FORMALIZED MATHEMATICS Volume 9, Number 3, 2001 University of Białystok

The Canonical Formulae

Andrzej Trybulec University of Białystok

MML Identifier: HILBERT3.

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

1. Preliminaries

One can prove the following propositions:

(1) For every integer i holds i is even iff i − 1 is odd.

(2) For every integer i holds i is odd iff i − 1 is even.

(3) Let X be a trivial set and x be a set. Suppose x ∈ X. Let f be a function from X into X. Then x is a fixpoint of f.

Let A, B, C be sets. Note that every function from A into C^{B} is function yielding.

One can prove the following three propositions:

(4) For every function yielding function f holds Sub_{f} rngf = rngf.

(5) For all sets A, B, x and for every function f such that x ∈ A and f ∈ B^{A} holds f(x) ∈ B.

(6) For all sets A, B, C such that if C = ∅, then B = ∅ or A = ∅ and for every function f from A into C^{B} holds dom_{κ} f(κ) = A ⟼→ B.

Let us note that ∅ is function yielding. In the sequel n is a natural number and p, q, r are elements of HP-WFF. Next we state the proposition

(7) For every set x holds ∅(x) = ∅.

Let A be a set and let B be a functional set. One can verify that every function from A into B is function yielding.

One can prove the following propositions:

(8) For every set X and for every subset A of X holds [0 ⟼→ 1, 1 ⟼→ 0] · χ_{A,X} = χ_{A^{c},X}.

(9) For every set X and for every subset A of X holds [0 ⟼→ 1, 1 ⟼→ 0] · χ_{A^{c},X} = χ_{A,X}.

(10) For all sets a, b, x, y, x′, y′ such that a ≠ b and [a ⟼→ x, b ⟼→ y] = [a ⟼→ x′, b ⟼→ y′] holds x = x′ and y = y′.

(11) For all sets a, b, x, y, X, Y such that a ≠ b and x ∈ X and y ∈ Y holds [a ⟼→ x, b ⟼→ y] ∈ ∏[a ⟼→ X, b ⟼→ Y].

(12) For every non empty set D and for every function f from 2 into D there exist elements d_{1}, d_{2} of D such that f = [0 ⟼→ d_{1}, 1 ⟼→ d_{2}].

(13) For all sets a, b, c, d such that a ≠ b holds [a ⟼→ c, b ⟼→ d] · [a ⟼→ b, b ⟼→ a] = [a ⟼→ d, b ⟼→ c].

(14) For all sets a, b, c, d and for every function f such that a ≠ b and c ∈ domf and d ∈ domf holds f ·[a ⟼→ c, b ⟼→ d] = [a ⟼→ f(c), b ⟼→ f(d)].

2. The Cartesian Product of Functions and the Frege Function

Let f, g be one-to-one functions. Note that [: f, g:] is one-to-one. We now state a number of propositions:

(15) Let A, B be non empty sets, C, D be sets, f be a function from C into A, and g be a function from D into B. Then π_{1}(A×B)·[: f, g:] = f ·π_{1}(C×D).

(16) Let A, B be non empty sets, C, D be sets, f be a function from C into A, and g be a function from D into B. Then π_{2}(A×B)·[: f, g:] = g·π_{2}(C ×D).

(17) For every function g holds ∅ ↫ g = ∅.

(18) For every function yielding function f and for all functions g, h holds f ↫ g · h = (f · h) ↫ (g · h).

(19) Let C be a set, A be a non empty set, f be a function from A into C^{(∅ qua set)}, and g be a function from A into ∅. Then rng(f ↫ g) = {∅}.

(20) Let A, B, C be sets such that if B = ∅, then A = ∅. Let f be a function from A into C^{B} and g be a function from A into B. Then rng(f ↫ g) ⊆ C.

(21) For all sets A, B, C such that if C = ∅, then B = ∅ or A = ∅ and for every function f from A into C^{B} holds domFrege(f) = B^{A}.

(22) Frege(∅) = {∅} ⟼→ ∅.

(23) For all sets A, B, C such that if C = ∅, then B = ∅ or A = ∅ and for every function f from A into C^{B} holds rngFrege(f) ⊆ C^{A}.

(24) Let A, B, C be sets such that if C = ∅, then B = ∅ or A = ∅. Let f be a function from A into C^{B}. Then Frege(f) is a function from B^{A} into C^{A}.

3. About Permutations

Let A be a set. Observe that every permutation of A is one-to-one. The following proposition is true

(25) For all sets A, B and for every permutation P of A and for every per-

mutation Q of B holds [: P, Q:] is permutation-like. Let A, B be non empty sets, let P be a permutation of A, and let Q be a

function from B into B. The functor P ⇒ Q yielding a function from B^{A} into B^{A} is defined as follows:

(Def. 1) For every function f from A into B holds (P ⇒ Q)(f) = Q · f · P^{−1} .

Let A, B be non empty sets, let P be a permutation of A, and let Q be a permutation of B. Observe that P ⇒ Q is permutation-like.

Next we state three propositions:

(26) Let A, B be non empty sets, P be a permutation of A, Q be a permutation of B, and f be a function from A into B. Then (P ⇒ Q)^{−1} (f) = Q^{−1} · f · P.

(27) For all non empty sets A, B and for every permutation P of A and for every permutation Q of B holds (P ⇒ Q)^{−1} = P^{−1} ⇒ Q^{−1} .

(28) Let A, B, C be non empty sets, f be a function from A into C^{B}, g be a function from A into B, P be a permutation of B, and Q be a permutation of C. Then ((P ⇒ Q) · f) ↫ (P · g) = Q · f ↫ g.

4. Set Valuations

A SetValuation is a non-empty many sorted set indexed by ℕ. In the sequel V denotes a SetValuation. Let us consider V. The functor SetValV yielding a many sorted set indexed

by HP-WFF is defined by the conditions (Def. 2). (Def. 2)(i) (SetValV)(VERUM) = 1,

(ii) for every n holds (SetValV)(propn) = V (n), and (iii) for all p, q holds (SetValV)(p ∧ q) = [: (SetValV)(p), (SetValV)(q) :]

and (SetValV)(p ⇒ q) = (SetValV)(q)^{(SetVal V)(p)}. Let us consider V, p. The functor SetVal(V, p) is defined as follows:

(Def. 3) SetVal(V, p) = (SetValV)(p).

Let us consider V, p. One can check that SetVal(V, p) is non empty. Next we state four propositions:

(29) SetVal(V, VERUM) = 1.

(30) SetVal(V, propn) = V (n).

(31) SetVal(V, p ∧ q) = [: SetVal(V, p), SetVal(V, q) :].

(32) SetVal(V, p ⇒ q) = (SetVal(V, q))^{SetVal(V,p)}.

Let us consider V, p, q. Observe that SetVal(V, p ⇒ q) is functional. Let us consider V, p, q, r. Note that every element of SetVal(V, p ⇒ (q ⇒ r))

is function yielding.

Let us consider V, p, q, r. One can check that there exists a function from SetVal(V, p ⇒ q) into SetVal(V, p ⇒ r) which is function yielding and there exists an element of SetVal(V, p ⇒ (q ⇒ r)) which is function yielding.

5. Permuting Set Valuations

Let us consider V. A function is called a permutation of V if: (Def. 4) domit = ℕ and for every n holds it(n) is a permutation of V (n).

In the sequel P is a permutation of V. Let us consider V, P. The functor PermP yielding a many sorted function

from SetValV into SetValV is defined by the conditions (Def. 5). (Def. 5)(i) (PermP)(VERUM) = id_{1},

(ii) for every n holds (PermP)(propn) = P (n), and (iii) for all p, q there exists a permutation p′ of SetVal(V, p) and there

exists a permutation q′ of SetVal(V, q) such that p′ = (PermP)(p) and q′= (PermP)(q) and (PermP)(p ∧ q) = [: p′, q′:] and (PermP)(p ⇒ q) = p′⇒ q′.

Let us consider V, P, p. The functor Perm(P, p) yields a function from SetVal(V, p) into SetVal(V, p) and is defined by:

(Def. 6) Perm(P, p) = (PermP)(p).

Next we state four propositions:

(33) Perm(P, VERUM) = id_{SetVal(V,VERUM )}.

(34) Perm(P, propn) = P (n).

(35) Perm(P, p ∧ q) = [: Perm(P, p), Perm(P, q) :].

(36) For every permutation p′ of SetVal(V, p) and for every permutation q′ of SetVal(V, q) such that p′ = Perm(P, p) and q′ = Perm(P, q) holds Perm(P, p ⇒ q) = p′⇒ q′.

Let us consider V, P, p. One can check that Perm(P, p) is permutation-like. We now state four propositions:

(37) For every function g from SetVal(V, p) into SetVal(V, q) holds (Perm(P, p ⇒ q))(g) = Perm(P, q) · g · (Perm(P, p))^{−1} .

(38) For every function g from SetVal(V, p) into SetVal(V, q) holds (Perm(P, p ⇒ q))^{−1} (g) = (Perm(P, q))^{−1} · g · Perm(P, p).

(39) For all functions f, g from SetVal(V, p) into SetVal(V, q) such that f = (Perm(P, p ⇒ q))(g) holds Perm(P, q) · g = f · Perm(P, p).

(40) Let given V, P be a permutation of V, and x be a set. Suppose x is a fixpoint of Perm(P, p). Let f be a function. If f is a fixpoint of Perm(P, p ⇒ q), then f(x) is a fixpoint of Perm(P, q).

6. Canonical Formulae

Let us consider p. We say that p is canonical if and only if: (Def. 7) For every V there exists a set x such that for every permutation P of V

holds x is a fixpoint of Perm(P, p). Let us observe that VERUM is canonical. Next we state several propositions:

(41) p ⇒ (q ⇒ p) is canonical.

(42) (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) is canonical.

(43) p ∧ q ⇒ p is canonical.

(44) p ∧ q ⇒ q is canonical.

(45) p ⇒ (q ⇒ p ∧ q) is canonical.

(46) If p is canonical and p ⇒ q is canonical, then q is canonical.

(47) If p ∈ HP TAUT, then p is canonical.

Let us observe that there exists an element of HP-WFF which is canonical.

7. Pseudo-Canonical Formulae

Let us consider p. We say that p is pseudo-canonical if and only if: (Def. 8) For every V and for every permutation P of V holds there exists a set

which is a fixpoint of Perm(P, p). Let us observe that every element of HP-WFF which is canonical is also

pseudo-canonical.

One can prove the following propositions:

(48) p ⇒ (q ⇒ p) is pseudo-canonical.

(49) (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) is pseudo-canonical.

446 andrzej trybulec

(50) p ∧ q ⇒ p is pseudo-canonical.

(51) p ∧ q ⇒ q is pseudo-canonical.

(52) p ⇒ (q ⇒ p ∧ q) is pseudo-canonical.

(53) If p is pseudo-canonical and p ⇒ q is pseudo-canonical, then q is pseudocanonical.

(54) Let given p, q, given V, and P be a permutation of V. Suppose there exists a set which is a fixpoint of Perm(P, p) and there exists no set which is a fixpoint of Perm(P, q). Then p ⇒ q is not pseudo-canonical.

(55) ((prop0 ⇒ prop1) ⇒ prop0) ⇒ prop0 is not pseudo-canonical.

References

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the canonical formulae 447

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

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Received July 4, 2000 448 andrzej trybulec

The Incompleteness of the Lattice of Substitutions

Adam Grabowski University of Białystok

Summary. In [11] we proved that the lattice of substitutions, as defined in [9], is a Heyting lattice (i.e. it is pseudo-complemented and it has the zero element). We show that the lattice needs not to be complete. Obviously, the example has to be infinite, namely we can take the set of natural numbers as variables and a singleton as a set of constants. The incompleteness has been shown for lattices of substitutions defined in terms of [22] and relational structures [18].

MML Identifier: HEYTING3.

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

1. Preliminaries

The scheme SSubsetUniq deals with a relational structure A and a unary predicate P, and states that:

Let A_{1}, A_{2} be subsets of A. Suppose for every set x holds x ∈ A_{1}

iff P[x] and for every set x holds x ∈ A_{2} iff P[x]. Then A_{1} = A_{2}

for all values of the parameters.

Let A, x be sets. Observe that [: A, {x}:] is function-like.

Next we state a number of propositions:

(1) For every odd natural number n holds 1 n.

(2) For every finite non empty subset X of ℕ holds maxX ∈ X.

(3) For every finite non empty subset X of ℕ there exists a natural number n such that X ⊆ Segn ∪ {0}.

(4) For every finite subset X of ℕ there exists an odd natural number k such that k ∉ X.

(5) Let k be a natural number and X be a finite non empty subset of [: ℕ, {k}:]. Then there exists a non empty natural number n such that X ⊆ [: Segn ∪ {0}, {k}:].

(6) Let m be a natural number and X be a finite non empty subset of [: ℕ, {m}:]. Then there exists a non empty natural number k such that ⟨⟨2·k+1, m⟩⟩ ∉ X.

(7) Let m be a natural number and X be a finite subset of [: ℕ, {m}:]. Then there exists a natural number k such that for every natural number l such that l k holds ⟨⟨l, m⟩⟩ ∉ X.

(8) For every upper-bounded lattice L holds ⊤_{L} = ⊤_{Poset(L)}.

(9) For every lower-bounded lattice L holds ⊥_{L} = ⊥_{Poset(L)}.

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

2. Poset of Substitutions

Next we state four propositions:

(11) For every set V and for every finite set C and for all elements A, B of Fin(V ˙→ C) such that A = ∅ and B ≠ ∅ holds B ↣ A = ∅.

(12) For all sets V, V′, C, C′ such that V ⊆ V′ and C ⊆ C′ holds SubstitutionSet(V, C) ⊆ SubstitutionSet(V′, C′).

(13) Let V, V′, C, C′ be sets, A be an element of Fin(V ˙→ C), and B be an element of Fin(V′ ˙→ C′). If V ⊆ V′and C ⊆ C′and A = B, then μA = μB.

(14) Let V, V′, C, C′be sets. Suppose V ⊆ V′and C ⊆ C′. Then the join operation of SubstLatt(V, C) = (the join operation of SubstLatt(V′, C′))↾[: the carrier of SubstLatt(V, C), the carrier of SubstLatt(V, C) :].

Let V, C be sets. The functor SubstPoset(V, C) yields a relational structure and is defined as follows:

(Def. 1) SubstPoset(V, C) = Poset(SubstLatt(V, C)).

Let V, C be sets. One can verify that SubstPoset(V, C) has l.u.b.’s and g.l.b.’s.

Let V, C be sets. One can verify that SubstPoset(V, C) is reflexive antisymmetric and transitive.

One can prove the following propositions:

(15) Let V, C be sets and a, b be elements of SubstPoset(V, C). Then a b if and only if for every set x such that x ∈ a there exists a set y such that y ∈ b and y ⊆ x.

(16) For all sets V, V′, C, C′ such that V ⊆ V′ and C ⊆ C′ holds

SubstPoset(V, C) is a full relational substructure of SubstPoset(V′, C′). Let n, k be natural numbers. The functor PF_{A}(n, k) yields an element of

ℕ ˙→ {k} and is defined as follows: (Def. 2) For every set x holds x ∈ PF_{A}(n, k) iff there exists an odd natural

number m such that m 2 · n and ⟨⟨m, k⟩⟩= x or ⟨⟨2 · n, k⟩⟩= x. Let n, k be natural numbers. One can verify that PF_{A}(n, k) is finite. Let n, k be natural numbers. The functor PF_{C}(n, k) yielding an element of

ℕ ˙→ {k} is defined by: (Def. 3) For every set x holds x ∈ PF_{C}(n, k) iff there exists an odd natural

number m such that m 2 · n + 1 and ⟨⟨m, k⟩⟩= x. Let n, k be natural numbers. Note that PF_{C}(n, k) is finite. The following four propositions are true:

(17) For all natural numbers n, k holds ⟨⟨2 · n + 1, k⟩⟩∈ PF_{C}(n, k).

(18) For all natural numbers n, k holds PF_{C}(n, k) ∩ {⟨⟨2 · n + 3, k⟩⟩} = ∅.

(19) For all natural numbers n, k holds PF_{C}(n+1, k) = PF_{C}(n, k)∪{⟨⟨2·n+3, k⟩⟩}.

(20) For all natural numbers n, k holds PF_{C}(n, k) ⊂ PF_{C}(n + 1, k). Let n, k be natural numbers. One can verify that PF_{A}(n, k) is non empty. Next we state three propositions:

(21) For all natural numbers n, m, k holds PF_{A}(n, m) ⊈ PF_{C}(k, m).

(22) For all natural numbers n, m, k such that n k holds PF_{C}(n, m) ⊆ PF_{C}(k, m).

(23) For every natural number n holds PF_{A}(1, n) = {⟨⟨1, n⟩⟩, ⟨⟨2, n⟩⟩}. Let n, k be natural numbers. The functor PF_{B}(n, k) yields an element of

Fin(ℕ ˙→ {k}) and is defined as follows: (Def. 4) For every set x holds x ∈ PF_{B}(n, k) iff there exists a non empty natural

number m such that m n and x = PF_{A}(m, k) or x = PF_{C}(n, k). The following propositions are true:

(24) For all natural numbers n, k and for every set x such that x ∈ PF_{B}(n + 1, k) there exists a set y such that y ∈ PF_{B}(n, k) and y ⊆ x.

(25) For all natural numbers n, k holds PF_{C}(n, k) ∉ PF_{B}(n + 1, k).

(26) For all natural numbers n, m, k such that PF_{A}(n, m) ⊆ PF_{A}(k, m) holds n = k.

(27) For all natural numbers n, m, k holds PF_{C}(n, m) ⊆ PF_{A}(k, m) iff n < k.

3. The Incompleteness

The following proposition is true

(28) For all natural numbers n, k holds PF_{B}(n, k) is an element of

SubstPoset(ℕ, {k}). Let k be a natural number. The functor PF_{D}(k) yielding a subset of SubstPoset(ℕ, {k}) is defined as follows:

(Def. 5) For every set x holds x ∈ PF_{D}(k) iff there exists a non empty natural

number n such that x = PF_{B}(n, k). The following propositions are true:

(29) For every natural number k holds PF_{B}(1, k) = {PF_{A}(1, k), PF_{C}(1, k)}.

(30) For every natural number k holds PF_{B}(1, k) ≠ {∅}.

Let k be a natural number. Note that PF_{B}(1, k) is non empty. We now state four propositions:

(31) For all natural numbers n, k holds {PF_{A}(n, k)} is an element of SubstPoset(ℕ, {k}).

(32) Let k be a natural number, V, X be sets, and a be an element of SubstPoset(V, {k}). If X ∈ a, then X is a finite subset of [: V, {k}:].

(33) Let m be a natural number and a be an element of SubstPoset(ℕ, {m}). Suppose PF_{D}(m) a. Let X be a non empty set. If X ∈ a, then it is not true that for every natural number n such that ⟨⟨n, m⟩⟩∈ X holds n is odd.

(34) Let k be a natural number, a, b be elements of SubstPoset(ℕ, {k}), and

X be a subset of SubstPoset(ℕ, {k}). If a X and b X, then a ⊔ b X. Let k be a natural number. Note that there exists an element of SubstPoset(ℕ, {k}) which is non empty. One can prove the following propositions:

(35) For every natural number n and for every element a of SubstPoset(ℕ, {n}) such that ∅ ∈ a holds a = {∅}.

(36) Let k be a natural number and a be a non empty element of SubstPoset(ℕ, {k}). If a ≠ {∅}, then there exists a finite function f such that f ∈ a and f ≠ ∅.

(37) Let k be a natural number, a be a non empty element of SubstPoset(ℕ, {k}), and a′ be an element of Fin(ℕ ˙→ {k}). If a ≠ {∅} and a = a′, then Involveda′ is a finite non empty subset of ℕ.

(38) Let k be a natural number, a be an element of SubstPoset(ℕ, {k}), a′ be an element of Fin(ℕ ˙→ {k}), and B be a finite non empty subset of ℕ. Suppose B = Involveda′ and a′ = a. Let X be a set. If X ∈ a, then for every natural number l such that l > maxB + 1 holds ⟨⟨l, k⟩⟩ ∉ X.

the incompleteness of the lattice of … 453

(39) For every natural number k holds ⊤_{SubstPoset (ℕ,{k})} = {∅}.

(40) For every natural number k holds ⊥_{SubstPoset (ℕ,{k})} = ∅.

(41) For every natural number k and for all elements a, b of SubstPoset(ℕ, {k}) such that a b and a = {∅} holds b = {∅}.

(42) For every natural number k and for all elements a, b of SubstPoset(ℕ, {k}) such that a b and b = ∅ holds a = ∅.

(43) For every natural number m and for every element a of SubstPoset(ℕ, {m}) such that PF_{D}(m) a holds a ≠ {∅}.

Let m be a natural number. One can verify that SubstPoset(ℕ, {m}) is non complete.

Let m be a natural number. One can check that SubstLatt(ℕ, {m}) is non complete.

References

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

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

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

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

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

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

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

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

[9] Adam Grabowski. Lattice of substitutions. Formalized Mathematics, 6(3):359–361, 1997.

[10] Adam Grabowski. Lattice of substitutions is a Heyting algebra. Formalized Mathematics, 7(2):323–327, 1998.

[11] Adam Grabowski. The incompleteness of the lattice of substitutions. Formalized Mathematics, 9(3):449–454, 2001.

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

[13] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

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

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

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

[17] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

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

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

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

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

454 adam grabowski

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

222, 1990.

Received July 17, 2000

Trigonometric Form of Complex Numbers

Robert Milewski University of Białystok

MML Identifier: COMPTRIG.

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

1. Preliminaries

One can prove the following propositions:

(1) Let F be an add-associative right zeroed right complementable left distributive non empty double loop structure and x be an element of the carrier of F. Then 0_{F} · x = 0_{F} .

(2) Let F be an add-associative right zeroed right complementable right distributive non empty double loop structure and x be an element of the carrier of F. Then x · 0_{F} = 0_{F} .

The scheme Regr without 0 concerns a unary predicate P, and states that:

P[1] provided the parameters meet the following conditions:

• There exists a non empty natural number k such that P[k], and

• For every non empty natural number k such that k ≠ 1 and P[k] there exists a non empty natural number n such that n < k and P[n].

One can prove the following propositions:

(3) For every element z of ℂ holds ℜ(z) − |z|.

(4) For every element z of ℂ holds ℑ(z) − |z|.

(5) For every element z of the carrier of ℂ_{F} holds ℜ(z) − |z|.

(6) For every element z of the carrier of ℂ_{F} holds ℑ(z) − |z|.

(7) For every element z of the carrier of ℂ_{F} holds |z|^{2} = ℜ(z)^{2} + ℑ(z)^{2}.

(8) For all real numbers x_{1}, x_{2}, y_{1}, y_{2} such that x_{1} + x_{2}i_{ℂ_{F}} = y_{1} + y_{2}i_{ℂ_{F}} holds x_{1} = y_{1} and x_{2} = y_{2}.

(9) For every element z of the carrier of ℂ_{F} holds z = ℜ(z) + ℑ(z)i_{ℂ_{F}}.

(10) 0_{ℂ_{F}} = 0 + 0i_{ℂ_{F}}.

(11) 0_{ℂ_{F}} = the zero of ℂ_{F}.

(12) For every unital non empty groupoid L and for every element x of the carrier of L holds power_{L}(x, 1) = x.

(13) For every unital non empty groupoid L and for every element x of the carrier of L holds power_{L}(x, 2) = x · x.

(14) Let L be an add-associative right zeroed right complementable right distributive unital non empty double loop structure and n be a natural number. If n > 0, then power_{L}(0_{L}, n) = 0_{L}.

(15) Let L be an associative commutative unital non empty groupoid, x, y be elements of the carrier of L, and n be a natural number. Then power_{L}(x·y,

n) = power_{L}(x, n) · power_{L}(y, n).

(16) For every real number x such that x > 0 and for every natural number n holds power_{ℂ_{F}}(x + 0i_{ℂ_{F}}, n) = x^{n} + 0i_{ℂ_{F}}.

(17) For every real number x and for every natural number n such that x 0 and n ≠ 0 holds ^{n} \sqrt{x}^{n} = x.

2. Sinus and Cosinus Properties

One can prove the following propositions: (20)^{1} π + \frac{π}{2} = \frac{3}{2} · π and \frac{3}{2} · π + \frac{π}{2} = 2 · π and \frac{3}{2} · π − π = \frac{π}{2} .

(21) 0 < \frac{π}{2} and \frac{π}{2} < π and 0 < π and − \frac{π}{2} < \frac{π}{2} and π < 2 · π and \frac{π}{2} < \frac{3}{2} · π and − \frac{π}{2} < 0 and 0 < 2 · π and π < \frac{3}{2} · π and \frac{3}{2} · π < 2 · π and 0 < \frac{3}{2} · π.

(22) For all real numbers a, b, c, x such that x ∈ ]a, c[ holds x ∈ ]a, b[ or x = b or x ∈ ]b, c[.

(23) For every real number x such that x ∈ ]0, π[ holds sin(x) > 0.

(24) For every real number x such that x ∈ [0, π] holds sin(x) 0.

(25) For every real number x such that x ∈ ]π, 2 · π[ holds sin(x) < 0.

(26) For every real number x such that x ∈ [π, 2 · π] holds sin(x) 0.

(27) For every real number x such that x ∈ ]− \frac{π}{2} , \frac{π}{2} [ holds cos(x) > 0.

(28) For every real number x such that x ∈ [− \frac{π}{2} , \frac{π}{2} ] holds cos(x) 0.

1The notation of π has been changed, previously ’Pai’. The propositions (18) and (19) have been removed.

(29) For every real number x such that x ∈ ]\frac{π}{2} , \frac{3}{2} · π[ holds cos(x) < 0.

(30) For every real number x such that x ∈ [\frac{π}{2} , \frac{3}{2} · π] holds cos(x) 0.

(31) For every real number x such that x ∈ ]\frac{3}{2} · π, 2 · π[ holds cos(x) > 0.

(32) For every real number x such that x ∈ [\frac{3}{2} · π, 2 · π] holds cos(x) 0.

(33) For every real number x such that 0 x and x < 2 · π and sinx = 0 holds x = 0 or x = π.

(34) For every real number x such that 0 x and x < 2 · π and cosx = 0 holds x = \frac{π}{2} or x = \frac{3}{2} · π.

(35) sin is increasing on ]− \frac{π}{2} , \frac{π}{2} [.

(36) sin is decreasing on ]\frac{π}{2} , \frac{3}{2} · π[.

(37) cos is decreasing on ]0, π[.

(38) cos is increasing on ]π, 2 · π[.

(39) sin is increasing on [− \frac{π}{2} , \frac{π}{2} ].

(40) sin is decreasing on [\frac{π}{2} , \frac{3}{2} · π].

(41) cos is decreasing on [0, π].

(42) cos is increasing on [π, 2 · π].

(43) sin is continuous on ℝ and for all real numbers x, y holds sin is continuous on [x, y] and sin is continuous on ]x, y[.

(44) cos is continuous on ℝ and for all real numbers x, y holds cos is continuous on [x, y] and cos is continuous on ]x, y[.

(45) For every real number x holds sin(x) ∈ [− 1, 1] and cos(x) ∈ [− 1, 1].

(46) rngsin = [− 1, 1].

(47) rngcos = [− 1, 1].

(48) rng(sin↾[− \frac{π}{2} , \frac{π}{2} ]) = [− 1, 1].

(49) rng(sin↾[\frac{π}{2} , \frac{3}{2} · π]) = [− 1, 1].

(50) rng(cos↾[0, π]) = [− 1, 1].

(51) rng(cos↾[π, 2 · π]) = [− 1, 1].

3. Argument of Complex Number

Let z be an element of the carrier of ℂ_{F}. The functor Argz yielding a real number is defined as follows:

(Def. 1)(i) z = |z| · cosArgz + (|z| · sinArgz)i_{ℂ_{F}} and 0 Argz and Argz < 2 · π

if z ≠ 0_{ℂ_{F}}, (ii) Argz = 0, otherwise.

One can prove the following propositions:

(52) For every element z of the carrier of ℂ_{F} holds 0 Argz and Argz < 2·π.

(53) For every real number x such that x 0 holds Argx + 0i_{ℂ_{F}} = 0.

(54) For every real number x such that x < 0 holds Argx + 0i_{ℂ_{F}} = π.

(55) For every real number x such that x > 0 holds Arg0 + xi_{ℂ_{F}} = \frac{π}{2} .

(56) For every real number x such that x < 0 holds Arg0 + xi_{ℂ_{F}} = \frac{3}{2} · π.

(57) Arg1_{ℂ_{F}} = 0.

(58) Argi_{ℂ_{F}} = \frac{π}{2} .

