, (ii) s leads to final state of M, and (iii) the initial state of M ∉ the final states of M.

Then there exists t which is a result of s in M.

(21) Suppose M is calculating type and s leads to final state of M. Let t_{1}, t_{2} be elements of O. If t_{1} is a result of s in M and t_{2} is a result of s in M, then t_{1} = t_{2}.

(22) Let X be a non empty set, f be a binary operation on X, and M be a non empty Moore state machine over [: X, X:] and X ∪ {X} with final states. Suppose that

(i) M is calculating type, (ii) the carrier of M = X ∪ {X}, (iii) the final states of M = X, (iv) the initial state of M = X,

(v) the output function of M = id_{the carrier of M}, and (vi) for all elements x, y of X holds (the transition of M)(⟨⟨the initial state

of M, ⟨⟨x, y⟩⟩⟩⟩) = f(x, y). Then M is halting and for all elements x, y of X holds f(x, y) is a result of ⟨⟨x, y⟩⟩ in M.

(23) Let M be a non empty Moore state machine over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that M is calculating type and the carrier of M = ℝ∪{ℝ} and the final states of M = ℝ and the initial state of M = ℝ and the output function of M = id_{the carrier of M} and for all x, y such that x y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = x and for all x, y such that x < y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = y. Let x, y be elements of ℝ. Then max(x, y) is a result of ⟨⟨x, y⟩⟩ in M.

(24) Let M be a non empty Moore state machine over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that M is calculating type and the carrier of M = ℝ∪{ℝ} and the final states of M = ℝ and the initial state of M = ℝ and the output function of M = id_{the carrier of M} and for all x, y such that x < y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = x and for all x, y such that x y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = y. Let x, y be elements of ℝ. Then min(x, y) is a result of ⟨⟨x, y⟩⟩ in M.

(25) Let M be a non empty Moore state machine over [: ℝ, ℝ:] and ℝ ∪ {ℝ}

with final states. Suppose that

(i) M is calculating type, (ii) the carrier of M = ℝ ∪ {ℝ}, (iii) the final states of M = ℝ, (iv) the initial state of M = ℝ,

(v) the output function of M = id_{the carrier of M}, and (vi) for all x, y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x,

y⟩⟩⟩⟩) = x + y. Let x, y be elements of ℝ. Then x + y is a result of ⟨⟨x, y⟩⟩ in M.

(26) Let M be a non empty Moore state machine over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that M is calculating type and the carrier of M = ℝ∪{ℝ} and the final states of M = ℝ and the initial state of M = ℝ and the output function of M = id_{the carrier of M} and for all x, y such that x > 0 or y > 0 holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = 1 and for all x, y such that x = 0 or y = 0 but x 0 but y 0 holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = 0 and for all x, y such that x < 0 and y < 0 holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = − 1. Let x, y be elements of ℝ. Then max(sgnx, sgny) is a result of ⟨⟨x, y⟩⟩ in M.

Let us consider I, O. Note that there exists a non empty Moore state machine over I and O with final states which is calculating type and halting.

Let us consider I. Observe that there exists a non empty state machine over I with final states which is calculating type and halting.

Let us consider I, O, let M be a calculating type halting non empty Moore state machine over I and O with final states, and let us consider s. The functor Result(s, M) yields an element of O and is defined as follows:

(Def. 9) Result(s, M) is a result of s in M.

Next we state several propositions:

(27) For every function f from {0, 1} into O holds Result(s, I -TwoStatesMooreSM(0, 1, f)) = f(1).

(28) Let M be a calculating type halting non empty Moore state machine over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that (i) the carrier of M = ℝ ∪ {ℝ}, (ii) the final states of M = ℝ, (iii) the initial state of M = ℝ, (iv) the output function of M = id_{the carrier of M},

(v) for all x, y such that x y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = x, and

(vi) for all x, y such that x < y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = y. Let x, y be elements of ℝ. Then Result(⟨⟨x, y⟩⟩, M) = max(x, y).

(29) Let M be a calculating type halting non empty Moore state machine

over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that

(i) the carrier of M = ℝ ∪ {ℝ}, (ii) the final states of M = ℝ, (iii) the initial state of M = ℝ, (iv) the output function of M = id_{the carrier of M},

(v) for all x, y such that x < y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = x, and

(vi) for all x, y such that x y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = y. Let x, y be elements of ℝ. Then Result(⟨⟨x, y⟩⟩, M) = min(x, y).