(59) For every element z of the carrier of ℂ_{F} holds Argz ∈ ]0, \frac{π}{2} [ iff ℜ(z) > 0 and ℑ(z) > 0.

(60) For every element z of the carrier of ℂ_{F} holds Argz ∈ ]\frac{π}{2} , π[ iff ℜ(z) < 0 and ℑ(z) > 0.

(61) For every element z of the carrier of ℂ_{F} holds Argz ∈ ]π, \frac{3}{2} ·π[ iff ℜ(z) < 0 and ℑ(z) < 0.

(62) For every element z of the carrier of ℂ_{F} holds Argz ∈ ]\frac{3}{2} · π, 2 · π[ iff ℜ(z) > 0 and ℑ(z) < 0.

(63) For every element z of the carrier of ℂ_{F} such that ℑ(z) > 0 holds sinArgz > 0.

(64) For every element z of the carrier of ℂ_{F} such that ℑ(z) < 0 holds sinArgz < 0.

(65) For every element z of the carrier of ℂ_{F} such that ℑ(z) 0 holds sinArgz 0.

(66) For every element z of the carrier of ℂ_{F} such that ℑ(z) 0 holds sinArgz 0.

(67) For every element z of the carrier of ℂ_{F} such that ℜ(z) > 0 holds cosArgz > 0.

(68) For every element z of the carrier of ℂ_{F} such that ℜ(z) < 0 holds cosArgz < 0.

(69) For every element z of the carrier of ℂ_{F} such that ℜ(z) 0 holds cosArgz 0.

(70) For every element z of the carrier of ℂ_{F} such that ℜ(z) 0 and z ≠ 0_{ℂ_{F}} holds cosArgz 0.

(71) For every real number x and for every natural number n holds power_{ℂ_{F}}(cosx + sinxi_{ℂ_{F}}, n) = cosn · x + sinn · xi_{ℂ_{F}}.

(72) Let z be an element of the carrier of ℂ_{F} and n be a natural number. If z ≠ 0_{ℂ_{F}} or n ≠ 0, then power_{ℂ_{F}}(z, n) = |z|^{n}·cosn·Argz+(|z|^{n}·sinn·Argz)i_{ℂ_{F}}.

(73) For every real number x and for all natural numbers n, k such that n ≠ 0 holds power_{ℂ_{F}}(cos \frac{x+2·π·k}{n}+ sin \frac{x+2·π·k}{n}i_{ℂ_{F}}, n) = cosx + sinxi_{ℂ_{F}}.

trigonometric form of complex numbers 459

(74) Let z be an element of the carrier of ℂ_{F} and n, k be natural numbers. If n ≠ 0, then z = power_{ℂ_{F}}( ^{n} \sqrt{|z|}· cos \frac{Arg z+2·π·k}{n}+ ( ^{n} \sqrt{|z|}· sin \frac{Arg z+2·π·k}{n})i_{ℂ_{F}}, n).

Let x be an element of the carrier of ℂ_{F} and let n be a non empty natural number. An element of ℂ_{F} is called a root of n, x if:

(Def. 2) power_{ℂ_{F}}(it, n) = x. We now state four propositions:

(75) Let x be an element of the carrier of ℂ_{F}, n be a non empty natural number, and k be a natural number. Then ^{n} \sqrt{|x|}· cos \frac{Arg x+2·π·k}{n}+ ( ^{n} \sqrt{|x|}· sin \frac{Arg x+2·π·k}{n})i_{ℂ_{F}} is a root of n, x.

(76) For every element x of the carrier of ℂ_{F} and for every root v of 1, x holds v = x.

(77) For every non empty natural number n and for every root v of n, 0_{ℂ_{F}} holds v = 0_{ℂ_{F}}.

(78) Let n be a non empty natural number, x be an element of the carrier of ℂ_{F}, and v be a root of n, x. If v = 0_{ℂ_{F}}, then x = 0_{ℂ_{F}}.

References

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

[2] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

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

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

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

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

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

[8] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2):265–269, 2001.

[9] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Formalized Mathematics, 9(2):363–371, 2001.

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

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

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

[13] Wojciech Skaba and Michał Muzalewski. From double loops to fields. Formalized Mathematics, 2(1):185–191, 1991.

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

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

460 robert milewski

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

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

Received July 21, 2000

Fundamental Theorem of Algebra^{1}

Robert Milewski University of Białystok

MML Identifier: POLYNOM5.

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

1. Preliminaries

The following propositions are true:

(1) For all natural numbers n, m such that n ≠ 0 and m ≠ 0 holds (n · m − n − m) + 1 0.

(2) For all real numbers x, y such that y > 0 holds \frac{min(x,y)}{max(x,y)} 1.

(3) For all real numbers x, y such that for every real number c such that c > 0 and c < 1 holds c · x y holds y 0.

(4) Let p be a finite sequence of elements of ℝ. Suppose that for every natural number n such that n ∈ domp holds p(n) 0. Let i be a natural number. If i ∈ domp, then ∑ p p(i).

(5) For all real numbers x, y holds − (x + yi_{ℂ_{F}}) = − x + (− y)i_{ℂ_{F}}.

(6) For all real numbers x_{1}, y_{1}, x_{2}, y_{2} holds (x_{1} + y_{1}i_{ℂ_{F}}) − (x_{2} + y_{2}i_{ℂ_{F}}) = (x_{1} − x_{2}) + (y_{1} − y_{2})i_{ℂ_{F}}.

(7) Let L be a commutative associative left unital distributive field-like non empty double loop structure and f, g, h be elements of the carrier of L. If h ≠ 0_{L}, then if h · g = h · f or g · h = f · h, then g = f.

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

In this article we present several logical schemes. The scheme ExDHGrStrSeq deals with a non empty groupoid A and a unary functor F yielding an element of the carrier of A, and states that:

There exists a sequence S of A such that for every natural number n holds S(n) = F(n)

for all values of the parameters.

The scheme ExDdoubleLoopStrSeq deals with a non empty double loop structure A and a unary functor F yielding an element of the carrier of A, and states that:

There exists a sequence S of A such that for every natural number n holds S(n) = F(n)

for all values of the parameters.

Next we state the proposition

(8) For every element z of the carrier of ℂ_{F} such that z ≠ 0_{ℂ_{F}} and for every natural number n holds |power_{ℂ_{F}}(z, n)| = |z|^{n}.

Let p be a finite sequence of elements of the carrier of ℂ_{F}. The functor |p| yields a finite sequence of elements of ℝ and is defined by:

(Def. 1) len|p| = lenp and for every natural number n such that n ∈ domp holds

|p|_{n} = |p_{n}|. We now state several propositions:

(9) |ε_{(the carrier of ℂ_{F})}| = ε_{ℝ}.

(10) For every element x of the carrier of ℂ_{F} holds |⟨x⟩| = ⟨|x|⟩.

(11) For all elements x, y of the carrier of ℂ_{F} holds |⟨x, y⟩| = ⟨|x|, |y|⟩.

(12) For all elements x, y, z of the carrier of ℂ_{F} holds |⟨x, y, z⟩| = ⟨|x|, |y|, |z|⟩.

(13) For all finite sequences p, q of elements of the carrier of ℂ_{F} holds |p^{⌢}q| = |p|^{⌢} |q|.

(14) Let p be a finite sequence of elements of the carrier of ℂ_{F} and x be an element of the carrier of ℂ_{F}. Then |p^{⌢} ⟨x⟩| = |p|^{⌢} ⟨|x|⟩ and |⟨x⟩^{⌢} p| = ⟨|x|⟩^{⌢} |p|.

(15) For every finite sequence p of elements of the carrier of ℂ_{F} holds | ∑ p| ∑ |p|.

2. Operations on Polynomials

Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure, let p be a Polynomial of L, and let n be a natural number. The functor p^{n} yields a sequence of L and is defined by:

(Def. 2) p^{n} = power_{Polynom-Ring L}(p, n).

Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure, let p be a Polynomial of L, and let n be a natural number. One can verify that p^{n} is finite-Support.

One can prove the following propositions:

(16) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and p be a Polynomial of L. Then p^{0} = 1. L.

(17) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and p be a Polynomial of L. Then p^{1} = p.

(18) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and p be a Polynomial of L. Then p^{2} = p ∗ p.

(19) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and p be a Polynomial of L. Then p^{3} = p ∗ p ∗ p.

(20) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure, p be a Polynomial of L, and n be a natural number. Then p^{n+1} = p^{n} ∗ p.

(21) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and n be a natural number. Then (0. L)^{n+1} = 0. L.

(22) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and n be a natural number. Then (1. L)^{n} = 1. L.

(23) Let L be a field, p be a Polynomial of L, x be an element of the carrier of L, and n be a natural number. Then eval(p^{n}, x) = power_{L}(eval(p, x), n).

(24) Let L be a field and p be a Polynomial of L. If lenp ≠ 0, then for every

natural number n holds len(p^{n}) = (n · lenp − n) + 1. Let L be a non empty groupoid, let p be a sequence of L, and let v be an

element of the carrier of L. The functor v ·p yields a sequence of L and is defined by:

(Def. 3) For every natural number n holds (v · p)(n) = v · p(n).

Let L be an add-associative right zeroed right complementable right distributive non empty double loop structure, let p be a Polynomial of L, and let v be an element of the carrier of L. Observe that v · p is finite-Support.

We now state several propositions:

(25) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure and p be a Polynomial of L. Then len(0_{L} · p) = 0.

(26) Let L be an add-associative right zeroed right complementable left unital commutative associative distributive field-like non empty double loop structure, p be a Polynomial of L, and v be an element of the carrier of L. If v ≠ 0_{L}, then len(v · p) = lenp.

(27) Let L be an add-associative right zeroed right complementable left distributive non empty double loop structure and p be a sequence of L. Then 0_{L} · p = 0. L.

(28) For every left unital non empty multiplicative loop structure L and for every sequence p of L holds 1_{L} · p = p.

(29) Let L be an add-associative right zeroed right complementable right distributive non empty double loop structure and v be an element of the carrier of L. Then v · 0. L = 0. L.

(30) Let L be an add-associative right zeroed right complementable right unital right distributive non empty double loop structure and v be an element of the carrier of L. Then v · 1. L = ⟨v⟩.

(31) Let L be an add-associative right zeroed right complementable left unital distributive commutative associative field-like non empty double loop structure, p be a Polynomial of L, and v, x be elements of the carrier of L. Then eval(v · p, x) = v · eval(p, x).

(32) Let L be an add-associative right zeroed right complementable right distributive unital non empty double loop structure and p be a Polynomial of L. Then eval(p, 0_{L}) = p(0).

Let L be a non empty zero structure and let z_{0}, z_{1} be elements of the carrier of L. The functor ⟨z_{0}, z_{1}⟩ yields a sequence of L and is defined by:

(Def. 4) ⟨z_{0}, z_{1}⟩ = 0. L +·(0, z_{0}) +·(1, z_{1}).

The following propositions are true:

(33) Let L be a non empty zero structure and z_{0} be an element of the carrier of L. Then ⟨z_{0}⟩(0) = z_{0} and for every natural number n such that n 1 holds ⟨z_{0}⟩(n) = 0_{L}.

(34) For every non empty zero structure L and for every element z_{0} of the carrier of L such that z_{0} ≠ 0_{L} holds len⟨z_{0}⟩ = 1.

(35) For every non empty zero structure L holds ⟨0_{L}⟩ = 0. L.

(36) Let L be an add-associative right zeroed right complementable distributive commutative associative left unital field-like non empty double loop structure and x, y be elements of the carrier of L. Then ⟨x⟩ ∗ ⟨y⟩ = ⟨x · y⟩.

(37) Let L be an Abelian add-associative right zeroed right complementable right unital associative commutative distributive field-like non empty double loop structure, x be an element of the carrier of L, and n be a natural number. Then ⟨x⟩^{n} = ⟨power_{L}(x, n)⟩.

(38) Let L be an add-associative right zeroed right complementable unital non empty double loop structure and z_{0}, x be elements of the carrier of L. Then eval(⟨z_{0}⟩, x) = z_{0}.

(39) Let L be a non empty zero structure and z_{0}, z_{1} be elements of the carrier of L. Then ⟨z_{0}, z_{1}⟩(0) = z_{0} and ⟨z_{0}, z_{1}⟩(1) = z_{1} and for every natural number n such that n 2 holds ⟨z_{0}, z_{1}⟩(n) = 0_{L}.

Let L be a non empty zero structure and let z_{0}, z_{1} be elements of the carrier of L. One can verify that ⟨z_{0}, z_{1}⟩ is finite-Support.

The following propositions are true:

(40) For every non empty zero structure L and for all elements z_{0}, z_{1} of the carrier of L holds len⟨z_{0}, z_{1}⟩ 2.

(41) For every non empty zero structure L and for all elements z_{0}, z_{1} of the carrier of L such that z_{1} ≠ 0_{L} holds len⟨z_{0}, z_{1}⟩ = 2.

(42) For every non empty zero structure L and for every element z_{0} of the carrier of L such that z_{0} ≠ 0_{L} holds len⟨z_{0}, 0_{L}⟩ = 1.

(43) For every non empty zero structure L holds ⟨0_{L}, 0_{L}⟩ = 0. L.

(44) For every non empty zero structure L and for every element z_{0} of the carrier of L holds ⟨z_{0}, 0_{L}⟩ = ⟨z_{0}⟩.

(45) Let L be an add-associative right zeroed right complementable left distributive unital non empty double loop structure and z_{0}, z_{1}, x be elements of the carrier of L. Then eval(⟨z_{0}, z_{1}⟩, x) = z_{0} + z_{1} · x.

(46) Let L be an add-associative right zeroed right complementable left distributive unital non empty double loop structure and z_{0}, z_{1}, x be elements of the carrier of L. Then eval(⟨z_{0}, 0_{L}⟩, x) = z_{0}.

(47) Let L be an add-associative right zeroed right complementable left distributive unital non empty double loop structure and z_{0}, z_{1}, x be elements of the carrier of L. Then eval(⟨0_{L}, z_{1}⟩, x) = z_{1} · x.

(48) Let L be an add-associative right zeroed right complementable left distributive well unital non empty double loop structure and z_{0}, z_{1}, x be elements of the carrier of L. Then eval(⟨z_{0}, 1_{L}⟩, x) = z_{0} + x.

(49) Let L be an add-associative right zeroed right complementable left distributive well unital non empty double loop structure and z_{0}, z_{1}, x be elements of the carrier of L. Then eval(⟨0_{L}, 1_{L}⟩, x) = x.

3. Substitution in Polynomials

Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and let p, q be Polynomials of L. The functor p[q] yielding a Polynomial of L is defined by the condition (Def. 5).

(Def. 5) There exists a finite sequence F of elements of the carrier of Polynom-RingL such that p[q] = ∑ F and lenF = lenp and for every natural number n such that n ∈ domF holds F (n) = p(n − ′1) · q^{n− ′1}.

One can prove the following propositions:

(50) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and p be a Polynomial of L. Then (0. L)[p] = 0. L.

(51) Let L be an Abelian add-associative right zeroed right complementable right unital commutative distributive non empty double loop structure and p be a Polynomial of L. Then p[0. L] = ⟨p(0)⟩.

(52) Let L be an Abelian add-associative right zeroed right complementable right unital associative commutative distributive field-like non empty double loop structure, p be a Polynomial of L, and x be an element of the carrier of L. Then len(p[⟨x⟩]) 1.

(53) For every field L and for all Polynomials p, q of L such that lenp ≠ 0 and lenq > 1 holds len(p[q]) = (lenp · lenq − lenp − lenq) + 2.

(54) Let L be a field, p, q be Polynomials of L, and x be an element of the carrier of L. Then eval(p[q], x) = eval(p, eval(q, x)).

4. Fundamental Theorem of Algebra

Let L be a unital non empty double loop structure, let p be a Polynomial of L, and let x be an element of the carrier of L. We say that x is a root of p if and only if:

(Def. 6) eval(p, x) = 0_{L}.

Let L be a unital non empty double loop structure and let p be a Polynomial of L. We say that p has roots if and only if:

(Def. 7) There exists an element x of the carrier of L such that x is a root of p.

The following proposition is true

(55) For every unital non empty double loop structure L holds 0. L has roots.

Let L be a unital non empty double loop structure. One can verify that 0. L has roots.

The following proposition is true

(56) Let L be a unital non empty double loop structure and x be an element

of the carrier of L. Then x is a root of 0. L. Let L be a unital non empty double loop structure. One can verify that there

exists a Polynomial of L which has roots.

Let L be a unital non empty double loop structure. We say that L is algebraic-closed if and only if:

(Def. 8) For every Polynomial p of L such that lenp > 1 holds p has roots.

Let L be a unital non empty double loop structure and let p be a Polynomial of L. The functor Rootsp yields a subset of L and is defined by:

(Def. 9) For every element x of the carrier of L holds x ∈ Rootsp iff x is a root

of p. Let L be a commutative associative left unital distributive field-like non

empty double loop structure and let p be a Polynomial of L. The functor NormPolynomialp yielding a sequence of L is defined as follows:

(Def. 10) For every natural number n holds (NormPolynomialp)(n) = \frac{p(n)}{p(len p− ′1)}.

Let L be an add-associative right zeroed right complementable commutative associative left unital distributive field-like non empty double loop structure and let p be a Polynomial of L. Note that NormPolynomialp is finite-Support.

The following propositions are true:

(57) Let L be a commutative associative left unital distributive field-like non empty double loop structure and p be a Polynomial of L. If lenp ≠ 0, then (NormPolynomialp)(lenp − ′1) = 1_{L}.

(58) For every field L and for every Polynomial p of L such that lenp ≠ 0 holds lenNormPolynomialp = lenp.

(59) Let L be a field and p be a Polynomial of L. Suppose lenp ≠ 0. Let x be an element of the carrier of L. Then eval(NormPolynomialp, x) = \frac{eval(p,x)}{p(len p− ′1)}.

(60) Let L be a field and p be a Polynomial of L. Suppose lenp ≠ 0. Let x be an element of the carrier of L. Then x is a root of p if and only if x is a root of NormPolynomialp.

(61) For every field L and for every Polynomial p of L such that lenp ≠ 0 holds p has roots iff NormPolynomialp has roots.

(62) For every field L and for every Polynomial p of L such that lenp ≠ 0 holds Rootsp = RootsNormPolynomialp.

(63) id_{ℂ} is continuous on ℂ.

(64) For every element x of ℂ holds ℂ ⟼→ x is continuous on ℂ.

Let L be a unital non empty groupoid, let x be an element of the carrier of L, and let n be a natural number. The functor FPower(x, n) yields a map from L into L and is defined as follows:

(Def. 11) For every element y of the carrier of L holds (FPower(x, n))(y) = x ·

power_{L}(y, n). The following propositions are true:

(65) For every unital non empty groupoid L holds FPower(1_{L}, 1) = id_{the carrier of L}.

(66) FPower(1_{ℂ_{F}}, 2) = id_{ℂ} id_{ℂ}.

(67) For every unital non empty groupoid L and for every element x of the carrier of L holds FPower(x, 0) = (the carrier of L) ⟼→ x.

(68) For every element x of the carrier of ℂ_{F} there exists an element x_{1} of ℂ such that x = x_{1} and FPower(x, 1) = x_{1} id_{ℂ}.

(69) For every element x of the carrier of ℂ_{F} there exists an element x_{1} of ℂ such that x = x_{1} and FPower(x, 2) = x_{1} (id_{ℂ} id_{ℂ}).

(70) Let x be an element of the carrier of ℂ_{F} and n be a natural number. Then there exists a function f from ℂ into ℂ such that f = FPower(x, n) and FPower(x, n + 1) = f id_{ℂ}.

(71) Let x be an element of the carrier of ℂ_{F} and n be a natural number. Then there exists a function f from ℂ into ℂ such that f = FPower(x, n) and f is continuous on ℂ.

Let L be a unital non empty double loop structure and let p be a Polynomial of L. The functor Polynomial-Function(L, p) yields a map from L into L and is defined as follows:

(Def. 12) For every element x of the carrier of L holds

(Polynomial-Function(L, p))(x) = eval(p, x). The following propositions are true:

(72) For every Polynomial p of ℂ_{F} there exists a function f from ℂ into ℂ such that f = Polynomial-Function(ℂ_{F}, p) and f is continuous on ℂ.

(73) Let p be a Polynomial of ℂ_{F}. Suppose lenp > 2 and |p(lenp− ′1)| = 1. Let F be a finite sequence of elements of ℝ. Suppose lenF = lenp and for every natural number n such that n ∈ domF holds F (n) = |p(n − ′1)|. Let z be an element of the carrier of ℂ_{F}. If |z| > ∑ F, then | eval(p, z)| > |p(0)| + 1.

(74) Let p be a Polynomial of ℂ_{F}. Suppose lenp > 2. Then there exists an element z_{0} of the carrier of ℂ_{F} such that for every element z of the carrier of ℂ_{F} holds | eval(p, z)| | eval(p, z_{0})|.

(75) For every Polynomial p of ℂ_{F} such that lenp > 1 holds p has roots.

Let us note that ℂ_{F} is algebraic-closed.

fundamental theorem of algebra 469

Let us mention that there exists a left unital right unital non empty double loop structure which is algebraic-closed, add-associative, right zeroed, right complementable, Abelian, commutative, associative, distributive, field-like, and non degenerated.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

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

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

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

[5] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

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

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

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

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

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

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

[12] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2):265–269, 2001.

[13] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Formalized Mathematics, 9(2):363–371, 2001.

[14] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391–395, 2001.

[15] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.

[16] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex sequence and continuity of complex function. Formalized Mathematics, 9(1):185–190, 2001.

[17] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[18] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97–104, 1991.

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

[20] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

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

[22] Wojciech Skaba and Michał Muzalewski. From double loops to fields. Formalized Mathematics, 2(1):185–191, 1991.

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

[24] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1(5):979–981, 1990.

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

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

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

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470 robert milewski

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1990.

Received August 21, 2000

On Replace Function and Swap Function for Finite Sequences

Hiroshi Yamazaki Shinshu University

Nagano Yoshinori Fujisawa Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this article, we show the property of the Replace Function and the Swap Function of finite sequences. In the first section, we prepared some useful theorems for finite sequences. In the second section, we defined the Replace function and proved some theorems about the function. This function replaces an element of a sequence by another value. In the third section, we defined the Swap function and proved some theorems about the function. This function swaps two elements of a sequence. In the last section, we show the property of composed functions of the Replace Function and the Swap Function.

MML Identifier: FINSEQ 7.

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

1. Some Basic Theorems

For simplicity, we adopt the following rules: D denotes a non empty set, f, g, h denote finite sequences of elements of D, p, p_{1}, p_{2}, p_{3}, q denote elements of D, and i, j, k, l, n denote natural numbers.

One can prove the following propositions:

(1) If 1 i and j lenf and i < j, then f = (f↾(i − ′1))^{⌢} ⟨f(i)⟩^{⌢} (f_{⇂i}↾(j − ′ i − ′1))^{⌢} ⟨f(j)⟩^{⌢} (f_{⇂j}).

(2) If leng = lenh and leng < i and i len(g^{⌢}f), then (g^{⌢}f)(i) = (h^{⌢}f)(i).

(3) If 1 i and i lenf, then f(i) = (g^{⌢} f)(leng + i).

(4) If i ∈ dom(f_{⇂n}), then f_{⇂n}(i) = f(n + i).

2. Definition of Replace Function and its Properties

Let D be a non empty set, let f be a finite sequence of elements of D, let i be a natural number, and let p be an element of D. Then f +·(i, p) is a finite sequence of elements of D and it can be characterized by the condition:

(Def. 1) f +·(i, p) = \{ \begin{matrix}(f↾(i − ′1))^{⌢} ⟨p⟩^{⌢} (f_{⇂i}), if 1 i and i lenf, \\ f, otherwise. \end{matrix}

We introduce Replace(f, i, p) as a synonym of f +·(i, p).

The following propositions are true:

(5) Replace(f, 0, p) = f.

(6) If i > lenf, then Replace(f, i, p) = f.

(7) lenReplace(f, i, p) = lenf.

(8) rngReplace(f, i, p) ⊆ rngf ∪ {p}.

(9) If 1 i and i lenf, then p ∈ rngReplace(f, i, p).

(10) If 1 i and i lenf, then (Replace(f, i, p))_{i} = p.

(11) If 1 i and i lenf, then for every k such that 0 < k and k lenf − i holds (Replace(f, i, p))(i + k) = f_{⇂i}(k).

(12) If 1 k and k lenf and k ≠ i, then (Replace(f, i, p))_{k} = f_{k}.

(13) If 1 i and i < j and j lenf, then Replace(Replace(f, j, q), i, p) = (f↾(i − ′1))^{⌢} ⟨p⟩^{⌢} (f_{⇂i}↾(j − ′i − ′1))^{⌢} ⟨q⟩^{⌢} (f_{⇂j}).

(14) Replace(⟨p⟩, 1, q) = ⟨q⟩.

(15) Replace(⟨p_{1}, p_{2}⟩, 1, q) = ⟨q, p_{2}⟩.

(16) Replace(⟨p_{1}, p_{2}⟩, 2, q) = ⟨p_{1}, q⟩.

(17) Replace(⟨p_{1}, p_{2}, p_{3}⟩, 1, q) = ⟨q, p_{2}, p_{3}⟩.

(18) Replace(⟨p_{1}, p_{2}, p_{3}⟩, 2, q) = ⟨p_{1}, q, p_{3}⟩.

(19) Replace(⟨p_{1}, p_{2}, p_{3}⟩, 3, q) = ⟨p_{1}, p_{2}, q⟩.

3. Definition of Swap Function and its Properties

Let D be a non empty set, let f be a finite sequence of elements of D, and let i, j be natural numbers. The functor Swap(f, i, j) yields a finite sequence of elements of D and is defined as follows:

(Def. 2) Swap(f, i, j) = \{ \begin{matrix}Replace(Replace(f, i, f_{j}), j, f_{i}), if 1 i and i lenf \\ and 1 j and j lenf, \\ f, otherwise. \end{matrix}

Next we state a number of propositions:

(20) lenSwap(f, i, j) = lenf.

(21) Swap(f, i, i) = f.

(22) Swap(Swap(f, i, j), j, i) = f.

(23) Swap(f, i, j) = Swap(f, j, i).

(24) rngSwap(f, i, j) = rngf.

(25) Swap(⟨p_{1}, p_{2}⟩, 1, 2) = ⟨p_{2}, p_{1}⟩.

(26) Swap(⟨p_{1}, p_{2}, p_{3}⟩, 1, 2) = ⟨p_{2}, p_{1}, p_{3}⟩.

(27) Swap(⟨p_{1}, p_{2}, p_{3}⟩, 1, 3) = ⟨p_{3}, p_{2}, p_{1}⟩.

(28) Swap(⟨p_{1}, p_{2}, p_{3}⟩, 2, 3) = ⟨p_{1}, p_{3}, p_{2}⟩.

(29) If 1 i and i < j and j lenf, then Swap(f, i, j) = (f↾(i − ′1))^{⌢} ⟨f_{j}⟩^{⌢} (f_{⇂i}↾(j − ′i − ′1))^{⌢} ⟨f_{i}⟩^{⌢} (f_{⇂j}).

(30) If 1 < i and i lenf, then Swap(f, 1, i) = ⟨f_{i}⟩^{⌢}(f_{⇂1}↾(i− ′2))^{⌢}⟨f_{1}⟩^{⌢}(f_{⇂i}).

(31) If 1 i and i < lenf, then Swap(f, i, lenf) = (f↾(i − ′1)) ^{⌢} ⟨f_{len f}⟩ ^{⌢} (f_{⇂i}↾(lenf − ′i − ′1))^{⌢} ⟨f_{i}⟩.

(32) If i ≠ k and j ≠ k and 1 k and k lenf, then (Swap(f, i, j))_{k} = f_{k}.

(33) If 1 i and i lenf and 1 j and j lenf, then (Swap(f, i, j))_{i} = f_{j} and (Swap(f, i, j))_{j} = f_{i}.

4. Properties of Combination Function of Replace Function and Swap Function

We now state four propositions:

(34) If 1 i and i lenf and 1 j and j lenf, then Replace(Swap(f, i, j), i, p) = Swap(Replace(f, j, p), i, j).

(35) If i ≠ k and j ≠ k and 1 i and i lenf and 1 j and j lenf and 1 k and k lenf, then Swap(Replace(f, k, p), i, j) = Replace(Swap(f, i, j), k, p).

(36) If i ≠ k and j ≠ k and 1 i and i lenf and 1 j and j lenf and 1 k and k lenf, then Swap(Swap(f, i, j), j, k) = Swap(Swap(f, i, k), i, j).

(37) Suppose i ≠ k and j ≠ k and l ≠ i and l ≠ j and 1 i and i lenf and 1 j and j lenf and 1 k and k lenf and 1 l and l lenf. Then Swap(Swap(f, i, j), k, l) = Swap(Swap(f, k, l), i, j).

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.

474 hiroshi yamazaki et al.

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

[4] 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.

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

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

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

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

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

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

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

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

The Correctness of the High Speed Array Multiplier Circuits

Hiroshi Yamazaki Shinshu University Nagano Katsumi Wasaki Shinshu University Nagano

Summary. This article introduces the verification of the correctness for the operations and the specification of the high speed array multiplier. We formalize the concepts of 2-by-2 and 3-by-3 bit Plain array multiplier, 3-by-3 Wallace tree multiplier circuit, and show that outputs of the array multiplier are equivalent to outputs of normal (sequencial) multiplier.

MML Identifier: GATE 5.

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

1. Preliminaries

Let x_{0}, x_{1}, y_{0}, y_{1} be sets. The functor MULT_{210}(x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined as follows:

(Def. 1) MULT_{210}(x_{1}, y_{1}, x_{0}, y_{0}) = AND2(x_{0}, y_{0}).

The functor MULT_{211}(x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined by: (Def. 2) MULT_{211}(x_{1}, y_{1}, x_{0}, y_{0}) = ADD1(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅).

The functor MULT_{212}(x_{1}, y_{1}, x_{0}, y_{0}) yielding a set is defined as follows: (Def. 3) MULT_{212}(x_{1}, y_{1}, x_{0}, y_{0}) = ADD2(∅, AND2(x_{1}, y_{1}), AND2(x_{1}, y_{0}),

AND2(x_{0}, y_{1}), ∅). The functor MULT_{213}(x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined as follows:

(Def. 4) MULT_{213}(x_{1}, y_{1}, x_{0}, y_{0}) = CARR2(∅, AND2(x_{1}, y_{1}), AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅).

We now state the proposition

(1) Let x_{0}, x_{1}, y_{0}, y_{1}, z_{0}, z_{1}, z_{2}, z_{3}, q_{0}, q_{1}, c_{1}, q_{11}, c_{11} be sets such that NE q_{0} iff NE AND2(x_{0}, y_{0}) and NE q_{1} iff NE XOR3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE c_{1} iff NE MAJ3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE q_{11} iff NE XOR3(AND2(x_{1}, y_{1}), ∅, c_{1}) and NE c_{11} iff NE MAJ3(AND2(x_{1}, y_{1}), ∅, c_{1}) and NE z_{0} iff NE q_{0} and NE z_{1} iff NE q_{1} and NE z_{2} iff NE q_{11} and NE z_{3} iff NE c_{11}. Then

(i) NE z_{0} iff NE MULT_{210}(x_{1}, y_{1}, x_{0}, y_{0}), (ii) NE z_{1} iff NE MULT_{211}(x_{1}, y_{1}, x_{0}, y_{0}), (iii) NE z_{2} iff NE MULT_{212}(x_{1}, y_{1}, x_{0}, y_{0}), and (iv) NE z_{3} iff NE MULT_{213}(x_{1}, y_{1}, x_{0}, y_{0}). Let x_{0}, x_{1}, x_{2}, y_{0}, y_{1} be sets. The functor MULT_{310}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yields

a set and is defined by: (Def. 5) MULT_{310}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = AND2(x_{0}, y_{0}).

The functor MULT_{311}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined as follows: (Def. 6) MULT_{311}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = ADD1(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅).

The functor MULT_{312}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined as follows: (Def. 7) MULT_{312}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = ADD2(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}),

AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅). The functor MULT_{313}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined as follows:

(Def. 8) MULT_{313}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = ADD3(∅, AND2(x_{2}, y_{1}), AND2(x_{2}, y_{0}),

AND2(x_{1}, y_{1}), AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅). The functor MULT_{314}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yielding a set is defined by:

(Def. 9) MULT_{314}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = CARR3(∅, AND2(x_{2}, y_{1}), AND2(x_{2}, y_{0}),

AND2(x_{1}, y_{1}), AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅). Let x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2} be sets. The functor MULT_{321}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0})

yields a set and is defined by: (Def. 10) MULT_{321}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = ADD1(MULT_{312}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}),

AND2(x_{0}, y_{2}), ∅). The functor MULT_{322}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined by:

(Def. 11) MULT_{322}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = ADD2(MULT_{313}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}),

AND2(x_{1}, y_{2}), MULT_{312}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), AND2(x_{0}, y_{2}), ∅). The functor MULT_{323}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yields a set and is defined as follows:

(Def. 12) MULT_{323}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = ADD3(MULT_{314}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), AND2(x_{2}, y_{2}), MULT_{313}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), AND2(x_{1}, y_{2}), MULT_{312}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), AND2(x_{0}, y_{2}), ∅).

The functor MULT_{324}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) yielding a set is defined as follows: (Def. 13) MULT_{324}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}) = CARR3(MULT_{314}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}),

AND2(x_{2}, y_{2}), MULT_{313}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), AND2(x_{1}, y_{2}), MULT_{312}(x_{2}, x_{1},

y_{1}, x_{0}, y_{0}), AND2(x_{0}, y_{2}), ∅). Next we state the proposition

(2) Let x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}, z_{0}, z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, q_{0}, q_{1}, q_{2}, c_{1}, c_{2}, q_{11}, q_{12}, c_{11}, c_{12}, q_{21}, q_{22}, c_{21}, c_{22} be sets such that NE q_{0} iff NE AND2(x_{0}, y_{0}) and NE q_{1} iff NE XOR3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE c_{1} iff NE MAJ3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE q_{2} iff NE XOR3(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}), ∅) and NE c_{2} iff NE MAJ3(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}), ∅) and NE q_{11} iff NE XOR3(q_{2}, AND2 (x_{0}, y_{2}), c_{1}) and NE c_{11} iff NE MAJ3(q_{2}, AND2(x_{0}, y_{2}), c_{1}) and NE q_{12} iff NE XOR3(AND2(x_{2}, y_{1}), AND2(x_{1}, y_{2}), c_{2}) and NE c_{12} iff NE MAJ3(AND2(x_{2}, y_{1}), AND2(x_{1}, y_{2}), c_{2}) and NE q_{21} iff NE XOR3(q_{12}, ∅, c_{11}) and NE c_{21} iff NE MAJ3(q_{12}, ∅, c_{11}) and NE q_{22} iff NE XOR3(AND2(x_{2}, y_{2}), c_{21}, c_{12}) and NE c_{22} iff NE MAJ3(AND2(x_{2}, y_{2}), c_{21}, c_{12}) and NE z_{0} iff NE q_{0} and NE z_{1} iff NE q_{1} and NE z_{2} iff NE q_{11} and NE z_{3} iff NE q_{21} and NE z_{4} iff NE q_{22} and NE z_{5} iff NE c_{22}. Then

(i) NE z_{0} iff NE MULT_{310}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (ii) NE z_{1} iff NE MULT_{311}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (iii) NE z_{2} iff NE MULT_{321}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (iv) NE z_{3} iff NE MULT_{322}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}),

(v) NE z_{4} iff NE MULT_{323}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}), and (vi) NE z_{5} iff NE MULT_{324}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}).

2. Logical Equivalence of Wallace Tree Multiplier

One can prove the following proposition

(3) Let x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}, z_{0}, z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, q_{0}, q_{1}, q_{2}, q_{3}, c_{1}, c_{2}, c_{3}, q_{11}, q_{12}, q_{13}, c_{11}, c_{12}, c_{13} be sets such that NE q_{0} iff NE AND2(x_{0}, y_{0}) and NE q_{1} iff NE XOR3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE c_{1} iff NE MAJ3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE q_{2} iff NE XOR3(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}), AND2(x_{0}, y_{2})) and NE c_{2} iff NE MAJ3(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}), AND2(x_{0}, y_{2})) and NE q_{3} iff NE XOR3(AND2(x_{2}, y_{1}), AND2(x_{1}, y_{2}), ∅) and NE c_{3} iff NE MAJ3(AND2(x_{2}, y_{1}), AND2(x_{1}, y_{2}), ∅) and NE q_{11} iff NE XOR3(q_{2}, c_{1}, ∅) and NE c_{11} iff NE MAJ3(q_{2}, c_{1}, ∅) and NE q_{12} iff NE XOR3(q_{3}, c_{2}, c_{11}) and NE c_{12} iff NE MAJ3(q_{3}, c_{2}, c_{11}) and NE q_{13} iff NE XOR3(AND2(x_{2}, y_{2}), c_{3}, c_{12}) and NE c_{13} iff NE MAJ3(AND2(x_{2}, y_{2}), c_{3}, c_{12}) and NE z_{0} iff NE q_{0} and NE z_{1} iff NE q_{1} and NE z_{2} iff NE q_{11} and NE z_{3} iff NE q_{12} and NE z_{4} iff NE q_{13} and NE z_{5} iff NE c_{13}. Then

(i) NE z_{0} iff NE MULT_{310}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (ii) NE z_{1} iff NE MULT_{311}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (iii) NE z_{2} iff NE MULT_{321}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (iv) NE z_{3} iff NE MULT_{322}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}),

(v) NE z_{4} iff NE MULT_{323}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}), and (vi) NE z_{5} iff NE MULT_{324}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}). Let a_{1}, b_{1}, c be sets. We introduce CLAADD1(a_{1}, b_{1}, c) as a synonym of

XOR3(a_{1}, b_{1}, c). We introduce CLACARR1(a_{1}, b_{1}, c) as a synonym of MAJ3(a_{1}, b_{1}, c). Let a_{1}, b_{1}, a_{2}, b_{2}, c be sets. The functor CLAADD2(a_{2}, b_{2}, a_{1}, b_{1}, c) yields a

set and is defined by: (Def. 16)^{1} CLAADD2(a_{2}, b_{2}, a_{1}, b_{1}, c) = XOR3(a_{2}, b_{2}, MAJ3(a_{1}, b_{1}, c)).

The functor CLACARR2(a_{2}, b_{2}, a_{1}, b_{1}, c) yields a set and is defined by: (Def. 17) CLACARR2(a_{2}, b_{2}, a_{1}, b_{1}, c) = OR2(AND2(a_{2}, b_{2}), AND2(OR2(a_{2}, b_{2}),

MAJ3(a_{1}, b_{1}, c))). Let a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, c be sets. The functor CLAADD3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1},

c) yielding a set is defined as follows: (Def. 18) CLAADD3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) = XOR3(a_{3}, b_{3}, CLACARR2(a_{2}, b_{2}, a_{1},

b_{1}, c)). The functor CLACARR3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yields a set and is defined by:

(Def. 19) CLACARR3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) = OR3(AND2(a_{3}, b_{3}), AND2(OR2(a_{3},

b_{3}), AND2(a_{2}, b_{2})), AND3(OR2(a_{3}, b_{3}), OR2(a_{2}, b_{2}), MAJ3(a_{1}, b_{1}, c))). Let a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, a_{4}, b_{4}, c be sets. The functor CLAADD4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yielding a set is defined by:

(Def. 20) CLAADD4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) = XOR3(a_{4}, b_{4}, CLACARR3(a_{3},

b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c)). The functor CLACARR4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yielding a set is defined as follows:

(Def. 21) CLACARR4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) = OR4(AND2(a_{4}, b_{4}), AND2 (OR2(a_{4}, b_{4}), AND2(a_{3}, b_{3})), AND3(OR2(a_{4}, b_{4}), OR2(a_{3}, b_{3}), AND2(a_{2}, b_{2})), AND4(OR2(a_{4}, b_{4}), OR2(a_{3}, b_{3}), OR2(a_{2}, b_{2}), MAJ3(a_{1}, b_{1}, c))).

One can prove the following proposition

(4) Let x_{0}, x_{1}, x_{2}, y_{0}, y_{1}, y_{2}, z_{0}, z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, q_{0}, q_{1}, q_{2}, q_{3}, c_{1}, c_{2}, c_{3} be sets such that NE q_{0} iff NE AND2(x_{0}, y_{0}) and NE q_{1} iff NE XOR3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE c_{1} iff NE MAJ3(AND2(x_{1}, y_{0}), AND2(x_{0}, y_{1}), ∅) and NE q_{2} iff NE XOR3(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}), AND2(x_{0}, y_{2})) and NE c_{2} iff NE MAJ3(AND2(x_{2}, y_{0}), AND2(x_{1}, y_{1}), AND2(x_{0}, y_{2})) and NE q_{3} iff NE XOR3(AND2(x_{2}, y_{1}), AND2(x_{1}, y_{2}), ∅) and NE c_{3} iff NE MAJ3(AND2(x_{2}, y_{1}), AND2(x_{1}, y_{2}), ∅) and NE z_{0} iff NE q_{0} and NE z_{1} iff NE q_{1} and NE z_{2} iff NE CLAADD1(q_{2}, c_{1}, ∅) and NE z_{3} iff NE CLAADD2(q_{3}, c_{2}, q_{2}, c_{1}, ∅) and NE z_{4} iff NE CLAADD3(AND2(x_{2}, y_{2}), c_{3}, q_{3}, c_{2}, q_{2}, c_{1}, ∅) and NE z_{5} iff NE CLACARR3(AND2(x_{2}, y_{2}), c_{3}, q_{3}, c_{2}, q_{2}, c_{1}, ∅). Then

1The definitions (Def. 14) and (Def. 15) have been removed.

(i) NE z_{0} iff NE MULT_{310}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (ii) NE z_{1} iff NE MULT_{311}(x_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (iii) NE z_{2} iff NE MULT_{321}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}), (iv) NE z_{3} iff NE MULT_{322}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}),

(v) NE z_{4} iff NE MULT_{323}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}), and (vi) NE z_{5} iff NE MULT_{324}(x_{2}, y_{2}, x_{1}, y_{1}, x_{0}, y_{0}).

References

[1] Yatsuka Nakamura. Logic gates and logical equivalence of adders. Formalized Mathematics, 8(1):35–45, 1999.

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

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

480 hiroshi yamazaki and katsumi wasaki

Some Lemmas for the Jordan Curve Theorem^{1}

Andrzej Trybulec University of Białystok

Summary. I present some miscellaneous simple facts that are still missing in the library. The only common feature is that, most of them, were needed as lemmas in the proof of the Jordan curve theorem.

MML Identifier: JCT MISC.

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

1. Preliminaries

The scheme NonEmpty deals with a non empty set A and a unary functor F yielding a set, and states that:

{F(a) : a ranges over elements of A} is non empty for all values of the parameters.

One can prove the following propositions:

(1) For all sets A, B, C such that A ⊆ B and A misses C holds A ⊆ B \ C.

(2) For all sets X, Y such that X meets ⋃ Y there exists a set Z such that Z ∈ Y and X meets Z.

(3) For all sets A, B and for every function f such that A ⊆ domf and f^{∘}A ⊆ B holds A ⊆ f^{−1} (B).

(4) For every function f and for all sets A, B such that A misses B holds f^{−1} (A) misses f^{−1} (B).

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(5) Let S, X be sets, f be a function from S into X, and A be a subset of X such that if X = ∅, then S = ∅. Then (f^{−1} (A))^{c} = f^{−1} (A^{c}).

(6) Let S be a 1-sorted structure, X be a non empty set, f be a function from the carrier of S into X, and A be a subset of X. Then − f^{−1} (A) = f^{−1} (A^{c}).

We use the following convention: i, j, m, n denote natural numbers and r, s, r_{0}, s_{0}, t denote real numbers.

Next we state several propositions:

(7) If m n, then n − ′(n − ′m) = m.

(8) For every real number r such that 1 r and i j holds r^{i} r^{j}.

(9) For all real numbers a, b such that r ∈ [a, b] and s ∈ [a, b] holds \frac{r+s}{2} ∈ [a, b].

(10) For every increasing sequence N_{1} of naturals and for all i, j such that i j holds N_{1}(i) N_{1}(j).

(11) ||r_{0} − s_{0}| − |r − s|| |r_{0} − r| + |s_{0} − s|.

(12) If t ∈ ]r, s[, then |t| < max(|r|, |s|).

Let A, B, C be non empty sets and let f be a function from A into [: B, C:]. Then pr1(f) is a function from A into B and it can be characterized by the condition:

(Def. 1) For every element x of A holds pr1(f)(x) = f(x)_{1}. Then pr2(f) is a function from A into C and it can be characterized by the condition:

(Def. 2) For every element x of A holds pr2(f)(x) = f(x)_{2}.

The scheme DoubleChoice deals with non empty sets A, B, C and a ternary predicate P, and states that:

There exists a function a from A into B and there exists a function b from A into C such that for every element i of A holds P[i, a(i), b(i)]

provided the parameters meet the following requirement:

• For every element i of A there exists an element a_{1} of B and there exists an element b_{1} of C such that P[i, a_{1}, b_{1}].

We now state the proposition

(13) Let S, T be non empty topological spaces and G be a subset of [: S, T :]. Suppose that for every point x of [: S, T :] such that x ∈ G there exists a subset G_{1} of S and there exists a subset G_{2} of T such that G_{1} is open and G_{2} is open and x ∈ [: G_{1}, G_{2} :] and [: G_{1}, G_{2} :] ⊆ G. Then G is open.

2. Topological Properties of Sets of Real Numbers

One can prove the following proposition

(14) For all compact subsets A, B of ℝ holds A ∩ B is compact.

Let A be a subset of ℝ. We say that A is connected if and only if: (Def. 3) For all real numbers r, s such that r ∈ A and s ∈ A holds [r, s] ⊆ A.

The following proposition is true

(15) Let T be a non empty topological space, f be a continuous real map of

T, and A be a subset of T. If A is connected, then f^{∘}A is connected. Let A, B be subsets of ℝ. The functor ρ(A, B) yielding a real number is

defined by: (Def. 4) There exists a subset X of ℝ such that X = {|r − s|; r ranges over

elements of ℝ, s ranges over elements of ℝ: r ∈ A ∧ s ∈ B} and ρ(A, B) = inf X.

Let us notice that the functor ρ(A, B) is commutative.

The following propositions are true:

(16) For all subsets A, B of ℝ and for all r, s such that r ∈ A and s ∈ B holds |r − s| ρ(A, B).

(17) For all subsets A, B of ℝ and for all non empty subsets C, D of ℝ such that C ⊆ A and D ⊆ B holds ρ(A, B) ρ(C, D).

(18) For all non empty compact subsets A, B of ℝ there exist real numbers r, s such that r ∈ A and s ∈ B and ρ(A, B) = |r − s|.

(19) For all non empty compact subsets A, B of ℝ holds ρ(A, B) 0.

(20) For all non empty compact subsets A, B of ℝ such that A misses B holds ρ(A, B) > 0.

(21) Let e, f be real numbers and A, B be compact subsets of ℝ. Suppose A misses B and A ⊆ [e, f] and B ⊆ [e, f]. Let S be a function from ℕ into 2^{ℝ}. Suppose that for every natural number i holds S(i) is connected and S(i) meets A and S(i) meets B. Then there exists a real number r such that r ∈ [e, f] and r ∉ A ∪ B and for every natural number i there exists a natural number k such that i k and r ∈ S(k).

References

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

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

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

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

484 andrzej trybulec

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mahlo and Inaccessible Cardinals

Josef Urban Charles University Praha

Summary. This article contains basic ordinal topology: closed unbounded and stationary sets and necessary theorems about them, completness of the centered system of Clubs of M, Mahlo and strongly Mahlo cardinals, the proof that (strongly) Mahlo is (strongly) inaccessible, and the proof that Rank of strongly inaccessible is a model of ZF.

MML Identifier: CARD LAR.

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

1. Clubs and Mahlo Cardinals

Let S be a set, let X be a set, and let Y be a subset of S. Then X ∩ Y is a subset of S.

Let us observe that every ordinal number which is cardinal and infinite is also limit.

Let us note that every ordinal number which is non empty and limit is also infinite.

Let us mention that every aleph which is non limit is also non countable.

Let us observe that there exists an aleph which is regular and non countable.

We use the following convention: A, B denote limit infinite ordinal numbers, B_{1}, B_{2}, B_{3}, C denote ordinal numbers, and X denotes a set.

Let us consider A, X. We say that X is unbounded in A if and only if: (Def. 1) X ⊆ A and supX = A.

We say that X is closed in A if and only if: (Def. 2) X ⊆ A and for every B such that B ∈ A holds if sup(X ∩ B) = B, then

B ∈ X. Let us consider A, X. We say that X is club in A if and only if:

(Def. 3) X is closed in A and X is unbounded in A.

Next we state the proposition

(1) X is club in A iff X is closed in A and X is unbounded in A. In the sequel X is a subset of A. Let us consider A, X. We say that X is unbounded if and only if:

(Def. 4) supX = A.

We introduce X is bounded as an antonym of X is unbounded. We say that X is closed if and only if:

(Def. 5) For every B such that B ∈ A holds if sup(X ∩ B) = B, then B ∈ X. We now state several propositions:

(2) X is club in A iff X is closed and unbounded.

(3) X ⊆ supX.

(4) Suppose X is non empty and for every B_{1} such that B_{1} ∈ X there exists B_{2} such that B_{2} ∈ X and B_{1} ∈ B_{2}. Then supX is limit infinite ordinal number.

(5) X is bounded iff there exists B_{1} such that B_{1} ∈ A and X ⊆ B_{1}.

(6) If sup(X ∩ B) ≠ B, then there exists B_{1} such that B_{1} ∈ B and X ∩ B ⊆ B_{1}.

(7) X is unbounded iff for every B_{1} such that B_{1} ∈ A there exists C such that C ∈ X and B_{1} ⊆ C.

(8) If X is unbounded, then X is non empty.

(9) If X is unbounded and B_{1} ∈ A, then there exists an element B_{3} of A such that B_{3} ∈ {B_{2}; B_{2} ranges over elements of A: B_{2} ∈ X ∧ B_{1} ∈ B_{2}}.

Let us consider A, X, B_{1}. Let us assume that X is unbounded. And let us assume that B_{1} ∈ A. The functor LBound(B_{1}, X) yields an element of X and is defined by:

(Def. 6) LBound(B_{1}, X) = inf{B_{2}; B_{2} ranges over elements of A: B_{2} ∈ X ∧ B_{1} ∈

B_{2}}. Next we state two propositions:

(10) If X is unbounded and B_{1} ∈ A, then LBound(B_{1}, X) ∈ X and B_{1} ∈ LBound(B_{1}, X).

(11) Ω_{A} is closed and unbounded.

Let A be a set, let X be a subset of A, and let Y be a set. Then X \ Y is a subset of A.

Next we state two propositions:

(12) If B_{1} ∈ A and X is closed and unbounded, then X \ B_{1} is closed and unbounded.

(13) If B_{1} ∈ A, then A \ B_{1} is closed and unbounded.

Let us consider A, X. We say that X is stationary if and only if: (Def. 7) For every subset Y of A such that Y is closed and unbounded holds

X ∩ Y is non empty. The following proposition is true

(14) For all subsets X, Y of A such that X is stationary and X ⊆ Y holds Y

is stationary. Let us consider A and let X be a set. We say that X is stationary in A if

and only if: (Def. 8) X ⊆ A and for every subset Y of A such that Y is closed and unbounded

holds X ∩ Y is non empty. One can prove the following proposition

(15) For all sets X, Y such that X is stationary in A and X ⊆ Y and Y ⊆ A

holds Y is stationary in A. Let X be a set and let S be a family of subsets of X. We see that the element

of S is a subset of X.

The following proposition is true

(16) If X is stationary, then X is unbounded.

Let us consider A, X. The functor limpointsX yields a subset of A and is defined as follows:

(Def. 9) limpointsX = {B_{1}; B_{1} ranges over elements of A: B_{1} is infinite and

limit ∧ sup(X ∩ B_{1}) = B_{1}}. We now state four propositions:

(17) If X ∩ B_{3} ⊆ B_{1}, then B_{3} ∩ limpointsX ⊆ succB_{1}.

(18) If X ⊆ B_{1}, then limpointsX ⊆ succB_{1}.

(19) limpointsX is closed.

(20) Suppose X is unbounded and limpointsX is bounded. Then there exists B_{1} such that B_{1} ∈ A and {succB_{2}; B_{2} ranges over elements of A: B_{2} ∈ X ∧ B_{1} ∈ succB_{2}} is club in A.

In the sequel M is a non countable aleph and X is a subset of M. Let us consider M. One can verify that there exists an element of M which

is cardinal and infinite. In the sequel N denotes a cardinal infinite element of M. Next we state several propositions:

(21) For every aleph M and for every subset X of M such that X is unbounded holds cf M \overline{\overline{X}} .

(22) For every family S of subsets of M such that every element of S is closed holds ⋂ S is closed.

(23) If ℵ_{0} < cf M, then for every function f from ℕ into X holds suprngf ∈ M.

(24) Suppose ℵ_{0} < cf M. Let S be a non empty family of subsets of M. If \overline{\overline{S}} < cf M and every element of S is closed and unbounded, then ⋂ S is closed and unbounded.

(25) If ℵ_{0} < cf M and X is unbounded, then for every B_{1} such that B_{1} ∈ M there exists B such that B ∈ M and B_{1} ∈ B and B ∈ limpointsX.

(26) If ℵ_{0} < cf M and X is unbounded, then limpointsX is unbounded.

Let us consider M. We say that M is Mahlo if and only if: (Def. 10) {N : N is regular} is stationary in M.

We say that M is strongly Mahlo if and only if: (Def. 11) {N : N is strongly inaccessible} is stationary in M.

We now state several propositions:

(27) If M is strongly Mahlo, then M is Mahlo.

(28) If M is Mahlo, then M is regular.

(29) If M is Mahlo, then M is limit.

(30) If M is Mahlo, then M is inaccessible.

(31) If M is strongly Mahlo, then M is strong limit.

(32) If M is strongly Mahlo, then M is strongly inaccessible.

2. Proof that Strongly Inaccessible is Model of ZF

We adopt the following convention: A denotes an ordinal number, x, y denote sets, and X, Y denote sets.

The following propositions are true:

(33) Suppose that for every x such that x ∈ X there exists y such that y ∈ X and x ⊆ y and y is a cardinal number. Then ⋃ X is a cardinal number.

(34) For every aleph M such that \overline{\overline{X}} < cf M and for every Y such that Y ∈ X holds \overline{\overline{Y}} < M holds \overline{\overline{⋃ X}} ∈ M.

(35) If M is strongly inaccessible and A ∈ M, then \overline{\overline{R_{A}}} < M.

(36) If M is strongly inaccessible, then \overline{\overline{R_{M}}} = M.

(37) If M is strongly inaccessible, then R_{M} is a Tarski class.

(38) For every non empty ordinal number A holds R_{A} is non empty. Let A be a non empty ordinal number. One can check that R_{A} is non empty. Next we state two propositions:

mahlo and inaccessible cardinals 489

(39) If M is strongly inaccessible, then R_{M} is a universal class.

(40) If M is strongly inaccessible, then R_{M} is model of ZF.

References

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

[2] Grzegorz Bancerek. Models and satisfiability. Formalized Mathematics, 1(1):191–199, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

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

[5] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.

[6] Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65–69, 1991.

[7] Grzegorz Bancerek. On powers of cardinals. Formalized Mathematics, 3(1):89–93, 1992.

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

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

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

[11] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

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

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

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

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

[16] Josef Urban. Basic facts about inaccessible and measurable cardinals. Formalized Mathematics, 9(2):323–329, 2001.

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

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

490 josef urban

Basic Properties of Extended Real Numbers

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. We introduce product, quotient and absolute value, and we prove some basic properties of extended real numbers.

MML Identifier: EXTREAL1.

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

1. Preliminaries

In this paper x, y, z denote extended real numbers and a denotes a real number.

One can prove the following propositions:

(1) If x ≠ +∞ and x ≠ −∞, then x is a real number.

(2) −∞ < +∞.

(3) If x < y, then x ≠ +∞ and y ≠ −∞.

(4) x = +∞ iff − x = −∞ and x = −∞ iff − x = +∞.

(5) If x ≠ +∞ or y ≠ −∞ and if x ≠ −∞ or y ≠ +∞, then x − − y = x + y.

(6) If x ≠ +∞ or y ≠ +∞ and if x ≠ −∞ or y ≠ −∞, then x + − y = x − y.

(7) If x ≠ −∞ and y ≠ +∞ and x y, then x ≠ +∞ and y ≠ −∞.

(8) Suppose x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ and y ≠ +∞ or z ≠ −∞ but y ≠ −∞ or z ≠ +∞ and x ≠ +∞ or z ≠ −∞ but x ≠ −∞ or z ≠ +∞. Then (x + y) + z = x + (y + z).

(9) If −∞ < x and x < +∞, then x + − x = 0\overline{ℝ} and − x + x = 0\overline{ℝ}.