(30) Let M be a calculating type halting non empty Moore state machine

over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that

(i) the carrier of M = ℝ ∪ {ℝ}, (ii) the final states of M = ℝ, (iii) the initial state of M = ℝ, (iv) the output function of M = id_{the carrier of M}, and

(v) for all x, y holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = x + y. Let x, y be elements of ℝ. Then Result(⟨⟨x, y⟩⟩, M) = x + y.

(31) Let M be a calculating type halting non empty Moore state machine over [: ℝ, ℝ:] and ℝ ∪ {ℝ} with final states. Suppose that the carrier of M = ℝ∪{ℝ} and the final states of M = ℝ and the initial state of M = ℝ and the output function of M = id_{the carrier of M} and for all x, y such that x > 0 or y > 0 holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = 1 and for all x, y such that x = 0 or y = 0 but x 0 but y 0 holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = 0 and for all x, y such that x < 0 and y < 0 holds (the transition of M)(⟨⟨the initial state of M, ⟨⟨x, y⟩⟩⟩⟩) = − 1. Let x, y be elements of ℝ. Then Result(⟨⟨x, y⟩⟩, M) = max(sgnx, sgny).

864 hisayoshi kunimune et al.

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Received December 3, 2001

Hierarchies and Classifications of Sets^{1}

Mariusz Giero University of Białystok

Summary. This article is a continuation of [2] article. Further properties of classification of sets are proved. The notion of hierarchy of a set is introduced. Properties of partitions and hierarchies are shown. The main theorem says that for each hierarchy there exists a classification which the union is equal to the considered hierarchy.

MML Identifier: TAXONOM2.

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

1. Tree and Classification of a Set

For simplicity, we follow the rules: A denotes a relational structure, X denotes a non empty set, P_{1}, P_{2}, P_{3}, Y, a, b, c, x denote sets, and S_{1} denotes a subset of Y.

Let us consider A. We say that A has superior elements if and only if: (Def. 1) There exists an element of A which is superior of the internal relation of

A. Let us consider A. We say that A has comparable down elements if and only

if: (Def. 2) For all elements x, y of A such that there exists an element z of A such

that z x and z y holds x y or y x. The following proposition is true (1) For every set a holds ⟨{{a}}, ⊆⟩ is non empty, reflexive, transitive, and

1This work has been partially supported by the European Community TYPES grant IST-1999-29001 and CALCULEMUS grant HPRN-CT-2000-00102.

antisymmetric and has superior elements and comparable down elements. Let us observe that there exists a relational structure which is non empty,

reflexive, transitive, antisymmetric, and strict and has superior elements and comparable down elements.

A tree is a poset with superior elements and comparable down elements. Next we state four propositions:

(2) For every equivalence relation E_{1} of X and for all sets x, y, z such that z ∈ [x]_{(E_{1})} and z ∈ [y]_{(E_{1})} holds [x]_{(E_{1})} = [y]_{(E_{1})}.

(3) For every partition P of X and for all sets x, y, z such that x ∈ P and y ∈ P and z ∈ x and z ∈ y holds x = y.

(4) For all sets C, x such that C is a classification of X and x ∈ ⋃ C holds x ⊆ X.

(5) For every set C such that C is a strong classification of X holds ⟨⋃ C, ⊆⟩ is a tree.

2. The Hierarchy of a Set

Let us consider Y. We say that Y is hierarchic if and only if: (Def. 3) For all sets x, y such that x ∈ Y and y ∈ Y holds x ⊆ y or y ⊆ x or x

misses y. One can verify that every set which is trivial is also hierarchic. Let us note that there exists a set which is non trivial and hierarchic. The following propositions are true:

(6) ∅ is hierarchic.

(7) {∅} is hierarchic. Let us consider Y. A family of subsets of Y is said to be a hierarchy of Y if:

(Def. 4) It is hierarchic.

Let us consider Y. We say that Y is mutually-disjoint if and only if: (Def. 5) For all sets x, y such that x ∈ Y and y ∈ Y and x ≠ y holds x misses y.

In the sequel H denotes a hierarchy of Y. Let us consider Y. Observe that there exists a family of subsets of Y which

is mutually-disjoint.

Next we state three propositions:

(8) ∅ is mutually-disjoint.