(10) If x ≠ +∞ or y ≠ +∞ and if x ≠ −∞ or y ≠ −∞, then x − y = x + − y.

(11) Suppose x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ and y ≠ +∞ or z ≠ +∞ but y ≠ −∞ or z ≠ −∞ and x + y ≠ +∞ or y − z ≠ −∞ but x + y ≠ −∞ or y − z ≠ +∞. Then (x + y) − z = x + (y − z).

2. Operations of Multiplication, Quotient and Absolute Value on Extended Real Numbers

Let x, y be extended real numbers. The functor x · y yields an extended real number and is defined by the conditions (Def. 1).

(Def. 1)(i) There exist real numbers a, b such that x = a and y = b and x·y = a·b,

or (ii) 0\overline{ℝ} < x and y = +∞ or 0\overline{ℝ} < y and x = +∞ or x < 0\overline{ℝ} and y = −∞

or y < 0\overline{ℝ} and x = −∞ but x · y = +∞, or (iii) x < 0\overline{ℝ} and y = +∞ or y < 0\overline{ℝ} and x = +∞ or 0\overline{ℝ} < x and y = −∞

or 0\overline{ℝ} < y and x = −∞ but x · y = −∞, or (iv) x = 0\overline{ℝ} or y = 0\overline{ℝ} but x · y = 0\overline{ℝ}. The following propositions are true:

(12) Let x, y be extended real numbers. Then

(i) there exist real numbers a, b such that x = a and y = b and x · y = a · b, or

(ii) 0\overline{ℝ} < x and y = +∞ or 0\overline{ℝ} < y and x = +∞ or x < 0\overline{ℝ} and y = −∞

or y < 0\overline{ℝ} and x = −∞ but x · y = +∞, or (iii) x < 0\overline{ℝ} and y = +∞ or y < 0\overline{ℝ} and x = +∞ or 0\overline{ℝ} < x and y = −∞

or 0\overline{ℝ} < y and x = −∞ but x · y = −∞, or (iv) x = 0\overline{ℝ} or y = 0\overline{ℝ} but x · y = 0\overline{ℝ}.

(13) For all extended real numbers x, y and for all real numbers a, b such that x = a and y = b holds x · y = a · b.

(14) For every extended real number x such that 0\overline{ℝ} < x holds +∞ · x = +∞ and x · +∞ = +∞ and −∞ · x = −∞ and x · −∞ = −∞.

(15) For every extended real number x such that x < 0\overline{ℝ} holds +∞ · x = −∞ and x · +∞ = −∞ and −∞ · x = +∞ and x · −∞ = +∞.

(16) For all extended real numbers x, y such that x = 0\overline{ℝ} holds x · y = 0\overline{ℝ} and y · x = 0\overline{ℝ}.

(17) For all extended real numbers x, y holds x · y = y · x.

Let x, y be extended real numbers. Let us notice that the functor x · y is commutative.

One can prove the following propositions:

(18) If x = a, then 0 < a iff 0\overline{ℝ} < x.

(19) If x = a, then a < 0 iff x < 0\overline{ℝ}.

(20) If 0\overline{ℝ} < x and 0\overline{ℝ} < y or x < 0\overline{ℝ} and y < 0\overline{ℝ}, then 0\overline{ℝ} < x · y.

(21) If 0\overline{ℝ} < x and y < 0\overline{ℝ} or x < 0\overline{ℝ} and 0\overline{ℝ} < y, then x · y < 0\overline{ℝ}.

(22) x · y = 0\overline{ℝ} iff x = 0\overline{ℝ} or y = 0\overline{ℝ}.

(23) (x · y) · z = x · (y · z).

(24) − 0\overline{ℝ} = 0\overline{ℝ}.

(25) 0\overline{ℝ} < x iff − x < 0\overline{ℝ} and x < 0\overline{ℝ} iff 0\overline{ℝ} < − x.

(26) − x · y= x · − y and − x · y= (− x) · y.

(27) If x ≠ +∞ and x ≠ −∞ and x · y = +∞, then y = +∞ or y = −∞.

(28) If x ≠ +∞ and x ≠ −∞ and x · y = −∞, then y = +∞ or y = −∞.

(29) If y ≠ +∞ or z ≠ −∞ but y ≠ −∞ or z ≠ +∞ and x ≠ +∞ and x ≠ −∞, then x · (y + z) = x · y + x · z.

(30) If y ≠ +∞ or z ≠ +∞ but y ≠ −∞ or z ≠ −∞ and x ≠ +∞ and

x ≠ −∞, then x · (y − z) = x · y − x · z. Let x, y be extended real numbers. Let us assume that x = −∞ or x = +∞

but y = −∞ or y = +∞ but y ≠ 0\overline{ℝ}. The functor \frac{x}{y} yielding an extended real number is defined by the conditions (Def. 2).

(Def. 2)(i) There exist real numbers a, b such that x = a and y = b and \frac{x}{y} = \frac{a}{b},

or (ii) x = +∞ and 0\overline{ℝ} < y or x = −∞ and y < 0\overline{ℝ} but \frac{x}{y} = +∞, or (iii) x = −∞ and 0\overline{ℝ} < y or x = +∞ and y < 0\overline{ℝ} but \frac{x}{y} = −∞, or (iv) y = −∞ or y = +∞ but \frac{x}{y} = 0\overline{ℝ}. The following four propositions are true:

(31) Let x, y be extended real numbers. Suppose x = −∞ or x = +∞ but

y = −∞ or y = +∞ but y ≠ 0\overline{ℝ}. Then

(i) there exist real numbers a, b such that x = a and y = b and \frac{x}{y} = \frac{a}{b}, or (ii) x = +∞ and 0\overline{ℝ} < y or x = −∞ and y < 0\overline{ℝ} but \frac{x}{y} = +∞, or (iii) x = −∞ and 0\overline{ℝ} < y or x = +∞ and y < 0\overline{ℝ} but \frac{x}{y} = −∞, or (iv) y = −∞ or y = +∞ but \frac{x}{y} = 0\overline{ℝ}.

(32) Let x, y be extended real numbers. Suppose y ≠ 0\overline{ℝ}. Let a, b be real numbers. If x = a and y = b, then \frac{x}{y} = \frac{a}{b}.

(33) For all extended real numbers x, y such that x ≠ −∞ but x ≠ +∞ but y = −∞ or y = +∞ holds \frac{x}{y} = 0\overline{ℝ}.

(34) For every extended real number x such that x ≠ −∞ and x ≠ +∞ and

x ≠ 0\overline{ℝ} holds \frac{x}{x} = 1. Let x be an extended real number. The functor |x| yielding an extended real

number is defined as follows: (Def. 3) |x| = \{ \begin{matrix}x, if 0\overline{ℝ} x, \\ − x, otherwise. \end{matrix}

One can prove the following propositions:

(35) For every extended real number x such that 0\overline{ℝ} x holds |x| = x.

(36) For every extended real number x such that 0\overline{ℝ} < x holds |x| = x.

(37) For every extended real number x such that x < 0\overline{ℝ} holds |x| = − x.

(38) For all real numbers a, b holds \overline{ℝ}(a · b) = \overline{ℝ}(a) · \overline{ℝ}(b).

(39) For all real numbers a, b such that b ≠ 0 holds \overline{ℝ}(\frac{a}{b}) = \frac{\overline{ℝ}(a)}{\overline{ℝ}(b)}.

(40) For all extended real numbers x, y such that x y and x < +∞ and −∞ < y holds 0\overline{ℝ} y − x.

(41) For all extended real numbers x, y such that x < y and x < +∞ and −∞ < y holds 0\overline{ℝ} < y − x.

(42) If x y and 0\overline{ℝ} z, then x · z y · z.

(43) If x y and z 0\overline{ℝ}, then y · z x · z.

(44) If x < y and 0\overline{ℝ} < z and z ≠ +∞, then x · z < y · z.

(45) If x < y and z < 0\overline{ℝ} and z ≠ −∞, then y · z < x · z.

(46) Suppose x is a real number and y is a real number. Then x < y if and only if there exist real numbers p, q such that p = x and q = y and p < q.

(47) If x ≠ −∞ and y ≠ +∞ and x y and 0\overline{ℝ} < z, then \frac{x}{z} \frac{y}{z} .

(48) If x y and 0\overline{ℝ} < z and z ≠ +∞, then \frac{x}{z} \frac{y}{z} .

(49) If x ≠ −∞ and y ≠ +∞ and x y and z < 0\overline{ℝ}, then \frac{y}{z} \frac{x}{z}.

(50) If x y and z < 0\overline{ℝ} and z ≠ −∞, then \frac{y}{z} \frac{x}{z}.

(51) If x < y and 0\overline{ℝ} < z and z ≠ +∞, then \frac{x}{z} < \frac{y}{z} .

(52) If x < y and z < 0\overline{ℝ} and z ≠ −∞, then \frac{y}{z} < \frac{x}{z}.

References

[1] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[2] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[3] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21–26, 1996.

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

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

Definitions and Basic Properties of Measurable Functions

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article we introduce some definitions concerning measurable functions and prove related properties.

MML Identifier: MESFUNC1.

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

1. Cardinal Numbers of ℤ and ℚ

In this paper k is a natural number, r is a real number, i is an integer, and q is a rational number.

The subset ℤ_{−} of ℝ is defined as follows: (Def. 1) r ∈ ℤ_{−} iff there exists k such that r = − k.

Let us observe that ℤ_{−} is non empty. Next we state three propositions:

(1) ℕ ≈ ℤ_{−} .

(2) ℤ = ℤ_{−} ∪ ℕ.

(3) ℕ ≈ ℤ. ℤ is a subset of ℝ. Let n be a natural number. The functor ℚ(n) yields a subset of ℚ and is

defined as follows: (Def. 2) q ∈ ℚ(n) iff there exists i such that q = \frac{i}{n}.

Let n be a natural number. Observe that ℚ(n + 1) is non empty.

We now state two propositions:

(4) For every natural number n holds ℤ ≈ ℚ(n + 1).

(5) ℕ ≈ ℚ.

2. Basic Operations of Extended Real Valued Functions

Let C be a non empty set, let f be a partial function from C to \overline{ℝ}, and let x be a set. Then f(x) is an extended real number.

Let C be a non empty set and let f_{1}, f_{2} be partial functions from C to \overline{ℝ}. The functor f_{1} + f_{2} yielding a partial function from C to \overline{ℝ} is defined by:

(Def. 3) dom(f_{1} + f_{2}) = domf_{1} ∩ domf_{2} \ (f_{1}^{−1} ({−∞}) ∩ f_{2}^{−1} ({+∞}) ∪ f_{1}^{−1} ({+∞}) ∩ f_{2}^{−1} ({−∞})) and for every element c of C such that

c ∈ dom(f_{1} + f_{2}) holds (f_{1} + f_{2})(c) = f_{1}(c) + f_{2}(c).

The functor f_{1} − f_{2} yields a partial function from C to \overline{ℝ} and is defined by: (Def. 4) dom(f_{1} − f_{2}) = domf_{1} ∩ domf_{2} \ (f_{1}^{−1} ({+∞}) ∩ f_{2}^{−1} ({+∞}) ∪

f_{1}^{−1} ({−∞}) ∩ f_{2}^{−1} ({−∞})) and for every element c of C such that

c ∈ dom(f_{1} − f_{2}) holds (f_{1} − f_{2})(c) = f_{1}(c) − f_{2}(c).

The functor f_{1} f_{2} yields a partial function from C to \overline{ℝ} and is defined as follows: (Def. 5) dom(f_{1} f_{2}) = domf_{1} ∩ domf_{2} and for every element c of C such that

c ∈ dom(f_{1} f_{2}) holds (f_{1} f_{2})(c) = f_{1}(c) · f_{2}(c). Let C be a non empty set, let f be a partial function from C to \overline{ℝ}, and let

r be a real number. The functor r f yielding a partial function from C to \overline{ℝ} is defined as follows:

(Def. 6) dom(r f) = domf and for every element c of C such that c ∈ dom(r f)

holds (r f)(c) = \overline{ℝ}(r) · f(c). The following proposition is true

(6) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and r be a real number. Suppose r ≠ 0. Let c be an element of C. If c ∈ dom(r f), then f(c) = \frac{(r f)(c)}{\overline{ℝ}(r)}.

Let C be a non empty set and let f be a partial function from C to \overline{ℝ}. The functor − f yielding a partial function from C to \overline{ℝ} is defined by:

(Def. 7) dom(− f) = domf and for every element c of C such that c ∈ dom(− f)

holds (− f)(c) = − f(c). The extended real number \overline{1} is defined by:

(Def. 8) \overline{1} = 1.

Let C be a non empty set, let f be a partial function from C to \overline{ℝ}, and let r be a real number. The functor \frac{r}{f} yielding a partial function from C to \overline{ℝ} is defined by:

(Def. 9) dom(\frac{r}{f}) = domf \ f^{−1} (\{0\overline{ℝ}\}) and for every element c of C such that

c ∈ dom(\frac{r}{f}) holds (\frac{r}{f})(c) = \frac{\overline{ℝ}(r)}{f(c)}. One can prove the following proposition

(7) Let C be a non empty set and f be a partial function from C to \overline{ℝ}. Then dom(\frac{1}{f}) = domf \ f^{−1} (\{0\overline{ℝ}\}) and for every element c of C such that c ∈ dom(\frac{1}{f}) holds (\frac{1}{f})(c) = \frac{\overline{1}}{f(c)}.

Let C be a non empty set and let f be a partial function from C to \overline{ℝ}. The functor |f| yields a partial function from C to \overline{ℝ} and is defined as follows:

(Def. 10) dom|f| = domf and for every element c of C such that c ∈ dom|f|

holds |f|(c) = |f(c)|. We now state three propositions:

(8) For all extended real numbers x, y such that x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ holds x + y = y + x.

(9) For every non empty set C and for all partial functions f_{1}, f_{2} from C to \overline{ℝ} holds f_{1} + f_{2} = f_{2} + f_{1}.

(10) For every non empty set C and for all partial functions f_{1}, f_{2} from C to

\overline{ℝ} holds f_{1} f_{2} = f_{2} f_{1}. Let C be a non empty set and let f_{1}, f_{2} be partial functions from C to \overline{ℝ}.

Let us note that the functor f_{1} + f_{2} is commutative. Let us observe that the functor f_{1} f_{2} is commutative.

3. Level Sets

Next we state several propositions:

(11) For every real number r there exists a natural number n such that r n.

(12) For every real number r there exists a natural number n such that − n r.

(13) For all real numbers r, s such that r < s there exists a natural number n such that \frac{1}{n+1} < s − r.

(14) For all real numbers r, s such that for every natural number n holds r − \frac{1}{n+1} s holds r s.

(15) For every extended real number a such that for every real number r holds \overline{ℝ}(r) < a holds a = +∞.

(16) For every extended real number a such that for every real number r

holds a < \overline{ℝ}(r) holds a = −∞. Let X be a set, let S be a σ-field of subsets of X, and let A be a set. We say

that A is measurable on S if and only if: (Def. 11) A ∈ S.

One can prove the following proposition

(17) Let X, A be sets and S be a σ-field of subsets of X. Then A is measurable on S if and only if for every σ-measure M on S holds A is measurable w.r.t. M.

For simplicity, we use the following convention: X is a non empty set, x is an element of X, f, g are partial functions from X to \overline{ℝ}, S is a σ-field of subsets of X, F is a function from ℕ into S, A is a set, a is an extended real number, r, s are real numbers, and n is a natural number.

Let us consider X, f, a. The functor LE-dom(f, a) yielding a subset of X is defined by:

(Def. 12) x ∈ LE-dom(f, a) iff x ∈ domf and there exists an extended real number

y such that y = f(x) and y < a. The functor LEQ-dom(f, a) yielding a subset of X is defined by:

(Def. 13) x ∈ LEQ-dom(f, a) iff x ∈ domf and there exists an extended real

number y such that y = f(x) and y a. The functor GT-dom(f, a) yields a subset of X and is defined as follows:

(Def. 14) x ∈ GT-dom(f, a) iff x ∈ domf and there exists an extended real num-

ber y such that y = f(x) and a < y. The functor GTE-dom(f, a) yields a subset of X and is defined as follows:

(Def. 15) x ∈ GTE-dom(f, a) iff x ∈ domf and there exists an extended real

number y such that y = f(x) and a y. The functor EQ-dom(f, a) yielding a subset of X is defined as follows:

(Def. 16) x ∈ EQ-dom(f, a) iff x ∈ domf and there exists an extended real number

y such that y = f(x) and a = y. One can prove the following propositions:

(18) For all X, S, f, A, a such that A ⊆ domf holds A ∩ GTE-dom(f, a) = A \ A ∩ LE-dom(f, a).

(19) For all X, S, f, A, a such that A ⊆ domf holds A ∩ GT-dom(f, a) = A \ A ∩ LEQ-dom(f, a).

(20) For all X, S, f, A, a such that A ⊆ domf holds A ∩ LEQ-dom(f, a) = A \ A ∩ GT-dom(f, a).

(21) For all X, S, f, A, a such that A ⊆ domf holds A ∩ LE-dom(f, a) = A \ A ∩ GTE-dom(f, a).

(22) For all X, S, f, A, a holds A ∩ EQ-dom(f, a) = A ∩ GTE-dom(f, a) ∩ LEQ-dom(f, a).

(23) For all X, S, F, f, A, r such that for every n holds F (n) = A ∩ GT-dom(f, \overline{ℝ}(r − \frac{1}{n+1})) holds A ∩ GTE-dom(f, \overline{ℝ}(r)) = ⋂ rngF.

(24) For all X, S, F, f, A and for every real number r such that for every n holds F (n) = A ∩ LE-dom(f, \overline{ℝ}(r + \frac{1}{n+1})) holds A ∩ LEQ-dom(f, \overline{ℝ}(r)) =

⋂ rngF.

(25) For all X, S, F, f, A and for every real number r such that for every n holds F (n) = A ∩ LEQ-dom(f, \overline{ℝ}(r − \frac{1}{n+1})) holds A ∩ LE-dom(f, \overline{ℝ}(r)) = ⋃ rngF.

(26) For all X, S, F, f, A, r such that for every n holds F (n) = A ∩ GTE-dom(f, \overline{ℝ}(r + \frac{1}{n+1})) holds A ∩ GT-dom(f, \overline{ℝ}(r)) = ⋃ rngF.

(27) For all X, S, F, f, A such that for every n holds F (n) = A ∩ GT-dom(f, \overline{ℝ}(n)) holds A ∩ EQ-dom(f, +∞) = ⋂ rngF.

(28) For all X, S, F, f, A such that for every n holds F (n) = A ∩ LE-dom(f, \overline{ℝ}(n)) holds A ∩ LE-dom(f, +∞) = ⋃ rngF.

(29) For all X, S, F, f, A such that for every n holds F (n) = A ∩ LE-dom(f, \overline{ℝ}(− n)) holds A ∩ EQ-dom(f, −∞) = ⋂ rngF.

(30) For all X, S, F, f, A such that for every n holds F (n) = A ∩ GT-dom(f, \overline{ℝ}(− n)) holds A ∩ GT-dom(f, −∞) = ⋃ rngF.

4. Measurable Functions

Let X be a non empty set, let S be a σ-field of subsets of X, let f be a partial function from X to \overline{ℝ}, and let A be an element of S. We say that f is measurable on A if and only if:

(Def. 17) For every real number r holds A ∩ LE-dom(f, \overline{ℝ}(r)) is measurable on S. In the sequel A, B are elements of S. Next we state a number of propositions:

(31) Let given X, S, f, A. Suppose A ⊆ domf. Then f is measurable on A if and only if for every real number r holds A ∩ GTE-dom(f, \overline{ℝ}(r)) is measurable on S.

(32) Let given X, S, f, A. Then f is measurable on A if and only if for every real number r holds A ∩ LEQ-dom(f, \overline{ℝ}(r)) is measurable on S.

(33) Let given X, S, f, A. Suppose A ⊆ domf. Then f is measurable on A if and only if for every real number r holds A ∩ GT-dom(f, \overline{ℝ}(r)) is measurable on S.

(34) For all X, S, f, A, B such that B ⊆ A and f is measurable on A holds f is measurable on B.

(35) For all X, S, f, A, B such that f is measurable on A and f is measurable on B holds f is measurable on A ∪ B.

(36) For all X, S, f, A, r, s such that f is measurable on A and A ⊆ domf holds A ∩ GT-dom(f, \overline{ℝ}(r)) ∩ LE-dom(f, \overline{ℝ}(s)) is measurable on S.

(37) For all X, S, f, A such that f is measurable on A and A ⊆ domf holds A ∩ EQ-dom(f, +∞) is measurable on S.

500 noboru endou et al.

(38) For all X, S, f, A such that f is measurable on A holds A ∩ EQ-dom(f, −∞) is measurable on S.

(39) For all X, S, f, A such that f is measurable on A and A ⊆ domf holds A ∩ GT-dom(f, −∞) ∩ LE-dom(f, +∞) is measurable on S.

(40) Let given X, S, f, g, A, r. Suppose f is measurable on A and g is measurable on A and A ⊆ domg. Then A ∩ LE-dom(f, \overline{ℝ}(r)) ∩ GT-dom(g, \overline{ℝ}(r)) is measurable on S.

(41) For all X, S, f, A, r such that f is measurable on A and A ⊆ domf holds r f is measurable on A.

References

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[6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.

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Gauges and Cages. Part I^{1}

Artur Korniłowicz University of Białystok

Robert Milewski University of Białystok

Adam Naumowicz University of Białystok

Andrzej Trybulec University of Białystok

MML Identifier: JORDAN1A.

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

1. Preliminaries

For simplicity, we use the following convention: i, i_{1}, i_{2}, j, j_{1}, j_{2}, k, m, n, t denote natural numbers, D denotes a non empty subset of E_{T}^{2}, E denotes a compact non vertical non horizontal subset of E_{T}^{2}, C denotes a compact connected non vertical non horizontal subset of E_{T}^{2}, G denotes a Go-board, p, q, x denote points of E_{T}^{2}, and r, s denote real numbers.

The following propositions are true:

(1) For all real numbers s_{1}, s_{3}, s_{4}, l such that s_{1} s_{3} and s_{1} s_{4} and 0 l and l 1 holds s_{1} (1 − l) · s_{3} + l · s_{4}.

(2) For all real numbers s_{1}, s_{3}, s_{4}, l such that s_{3} s_{1} and s_{4} s_{1} and 0 l and l 1 holds (1 − l) · s_{3} + l · s_{4} s_{1}.

(3) If n > 0, then m^{n} mod m = 0.

(4) If j > 0 and i mod j = 0, then i ÷ j = \frac{i}{j} .

(5) If n > 0, then i^{n} ÷ i = \frac{i^{n}}{i} .

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(6) If 0 < n and 1 < r, then 1 < r^{n}.

(7) If r > 1 and m > n, then r^{m} > r^{n}.

(8) Let T be a non empty topological space, A be a subset of T, and B, C be subsets of the carrier of T. If A is connected and C is a component of B and A ∩ C ≠ ∅ and A ⊆ B, then A ⊆ C.

Let f be a finite sequence. The functor Centerf yields a natural number and is defined as follows:

(Def. 1) Centerf = (lenf ÷ 2) + 1. The following two propositions are true:

(9) For every finite sequence f such that lenf is odd holds lenf = 2 · Centerf − 1.

(10) For every finite sequence f such that lenf is even holds lenf = 2 · Centerf − 2.

2. Some Subsets of the Plane

One can check the following observations:

∗ there exists a subset of E_{T}^{2} which is compact, non vertical, non horizontal, and non empty and satisfies conditions of simple closed curve,

∗ there exists a subset of E_{T}^{2} which is compact, non empty, and horizontal, and

∗ there exists a subset of E_{T}^{2} which is compact, non empty, and vertical. The following propositions are true:

(11) If p ∈ N-mostD, then p_{2} = N-boundD.

(12) If p ∈ E-mostD, then p_{1} = E-boundD.

(13) If p ∈ S-mostD, then p_{2} = S-boundD.

(14) If p ∈ W-mostD, then p_{1} = W-boundD.

(15) BDDD misses D.

(16) For every compact non empty subset S of E_{T}^{2} satisfying conditions of simple closed curve holds LowerArcS ⊆ S and UpperArcS ⊆ S.

(17) p ∈ VerticalLinep_{1}.

(18) [r, s] ∈ VerticalLiner.

(19) For every subset A of E_{T}^{2} such that A ⊆ VerticalLines holds A is vertical.

(20) (proj2)([r, s]) = s and (proj1)([r, s]) = r.

(21) If p_{1} = q_{1} and r ∈ [(proj2)(p), (proj2)(q)], then [p_{1}, r] ∈ L(p, q).

(22) If p_{2} = q_{2} and r ∈ [(proj1)(p), (proj1)(q)], then [r, p_{2}] ∈ L(p, q).

(23) If p ∈ VerticalLines and q ∈ VerticalLines, then L(p, q) ⊆

VerticalLines. Let S be a non empty subset of E_{T}^{2} satisfying conditions of simple closed

curve. Observe that LowerArcS is non empty and compact and UpperArcS is non empty and compact.

We now state several propositions:

(24) For all subsets A, B of E_{T}^{2} such that A meets B holds (proj2)^{∘}A meets (proj2)^{∘}B.

(25) For all subsets A, B of E_{T}^{2} such that A misses B and A ⊆ VerticalLines and B ⊆ VerticalLines holds (proj2)^{∘}A misses (proj2)^{∘}B.

(26) For every closed subset S of E_{T}^{2} such that S is Bounded holds (proj2)^{∘}S is closed.

(27) For every subset S of E_{T}^{2} such that S is Bounded holds (proj2)^{∘}S is bounded.

(28) For every compact subset S of E_{T}^{2} holds (proj2)^{∘}S is compact. In this article we present several logical schemes. The scheme TRSubsetEx

deals with a natural number A and a unary predicate P, and states that: There exists a subset A of E_{T}^{A} such that for every point p of E_{T}^{A} holds p ∈ A iff P[p]

for all values of the parameters.

The scheme TRSubsetUniq deals with a natural number A and a unary predicate P, and states that:

Let A, B be subsets of E_{T}^{A}. Suppose for every point p of E_{T}^{A} holds p ∈ A iff P[p] and for every point p of E_{T}^{A} holds p ∈ B iff P[p]. Then A = B

for all values of the parameters.

Let p be a point of E_{T}^{2}. The functor NorthHalflinep yielding a subset of E_{T}^{2} is defined as follows:

(Def. 2) For every point x of E_{T}^{2} holds x ∈ NorthHalflinep iff x_{1} = p_{1} and x_{2} p_{2}.

The functor EastHalflinep yielding a subset of E_{T}^{2} is defined as follows: (Def. 3) For every point x of E_{T}^{2} holds x ∈ EastHalflinep iff x_{1} p_{1} and x_{2} = p_{2}.

The functor SouthHalflinep yielding a subset of E_{T}^{2} is defined as follows: (Def. 4) For every point x of E_{T}^{2} holds x ∈ SouthHalflinep iff x_{1} = p_{1} and x_{2} p_{2}.

The functor WestHalflinep yields a subset of E_{T}^{2} and is defined by: (Def. 5) For every point x of E_{T}^{2} holds x ∈ WestHalflinep iff x_{1} p_{1} and x_{2} = p_{2}.

The following propositions are true:

(29) NorthHalflinep = {q; q ranges over points of E_{T}^{2}: q_{1} = p_{1} ∧ q_{2} p_{2}}.

(30) NorthHalflinep = {[p_{1}, r]; r ranges over elements of ℝ: r p_{2}}.

(31) EastHalflinep = {q; q ranges over points of E_{T}^{2}: q_{1} p_{1} ∧ q_{2} = p_{2}}.

(32) EastHalflinep = {[r, p_{2}]; r ranges over elements of ℝ: r p_{1}}.

(33) SouthHalflinep = {q; q ranges over points of E_{T}^{2}: q_{1} = p_{1} ∧ q_{2} p_{2}}.

(34) SouthHalflinep = {[p_{1}, r]; r ranges over elements of ℝ: r p_{2}}.

(35) WestHalflinep = {q; q ranges over points of E_{T}^{2}: q_{1} p_{1} ∧ q_{2} = p_{2}}.

(36) WestHalflinep = {[r, p_{2}]; r ranges over elements of ℝ: r p_{1}}. Let p be a point of E_{T}^{2}. One can check the following observations:

∗ NorthHalflinep is non empty and convex,

∗ EastHalflinep is non empty and convex,

∗ SouthHalflinep is non empty and convex, and

∗ WestHalflinep is non empty and convex.

3. Goboards

We now state a number of propositions:

(37) If 1 i and i lenG and 1 j and j widthG, then G_{i,j} ∈ L(G_{i,1}, G_{i,width G}).

(38) If 1 i and i lenG and 1 j and j widthG, then G_{i,j} ∈ L(G_{1,j}, G_{len G,j}).

(39) If 1 j_{1} and j_{1} widthG and 1 j_{2} and j_{2} widthG and 1 i_{1} and i_{1} i_{2} and i_{2} lenG, then (G_{i_{1},j_{1}})_{1} (G_{i_{2},j_{2}})_{1}.

(40) If 1 i_{1} and i_{1} lenG and 1 i_{2} and i_{2} lenG and 1 j_{1} and j_{1} j_{2} and j_{2} widthG, then (G_{i_{1},j_{1}})_{2} (G_{i_{2},j_{2}})_{2}.

(41) Let f be a non constant standard special circular sequence. Suppose f is a sequence which elements belong to G and 1 t and t lenG. Then (G_{t,width G})_{2} N-bound L̃(f).

(42) Let f be a non constant standard special circular sequence. Suppose f is a sequence which elements belong to G and 1 t and t widthG. Then (G_{1,t})_{1} W-bound L̃(f).

(43) Let f be a non constant standard special circular sequence. Suppose f is a sequence which elements belong to G and 1 t and t lenG. Then (G_{t,1})_{2} S-bound L̃(f).

(44) Let f be a non constant standard special circular sequence. Suppose f is a sequence which elements belong to G and 1 t and t widthG. Then (G_{len G,t})_{1} E-bound L̃(f).

(45) If i lenG and j widthG, then cell(G, i, j) is non empty.

(46) If i lenG and j widthG, then cell(G, i, j) is connected.

(47) If i lenG, then cell(G, i, 0) is not Bounded.

(48) If i lenG, then cell(G, i, widthG) is not Bounded.

4. Gauges

One can prove the following propositions:

(49) widthGauge(D, n) = 2^{n} + 3.

(50) If i < j, then lenGauge(D, i) < lenGauge(D, j).

(51) If i j, then lenGauge(D, i) lenGauge(D, j).

(52) If m n and 1 < i and i < lenGauge(D, m), then 1 < 2^{n− ′m} · (i − 2) + 2 and 2^{n− ′m} · (i − 2) + 2 < lenGauge(D, n).

(53) If m n and 1 < i and i < widthGauge(D, m), then 1 < 2^{n− ′m}·(i−2)+2 and 2^{n− ′m} · (i − 2) + 2 < widthGauge(D, n).

(54) Suppose m n and 1 < i and i < lenGauge(D, m) and 1 < j and j < widthGauge(D, m). Let i_{1}, j_{1} be natural numbers. If i_{1} = 2^{n− ′m}·(i−2)+2 and j_{1} = 2^{n− ′m} · (j − 2) + 2, then (Gauge(D, m))_{i,j} = (Gauge(D, n))_{i_{1},j_{1}}.

(55) If m n and 1 < i and i+1 < lenGauge(D, m), then 1 < 2^{n− ′m}·(i−1)+2 and 2^{n− ′m} · (i − 1) + 2 lenGauge(D, n).

(56) If m n and 1 < i and i + 1 < widthGauge(D, m), then 1 < 2^{n− ′m} · (i − 1) + 2 and 2^{n− ′m} · (i − 1) + 2 widthGauge(D, n).

(57) If 1 i and i lenGauge(D, n) and 1 j and j lenGauge(D, m) and n > 0 and m > 0 or n = 0 and m = 0, then ((Gauge(D, n))_{Center Gauge(D,n),i})_{1} = ((Gauge(D, m))_{Center Gauge(D,m),j})_{1}.

(58) If 1 i and i lenGauge(D, n) and 1 j and j lenGauge(D, m) and n > 0 and m > 0 or n = 0 and m = 0, then ((Gauge(D, n))_{i,Center Gauge(D,n)})_{2} = ((Gauge(D, m))_{j,Center Gauge(D,m)})_{2}.

(59) If 1 i and i lenGauge(C, 1), then ((Gauge(C, 1))_{Center Gauge(C,1),i})_{1} = \frac{W-bound C+E-bound C}{2} .

(60) If 1 i and i lenGauge(C, 1), then ((Gauge(C, 1))_{i,Center Gauge(C,1)})_{2} = \frac{S-bound C+N-bound C}{2} .

(61) If 1 i and i lenGauge(E, n) and 1 j and j lenGauge(E, m) and m n, then ((Gauge(E, n))_{i,len Gauge(E,n)})_{2} ((Gauge(E, m))_{j,len Gauge(E,m)})_{2}.

(62) If 1 i and i lenGauge(E, n) and 1 j and j lenGauge(E, m) and m n, then ((Gauge(E, n))_{len Gauge(E,n),i})_{1} ((Gauge(E, m))_{len Gauge(E,m),j})_{1}.

(63) If 1 i and i lenGauge(E, n) and 1 j and j lenGauge(E, m) and m n, then ((Gauge(E, m))_{1,j})_{1} ((Gauge(E, n))_{1,i})_{1}.

(64) If 1 i and i lenGauge(E, n) and 1 j and j lenGauge(E, m) and m n, then ((Gauge(E, m))_{j,1})_{2} ((Gauge(E, n))_{i,1})_{2}.

(65) If 1 m and m n, then L((Gauge(E, n))_{Center Gauge(E,n),1}, (Gauge(E, n))_{Center Gauge(E,n),len Gauge(E,n)}) ⊆ L((Gauge(E, m))_{Center Gauge(E,m),1}, (Gauge(E, m))_{Center Gauge(E,m),len Gauge(E,m)}).

(66) If 1 m and m n and 1 j and j widthGauge(E, n), then L((Gauge(E, n))_{Center Gauge(E,n),1}, (Gauge(E, n))_{Center Gauge(E,n),j}) ⊆ L((Gauge(E, m))_{Center Gauge(E,m),1}, (Gauge(E, n))_{Center Gauge(E,n),j}).

(67) If 1 m and m n and 1 j and j widthGauge(E, n), then L((Gauge(E, m))_{Center Gauge(E,m),1}, (Gauge(E, n))_{Center Gauge(E,n),j}) ⊆ L((Gauge(E, m))_{Center Gauge(E,m),1}, (Gauge(E, m))_{Center Gauge(E,m),len Gauge(E,m)}).

(68) Suppose m n and 1 < i and i + 1 < lenGauge(E, m) and 1 < j and j + 1 < widthGauge(E, m). Let i_{1}, j_{1} be natural numbers. Suppose 2^{n− ′m} · (i − 2) + 2 i_{1} and i_{1} < 2^{n− ′m} · (i − 1) + 2 and 2^{n− ′m} · (j − 2) + 2 j_{1} and j_{1} < 2^{n− ′m} · (j − 1) + 2. Then cell(Gauge(E, n), i_{1}, j_{1}) ⊆ cell(Gauge(E, m), i, j).

(69) Suppose m n and 3 i and i < lenGauge(E, m) and 1 < j and j + 1 < widthGauge(E, m). Let i_{1}, j_{1} be natural numbers. If i_{1} = 2^{n− ′m} · (i − 2) + 2 and j_{1} = 2^{n− ′m} · (j − 2) + 2, then cell(Gauge(E, n), i_{1} − ′1, j_{1}) ⊆ cell(Gauge(E, m), i − ′1, j).

(70) If i lenGauge(C, n), then cell(Gauge(C, n), i, 0) ⊆ UBDC.

(71) If i lenGauge(E, n), then cell(Gauge(E, n), i, widthGauge(E, n)) ⊆ UBDE.

5. Cages

The following propositions are true:

(72) If p ∈ C, then NorthHalflinep meets L̃(Cage(C, n)).

(73) If p ∈ C, then EastHalflinep meets L̃(Cage(C, n)).

(74) If p ∈ C, then SouthHalflinep meets L̃(Cage(C, n)).

(75) If p ∈ C, then WestHalflinep meets L̃(Cage(C, n)).

(76) There exist k, t such that 1 k and k < lenCage(C, n) and 1 t and t widthGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{1,t}.

(77) There exist k, t such that 1 k and k < lenCage(C, n) and 1 t and t lenGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{t,1}.

(78) There exist k, t such that 1 k and k < lenCage(C, n) and 1 t and t widthGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{len Gauge(C,n),t}.

(79) If 1 k and k lenCage(C, n) and 1 t and t lenGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{t,width Gauge(C,n)}, then (Cage(C, n))_{k} ∈ N-most L̃(Cage(C, n)).

(80) If 1 k and k lenCage(C, n) and 1 t and t widthGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{1,t}, then (Cage(C, n))_{k} ∈ W-most L̃(Cage(C, n)).

(81) If 1 k and k lenCage(C, n) and 1 t and t lenGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{t,1}, then (Cage(C, n))_{k} ∈ S-most L̃(Cage(C, n)).

(82) If 1 k and k lenCage(C, n) and 1 t and t widthGauge(C, n) and (Cage(C, n))_{k} = (Gauge(C, n))_{len Gauge(C,n),t}, then (Cage(C, n))_{k} ∈ E-most L̃(Cage(C, n)).

(83) W-bound L̃(Cage(C, n)) = W-boundC − \frac{E-bound C−W-bound C}{2^{n}} .

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

(85) E-bound L̃(Cage(C, n)) = E-boundC + \frac{E-bound C−W-bound C}{2^{n}} .

(86) N-bound L̃(Cage(C, n)) + S-bound L̃(Cage(C, n)) = N-bound L̃(Cage(C, m)) + S-bound L̃(Cage(C, m)).

(87) E-bound L̃(Cage(C, n))+W-bound L̃(Cage(C, n)) = E-bound L̃(Cage(C, m)) + W-bound L̃(Cage(C, m)).

(88) If i < j, then E-bound L̃(Cage(C, j)) < E-bound L̃(Cage(C, i)).

(89) If i < j, then W-bound L̃(Cage(C, i)) < W-bound L̃(Cage(C, j)).

(90) If i < j, then S-bound L̃(Cage(C, i)) < S-bound L̃(Cage(C, j)).

(91) If 1 i and i lenGauge(C, n), then N-bound L̃(Cage(C, n)) = ((Gauge(C, n))_{i,len Gauge(C,n)})_{2}.

(92) If 1 i and i lenGauge(C, n), then E-bound L̃(Cage(C, n)) = ((Gauge(C, n))_{len Gauge(C,n),i})_{1}.

(93) If 1 i and i lenGauge(C, n), then S-bound L̃(Cage(C, n)) = ((Gauge(C, n))_{i,1})_{2}.

(94) If 1 i and i lenGauge(C, n), then W-bound L̃(Cage(C, n)) = ((Gauge(C, n))_{1,i})_{1}.

(95) If x ∈ C and p ∈ NorthHalflinex ∩ L̃(Cage(C, n)), then p_{2} > x_{2}.

(96) If x ∈ C and p ∈ EastHalflinex ∩ L̃(Cage(C, n)), then p_{1} > x_{1}.

(97) If x ∈ C and p ∈ SouthHalflinex ∩ L̃(Cage(C, n)), then p_{2} < x_{2}.

(98) If x ∈ C and p ∈ WestHalflinex ∩ L̃(Cage(C, n)), then p_{1} < x_{1}.

(99) If x ∈ N-mostC and p ∈ NorthHalflinex and 1 i and i < lenCage(C, n) and p ∈ L(Cage(C, n), i), then L(Cage(C, n), i) is horizontal.

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(100) If x ∈ E-mostC and p ∈ EastHalflinex and 1 i and i < lenCage(C, n) and p ∈ L(Cage(C, n), i), then L(Cage(C, n), i) is vertical.

(101) If x ∈ S-mostC and p ∈ SouthHalflinex and 1 i and i < lenCage(C, n) and p ∈ L(Cage(C, n), i), then L(Cage(C, n), i) is horizontal.

(102) If x ∈ W-mostC and p ∈ WestHalflinex and 1 i and i < lenCage(C, n) and p ∈ L(Cage(C, n), i), then L(Cage(C, n), i) is vertical.

(103) If x ∈ N-mostC and p ∈ NorthHalflinex ∩ L̃(Cage(C, n)), then p_{2} = N-bound L̃(Cage(C, n)).

(104) If x ∈ E-mostC and p ∈ EastHalflinex ∩ L̃(Cage(C, n)), then p_{1} = E-bound L̃(Cage(C, n)).

(105) If x ∈ S-mostC and p ∈ SouthHalflinex ∩ L̃(Cage(C, n)), then p_{2} = S-bound L̃(Cage(C, n)).

(106) If x ∈ W-mostC and p ∈ WestHalflinex ∩ L̃(Cage(C, n)), then p_{1} = W-bound L̃(Cage(C, n)).

(107) If x ∈ N-mostC, then there exists a point p of E_{T}^{2} such that NorthHalflinex ∩ L̃(Cage(C, n)) = {p}.

(108) If x ∈ E-mostC, then there exists a point p of E_{T}^{2} such that EastHalflinex ∩ L̃(Cage(C, n)) = {p}.

(109) If x ∈ S-mostC, then there exists a point p of E_{T}^{2} such that SouthHalflinex ∩ L̃(Cage(C, n)) = {p}.

(110) If x ∈ W-mostC, then there exists a point p of E_{T}^{2} such that WestHalflinex ∩ L̃(Cage(C, n)) = {p}.

References

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

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

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Some Properties of Extended Real Numbers Operations: abs, min and max

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article, we extend some properties concerning real numbers to extended real numbers. Almost all properties included in this article are extended properties of other articles: [9], [6], [8], [10] and [7].

MML Identifier: EXTREAL2.

The terminology and notation used in this paper are introduced in the following papers: [8], [4], [3], [5], [10], [11], [1], and [2].

1. Preliminaries

We follow the rules: x, y, w, z denote extended real numbers and a, b denote real numbers.

The following propositions are true:

(1) If x ≠ +∞ or y ≠ −∞ and if x ≠ −∞ or y ≠ +∞, then x + y = y + x.

(2) If x ≠ +∞ and x ≠ −∞, then there exists y such that x + y = 0\overline{ℝ}.

(3) If x ≠ +∞ and x ≠ −∞ and x ≠ 0\overline{ℝ}, then there exists y such that x · y = \overline{1} .

(4) \overline{1} ·x = x and x · \overline{1} = x and \overline{ℝ}(1) · x = x and x · \overline{ℝ}(1) = x.

(5) 0\overline{ℝ} − x = − x.

(6) x ≠ 0\overline{ℝ} iff − x ≠ 0\overline{ℝ}.

(7) If 0\overline{ℝ} x and 0\overline{ℝ} y, then 0\overline{ℝ} x + y.

(8) If 0\overline{ℝ} x and 0\overline{ℝ} < y or 0\overline{ℝ} < x and 0\overline{ℝ} y, then 0\overline{ℝ} < x + y.

(9) If x 0\overline{ℝ} and y 0\overline{ℝ}, then x + y 0\overline{ℝ}.

(10) If x 0\overline{ℝ} and y < 0\overline{ℝ} or x < 0\overline{ℝ} and y 0\overline{ℝ}, then x + y < 0\overline{ℝ}.

(11) If z ≠ +∞ and z ≠ −∞ and x + z = y, then x = y − z.

(12) If x ≠ +∞ and x ≠ −∞ and x ≠ 0\overline{ℝ}, then x · \frac{\overline{1}}{x} = \overline{1} and \frac{\overline{1}}{x} · x = \overline{1} .

(13) If x ≠ +∞ and x ≠ −∞, then x − x = 0\overline{ℝ}.

(14) If x ≠ +∞ or y ≠ −∞ and if x ≠ −∞ or y ≠ +∞, then − (x + y) = − x + − y and − (x + y) = − y − x and − (x + y) = − x − y.

(15) If x ≠ +∞ or y ≠ +∞ and if x ≠ −∞ or y ≠ −∞, then − (x − y) = − x + y and − (x − y) = y − x.

(16) If x ≠ +∞ or y ≠ +∞ and if x ≠ −∞ or y ≠ −∞, then − (− x + y) = x − y and − (− x + y) = x + − y.

(17) If x = +∞ and 0\overline{ℝ} < y and y < +∞ or x = −∞ and y < 0\overline{ℝ} and −∞ < y, then \frac{x}{y} = +∞.

(18) If x = +∞ and y < 0\overline{ℝ} and −∞ < y or x = −∞ and 0\overline{ℝ} < y and y < +∞, then \frac{x}{y} = −∞.

(19) If −∞ < y and y < +∞ and y ≠ 0\overline{ℝ}, then \frac{x·y}{y} = x and x · \frac{y}{y} = x.

(20) \overline{1} < +∞ and −∞ < \overline{1} .

(21) If x = +∞ or x = −∞, then for every y such that y ∈ ℝ holds x+y ≠ 0\overline{ℝ}.

(22) If x = +∞ or x = −∞, then for every y holds x · y ≠ \overline{1} .

(23) If x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ and x + y < +∞, then x ≠ +∞ and y ≠ +∞.

(24) If x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ and −∞ < x + y, then x ≠ −∞ and y ≠ −∞.

(25) If x ≠ +∞ or y ≠ +∞ but x ≠ −∞ or y ≠ −∞ and x − y < +∞, then x ≠ +∞ and y ≠ −∞.

(26) If x ≠ +∞ or y ≠ +∞ but x ≠ −∞ or y ≠ −∞ and −∞ < x − y, then x ≠ −∞ and y ≠ +∞.

(27) If x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ and x + y < z, then x ≠ +∞ and y ≠ +∞ and z ≠ −∞ and x < z − y.

(28) If z ≠ +∞ or y ≠ +∞ but z ≠ −∞ or y ≠ −∞ and x < z − y, then x ≠ +∞ and y ≠ +∞ and z ≠ −∞ and x + y < z.

(29) If x ≠ +∞ or y ≠ +∞ but x ≠ −∞ or y ≠ −∞ and x − y < z, then x ≠ +∞ and y ≠ −∞ and z ≠ −∞ and x < z + y.

(30) If z ≠ +∞ or y ≠ −∞ but z ≠ −∞ or y ≠ +∞ and x < z + y, then x ≠ +∞ and y ≠ −∞ and z ≠ −∞ and x − y < z.

(31) If x ≠ +∞ or y ≠ −∞ and x ≠ −∞ or y ≠ +∞ and y ≠ +∞ or z ≠ +∞ and y ≠ −∞ or z ≠ −∞ and x + y z, then y ≠ +∞ and x z − y.

(32) If x = +∞ and y = −∞ and x = −∞ and y = +∞ and y = +∞ and z = +∞ and y = −∞ and z = −∞ and x z − y, then y ≠ +∞ and x + y z.

(33) If x ≠ +∞ or y ≠ +∞ and x ≠ −∞ or y ≠ −∞ and y ≠ +∞ or z ≠ −∞ and y ≠ −∞ or z ≠ +∞ and x − y z, then y ≠ −∞ and x z + y.

(34) If x = +∞ and y = +∞ and x = −∞ and y = −∞ and y = −∞ and z = +∞ and x z + y, then y ≠ −∞ and x − y z.

(35) If x ≠ +∞ and y ≠ +∞, then x + y ≠ +∞.

(36) If x ≠ −∞ and y ≠ −∞, then x + y ≠ −∞.

(37) If x ≠ +∞ and y ≠ −∞, then x − y ≠ +∞.

(38) If x ≠ −∞ and y ≠ +∞, then x − y ≠ −∞.

(39) Suppose x = +∞ and y = −∞ and x = −∞ and y = +∞ and y = +∞ and z = +∞ and y = −∞ and z = −∞ and x = +∞ and z = +∞ and x = −∞ and z = −∞. Then (x + y) − z = x + (y − z).

(40) Suppose x = +∞ and y = +∞ and x = −∞ and y = −∞ and y = +∞ and z = −∞ and y = −∞ and z = +∞ and x = +∞ and z = +∞ and x = −∞ and z = −∞. Then x − y − z = x − (y + z).

(41) Suppose x = +∞ and y = +∞ and x = −∞ and y = −∞ and y = +∞ and z = +∞ and y = −∞ and z = −∞ and x = +∞ and z = −∞ and x = −∞ and z = +∞. Then (x − y) + z = x − (y − z).

(42) If x · y ≠ +∞ and x · y ≠ −∞, then x is a real number or y is a real number.

(43) 0\overline{ℝ} < x and 0\overline{ℝ} < y or x < 0\overline{ℝ} and y < 0\overline{ℝ} iff 0\overline{ℝ} < x · y.

(44) 0\overline{ℝ} < x and y < 0\overline{ℝ} or x < 0\overline{ℝ} and 0\overline{ℝ} < y iff x · y < 0\overline{ℝ}.

(45) 0\overline{ℝ} x or 0\overline{ℝ} < x but 0\overline{ℝ} y or 0\overline{ℝ} < y or x 0\overline{ℝ} or x < 0\overline{ℝ} but y 0\overline{ℝ} or y < 0\overline{ℝ} iff 0\overline{ℝ} x · y.

(46) x 0\overline{ℝ} or x < 0\overline{ℝ} but 0\overline{ℝ} y or 0\overline{ℝ} < y or 0\overline{ℝ} x or 0\overline{ℝ} < x but y 0\overline{ℝ} or y < 0\overline{ℝ} iff x · y 0\overline{ℝ}.

(47) x − y iff y − x and − x y iff − y x.

(48) x < − y iff y < − x and − x < y iff − y < x.

2. Basic Properties of abs for Extended Real Numbers

One can prove the following propositions:

(49) If x = a, then |x| = |a|.

(50) |x| = x or |x| = − x.

(51) 0\overline{ℝ} |x|.

(52) If x ≠ 0\overline{ℝ}, then 0\overline{ℝ} < |x|.

(53) x = 0\overline{ℝ} iff |x| = 0\overline{ℝ}.

(54) If |x| = − x and x ≠ 0\overline{ℝ}, then x < 0\overline{ℝ}.

(55) If x 0\overline{ℝ}, then |x| = − x.

(56) |x · y| = |x| · |y|.

(57) − |x| x and x |x|.

(58) If |x| < y, then − y < x and x < y.

(59) If − y < x and x < y, then 0\overline{ℝ} < y and |x| < y.

(60) − y x and x y iff |x| y.

(61) If x ≠ +∞ or y ≠ −∞ and if x ≠ −∞ or y ≠ +∞, then |x+y| |x|+|y|.

(62) If −∞ < x and x < +∞ and x ≠ 0\overline{ℝ}, then |x| · |\frac{\overline{1}}{x}| = \overline{1} .

(63) If x = +∞ or x = −∞, then |x| · |\frac{\overline{1}}{x}| = 0\overline{ℝ}.

(64) If x ≠ 0\overline{ℝ}, then |\frac{\overline{1}}{x}| = \frac{\overline{1}}{|x|}.

(65) If x = −∞ or x = +∞ and if y = −∞ or y = +∞ and if y ≠ 0\overline{ℝ}, then |\frac{x}{y}| = \frac{|x|}{|y|}.

(66) |x| = |− x|.

(67) If x = +∞ or x = −∞, then |x| = +∞.

(68) If x is a real number or y is a real number, then |x| − |y| |x − y|.

(69) If x ≠ +∞ or y ≠ +∞ and if x ≠ −∞ or y ≠ −∞, then |x−y| |x|+|y|.

(70) ||x|| = |x|.

(71) If x ≠ +∞ or y ≠ −∞ but x ≠ −∞ or y ≠ +∞ and |x| z and |y| w, then |x + y| z + w.

(72) If x is a real number or y is a real number, then ||x| − |y|| |x − y|.

(73) If 0\overline{ℝ} x · y, then |x + y| = |x| + |y|.

3. Definitions of min, max for Extended Real Numbers and their Basic Properties

Next we state the proposition

(74) If x = a and y = b, then b < a iff y < x and b a iff y x. Let us consider x, y. The functor min(x, y) yielding an extended real number

is defined by:

(Def. 1) min(x, y) = \{ \begin{matrix}x, if x y, \\ y, otherwise. \end{matrix}

The functor max(x, y) yielding an extended real number is defined as follows:

(Def. 2) max(x, y) = \{ \begin{matrix}x, if y x, \\ y, otherwise. \end{matrix}

One can prove the following propositions:

(75) If x = −∞ or y = −∞, then min(x, y) = −∞.

(76) If x = +∞ or y = +∞, then max(x, y) = +∞.

(77) Let x, y be extended real numbers and a, b be real numbers. If x = a and y = b, then min(x, y) = min(a, b) and max(x, y) = max(a, b).

(78) If y x, then min(x, y) = y.

(79) If y ≰ x, then min(x, y) = x.

(80) If x ≠ +∞ and y ≠ +∞ and x ≠ +∞ or y ≠ +∞ but x ≠ −∞ or y ≠ −∞, then min(x, y) = \frac{(x+y)−|x−y|}{\overline{ℝ}(2)} .

(81) min(x, y) x and min(y, x) x.

(82) min(x, x) = x.

(83) min(x, y) = min(y, x).

(84) min(x, y) = x or min(x, y) = y.

(85) x y and x z iff x min(y, z).

(86) If min(x, y) = x, then x y.

(87) If min(x, y) = y, then y x.

(88) min(x, min(y, z)) = min(min(x, y), z).

(89) If x y, then max(x, y) = y.

(90) If x ≰ y, then max(x, y) = x.

(91) If x ≠ −∞ and y ≠ −∞ and x ≠ +∞ or y ≠ +∞ but x ≠ −∞ or y ≠ −∞, then max(x, y) = \frac{x+y+|x−y|}{\overline{ℝ}(2)} .

(92) x max(x, y) and x max(y, x).

(93) max(x, x) = x.

(94) max(x, y) = max(y, x).

(95) max(x, y) = x or max(x, y) = y.

(96) y x and z x iff max(y, z) x.

(97) If max(x, y) = x, then y x.

(98) If max(x, y) = y, then x y.

(99) max(x, max(y, z)) = max(max(x, y), z).

(100) If x ≠ +∞ or y ≠ −∞ and if x ≠ −∞ or y ≠ +∞, then min(x, y) + max(x, y) = x + y.

(101) max(x, min(x, y)) = x and max(min(x, y), x) = x and max(min(y, x), x) = x and max(x, min(y, x)) = x.

(102) min(x, max(x, y)) = x and min(max(x, y), x) = x and min(max(y, x), x) = x and min(x, max(y, x)) = x.

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(103) min(x, max(y, z)) = max(min(x, y), min(x, z)) and min(max(y, z), x) = max(min(y, x), min(z, x)).

(104) max(x, min(y, z)) = min(max(x, y), max(x, z)) and max(min(y, z), x) = min(max(y, x), max(z, x)).

References

[1] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[2] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[3] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21–26, 1996.

[4] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491–494, 2001.

[5] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495–500, 2001.

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

[7] Andrzej Kondracki. Equalities and inequalities in real numbers. Formalized Mathematics, 2(1):49–63, 1991.

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

[9] Andrzej Trybulec. Built-in concepts. Formalized Mathematics, 1(1):13–15, 1990.

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

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

The Concept of Fuzzy Relation and Basic Properties of its Operation

Takashi Mitsuishi Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. This article introduces the fuzzy relation. This is the expansion of usual relation, and the value is given at the fuzzy value. At first, the definition of the fuzzy relation characterized by membership function is described. Next, the definitions of the zero relation and universe relation and basic operations of these relations are shown.

MML Identifier: FUZZY 3.

The papers [8], [1], [5], [9], [3], [4], [6], [7], and [2] provide the terminology and notation for this paper.

1. Definition of Fuzzy Relation

In this paper C_{1}, C_{2} are non empty sets.

Let us consider C_{1}, C_{2}. A partial function from [: C_{1}, C_{2} :] to ℝ is said to be a Membership function of C_{1}, C_{2} if:

(Def. 1) domit = [: C_{1}, C_{2} :] and rngit ⊆ [0, 1]. The following proposition is true

(1) χ_{[: C_{1}, C_{2} :],[: C_{1}, C_{2} :]} is a Membership function of C_{1}, C_{2}.

Let C_{1}, C_{2} be non empty sets and let h be a Membership function of C_{1}, C_{2}. A set is called a fuzzy relation of C_{1}, C_{2}, h if:

(Def. 2) It = [: [: C_{1}, C_{2} :], h^{∘}[: C_{1}, C_{2} :] :].

Let C_{1}, C_{2} be non empty sets, let h, g be Membership functions of C_{1}, C_{2}, let A be a fuzzy relation of C_{1}, C_{2}, h, and let B be a fuzzy relation of C_{1}, C_{2}, g. The predicate A = B is defined by:

(Def. 3) For every element c of [: C_{1}, C_{2} :] holds h(c) = g(c).

Let C_{1}, C_{2} be non empty sets, let h, g be Membership functions of C_{1}, C_{2}, let A be a fuzzy relation of C_{1}, C_{2}, h, and let B be a fuzzy relation of C_{1}, C_{2}, g. The predicate A ⊆ B is defined by:

(Def. 4) For every element c of [: C_{1}, C_{2} :] holds h(c) g(c).