(9) {∅} is mutually-disjoint.

(10) {a} is mutually-disjoint.

Let us consider Y and let F be a family of subsets of Y. We say that F is T_{3} if and only if the condition (Def. 6) is satisfied.

(Def. 6) Let A be a subset of Y. Suppose A ∈ F. Let x be an element of Y. If x ∉ A, then there exists a subset B of Y such that x ∈ B and B ∈ F and A misses B.

We now state the proposition

(11) For every family F of subsets of Y such that F = ∅ holds F is T_{3}. Let us consider Y. One can verify that there exists a hierarchy of Y which

is covering and T_{3}.

Let us consider Y and let F be a family of subsets of Y. We say that F is lower-bounded if and only if the condition (Def. 7) is satisfied.

(Def. 7) Let B be a set. Suppose B ≠ ∅ and B ⊆ F and for all a, b such that a ∈ B and b ∈ B holds a ⊆ b or b ⊆ a. Then there exists c such that c ∈ F and c ⊆ ⋂ B.

Next we state the proposition

(12) Let B be a mutually-disjoint family of subsets of Y. Suppose that for every set b such that b ∈ B holds S_{1} misses b and Y ≠ ∅. Then B∪{S_{1}} is a mutually-disjoint family of subsets of Y and if S_{1} ≠ ∅, then ⋃(B ∪{S_{1}}) ≠ ⋃ B.

Let us consider Y and let F be a family of subsets of Y. We say that F has maximum elements if and only if the condition (Def. 8) is satisfied.

(Def. 8) Let S be a subset of Y. Suppose S ∈ F. Then there exists a subset T of Y such that S ⊆ T and T ∈ F and for every subset V of Y such that T ⊆ V and V ∈ F holds V = Y.

3. Some Properties of Partitions, Hierarchies and Classifications of Sets

The following propositions are true:

(13) For every covering hierarchy H of Y such that H has maximum elements there exists a partition P of Y such that P ⊆ H.

(14) Let H be a covering hierarchy of Y and B be a mutually-disjoint family of subsets of Y. Suppose B ⊆ H and for every mutually-disjoint family C of subsets of Y such that C ⊆ H and ⋃ B ⊆ ⋃ C holds B = C. Then B is a partition of Y.

(15) Let H be a covering T_{3} hierarchy of Y. Suppose H is lower-bounded and ∅ ∉ H. Let A be a subset of Y and B be a mutually-disjoint family of subsets of Y. Suppose that

(i) A ∈ B, (ii) B ⊆ H, and (iii) for every mutually-disjoint family C of subsets of Y such that A ∈ C

and C ⊆ H and ⋃ B ⊆ ⋃ C holds ⋃ B = ⋃ C. Then B is a partition of Y.

(16) Let H be a covering T_{3} hierarchy of Y. Suppose H is lower-bounded and ∅ ∉ H. Let A be a subset of Y and B be a mutually-disjoint family of subsets of Y. Suppose A ∈ B and B ⊆ H and for every mutually-disjoint family C of subsets of Y such that A ∈ C and C ⊆ H and B ⊆ C holds B = C. Then B is a partition of Y.

(17) Let H be a covering T_{3} hierarchy of Y. Suppose H is lower-bounded and ∅ ∉ H. Let A be a subset of Y. If A ∈ H, then there exists a partition P of Y such that A ∈ P and P ⊆ H.

(18) Let h be a non empty set, P_{4} be a partition of X, and h_{1} be a set. Suppose h_{1} ∈ P_{4} and h ⊆ h_{1}. Let P_{6} be a partition of X. Suppose h ∈ P_{6} and for every x such that x ∈ P_{6} holds x ⊆ h_{1} or h_{1} ⊆ x or h_{1} misses x. Let P_{5} be a set. Suppose that for every a holds a ∈ P_{5} iff a ∈ P_{6} and a ⊆ h_{1}. Then P_{5} ∪ (P_{4} \ {h_{1}}) is a partition of X and P_{5} ∪ (P_{4} \ {h_{1}}) is finer than P_{4}.