For simplicity, we adopt the following rules: f, g, h, h_{1} denote Membership functions of C_{1}, C_{2}, A denotes a fuzzy relation of C_{1}, C_{2}, f, B denotes a fuzzy relation of C_{1}, C_{2}, g, D denotes a fuzzy relation of C_{1}, C_{2}, h, and D_{1} denotes a fuzzy relation of C_{1}, C_{2}, h_{1}.

The following three propositions are true:

(2) A = B iff A ⊆ B and B ⊆ A.

(3) A ⊆ A.

(4) If A ⊆ B and B ⊆ D, then A ⊆ D.

2. Intersection, Union and Complement

Let C_{1}, C_{2} be non empty sets and let h, g be Membership functions of C_{1}, C_{2}. The functor min(h, g) yielding a Membership function of C_{1}, C_{2} is defined as follows:

(Def. 5) For every element c of [: C_{1}, C_{2} :] holds (min(h, g))(c) = min(h(c), g(c)).

Let C_{1}, C_{2} be non empty sets and let h, g be Membership functions of C_{1}, C_{2}. The functor max(h, g) yields a Membership function of C_{1}, C_{2} and is defined by:

(Def. 6) For every element c of [: C_{1}, C_{2} :] holds (max(h, g))(c) = max(h(c), g(c)).

Let C_{1}, C_{2} be non empty sets and let h be a Membership function of C_{1}, C_{2}. The functor 1-minush yields a Membership function of C_{1}, C_{2} and is defined as follows:

(Def. 7) For every element c of [: C_{1}, C_{2} :] holds (1-minush)(c) = 1 − h(c).

Let C_{1}, C_{2} be non empty sets, let h, g be Membership functions of C_{1}, C_{2}, let A be a fuzzy relation of C_{1}, C_{2}, h, and let B be a fuzzy relation of C_{1}, C_{2}, g. The functor A ∩ B yields a fuzzy relation of C_{1}, C_{2}, min(h, g) and is defined as follows:

(Def. 8) A ∩ B = [: [: C_{1}, C_{2} :], (min(h, g))^{∘}[: C_{1}, C_{2} :] :].

Let C_{1}, C_{2} be non empty sets, let h, g be Membership functions of C_{1}, C_{2}, let A be a fuzzy relation of C_{1}, C_{2}, h, and let B be a fuzzy relation of C_{1}, C_{2}, g. The functor A ∪ B yielding a fuzzy relation of C_{1}, C_{2}, max(h, g) is defined by:

(Def. 9) A ∪ B = [: [: C_{1}, C_{2} :], (max(h, g))^{∘}[: C_{1}, C_{2} :] :].

Let C_{1}, C_{2} be non empty sets, let h be a Membership function of C_{1}, C_{2}, and let A be a fuzzy relation of C_{1}, C_{2}, h. The functor A^{c} yielding a fuzzy relation of C_{1}, C_{2}, 1-minush is defined as follows:

(Def. 10) A^{c} = [: [: C_{1}, C_{2} :], (1-minush)^{∘}[: C_{1}, C_{2} :] :]. The following propositions are true:

(5) For every element x of [: C_{1}, C_{2} :] holds min(h(x), g(x)) = (min(h, g))(x) and max(h(x), g(x)) = (max(h, g))(x).

(6) max(h, h) = h and min(h, h) = h and max(h, h) = min(h, h) and min(f, g) = min(g, f) and max(f, g) = max(g, f).

(7) f = g iff A = B.

(8) A ∩ A = A and A ∪ A = A.

(9) A ∩ B = B ∩ A and A ∪ B = B ∪ A.

(10) max(max(f, g), h) = max(f, max(g, h)) and min(min(f, g), h) = min(f, min(g, h)).

(11) (A ∪ B) ∪ D = A ∪ (B ∪ D).

(12) (A ∩ B) ∩ D = A ∩ (B ∩ D).

(13) max(f, min(f, g)) = f and min(f, max(f, g)) = f.

(14) A ∪ A ∩ B = A and A ∩ (A ∪ B) = A.

(15) min(f, max(g, h)) = max(min(f, g), min(f, h)) and max(f, min(g, h)) = min(max(f, g), max(f, h)).

(16) A ∪ B ∩ D = (A ∪ B) ∩ (A ∪ D) and A ∩ (B ∪ D) = A ∩ B ∪ A ∩ D.

(17) 1-minus1-minush = h.

(18) (A^{c})^{c} = A.

(19) 1-minusmax(f, g) = min(1-minusf, 1-minusg) and 1-minusmin(f, g) = max(1-minusf, 1-minusg).

(20) (A ∪ B)^{c} = A^{c} ∩ B^{c} and (A ∩ B)^{c} = A^{c} ∪ B^{c}.

(21) A ⊆ A ∪ B.

(22) If A ⊆ D and B ⊆ D, then A ∪ B ⊆ D.

(23) If A ⊆ B, then A ∪ D ⊆ B ∪ D.

(24) If A ⊆ B and D ⊆ D_{1}, then A ∪ D ⊆ B ∪ D_{1}.

(25) If A ⊆ B, then A ∪ B = B.

(26) A ∩ B ⊆ A.

(27) A ∩ B ⊆ A ∪ B.

(28) If D ⊆ A and D ⊆ B, then D ⊆ A ∩ B.

(29) For all elements a, b, c, d of ℝ such that a b and c d holds min(a, c) min(b, d).

(30) For all elements a, b, c of ℝ such that a b holds min(a, c) min(b, c).

(31) If A ⊆ B, then A ∩ D ⊆ B ∩ D.

(32) If A ⊆ B and D ⊆ D_{1}, then A ∩ D ⊆ B ∩ D_{1}.

(33) If A ⊆ B, then A ∩ B = A.

(34) If A ∩ B ∪ A ∩ D = A, then A ⊆ B ∪ D.

(35) A = B ∪ D iff B ⊆ A and D ⊆ A and for all h_{1}, D_{1} such that B ⊆ D_{1} and D ⊆ D_{1} holds A ⊆ D_{1}.

(36) A = B ∩ D iff A ⊆ B and A ⊆ D and for all h_{1}, D_{1} such that D_{1} ⊆ B and D_{1} ⊆ D holds D_{1} ⊆ A.

(37) A ⊆ B iff B^{c} ⊆ A^{c}.

(38) If A ⊆ B^{c}, then B ⊆ A^{c}.

(39) If A^{c} ⊆ B, then B^{c} ⊆ A.

(40) (A ∪ B)^{c} ⊆ A^{c} and (A ∪ B)^{c} ⊆ B^{c}.

(41) A^{c} ⊆ (A ∩ B)^{c} and B^{c} ⊆ (A ∩ B)^{c}.

3. Exclusive Sum

Let C_{1}, C_{2} be non empty sets, let h, g be Membership functions of C_{1}, C_{2}, let A be a fuzzy relation of C_{1}, C_{2}, h, and let B be a fuzzy relation of C_{1}, C_{2}, g. The functor A−. B yields a fuzzy relation of C_{1}, C_{2}, max(min(h, 1-minusg),

min(1-minush, g)) and is defined by: (Def. 11) A−. B = [: [: C_{1}, C_{2} :], (max(min(h, 1-minusg), min(1-minush, g)))^{∘}[: C_{1},

C_{2} :] :]. The following propositions are true:

(42) A−. B = A ∩ B^{c} ∪ A^{c} ∩ B.

(43) A−. B = B−. A.

4. Zero Relation and Universe Relation

Let C_{1}, C_{2} be non empty sets. A set is called a zero relation of C_{1}, C_{2} if: (Def. 12) It = [: [: C_{1}, C_{2} :], (χ_{∅,[: C_{1}, C_{2} :]})^{∘}[: C_{1}, C_{2} :] :].

Let C_{1}, C_{2} be non empty sets. A set is called a universe relation of C_{1}, C_{2} if:

(Def. 13) It = [: [: C_{1}, C_{2} :], (χ_{[: C_{1}, C_{2} :],[: C_{1}, C_{2} :]})^{∘}[: C_{1}, C_{2} :] :].

In the sequel X is a universe relation of C_{1}, C_{2} and O is a zero relation of C_{1}, C_{2}.

The following proposition is true

(44) χ_{∅,[: C_{1}, C_{2} :]} is a Membership function of C_{1}, C_{2}.

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

(Def. 14) Zmf(C_{1}, C_{2}) = χ_{∅,[: C_{1}, C_{2} :]}.

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

(Def. 15) Umf(C_{1}, C_{2}) = χ_{[: C_{1}, C_{2} :],[: C_{1}, C_{2} :]}.

Next we state four propositions:

(45) Let h be a Membership function of C_{1}, C_{2}. If h = χ_{[: C_{1}, C_{2} :],[: C_{1}, C_{2} :]}, then [: [: C_{1}, C_{2} :], (χ_{[: C_{1}, C_{2} :],[: C_{1}, C_{2} :]})^{∘}[: C_{1}, C_{2} :] :] is a fuzzy relation of C_{1}, C_{2}, h.

(46) For every Membership function h of C_{1}, C_{2} such that h = χ_{∅,[: C_{1}, C_{2} :]} holds [: [: C_{1}, C_{2} :], (χ_{∅,[: C_{1}, C_{2} :]})^{∘}[: C_{1}, C_{2} :] :] is a fuzzy relation of C_{1}, C_{2}, h.

(47) O is a fuzzy relation of C_{1}, C_{2}, Zmf(C_{1}, C_{2}).

(48) X is a fuzzy relation of C_{1}, C_{2}, Umf(C_{1}, C_{2}).

Let C_{1}, C_{2} be non empty sets. We see that the zero relation of C_{1}, C_{2} is a fuzzy relation of C_{1}, C_{2}, Zmf(C_{1}, C_{2}).

Let C_{1}, C_{2} be non empty sets. We see that the universe relation of C_{1}, C_{2} is a fuzzy relation of C_{1}, C_{2}, Umf(C_{1}, C_{2}).

In the sequel X denotes a universe relation of C_{1}, C_{2} and O denotes a zero relation of C_{1}, C_{2}.

Next we state a number of propositions:

(49) Let a, b be elements of ℝ and f be a partial function from [: C_{1}, C_{2} :] to ℝ. Suppose rngf ⊆ [a, b] and domf ≠ ∅ and a b. Let x be an element of [: C_{1}, C_{2} :]. If x ∈ domf, then a f(x) and f(x) b.

(50) O ⊆ A.

(51) A ⊆ X.

(52) For every element x of [: C_{1}, C_{2} :] and for every Membership function h of C_{1}, C_{2} holds (Zmf(C_{1}, C_{2}))(x) h(x) and h(x) (Umf(C_{1}, C_{2}))(x).

(53) max(f, Umf(C_{1}, C_{2})) = Umf(C_{1}, C_{2}) and min(f, Umf(C_{1}, C_{2})) = f and max(f, Zmf(C_{1}, C_{2})) = f and min(f, Zmf(C_{1}, C_{2})) = Zmf(C_{1}, C_{2}).

(54) A ∪ X = X and A ∩ X = A.

(55) A ∪ O = A and A ∩ O = O.

(56) If A ⊆ B and A ⊆ D and B ∩ D = O, then A = O.

(57) If A ⊆ B and B ∩ D = O, then A ∩ D = O.

(58) If A ⊆ O, then A = O.

(59) A ∪ B = O iff A = O and B = O.

(60) If A ⊆ B ∪ D and A ∩ D = O, then A ⊆ B.

(61) 1-minusZmf(C_{1}, C_{2}) = Umf(C_{1}, C_{2}) and 1-minusUmf(C_{1}, C_{2}) = Zmf(C_{1}, C_{2}).

(62) O^{c} = X and X^{c} = O.

(63) A−. O = A and O−. A = A.

(64) A−. X = A^{c} and X−. A = A^{c}.

(65) For every element c of [: C_{1}, C_{2} :] such that f(c) h(c) holds (max(f, min(g, h)))(c) = (min(max(f, g), h))(c).

(66) If A ⊆ D, then A ∪ B ∩ D = (A ∪ B) ∩ D.

Let C_{1}, C_{2} be non empty sets, let f, g be Membership functions of C_{1}, C_{2}, let A be a fuzzy relation of C_{1}, C_{2}, f, and let B be a fuzzy relation of C_{1}, C_{2}, g. The functor A \ B yielding a fuzzy relation of C_{1}, C_{2}, min(f, 1-minusg) is defined by:

(Def. 16) A \ B = [: [: C_{1}, C_{2} :], (min(f, 1-minusg))^{∘}[: C_{1}, C_{2} :] :].

One can prove the following propositions:

(67) A \ B = A ∩ B^{c}.

(68) 1-minusmin(f, 1-minusg) = max(1-minusf, g).

(69) (A \ B)^{c} = A^{c} ∪ B.

(70) For every element c of [: C_{1}, C_{2} :] such that f(c) g(c) holds (min(f, 1-minush))(c) (min(g, 1-minush))(c).

(71) If A ⊆ B, then A \ D ⊆ B \ D.

(72) For every element c of [: C_{1}, C_{2} :] such that f(c) g(c) holds (min(h, 1-minusg))(c) (min(h, 1-minusf))(c).

(73) If A ⊆ B, then D \ B ⊆ D \ A.

(74) For every element c of [: C_{1}, C_{2} :] such that f(c) g(c) and h(c) h_{1}(c) holds (min(f, 1-minush_{1}))(c) (min(g, 1-minush))(c).

(75) If A ⊆ B and D ⊆ D_{1}, then A \ D_{1} ⊆ B \ D.

(76) For every element c of [: C_{1}, C_{2} :] holds (min(f, 1-minusg))(c) f(c).

(77) A \ B ⊆ A.

(78) For every element c of [: C_{1}, C_{2} :] holds (min(f, 1-minusg))(c) (max(min(f, 1-minusg), min(1-minusf, g)))(c).

(79) A \ B ⊆ A−. B.

(80) A \ O = A.

(81) O \ A = O.

(82) For every element c of [: C_{1}, C_{2} :] holds (min(f, 1-minusg))(c) (min(f, 1-minusmin(f, g)))(c).

(83) A \ B ⊆ A \ A ∩ B.

(84) For every element c of [: C_{1}, C_{2} :] holds (max(min(f, g), min(f, 1-minusg))) (c) f(c).

(85) For every element c of [: C_{1}, C_{2} :] holds (max(f, min(g, 1-minusf)))(c) (max(f, g))(c).

(86) A ∪ (B \ A) ⊆ A ∪ B.

(87) A ∩ B ∪ (A \ B) ⊆ A.

(88) min(f, 1-minusmin(g, 1-minush)) = max(min(f, 1-minusg), min(f, h)).

(89) A \ (B \ D) = (A \ B) ∪ A ∩ D.

(90) For every element c of [: C_{1}, C_{2} :] holds (min(f, g))(c) (min(f, 1-minusmin(f, 1-minusg)))(c).

(91) A ∩ B ⊆ A \ (A \ B).

(92) For every element c of [: C_{1}, C_{2} :] holds (min(f, 1-minusg))(c) (min(max(f, g), 1-minusg))(c).

(93) A \ B ⊆ (A ∪ B) \ B.

(94) min(f, 1-minusmax(g, h)) = min(min(f, 1-minusg), min(f, 1-minush)).

(95) A \ (B ∪ D) = (A \ B) ∩ (A \ D).

(96) min(f, 1-minusmin(g, h)) = max(min(f, 1-minusg), min(f, 1-minush)).

(97) A \ B ∩ D = (A \ B) ∪ (A \ D).

(98) min(min(f, 1-minusg), 1-minush) = min(f, 1-minusmax(g, h)).

(99) A \ B \ D = A \ (B ∪ D).

(100) For every element c of [: C_{1}, C_{2} :] holds (min(max(f, g), 1-minusmin(f, g)))(c) (max(min(f, 1-minusg), min(g, 1-minusf)))(c).

(101) (A \ B) ∪ (B \ A) ⊆ (A ∪ B) \ A ∩ B.

(102) min(max(f, g), 1-minush) = max(min(f, 1-minush), min(g, 1-minush)).

(103) (A ∪ B) \ D = (A \ D) ∪ (B \ D).

(104) For every element c of [: C_{1}, C_{2} :] such that (min(f, 1-minusg))(c) h(c) and (min(g, 1-minusf))(c) h(c) holds (max(min(f, 1-minusg), min(1-minusf, g)))(c) h(c).

(105) If A \ B ⊆ D and B \ A ⊆ D, then A−. B ⊆ D.

(106) A ∩ (B \ D) = A ∩ B \ D.

(107) For every element c of [: C_{1}, C_{2} :] holds (min(f, min(g, 1-minush)))(c) (min(min(f, g), 1-minusmin(f, h)))(c).

(108) A ∩ (B \ D) ⊆ A ∩ B \ A ∩ D.

(109) For every element c of [: C_{1}, C_{2} :] holds (min(max(f, g), 1-minusmin(f, g)))(c) (max(min(f, 1-minusg), min(1-minusf, g)))(c).

(110) A−. B ⊆ (A ∪ B) \ A ∩ B.

(111) For every element c of [: C_{1}, C_{2} :] holds (max(min(f, g), 1-minusmax(f, g)))(c) (1-minusmax(min(f, 1-minusg), min(1-minusf, g)))(c).

(112) A ∩ B ∪ (A ∪ B)^{c} ⊆ (A−. B)^{c}.

(113) min(max(min(f, 1-minusg), min(1-minusf, g)), 1-minush) = max(min(f, 1-minusmax(g, h)), min(g, 1-minusmax(f, h))).

(114) (A−. B) \ D = (A \ (B ∪ D)) ∪ (B \ (A ∪ D)).

524 takashi mitsuishi et al.

(115) For every element c of [: C_{1}, C_{2} :] holds (min(f, 1-minusmax(min(g, 1-minush), min(1-minusg, h))))(c) (max(min(f, 1-minusmax(g, h)), min(min(f, g), h)))(c).

(116) (A \ (B ∪ D)) ∪ A ∩ B ∩ D ⊆ A \ (B−. D).

(117) For every element c of [: C_{1}, C_{2} :] such that f(c) g(c) holds g(c) (max(f, min(g, 1-minusf)))(c).

(118) If A ⊆ B, then A ∪ (B \ A) ⊆ B.

(119) For every element c of [: C_{1}, C_{2} :] holds (max(f, g))(c) (max(max(min(f, 1-minusg), min(1-minusf, g)), min(f, g)))(c).

(120) (A−. B) ∪ A ∩ B ⊆ A ∪ B.

(121) If min(f, 1-minusg) = Zmf(C_{1}, C_{2}), then for every element c of [: C_{1}, C_{2} :] holds f(c) g(c).

(122) If A \ B = O, then A ⊆ B.

(123) If min(f, g) = Zmf(C_{1}, C_{2}), then min(f, 1-minusg) = f.

(124) If A ∩ B = O, then A \ B = A.

References

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

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

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

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

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

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

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

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

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

The Measurability of Extended Real Valued Functions

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article we prove the measurablility of some extended real valued functions which are f+g, f – g and so on. Moreover, we will define the simple function which are defined on the sigma field. It will play an important role for the Lebesgue integral theory.

MML Identifier: MESFUNC2.

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

1. Finite Valued Function

For simplicity, we adopt the following rules: X is a non empty set, x is an element of X, f, g are partial functions from X to \overline{ℝ}, S is a σ-field of subsets of X, F is a function from ℚ into S, p is a rational number, r is a real number, n, m are natural numbers, and A, B are elements of S.

Let us consider X and let us consider f. We say that f is finite if and only if:

(Def. 1) For every x such that x ∈ domf holds |f(x)| < +∞. Next we state three propositions:

(1) f = 1 f.

(2) For all f, g, A such that f is finite or g is finite holds dom(f + g) = domf ∩ domg and dom(f − g) = domf ∩ domg.

(3) Let given f, g, F, r, A. Suppose f is finite and g is finite and for every p holds F (p) = A ∩ LE-dom(f, \overline{ℝ}(p)) ∩ (A ∩ LE-dom(g, \overline{ℝ}(r − p))). Then A ∩ LE-dom(f + g, \overline{ℝ}(r)) = ⋃ rngF.

2. Measurability of f + g and f − g

The following propositions are true:

(4) There exists a function F from ℕ into ℚ such that F is one-to-one and domF = ℕ and rngF = ℚ.

(5) Let X, Y, Z be non empty sets and F be a function from X into Z. If X ≈ Y, then there exists a function G from Y into Z such that rngF = rngG.

(6) Let given S, f, g, A. Suppose f is measurable on A and g is measurable on A. Then there exists a function F from ℚ into S such that for every rational number p holds F (p) = A ∩ LE-dom(f, \overline{ℝ}(p)) ∩ (A ∩ LE-dom(g, \overline{ℝ}(r − p))).

(7) Let given f, g, A. Suppose f is finite and g is finite and f is measurable on A and g is measurable on A. Then f + g is measurable on A.

(8) For all sets E, F, G and for every partial function f from E to F holds f^{−1} (G) ⊆ E.

(9) For every non empty set C and for all partial functions f_{1}, f_{2} from C to \overline{ℝ} holds f_{1} − f_{2} = f_{1} + − f_{2}.

(10) For every real number r holds \overline{ℝ}(− r) = − \overline{ℝ}(r).

(11) For every non empty set C and for every partial function f from C to \overline{ℝ} holds − f = (− 1) f.

(12) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and r be a real number. If f is finite, then r f is finite.

(13) Let given f, g, A. Suppose f is finite and g is finite and f is measurable on A and g is measurable on A and A ⊆ domg. Then f − g is measurable on A.

3. Definitions of Extended Real Valued Functions max_{+}(f) and max_{−} (f) and their Basic Properties

Let C be a non empty set and let f be a partial function from C to \overline{ℝ}. The functor max_{+}(f) yields a partial function from C to \overline{ℝ} and is defined as follows:

(Def. 2) dommax_{+}(f) = domf and for every element x of C such that x ∈ dommax_{+}(f) holds (max_{+}(f))(x) = max(f(x), 0\overline{ℝ}).

The functor max_{−} (f) yielding a partial function from C to \overline{ℝ} is defined by: (Def. 3) dommax_{−} (f) = domf and for every element x of C such that x ∈

dommax_{−} (f) holds (max_{−} (f))(x) = max(− f(x), 0\overline{ℝ}). The following propositions are true:

(14) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf, then 0\overline{ℝ} (max_{+}(f))(x).

(15) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf, then 0\overline{ℝ} (max_{−} (f))(x).

(16) For every non empty set C and for every partial function f from C to \overline{ℝ} holds max_{−} (f) = max_{+}(− f).

(17) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf and 0\overline{ℝ} < (max_{+}(f))(x), then (max_{−} (f))(x) = 0\overline{ℝ}.

(18) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf and 0\overline{ℝ} < (max_{−} (f))(x), then (max_{+}(f))(x) = 0\overline{ℝ}.

(19) For every non empty set C and for every partial function f from C to \overline{ℝ} holds domf = dom(max_{+}(f) − max_{−} (f)) and domf = dom(max_{+}(f) + max_{−} (f)).

(20) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf, then (max_{+}(f))(x) = f(x) or (max_{+}(f))(x) = 0\overline{ℝ} but (max_{−} (f))(x) = − f(x) or (max_{−} (f))(x) = 0\overline{ℝ}.

(21) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf and (max_{+}(f))(x) = f(x), then (max_{−} (f))(x) = 0\overline{ℝ}.

(22) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf and (max_{+}(f))(x) = 0\overline{ℝ}, then (max_{−} (f))(x) = − f(x).

(23) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf and (max_{−} (f))(x) = − f(x), then (max_{+}(f))(x) = 0\overline{ℝ}.

(24) Let C be a non empty set, f be a partial function from C to \overline{ℝ}, and x be an element of C. If x ∈ domf and (max_{−} (f))(x) = 0\overline{ℝ}, then (max_{+}(f))(x) = f(x).

(25) For every non empty set C and for every partial function f from C to \overline{ℝ} holds f = max_{+}(f) − max_{−} (f).

(26) For every non empty set C and for every partial function f from C to \overline{ℝ} holds |f| = max_{+}(f) + max_{−} (f).

4. Measurability of max_{+}(f), max_{−} (f) and |f|

Next we state three propositions:

(27) If f is measurable on A, then max_{+}(f) is measurable on A.

(28) If f is measurable on A and A ⊆ domf, then max_{−} (f) is measurable on A.

(29) For all f, A such that f is measurable on A and A ⊆ domf holds |f| is measurable on A.

5. Definition and Measurability of Characteristic Function

One can prove the following proposition

(30) For all sets A, X holds rng(χ_{A,X}) ⊆ \{0\overline{ℝ}, \overline{1}\}.

Let A, X be sets. Then χ_{A,X} is a partial function from X to \overline{ℝ}.

Next we state two propositions:

(31) χ_{A,X} is finite.

(32) χ_{A,X} is measurable on B.

6. Definition and Measurability of Simple Function

Let X be a set and let S be a σ-field of subsets of X. One can check that there exists a finite sequence of elements of S which is disjoint valued.

Let X be a set and let S be a σ-field of subsets of X. A finite sequence of separated subsets of S is a disjoint valued finite sequence of elements of S.

The following propositions are true:

(33) Suppose F is a finite sequence of separated subsets of S. Then there exists a sequence G of separated subsets of S such that ⋃ rngF = ⋃ rngG and for every n such that n ∈ domF holds F (n) = G(n) and for every m such that m ∉ domF holds G(m) = ∅.

(34) If F is a finite sequence of separated subsets of S, then ⋃ rngF ∈ S. Let X be a non empty set, let S be a σ-field of subsets of X, and let f be a

partial function from X to \overline{ℝ}. We say that f is simple function in S if and only if the conditions (Def. 5) are satisfied.

(Def. 5)^{1}(i) f is finite, and

(ii) there exists a finite sequence F of separated subsets of S such that domf = ⋃ rngF and for every natural number n and for all elements x, y of X such that n ∈ domF and x ∈ F (n) and y ∈ F (n) holds f(x) = f(y).

1The definition (Def. 4) has been removed.

the measurability of extended real valued … 529

One can prove the following propositions:

(35) If f is finite, then rngf is a subset of ℝ.

(36) Suppose F is a finite sequence of separated subsets of S. Let given n. Then F ↾ Segn is a finite sequence of separated subsets of S.

(37) If f is simple function in S, then f is measurable on A.

References

[1] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

[3] Józef Białas. Completeness of the σ-additive measure. Measure theory. Formalized Mathematics, 2(5):689–693, 1991.

[4] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[5] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[6] Józef Białas. Several properties of the σ-additive measure. Formalized Mathematics, 2(4):493–497, 1991.

[7] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.

[8] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21–26, 1996.

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

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

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

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

[13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495–500, 2001.

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

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

[16] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745–749, 1990.

[17] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

530 noboru endou et al.

Some Properties of Cells and Arcs^{1}

Robert Milewski University of Białystok

Andrzej Trybulec University of Białystok

Artur Korniłowicz University of Białystok

Adam Naumowicz University of Białystok

MML Identifier: JORDAN1B.

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

For simplicity, we adopt the following convention: E denotes a compact non vertical non horizontal subset of E_{T}^{2}, C denotes a compact connected non vertical non horizontal subset of E_{T}^{2}, G denotes a Go-board, i, j, m, n denote natural numbers, and p denotes a point of E_{T}^{2}.

Let us observe that every simple closed curve is non vertical and non horizontal.

Let T be a non empty topological space. Note that there exists a union of components of T which is non empty.

The following propositions are true:

(1) Let T be a non empty topological space and A be a non empty union of components of T. If A is connected, then A is a component of T.

(2) For every finite sequence f holds f is empty iff Rev(f) is empty.

(3) Let D be a non empty set, f be a finite sequence of elements of D, and given i, j. If 1 i and i lenf and 1 j and j lenf, then mid(f, i, j) is non empty.

(4) Let f be a non empty finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If 1 lenf and p ∈ L̃(f), then (⇂ f, p)(1) = f(1).

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

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(6) For every simple closed curve P holds W-maxP ≠ E-maxP.

(7) Let D be a non empty set and f be a finite sequence of elements of D. If 1 i and i < lenf, then (mid(f, i, lenf − ′1))^{⌢} ⟨f_{len f}⟩ = mid(f, i, lenf).

(8) For all points p, q of E_{T}^{2} such that p ≠ q and L(p, q) is vertical holds ⟨p, q⟩ is a special sequence.

(9) For all points p, q of E_{T}^{2} such that p ≠ q and L(p, q) is horizontal holds ⟨p, q⟩ is a special sequence.

(10) Let p, q be finite sequences of elements of E_{T}^{2} and v be a point of E_{T}^{2}. If p is in the area of q, then p_{⟲}^{v} is in the area of q.

(11) For every non trivial finite sequence p of elements of E_{T}^{2} and for every point v of E_{T}^{2} holds p_{⟲}^{v} is in the area of p.

(12) For every finite sequence f holds Centerf 1.

(13) For every finite sequence f such that lenf 1 holds Centerf lenf.

(14) CenterG lenG.

(15) For every finite sequence f such that lenf 2 holds Centerf > 1.

(16) For every finite sequence f such that lenf 3 holds Centerf < lenf.

(17) CenterGauge(E, n) = 2^{n− ′1} + 2.

(18) E ⊆ cell(Gauge(E, 0), 2, 2).

(19) cell(Gauge(E, 0), 2, 2) ⊈ BDDE.

(20) (Gauge(C, 1))_{Center Gauge(C,1),1} = [\frac{W-bound C+E-bound C}{2} , S-bound L̃(Cage(C, 1))].

(21) (Gauge(C, 1))_{Center Gauge(C,1),len Gauge(C,1)} = [\frac{W-bound C+E-bound C}{2} , N-bound L̃(Cage(C, 1))].

(22) If 1 j and j < widthG and 1 m and m lenG and 1 n and n widthG and p ∈ cell(G, lenG, j) and p_{1} = (G_{m,n})_{1}, then lenG = m.

(23) Suppose 1 i and i lenG and 1 j and j < widthG and 1 m and m lenG and 1 n and n widthG and p ∈ cell(G, i, j) and p_{1} = (G_{m,n})_{1}. Then i = m or i = m − ′1.

(24) If 1 i and i < lenG and 1 m and m lenG and 1 n and n widthG and p ∈ cell(G, i, widthG) and p_{2} = (G_{m,n})_{2}, then widthG = n.

(25) Suppose 1 i and i < lenG and 1 j and j widthG and 1 m and m lenG and 1 n and n widthG and p ∈ cell(G, i, j) and p_{2} = (G_{m,n})_{2}. Then j = n or j = n − ′1.

(26) For every simple closed curve C and for every real number r such that W-boundC r and r E-boundC holds L([r, S-boundC], [r, N-boundC]) meets UpperArcC.

(27) For every simple closed curve C and for every real number r such that W-boundC r and r E-boundC holds L([r, S-boundC], [r,

N-boundC]) meets LowerArcC.

(28) Let C be a simple closed curve and i be a natural number. If 1 < i and i < lenGauge(C, n), then L((Gauge(C, n))_{i,1}, (Gauge(C, n))_{i,len Gauge(C,n)}) meets UpperArcC.

(29) Let C be a simple closed curve and i be a natural number. If 1 < i and i < lenGauge(C, n), then L((Gauge(C, n))_{i,1}, (Gauge(C, n))_{i,len Gauge(C,n)}) meets LowerArcC.

(30) For every simple closed curve C holds L((Gauge(C, n))_{Center Gauge(C,n),1}, (Gauge(C, n))_{Center Gauge(C,n),len Gauge(C,n)}) meets UpperArcC.

(31) For every simple closed curve C holds L((Gauge(C, n))_{Center Gauge(C,n),1}, (Gauge(C, n))_{Center Gauge(C,n),len Gauge(C,n)}) meets LowerArcC.

(32) 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 L((Gauge(C, n))_{i,1}, (Gauge(C, n))_{i,len Gauge(C,n)}) meets UpperArc L̃(Cage(C, n)).

(33) 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 L((Gauge(C, n))_{i,1}, (Gauge(C, n))_{i,len Gauge(C,n)}) meets LowerArc L̃(Cage(C, n)).

(34) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds L((Gauge(C, n))_{Center Gauge(C,n),1}, (Gauge(C, n))_{Center Gauge(C,n),len Gauge(C,n)}) meets UpperArc L̃(Cage(C, n)).

(35) For every compact connected non vertical non horizontal subset C of E_{T}^{2} holds L((Gauge(C, n))_{Center Gauge(C,n),1}, (Gauge(C, n))_{Center Gauge(C,n),len Gauge(C,n)}) meets LowerArc L̃(Cage(C, n)).

(36) If j widthG, then cell(G, 0, j) is not Bounded.

(37) If i widthG, then cell(G, lenG, i) is not Bounded.

(38) If j widthGauge(C, n), then cell(Gauge(C, n), 0, j) ⊆ UBDC.

(39) If j lenGauge(E, n), then cell(Gauge(E, n), lenGauge(E, n), j) ⊆ UBDE.

(40) If i lenGauge(C, n) and j widthGauge(C, n) and cell(Gauge(C, n), i, j) ⊆ BDDC, then j > 1.

(41) If i lenGauge(C, n) and j widthGauge(C, n) and cell(Gauge(C, n), i, j) ⊆ BDDC, then i > 1.

(42) If i lenGauge(C, n) and j widthGauge(C, n) and cell(Gauge(C, n), i, j) ⊆ BDDC, then j + 1 < widthGauge(C, n).

(43) If i lenGauge(C, n) and j widthGauge(C, n) and cell(Gauge(C, n), i, j) ⊆ BDDC, then i + 1 < lenGauge(C, n).

534 robert milewski et al.

(44) If there exist i, j such that i lenGauge(C, n) and j widthGauge(C, n) and cell(Gauge(C, n), i, j) ⊆ BDDC, then n 1.

References

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

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

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

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

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

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

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

[17] Artur Korniłowicz, Robert Milewski, Adam Naumowicz, and Andrzej Trybulec. Gauges and cages. Part I. Formalized Mathematics, 9(3):501–509, 2001.

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

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

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536 robert milewski et al.

Formal Topological Spaces

Gang Liu

Tokyo University of Mercantile Marine

Yasushi Fuwa Shinshu University Nagano

Masayoshi Eguchi Tokyo University of Mercantile Marine

Summary. This article is divided into two parts. In the first part, we prove some useful theorems on finite topological spaces. In the second part, in order to consider a family of neighborhoods to a point in a space, we extend the notion of finite topological space and define a new topological space, which we call formal topological space. We show the relation between formal topological space struct (FMT Space Str) and the TopStruct by giving a function (NeighSp). And the following notions are introduced in formal topological spaces: boundary, closure, interior, isolated point, connected point, open set and close set, then some basic facts concerning them are proved. We will discuss the relation between the formal topological space and the finite topological space in future work.

MML Identifier: FINTOPO2.

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

1. Some Useful Theorems on Finite Topological Spaces

In this paper F_{1} denotes a non empty finite topology space and A denotes a subset of the carrier of F_{1}.

The following propositions are true:

(1) Let F_{1} be a non empty finite topology space and A, B be subsets of the carrier of F_{1}. If A ⊆ B, then A^{i} ⊆ B^{i}.

(2) A^{δ} = A^{b} ∩ (A^{i})^{c}.

(3) A^{δ} = A^{b} \ A^{i}.

(4) If A^{c} is open, then A is closed.

(5) If A^{c} is closed, then A is open.

Let F_{1} be a non empty finite topology space, let x be an element of the carrier of F_{1}, let y be an element of the carrier of F_{1}, and let A be a subset of the carrier of F_{1}. The functor P_{1}(x, y, A) yields an element of Boolean and is defined by:

(Def. 1) P_{1}(x, y, A) = \{ \begin{matrix}true, if y ∈ U(x) and y ∈ A, \\ false, otherwise. \end{matrix}

Let F_{1} be a non empty finite topology space, let x be an element of the carrier of F_{1}, let y be an element of the carrier of F_{1}, and let A be a subset of the carrier of F_{1}. The functor P_{2}(x, y, A) yielding an element of Boolean is defined as follows:

(Def. 2) P_{2}(x, y, A) = \{ \begin{matrix}true, if y ∈ U(x) and y ∈ A^{c}, \\ false, otherwise. \end{matrix}

We now state three propositions:

(6) Let x, y be elements of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then P_{1}(x, y, A) = true if and only if y ∈ U(x) and y ∈ A.

(7) Let x, y be elements of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then P_{2}(x, y, A) = true if and only if y ∈ U(x) and y ∈ A^{c}.

(8) Let x be an element of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then x ∈ A^{δ} if and only if there exist elements y_{1}, y_{2} of the carrier of F_{1} such that P_{1}(x, y_{1}, A) = true and P_{2}(x, y_{2}, A) = true.

Let F_{1} be a non empty finite topology space, let x be an element of the carrier of F_{1}, and let y be an element of the carrier of F_{1}. The functor P_{0}(x, y) yielding an element of Boolean is defined as follows:

(Def. 3) P_{0}(x, y) = \{ \begin{matrix}true, if y ∈ U(x), \\ false, otherwise. \end{matrix}

We now state three propositions:

(9) For all elements x, y of the carrier of F_{1} holds P_{0}(x, y) = true iff y ∈ U(x).

(10) Let x be an element of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then x ∈ A^{i} if and only if for every element y of the carrier of F_{1} such that P_{0}(x, y) = true holds P_{1}(x, y, A) = true.

(11) Let x be an element of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then x ∈ A^{b} if and only if there exists an element y_{1} of the carrier of F_{1} such that P_{1}(x, y_{1}, A) = true.

Let F_{1} be a non empty finite topology space, let x be an element of the carrier of F_{1}, and let A be a subset of the carrier of F_{1}. The functor P_{A}(x, A) yielding an element of Boolean is defined as follows:

(Def. 4) P_{A}(x, A) = \{ \begin{matrix}true, if x ∈ A, \\ false, otherwise. \end{matrix}

One can prove the following three propositions:

(12) Let x be an element of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then P_{A}(x, A) = true if and only if x ∈ A.

(13) Let x be an element of the carrier of F_{1} and A be a subset of the carrier

of F_{1}. Then x ∈ A^{δ_{i}} if and only if the following conditions are satisfied:

(i) there exist elements y_{1}, y_{2} of the carrier of F_{1} such that P_{1}(x, y_{1}, A) = true and P_{2}(x, y_{2}, A) = true, and

(ii) P_{A}(x, A) = true.

(14) Let x be an element of the carrier of F_{1} and A be a subset of the carrier

of F_{1}. Then x ∈ A^{δ_{o}} if and only if the following conditions are satisfied:

(i) there exist elements y_{1}, y_{2} of the carrier of F_{1} such that P_{1}(x, y_{1}, A) = true and P_{2}(x, y_{2}, A) = true, and

(ii) P_{A}(x, A) = false. Let F_{1} be a non empty finite topology space, let x be an element of the

carrier of F_{1}, and let y be an element of the carrier of F_{1}. The functor P_{e}(x, y) yielding an element of Boolean is defined by:

(Def. 5) P_{e}(x, y) = \{ \begin{matrix}true, if x = y, \\ false, otherwise. \end{matrix}

The following four propositions are true:

(15) For all elements x, y of the carrier of F_{1} holds P_{e}(x, y) = true iff x = y.

(16) Let x be an element of the carrier of F_{1} and A be a subset of the carrier

of F_{1}. Then x ∈ A^{s} if and only if the following conditions are satisfied:

(i) P_{A}(x, A) = true, and (ii) it is not true that there exists an element y of the carrier of F_{1} such

that P_{1}(x, y, A) = true and P_{e}(x, y) = false.

(17) Let x be an element of the carrier of F_{1} and A be a subset of the carrier

of F_{1}. Then x ∈ A^{n} if and only if the following conditions are satisfied:

(i) P_{A}(x, A) = true, and (ii) there exists an element y of the carrier of F_{1} such that P_{1}(x, y, A) = true

and P_{e}(x, y) = false.

(18) Let x be an element of the carrier of F_{1} and A be a subset of the carrier of F_{1}. Then x ∈ A^{f} if and only if there exists an element y of the carrier of F_{1} such that P_{A}(y, A) = true and P_{0}(y, x) = true.

2. Formal Topological Spaces

We introduce formal topological spaces which are extensions of 1-sorted structure and are systems

⟨ a carrier, a Neighbour-map ⟩, where the carrier is a set and the Neighbour-map is a function from the carrier into 2^{2^{^{the} ^{carrier}}} .

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

In the sequel T is a non empty topological structure, F_{2} is a non empty formal topological space, and x is an element of the carrier of F_{2}.

Let us consider F_{2} and let x be an element of the carrier of F_{2}. The functor U_{F} (x) yielding a subset of 2^{the carrier of F_{2}} is defined as follows:

(Def. 6) U_{F} (x) = (the Neighbour-map of F_{2})(x).

Next we state the proposition

(19) Let F_{2} be a non empty formal topological space and x be an element of

the carrier of F_{2}. Then U_{F} (x) = (the Neighbour-map of F_{2})(x). Let us consider T. The functor NeighSpT yielding a non empty strict formal

topological space is defined by the conditions (Def. 7). (Def. 7)(i) The carrier of NeighSpT = the carrier of T, and

(ii) for every point x of NeighSpT holds U_{F} (x) = {V; V ranges over subsets

of T: V ∈ the topology of T ∧ x ∈ V }. In the sequel A, B, W, V denote subsets of the carrier of F_{2}. Let I_{1} be a non empty formal topological space. We say that I_{1} is filled if

and only if: (Def. 8) For every element x of the carrier of I_{1} and for every subset D of the

carrier of I_{1} such that D ∈ U_{F} (x) holds x ∈ D. Let us consider F_{2} and let us consider A. The functor A^{F δ} yielding a subset

of the carrier of F_{2} is defined as follows: (Def. 9) A^{F δ} = {x : ⋀_{W} (W ∈ U_{F} (x) ⇒ W ∩ A ≠ ∅ ∧ W ∩ A^{c} ≠ ∅)}.

The following proposition is true

(20) x ∈ A^{F δ} iff for every W such that W ∈ U_{F} (x) holds W ∩ A ≠ ∅ and

W ∩ A^{c} ≠ ∅. Let us consider F_{2} and let us consider A. The functor A^{F_{b}} yielding a subset

of the carrier of F_{2} is defined as follows: (Def. 10) A^{F_{b}} = {x : ⋀_{W} (W ∈ U_{F} (x) ⇒ W ∩ A ≠ ∅)}.

One can prove the following proposition

(21) x ∈ A^{F_{b}} iff for every W such that W ∈ U_{F} (x) holds W ∩ A ≠ ∅.

Let us consider F_{2} and let us consider A. The functor A^{F_{i}} yielding a subset of the carrier of F_{2} is defined as follows:

(Def. 11) A^{F_{i}} = {x : ⋁_{V} (V ∈ U_{F} (x) ∧ V ⊆ A)}.

Next we state the proposition

(22) x ∈ A^{F_{i}} iff there exists V such that V ∈ U_{F} (x) and V ⊆ A.

Let us consider F_{2} and let us consider A. The functor A^{F_{s}} yields a subset of the carrier of F_{2} and is defined by:

(Def. 12) A^{F_{s}} = {x : x ∈ A ∧ ⋁_{V} (V ∈ U_{F} (x) ∧ (V \ {x}) ∩ A = ∅)}.

One can prove the following proposition

(23) x ∈ A^{F_{s}} iff x ∈ A and there exists V such that V ∈ U_{F} (x) and (V \

{x}) ∩ A = ∅. Let us consider F_{2} and let us consider A. The functor A^{F_{on}} yields a subset

of the carrier of F_{2} and is defined by: (Def. 13) A^{F_{on}} = A \ A^{F_{s}}.

We now state a number of propositions:

(24) x ∈ A^{F_{on}} iff x ∈ A and for every V such that V ∈ U_{F} (x) holds (V \ {x}) ∩ A ≠ ∅.

(25) Let F_{2} be a non empty formal topological space and A, B be subsets of the carrier of F_{2}. If A ⊆ B, then A^{F_{b}} ⊆ B^{F_{b}}.

(26) Let F_{2} be a non empty formal topological space and A, B be subsets of the carrier of F_{2}. If A ⊆ B, then A^{F_{i}} ⊆ B^{F_{i}}.

(27) A^{F_{b}} ∪ B^{F_{b}} ⊆ A ∪ B^{F_{b}}.

(28) A ∩ B^{F_{b}} ⊆ A^{F_{b}} ∩ B^{F_{b}}.

(29) A^{F_{i}} ∪ B^{F_{i}} ⊆ A ∪ B^{F_{i}}.

(30) A ∩ B^{F_{i}} ⊆ A^{F_{i}} ∩ B^{F_{i}}.

(31) Let F_{2} be a non empty formal topological space. Then the following

statements are equivalent

(i) for every element x of the carrier of F_{2} and for all subsets V_{1}, V_{2} of the carrier of F_{2} such that V_{1} ∈ U_{F} (x) and V_{2} ∈ U_{F} (x) there exists a subset W of the carrier of F_{2} such that W ∈ U_{F} (x) and W ⊆ V_{1} ∩ V_{2},

(ii) for all subsets A, B of the carrier of F_{2} holds A ∪ B^{F_{b}} = A^{F_{b}} ∪ B^{F_{b}}.

(32) Let F_{2} be a non empty formal topological space. Then the following

statements are equivalent

(i) for every element x of the carrier of F_{2} and for all subsets V_{1}, V_{2} of the carrier of F_{2} such that V_{1} ∈ U_{F} (x) and V_{2} ∈ U_{F} (x) there exists a subset W of the carrier of F_{2} such that W ∈ U_{F} (x) and W ⊆ V_{1} ∩ V_{2},

(ii) for all subsets A, B of the carrier of F_{2} holds A^{F_{i}} ∩ B^{F_{i}} = A ∩ B^{F_{i}}.

(33) Let F_{2} be a non empty formal topological space and A, B be subsets of the carrier of F_{2}. Suppose that for every element x of the carrier of F_{2} and for all subsets V_{1}, V_{2} of the carrier of F_{2} such that V_{1} ∈ U_{F} (x) and V_{2} ∈ U_{F} (x) there exists a subset W of the carrier of F_{2} such that W ∈ U_{F} (x) and W ⊆ V_{1} ∩ V_{2}. Then A ∪ B^{F δ} ⊆ A^{F δ} ∪ B^{F δ}.

(34) Let F_{2} be a non empty formal topological space. Suppose F_{2} is filled. Suppose that for all subsets A, B of the carrier of F_{2} holds A ∪ B^{F δ} = A^{F δ} ∪B^{F δ}. Let x be an element of the carrier of F_{2} and V_{1}, V_{2} be subsets of the carrier of F_{2}. Suppose V_{1} ∈ U_{F} (x) and V_{2} ∈ U_{F} (x). Then there exists a subset W of the carrier of F_{2} such that W ∈ U_{F} (x) and W ⊆ V_{1} ∩ V_{2}.

(35) Let x be an element of the carrier of F_{2} and A be a subset of the carrier

of F_{2}. Then x ∈ A^{F_{s}} if and only if the following conditions are satisfied:

(i) x ∈ A, and (ii) x ∉ A \ {x}^{F_{b}}.

(36) Let F_{2} be a non empty formal topological space. Then F_{2} is filled if and only if for every subset A of the carrier of F_{2} holds A ⊆ A^{F_{b}}.

(37) Let F_{2} be a non empty formal topological space. Then F_{2} is filled if and only if for every subset A of the carrier of F_{2} holds A^{F_{i}} ⊆ A.

(38) (A^{c}^{F_{b}})^{c} = A^{F_{i}}.

(39) (A^{c}^{F_{i}})^{c} = A^{F_{b}}.

(40) A^{F δ} = A^{F_{b}} ∩ A^{c}^{F_{b}}.

(41) A^{F δ} = A^{F_{b}} ∩ (A^{F_{i}})^{c}.

(42) A^{F δ} = A^{c}^{F δ}.

(43) A^{F δ} = A^{F_{b}} \ A^{F_{i}}.

Let us consider F_{2} and let us consider A. The functor A^{F δ_{i}} yields a subset of the carrier of F_{2} and is defined by:

(Def. 14) A^{F δ_{i}} = A ∩ A^{F δ}.

The functor A^{F δ_{o}} yields a subset of the carrier of F_{2} and is defined by: (Def. 15) A^{F δ_{o}} = A^{c} ∩ A^{F δ}.

The following proposition is true

(44) A^{F δ} = A^{F δ_{i}} ∪ A^{F δ_{o}}.

Let us consider F_{2} and let G be a subset of the carrier of F_{2}. We say that G is open if and only if:

(Def. 16) G = G^{F_{i}}.

We say that G is closed if and only if: (Def. 17) G = G^{F_{b}}.

Next we state four propositions:

(45) If A^{c} is open, then A is closed.

(46) If A^{c} is closed, then A is open.

formal topological spaces 543

(47) Let F_{2} be a non empty formal topological space and A, B be subsets of the carrier of F_{2}. Suppose F_{2} is filled. Suppose that for every element x of the carrier of F_{2} holds {x} ∈ U_{F} (x). Then A ∩ B^{F_{b}} = A^{F_{b}} ∩ B^{F_{b}}.

(48) Let F_{2} be a non empty formal topological space and A, B be subsets of the carrier of F_{2}. Suppose F_{2} is filled. Suppose that for every element x of the carrier of F_{2} holds {x} ∈ U_{F} (x). Then A^{F_{i}} ∪ B^{F_{i}} = A ∪ B^{F_{i}}.

References

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

[2] Hiroshi Imura and Masayoshi Eguchi. Finite topological spaces. Formalized Mathematics, 3(2):189–193, 1992.

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

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

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

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

544 gang liu et al.

Some Properties of Cells and Gauges^{1}

Adam Grabowski University of Białystok

Artur Korniłowicz University of Białystok

Andrzej Trybulec University of Białystok

MML Identifier: JORDAN1C.

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

We adopt the following convention: C denotes a simple closed curve, i, j, n denote natural numbers, and p denotes a point of E_{T}^{2}.

The following propositions are true:

(1) BDDC is Bounded.

(2) If ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, n) and ⟨⟨i + 1, j⟩⟩ ∈ the indices of Gauge(C, n), then ρ((Gauge(C, n))_{1,1}, (Gauge(C, n))_{2,1}) = |((Gauge(C, n))_{i+1,j})_{1} − ((Gauge(C, n))_{i,j})_{1}|.

(3) If ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, n) and ⟨⟨i, j + 1⟩⟩ ∈ the indices of Gauge(C, n), then ρ((Gauge(C, n))_{1,1}, (Gauge(C, n))_{1,2}) = |((Gauge(C, n))_{i,j+1})_{2} − ((Gauge(C, n))_{i,j})_{2}|.

(4) For every subset S of E_{T}^{2} such that S is Bounded holds (proj1)^{∘}S is bounded.

(5) Let C_{1} be a non empty compact subset of E_{T}^{2} and C_{2}, S be non empty subsets of E_{T}^{2}. If S = C_{1} ∪ C_{2} and (proj1)^{∘}C_{2} is non empty and lower bounded, then W-boundS = min(W-boundC_{1}, W-boundC_{2}).

(6) For every subset X of E_{T}^{2} such that p ∈ X and X is Bounded holds W-boundX p_{1} and p_{1} E-boundX and S-boundX p_{2} and p_{2} N-boundX.

(7) p ∈ WestHalflinep and p ∈ EastHalflinep.

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(8) WestHalflinep is non Bounded.

(9) EastHalflinep is non Bounded.

(10) NorthHalflinep is non Bounded.

(11) SouthHalflinep is non Bounded.

(12) If UBDC ≠ ∅, then UBDC is a component of C^{c}.

(13) For every connected subset W_{1} of E_{T}^{2} such that W_{1} is non Bounded and W_{1} ∩ C = ∅ holds W_{1} ⊆ UBDC.

(14) For every point p of E_{T}^{2} such that WestHalflinep ∩ C = ∅ holds WestHalflinep ⊆ UBDC.

(15) For every point p of E_{T}^{2} such that EastHalflinep ∩ C = ∅ holds EastHalflinep ⊆ UBDC.

(16) For every point p of E_{T}^{2} such that SouthHalflinep ∩ C = ∅ holds SouthHalflinep ⊆ UBDC.

(17) For every point p of E_{T}^{2} such that NorthHalflinep ∩ C = ∅ holds NorthHalflinep ⊆ UBDC.

(18) If BDDC ≠ ∅, then W-boundC W-boundBDDC.

(19) If BDDC ≠ ∅, then E-boundC E-boundBDDC.

(20) If BDDC ≠ ∅, then S-boundC S-boundBDDC.

(21) If BDDC ≠ ∅, then N-boundC N-boundBDDC.

(22) For every integer I such that p ∈ BDDC and I = ⌊\frac{p_{1} −W-bound C}{E-bound C−W-bound C} · 2^{n} + 2⌋ holds 1 < I.

(23) For every integer I such that p ∈ BDDC and I = ⌊\frac{p_{1} −W-bound C}{E-bound C−W-bound C} · 2^{n} + 2⌋ holds I + 1 lenGauge(C, n).

(24) For every integer J such that p ∈ BDDC and J = ⌊\frac{p_{2} −S-bound C}{N-bound C−S-bound C} · 2^{n} + 2⌋ holds 1 < J and J + 1 widthGauge(C, n).

(25) For every integer I such that I = ⌊\frac{p_{1} −W-bound C}{E-bound C−W-bound C} · 2^{n} + 2⌋ holds W-boundC + \frac{E-bound C−W-bound C}{2^{n}} · (I − 2) p_{1}.

(26) For every integer I such that I = ⌊\frac{p_{1} −W-bound C}{E-bound C−W-bound C} · 2^{n} + 2⌋ holds p_{1} < W-boundC + \frac{E-bound C−W-bound C}{2^{n}} · (I − 1).

(27) For every integer J such that J = ⌊\frac{p_{2} −S-bound C}{N-bound C−S-bound C} · 2^{n} + 2⌋ holds S-boundC + \frac{N-bound C−S-bound C}{2^{n}} · (J − 2) p_{2}.

(28) For every integer J such that J = ⌊\frac{p_{2} −S-bound C}{N-bound C−S-bound C} · 2^{n} + 2⌋ holds p_{2} < S-boundC + \frac{N-bound C−S-bound C}{2^{n}} · (J − 1).

(29) Let C be a closed subset of E_{T}^{2} and p be a point of E^{2}. If p ∈ BDDC, then there exists a real number r such that r > 0 and Ball(p, r) ⊆ BDDC.

(30) Let p, q be points of E_{T}^{2} and r be a real number. Suppose ρ((Gauge(C, n))_{1,1}, (Gauge(C, n))_{1,2}) < r and ρ((Gauge(C, n))_{1,1},

some properties of cells and gauges 547

(Gauge(C, n))_{2,1}) < r and p ∈ cell(Gauge(C, n), i, j) and q ∈ cell(Gauge(C, n), i, j) and 1 i and i + 1 lenGauge(C, n) and 1 j and j + 1 widthGauge(C, n). Then ρ(p, q) < 2 · r.

(31) If p ∈ BDDC, then p_{2} ≠ N-boundBDDC.

(32) If p ∈ BDDC, then p_{1} ≠ E-boundBDDC.

(33) If p ∈ BDDC, then p_{2} ≠ S-boundBDDC.

(34) If p ∈ BDDC, then p_{1} ≠ W-boundBDDC.

(35) Suppose p ∈ BDDC. Then there exist natural numbers n, i, j such that 1 < i and i < lenGauge(C, n) and 1 < j and j < widthGauge(C, n) and p_{1} ≠ ((Gauge(C, n))_{i,j})_{1} and p ∈ cell(Gauge(C, n), i, j) and cell(Gauge(C, n), i, j) ⊆ BDDC.

References

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[2] Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65–69, 1991.

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

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

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

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

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[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}. Simple closed curves. Formalized Mathematics, 2(5):663–664, 1991.

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

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

[13] Artur Korniłowicz. Properties of left and right components. Formalized Mathematics, 8(1):163–168, 1999.

[14] Artur Korniłowicz, Robert Milewski, Adam Naumowicz, and Andrzej Trybulec. Gauges and cages. Part I. Formalized Mathematics, 9(3):501–509, 2001.

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

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

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

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

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

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

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

548 adam grabowski et al.

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

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

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

Again on the Order on a Special Polygon^{1}

Andrzej Trybulec University of Białystok Yatsuka Nakamura Shinshu University Nagano

MML Identifier: SPRECT 5.

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

1. Preliminaries

For simplicity, we use the following convention: D denotes a non empty set, f denotes a finite sequence of elements of D, g denotes a circular finite sequence of elements of D, and p, p_{1}, p_{2}, p_{3}, q denote elements of D.

We now state several propositions:

(1) If q ∈ rng(f↾p ↫ f), then q ↫ f p ↫ f.

(2) If p ∈ rngf and q ∈ rngf and p ↫ f q ↫ f, then q ↫ (f :− p) = (q ↫ f − p ↫ f) + 1.

(3) If p ∈ rngf and q ∈ rngf and p ↫ f < q ↫ f, then p ↫ (f − : q) = p ↫ f.

(4) If p ∈ rngf and q ∈ rngf and p ↫ f q ↫ f, then q ↫ (f_{⟲}^{p}) = (q ↫ f − p ↫ f) + 1.