(19) Let h be a non empty set. Suppose h ⊆ X. Let P_{8} be a partition of X. Suppose there exists a set h_{2} such that h_{2} ∈ P_{8} and h_{2} ⊆ h and for every x such that x ∈ P_{8} holds x ⊆ h or h ⊆ x or h misses x. Let P_{7} be a set. Suppose that for every x holds x ∈ P_{7} iff x ∈ P_{8} and x misses h. Then

(i) P_{7} ∪ {h} is a partition of X, (ii) P_{8} is finer than P_{7} ∪ {h}, and (iii) for every partition P_{4} of X such that P_{8} is finer than P_{4} and for every

set h_{1} such that h_{1} ∈ P_{4} and h ⊆ h_{1} holds P_{7} ∪ {h} is finer than P_{4}.

(20) Let H be a covering T_{3} hierarchy of X. Suppose that (i) H is lower-bounded, (ii) ∅ ∉ H, and (iii) for every set C_{1} such that C_{1} ≠ ∅ and C_{1} ⊆ PARTITIONS(X) and for

all sets P_{9}, P_{10} such that P_{9} ∈ C_{1} and P_{10} ∈ C_{1} holds P_{9} is finer than P_{10} or P_{10} is finer than P_{9} there exist P_{1}, P_{2} such that P_{1} ∈ C_{1} and P_{2} ∈ C_{1} and for every P_{3} such that P_{3} ∈ C_{1} holds P_{3} is finer than P_{2} and P_{1} is finer than P_{3}. Then there exists a classification C of X such that ⋃ C = H.

Acknowledgments

I would like to thank Prof. Andrzej Trybulec for his help in the preparation of this article. hierarchies and classifications of sets 869

References

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

[2] Mariusz Giero and Roman Matuszewski. Lower tolerance. Preliminaries to Wroclaw taxonomy. Formalized Mathematics, 9(3):597–603, 2001.

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

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

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

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

[7] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

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

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

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

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

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

Received December 28, 2001

Index of MML Identifiers

AFINSQ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .825 AMI 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 AMI 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 COMPUT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .705 FSM 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 FUZZY 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 JGRAPH 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .697 JGRAPH 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .801 JORDAN1E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 JORDAN1F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 JORDAN1G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 JORDAN1H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 MSAFREE3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 PENCIL 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .795 POLYNOM6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 POLYNOM7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849 PYTHTRIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 ROBBINS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 SCMFSA10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 TAXONOM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 TOPREAL8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 TURING 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .721 WAYBEL33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 WAYBEL34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 YELLOW19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 YELLOW20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 YELLOW21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

Contents Formaliz. Math. 9 (4)

On the Instructions of SCM

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

Input and Output of Instructions

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

On the Instructions of SCM _{FSA}

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Robbins Algebras vs. Boolean Algebras

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

Properties of Fuzzy Relation

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

On Outside Fashoda Meet Theorem

By Yatsuka Nakamura  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

The Set of Primitive Recursive Functions

By Grzegorz Bancerek and Piotr Rudnicki  . . . . . . . . . . . . . . 705

Introduction to Turing Machines

By Jing-Chao Chen and Yatsuka Nakamura  . . . . . . . . . . . . . . 721

On the Characterizations of Compactness

By Grzegorz Bancerek et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

Compactness of Lim-inf Topology

By Grzegorz Bancerek and Noboru Endou  . . . . . . . . . . . . . . 739

Miscellaneous Facts about Functors

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

Categorial Background for Duality Theory

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

Duality Based on the Galois Connection. Part I

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

Continued on inside back cover

Yet Another Construction of Free Algebra

By Grzegorz Bancerek and Artur Korniłowicz  . . . . . . . . . 779

Upper and Lower Sequence of a Cage

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

On Polynomials with Coefficients in a Ring of Polynomials

By Barbara Dzienis  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

On Cosets in Segre’s Product of Partial Linear Spaces

By Adam Naumowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

On the Simple Closed Curve Property of the Circle and the Fa-

shoda Meet Theorem

By Yatsuka Nakamura  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

Pythagorean Triples

By Freek Wiedijk  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809

Some Remarks on Finite Sequences on Go-boards

By Adam Naumowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

Upper and Lower Sequence on the Cage. Part II

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

Zero-Based Finite Sequences

By Tetsuya Tsunetou et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825

More on the External Approximation of a Continuum

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

More on the Finite Sequences on the Plane

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843

More on Multivariate Polynomials: Monomials and Constant Po-

lynomials

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

On State Machines of Calculating Type

By Hisayoshi Kunimune et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857

Hierarchies and Classifications of Sets

By Mariusz Giero  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870