(5) If p_{1} ∈ rngf and p_{2} ∈ rngf and p_{3} ∈ rngf and p_{1} ↫ f p_{2} ↫ f and p_{2} ↫ f < p_{3} ↫ f, then p_{2} ↫ (f_{⟲}^{p_{1}}) < p_{3} ↫ (f_{⟲}^{p_{1}}).

(6) If p_{1} ∈ rngf and p_{2} ∈ rngf and p_{3} ∈ rngf and p_{1} ↫ f < p_{2} ↫ f and p_{2} ↫ f p_{3} ↫ f, then p_{2} ↫ (f_{⟲}^{p_{1}}) p_{3} ↫ (f_{⟲}^{p_{1}}).

(7) If p ∈ rngg and leng > 1, then p ↫ g < leng.

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

2. Ordering of Special Points on a Standard Special Sequence

We adopt the following rules: f denotes a non constant standard special circular sequence and p, p_{1}, p_{2}, p_{3}, q denote points of E_{T}^{2}.

The following propositions are true:

(8) f_{⇂1} is one-to-one.

(9) If 1 < q ↫ f and q ∈ rngf, then (π_{1}f) ↫ (f_{⟲}^{q}) = (lenf + 1) − q ↫ f.

(10) If p ∈ rngf and q ∈ rngf and p ↫ f < q ↫ f, then p ↫ (f_{⟲}^{q}) = (lenf + p ↫ f) − q ↫ f.

(11) If p_{1} ∈ rngf and p_{2} ∈ rngf and p_{3} ∈ rngf and p_{1} ↫ f < p_{2} ↫ f and p_{2} ↫ f < p_{3} ↫ f, then p_{3} ↫ (f_{⟲}^{p_{2}}) < p_{1} ↫ (f_{⟲}^{p_{2}}).

(12) If p_{1} ∈ rngf and p_{2} ∈ rngf and p_{3} ∈ rngf and p_{1} ↫ f < p_{2} ↫ f and p_{2} ↫ f < p_{3} ↫ f, then p_{1} ↫ (f_{⟲}^{p_{3}}) < p_{2} ↫ (f_{⟲}^{p_{3}}).

(13) If p_{1} ∈ rngf and p_{2} ∈ rngf and p_{3} ∈ rngf and p_{1} ↫ f p_{2} ↫ f and p_{2} ↫ f < p_{3} ↫ f, then p_{1} ↫ (f_{⟲}^{p_{3}}) p_{2} ↫ (f_{⟲}^{p_{3}}).

(14) (S-min L̃(f)) ↫ f < lenf.

(15) (S-max L̃(f)) ↫ f < lenf.

(16) (E-min L̃(f)) ↫ f < lenf.

(17) (E-max L̃(f)) ↫ f < lenf.

(18) (N-min L̃(f)) ↫ f < lenf.

(19) (N-max L̃(f)) ↫ f < lenf.

(20) (W-max L̃(f)) ↫ f < lenf.

(21) (W-min L̃(f)) ↫ f < lenf.

3. Ordering of Special Points on a Clockwise Oriented Sequence

In the sequel z is a clockwise oriented non constant standard special circular sequence.

Next we state a number of propositions:

(22) If π_{1}f = W-min L̃(f), then (W-min L̃(f)) ↫ f < (W-max L̃(f)) ↫ f.

(23) If π_{1}f = W-min L̃(f), then (W-max L̃(f)) ↫ f > 1.

(24) If π_{1}z = W-min L̃(z) and W-max L̃(z) ≠ N-min L̃(z), then (W-max L̃(z)) ↫ z < (N-min L̃(z)) ↫ z.

(25) If π_{1}z = W-min L̃(z), then (N-min L̃(z)) ↫ z < (N-max L̃(z)) ↫ z.

(26) If π_{1}z = W-min L̃(z) and N-max L̃(z) ≠ E-max L̃(z), then (N-max L̃(z)) ↫ z < (E-max L̃(z)) ↫ z.

(27) If π_{1}z = W-min L̃(z), then (E-max L̃(z)) ↫ z < (E-min L̃(z)) ↫ z.

(28) If π_{1}z = W-min L̃(z) and E-min L̃(z) ≠ S-max L̃(z), then (E-min L̃(z)) ↫ z < (S-max L̃(z)) ↫ z.

(29) If π_{1}z = W-min L̃(z) and S-min L̃(z) ≠ W-min L̃(z), then (S-max L̃(z)) ↫ z < (S-min L̃(z)) ↫ z.

(30) If π_{1}f = S-max L̃(f), then (S-max L̃(f)) ↫ f < (S-min L̃(f)) ↫ f.

(31) If π_{1}f = S-max L̃(f), then (S-min L̃(f)) ↫ f > 1.

(32) If π_{1}z = S-max L̃(z) and S-min L̃(z) ≠ W-min L̃(z), then (S-min L̃(z)) ↫ z < (W-min L̃(z)) ↫ z.

(33) If π_{1}z = S-max L̃(z), then (W-min L̃(z)) ↫ z < (W-max L̃(z)) ↫ z.

(34) If π_{1}z = S-max L̃(z) and W-max L̃(z) ≠ N-min L̃(z), then (W-max L̃(z)) ↫ z < (N-min L̃(z)) ↫ z.

(35) If π_{1}z = S-max L̃(z), then (N-min L̃(z)) ↫ z < (N-max L̃(z)) ↫ z.

(36) If π_{1}z = S-max L̃(z) and N-max L̃(z) ≠ E-max L̃(z), then (N-max L̃(z)) ↫ z < (E-max L̃(z)) ↫ z.

(37) If π_{1}z = S-max L̃(z) and E-min L̃(z) ≠ S-max L̃(z), then (E-max L̃(z)) ↫ z < (E-min L̃(z)) ↫ z.

(38) If π_{1}f = E-max L̃(f), then (E-max L̃(f)) ↫ f < (E-min L̃(f)) ↫ f.

(39) If π_{1}f = E-max L̃(f), then (E-min L̃(f)) ↫ f > 1.

(40) If π_{1}z = E-max L̃(z) and S-max L̃(z) ≠ E-min L̃(z), then (E-min L̃(z)) ↫ z < (S-max L̃(z)) ↫ z.

(41) If π_{1}z = E-max L̃(z), then (S-max L̃(z)) ↫ z < (S-min L̃(z)) ↫ z.

(42) If π_{1}z = E-max L̃(z) and S-min L̃(z) ≠ W-min L̃(z), then (S-min L̃(z)) ↫ z < (W-min L̃(z)) ↫ z.

(43) If π_{1}z = E-max L̃(z), then (W-min L̃(z)) ↫ z < (W-max L̃(z)) ↫ z.

(44) If π_{1}z = E-max L̃(z) and W-max L̃(z) ≠ N-min L̃(z), then (W-max L̃(z)) ↫ z < (N-min L̃(z)) ↫ z.

(45) If π_{1}z = E-max L̃(z) and N-max L̃(z) ≠ E-max L̃(z), then (N-min L̃(z)) ↫ z < (N-max L̃(z)) ↫ z.

(46) If π_{1}f = N-max L̃(f) and N-max L̃(f) ≠ E-max L̃(f), then (N-max L̃(f)) ↫ f < (E-max L̃(f)) ↫ f.

(47) If π_{1}z = N-max L̃(z), then (E-max L̃(z)) ↫ z < (E-min L̃(z)) ↫ z.

(48) If π_{1}z = N-max L̃(z) and E-min L̃(z) ≠ S-max L̃(z), then (E-min L̃(z)) ↫ z < (S-max L̃(z)) ↫ z.

(49) If π_{1}z = N-max L̃(z), then (S-max L̃(z)) ↫ z < (S-min L̃(z)) ↫ z.

(50) If π_{1}z = N-max L̃(z) and S-min L̃(z) ≠ W-min L̃(z), then (S-min L̃(z)) ↫ z < (W-min L̃(z)) ↫ z.

(51) If π_{1}z = N-max L̃(z), then (W-min L̃(z)) ↫ z < (W-max L̃(z)) ↫ z.

(52) If π_{1}z = N-max L̃(z) and N-min L̃(z) ≠ W-max L̃(z), then (W-max L̃(z)) ↫ z < (N-min L̃(z)) ↫ z.

(53) If π_{1}f = E-min L̃(f) and E-min L̃(f) ≠ S-max L̃(f), then (E-min L̃(f)) ↫ f < (S-max L̃(f)) ↫ f.

(54) If π_{1}z = E-min L̃(z), then (S-max L̃(z)) ↫ z < (S-min L̃(z)) ↫ z.

(55) If π_{1}z = E-min L̃(z) and S-min L̃(z) ≠ W-min L̃(z), then (S-min L̃(z)) ↫ z < (W-min L̃(z)) ↫ z.

(56) If π_{1}z = E-min L̃(z), then (W-min L̃(z)) ↫ z < (W-max L̃(z)) ↫ z.

(57) If π_{1}z = E-min L̃(z) and W-max L̃(z) ≠ N-min L̃(z), then (W-max L̃(z)) ↫ z < (N-min L̃(z)) ↫ z.

(58) If π_{1}z = E-min L̃(z), then (N-min L̃(z)) ↫ z < (N-max L̃(z)) ↫ z.

(59) If π_{1}z = E-min L̃(z) and E-max L̃(z) ≠ N-max L̃(z), then (N-max L̃(z)) ↫ z < (E-max L̃(z)) ↫ z.

(60) If π_{1}f = S-min L̃(f) and S-min L̃(f) ≠ W-min L̃(f), then (S-min L̃(f)) ↫ f < (W-min L̃(f)) ↫ f.

(61) If π_{1}z = S-min L̃(z), then (W-min L̃(z)) ↫ z < (W-max L̃(z)) ↫ z.

(62) If π_{1}z = S-min L̃(z) and W-max L̃(z) ≠ N-min L̃(z), then (W-max L̃(z)) ↫ z < (N-min L̃(z)) ↫ z.

(63) If π_{1}z = S-min L̃(z), then (N-min L̃(z)) ↫ z < (N-max L̃(z)) ↫ z.

(64) If π_{1}z = S-min L̃(z) and N-max L̃(z) ≠ E-max L̃(z), then (N-max L̃(z)) ↫ z < (E-max L̃(z)) ↫ z.

(65) If π_{1}z = S-min L̃(z), then (E-max L̃(z)) ↫ z < (E-min L̃(z)) ↫ z.

(66) If π_{1}z = S-min L̃(z) and S-max L̃(z) ≠ E-min L̃(z), then (E-min L̃(z)) ↫ z < (S-max L̃(z)) ↫ z.

(67) If π_{1}f = W-max L̃(f) and W-max L̃(f) ≠ N-min L̃(f), then (W-max L̃(f)) ↫ f < (N-min L̃(f)) ↫ f.

(68) If π_{1}z = W-max L̃(z), then (N-min L̃(z)) ↫ z < (N-max L̃(z)) ↫ z.

(69) If π_{1}z = W-max L̃(z) and N-max L̃(z) ≠ E-max L̃(z), then (N-max L̃(z)) ↫ z < (E-max L̃(z)) ↫ z.

(70) If π_{1}z = W-max L̃(z), then (E-max L̃(z)) ↫ z < (E-min L̃(z)) ↫ z.

(71) If π_{1}z = W-max L̃(z) and E-min L̃(z) ≠ S-max L̃(z), then (E-min L̃(z)) ↫ z < (S-max L̃(z)) ↫ z.

(72) If π_{1}z = W-max L̃(z), then (S-max L̃(z)) ↫ z < (S-min L̃(z)) ↫ z.

(73) If π_{1}z = W-max L̃(z) and W-min L̃(z) ≠ S-min L̃(z), then (S-min L̃(z)) ↫ z < (W-min L̃(z)) ↫ z.

References

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

again on the order on a special polygon 553

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

[7] 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.

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

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

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

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

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

[13] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

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

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

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

554 andrzej trybulec and yatsuka nakamura

Gauges and Cages. Part II^{1}

Artur Korniłowicz University of Białystok

Robert Milewski University of Białystok

MML Identifier: JORDAN1D.

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

1. Preliminaries

For simplicity, we use the following convention: a, b, i, k, m, n are natural numbers, r, s are real numbers, D is a non empty subset of E_{T}^{2}, and C is a compact connected non vertical non horizontal subset of E_{T}^{2}.

Next we state the proposition

(1) For all sets A, B such that for every set x such that x ∈ A there exists a set K such that K ⊆ B and x ⊆ ⋃ K holds ⋃ A ⊆ ⋃ B.

Let m be an even integer. Note that m + 2 is even. Let m be an odd integer. Observe that m + 2 is odd. Let m be a non empty natural number. Observe that 2^{m} is even. Let n be an even natural number and let m be a non empty natural number.

Note that n^{m} is even.

We now state several propositions:

(2) If r ≠ 0, then \frac{1}{r} · r^{i+1} = r^{i}.

(3) If \frac{r}{s} is an integer and s ≠ 0, then − ⌊\frac{r}{s}⌋ = ⌊\frac{− r}{s} ⌋+ 1.

(4) If \frac{r}{s} is an integer, then − ⌊\frac{r}{s}⌋ = ⌊\frac{− r}{s} ⌋.

(5) If n > 0 and k mod n ≠ 0, then − (k ÷ n) = (− k ÷ n) + 1.

(6) If n > 0 and k mod n = 0, then − (k ÷ n) = − k ÷ n.

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

2. Gauges and Cages

We now state a number of propositions:

(7) If 2 m and m < lenGauge(D, i) and 1 a and a lenGauge(D, i) and 1 b and b lenGauge(D, i + 1), then ((Gauge(D, i))_{m,a})_{1} = ((Gauge(D, i + 1))_{2·m− ′2,b})_{1}.

(8) If 2 n and n < lenGauge(D, i) and 1 a and a lenGauge(D, i) and 1 b and b lenGauge(D, i + 1), then ((Gauge(D, i))_{a,n})_{2} = ((Gauge(D, i + 1))_{b,2·n− ′2})_{2}.

(9) Let D be a compact non vertical non horizontal subset of E_{T}^{2}. Suppose 2 m and m+1 < lenGauge(D, i) and 2 n and n+1 < lenGauge(D, i). Then cell(Gauge(D, i), m, n) = cell(Gauge(D, i + 1), 2 · m − ′2, 2 · n − ′2) ∪ cell(Gauge(D, i + 1), 2 · m − ′1, 2 · n − ′2) ∪ cell(Gauge(D, i + 1), 2 · m − ′ 2, 2 · n − ′1) ∪ cell(Gauge(D, i + 1), 2 · m − ′1, 2 · n − ′1).

(10) Let D be a compact non vertical non horizontal subset of E_{T}^{2} and k be a natural number. Suppose 2 m and m + 1 < lenGauge(D, i) and 2 n and n + 1 < lenGauge(D, i). Then cell(Gauge(D, i), m, n) = ⋃{cell(Gauge(D, i + k), a, b); a ranges over natural numbers, b ranges over natural numbers: (2^{k} · m − 2^{k+1}) + 2 a ∧ a (2^{k} · m − 2^{k}) + 1 ∧ (2^{k} · n − 2^{k+1}) + 2 b ∧ b (2^{k} · n − 2^{k}) + 1}.

(11) There exists a natural number i such that 1 i and i < lenCage(C, n) and N-maxC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(12) There exists a natural number i such that 1 i and i < lenCage(C, n) and N-maxC ∈ rightcell(Cage(C, n), i).

(13) There exists a natural number i such that 1 i and i < lenCage(C, n) and E-minC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(14) There exists a natural number i such that 1 i and i < lenCage(C, n) and E-minC ∈ rightcell(Cage(C, n), i).

(15) There exists a natural number i such that 1 i and i < lenCage(C, n) and E-maxC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(16) There exists a natural number i such that 1 i and i < lenCage(C, n) and E-maxC ∈ rightcell(Cage(C, n), i).

(17) There exists a natural number i such that 1 i and i < lenCage(C, n) and S-minC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(18) There exists a natural number i such that 1 i and i < lenCage(C, n) and S-minC ∈ rightcell(Cage(C, n), i).

(19) There exists a natural number i such that 1 i and i < lenCage(C, n) and S-maxC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(20) There exists a natural number i such that 1 i and i < lenCage(C, n) and S-maxC ∈ rightcell(Cage(C, n), i).

(21) There exists a natural number i such that 1 i and i < lenCage(C, n) and W-minC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(22) There exists a natural number i such that 1 i and i < lenCage(C, n) and W-minC ∈ rightcell(Cage(C, n), i).

(23) There exists a natural number i such that 1 i and i < lenCage(C, n) and W-maxC ∈ right cell(Cage(C, n), i, Gauge(C, n)).

(24) There exists a natural number i such that 1 i and i < lenCage(C, n) and W-maxC ∈ rightcell(Cage(C, n), i).

(25) There exists a natural number i such that 1 i and i lenGauge(C, n) and N-min L̃(Cage(C, n)) = (Gauge(C, n))_{i,width Gauge(C,n)}.

(26) There exists a natural number i such that 1 i and i lenGauge(C, n) and N-max L̃(Cage(C, n)) = (Gauge(C, n))_{i,width Gauge(C,n)}.

(27) There exists a natural number i such that 1 i and i lenGauge(C, n) and (Gauge(C, n))_{i,width Gauge(C,n)} ∈ rngCage(C, n).

(28) There exists a natural number j such that 1 j and j widthGauge(C, n) and E-min L̃(Cage(C, n)) = (Gauge(C, n))_{len Gauge(C,n),j}.

(29) There exists a natural number j such that 1 j and j widthGauge(C, n) and E-max L̃(Cage(C, n)) = (Gauge(C, n))_{len Gauge(C,n),j}.

(30) There exists a natural number j such that 1 j and j widthGauge(C, n) and (Gauge(C, n))_{len Gauge(C,n),j} ∈ rngCage(C, n).

(31) There exists a natural number i such that 1 i and i lenGauge(C, n) and S-min L̃(Cage(C, n)) = (Gauge(C, n))_{i,1}.

(32) There exists a natural number i such that 1 i and i lenGauge(C, n) and S-max L̃(Cage(C, n)) = (Gauge(C, n))_{i,1}.

(33) There exists a natural number i such that 1 i and i lenGauge(C, n) and (Gauge(C, n))_{i,1} ∈ rngCage(C, n).

(34) There exists a natural number j such that 1 j and j widthGauge(C, n) and W-min L̃(Cage(C, n)) = (Gauge(C, n))_{1,j}.

(35) There exists a natural number j such that 1 j and j widthGauge(C, n) and W-max L̃(Cage(C, n)) = (Gauge(C, n))_{1,j}.

(36) There exists a natural number j such that 1 j and j widthGauge(C, n) and (Gauge(C, n))_{1,j} ∈ rngCage(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.

558 artur korniłowicz and robert milewski

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

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

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

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

[12] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573–577, 1997.

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

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

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

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

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

The Binomial Theorem for Algebraic Structures^{1}

Christoph Schwarzweller University of Tübingen

Summary. In this paper we prove the well-known binomial theorem for algebraic structures. In doing so we tried to be as modest as possible concerning the algebraic properties of the underlying structure. Consequently, we proved the binomial theorem for “commutative rings” in which the existence of an inverse with respect to addition is replaced by a weaker property of cancellation.

MML Identifier: BINOM.

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

1. Preliminaries

Let L be a non empty loop structure. We say that L is add-left-cancelable if and only if:

(Def. 1) For all elements a, b, c of L such that a + b = a + c holds b = c.

We say that L is add-right-cancelable if and only if: (Def. 2) For all elements a, b, c of L such that b + a = c + a holds b = c.

We say that L is add-cancelable if and only if: (Def. 3) For all elements a, b, c of L holds if a + b = a + c, then b = c and if

b + a = c + a, then b = c. One can check the following observations:

∗ there exists a non empty loop structure which is add-left-cancelable, ∗ there exists a non empty loop structure which is add-right-cancelable, and

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

∗ there exists a non empty loop structure which is add-cancelable.

Let us note that every non empty loop structure which is add-left-cancelable and add-right-cancelable is also add-cancelable and every non empty loop structure which is add-cancelable is also add-left-cancelable and add-right-cancelable.

One can verify that every non empty loop structure which is Abelian and add-right-cancelable is also add-left-cancelable and every non empty loop structure which is Abelian and add-left-cancelable is also add-right-cancelable.

Let us observe that every non empty loop structure which is right zeroed, right complementable, and add-associative is also add-right-cancelable.

Let us observe that there exists a non empty double loop structure which is Abelian, add-associative, left zeroed, right zeroed, commutative, associative, add-cancelable, distributive, and unital.

We now state two propositions:

(1) Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and a be an element of R. Then 0_{R} · a = 0_{R}.

(2) Let R be a left zeroed add-right-cancelable right distributive non empty double loop structure and a be an element of R. Then a · 0_{R} = 0_{R}.

In this article we present several logical schemes. The scheme Ind2 deals with a natural number A and a unary predicate P, and states that:

For every natural number i such that A i holds P[i] provided the following conditions are satisfied:

• P[A], and

• For every natural number j such that A j holds if P[j], then P[j + 1].

The scheme RecDef1 deals with a non empty set A, an element B of A, and a binary operation C on A, and states that:

There exists a function g from [: ℕ, A:] into A such that for every element a of A holds

g(0, a) = B and for every natural number n holds g(n + 1,

a) = C(a, g(n, a)) for all values of the parameters.

The scheme RecDef2 deals with a non empty set A, an element B of A, and a binary operation C on A, and states that:

There exists a function g from [: A, ℕ:] into A such that for every element a of A holds

g(a, 0) = B and for every natural number n holds g(a, n+1) = C(g(a, n), a)

for all values of the parameters.

2. On Finite Sequences

One can prove the following propositions:

(3) For every left zeroed non empty loop structure L and for every element a of L holds ∑⟨a⟩ = a.

(4) Let R be a left zeroed add-right-cancelable right distributive non empty double loop structure, a be an element of R, and p be a finite sequence of elements of the carrier of R. Then ∑(a · p) = a · ∑ p.

(5) Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure, a be an element of R, and p be a finite sequence of elements of the carrier of R. Then ∑(p · a) = ∑ p · a.

(6) Let R be a commutative non empty double loop structure, a be an element of R, and p be a finite sequence of elements of the carrier of R. Then ∑(p · a) = ∑(a · p).

Let R be a non empty loop structure and let p, q be finite sequences of elements of the carrier of R. Let us assume that domp = domq. The functor p + q yields a finite sequence of elements of the carrier of R and is defined by:

(Def. 4) dom(p + q) = domp and for every natural number i such that 1 i and

i len(p + q) holds (p + q)_{i} = p_{i} + q_{i}. The following proposition is true

(7) Let R be an Abelian right zeroed add-associative non empty loop structure and p, q be finite sequences of elements of the carrier of R. If domp = domq, then ∑(p + q) = ∑ p + ∑ q.

3. On Powers in Rings

Let R be a unital non empty groupoid, let a be an element of R, and let n be a natural number. The functor a^{n} yielding an element of R is defined as follows:

(Def. 5) a^{n} = power_{R}(a, n).

We now state several propositions:

(8) For every unital non empty groupoid R and for every element a of R holds a^{0} = 1_{R} and a^{1} = a.

(9) For every unital non empty groupoid R and for every element a of R and for every natural number n holds a^{n+1} = a^{n} · a.

(10) Let R be a unital associative commutative non empty groupoid, a, b be elements of R, and n be a natural number. Then (a · b)^{n} = a^{n} · b^{n}.

(11) Let R be a unital associative non empty groupoid, a be an element of R, and n, m be natural numbers. Then a^{n+m} = a^{n} · a^{m}.

(12) Let R be a unital associative non empty groupoid, a be an element of R, and n, m be natural numbers. Then (a^{n})^{m} = a^{n·m}.

4. On Natural Products in Rings

Let R be a non empty loop structure. The functor Nat-mult-leftR yielding a function from [: ℕ, the carrier of R:] into the carrier of R is defined by:

(Def. 6) For every element a of R holds (Nat-mult-leftR)(0, a) = 0_{R} and for every natural number n holds (Nat-mult-leftR)(n + 1, a) = a + (Nat-mult-leftR)(n, a).

The functor Nat-mult-rightR yields a function from [: the carrier of R, ℕ:] into the carrier of R and is defined by:

(Def. 7) For every element a of R holds (Nat-mult-rightR)(a, 0) = 0_{R} and for every natural number n holds (Nat-mult-rightR)(a, n + 1) = (Nat-mult-rightR)(a, n) + a.

Let R be a non empty loop structure, let a be an element of R, and let n be a natural number. The functor n · a yields an element of R and is defined by:

(Def. 8) n · a = (Nat-mult-leftR)(n, a).

The functor a · n yields an element of R and is defined as follows: (Def. 9) a · n = (Nat-mult-rightR)(a, n).

One can prove the following propositions:

(13) For every non empty loop structure R and for every element a of R holds 0 · a = 0_{R} and a · 0 = 0_{R}.

(14) For every right zeroed non empty loop structure R and for every element a of R holds 1 · a = a.

(15) For every left zeroed non empty loop structure R and for every element a of R holds a · 1 = a.

(16) Let R be a left zeroed add-associative non empty loop structure, a be an element of R, and n, m be natural numbers. Then (n+m)·a = n·a+m·a.

(17) Let R be a right zeroed add-associative non empty loop structure, a be an element of R, and n, m be natural numbers. Then a·(n+m) = a·n+a·m.

(18) Let R be a left zeroed right zeroed add-associative non empty loop structure, a be an element of R, and n be a natural number. Then n · a = a · n.

(19) Let R be an Abelian non empty loop structure, a be an element of R, and n be a natural number. Then n · a = a · n.

(20) Let R be a left zeroed right zeroed add-left-cancelable add-associative left distributive non empty double loop structure, a, b be elements of R, and n be a natural number. Then (n · a) · b = n · (a · b).

(21) Let R be a left zeroed right zeroed add-right-cancelable add-associative distributive non empty double loop structure, a, b be elements of R, and n be a natural number. Then b · (n · a) = (b · a) · n.

(22) Let R be a left zeroed right zeroed add-associative add-cancelable distributive non empty double loop structure, a, b be elements of R, and n be a natural number. Then (a · n) · b = a · (n · b).

5. The Binomial Theorem

Let k, n be natural numbers. Then \binom{n}{k} is a natural number. Let R be a unital non empty double loop structure, let a, b be elements of

R, and let n be a natural number. The functor ⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩ yields a finite sequence of elements of the carrier of R and is defined by the conditions (Def. 10).

(Def. 10)(i) len⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩ = n + 1, and

(ii) for all natural numbers i, l, m such that i ∈ dom⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩

and m = i − 1 and l = n − m holds ⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩_{i} = \binom{n}{m} · a^{l} · b^{m}. The following four propositions are true:

(23) For every right zeroed unital non empty double loop structure R and for all elements a, b of R holds ⟨\binom{0}{0}a^{0}b^{0}, …, \binom{0}{0}a^{0}b^{0}⟩ = ⟨1_{R}⟩.

(24) Let R be a right zeroed unital non empty double loop structure, a, b be elements of R, and n be a natural number. Then ⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩(1) = a^{n}.

(25) Let R be a right zeroed unital non empty double loop structure, a, b be elements of R, and n be a natural number. Then ⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩(n+ 1) = b^{n}.

(26) Let R be an Abelian add-associative left zeroed right zeroed commutative associative add-cancelable distributive unital non empty double loop structure, a, b be elements of R, and n be a natural number. Then (a + b)^{n} = ∑⟨\binom{n}{0}a^{0}b^{n}, …, \binom{n}{n}a^{n}b^{0}⟩.

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.

564 christoph schwarzweller

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 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 and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

[6] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[7] Wojciech Skaba and Michał Muzalewski. From double loops to fields. Formalized Mathematics, 2(1):185–191, 1991.

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

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

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

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

Ring Ideals

Jonathan Backer^{1} University of Alberta Edmonton

Piotr Rudnicki^{2} University of Alberta Edmonton

Christoph Schwarzweller^{3} University of Tübingen

Summary. We introduce the basic notions of ideal theory in rings. This includes left and right ideals, (finitely) generated ideals and some operations on ideals such as the addition of ideals and the radical of an ideal. In addition we introduce linear combinations to formalize the well-known characterization of generated ideals. Principal ideal domains and Noetherian rings are defined. The latter development follows [3], pages 144–145.

MML Identifier: IDEAL 1.

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

1. Preliminaries

Let us note that there exists a non empty loop structure which is addassociative, left zeroed, and right zeroed.

Let us observe that there exists a non empty double loop structure which is Abelian, left zeroed, right zeroed, add-cancelable, well unital, add-associative, associative, commutative, distributive, and non trivial.

One can prove the following proposition (1) Let V be an add-associative left zeroed right zeroed non empty loop

1Partially supported by NSERC grant OGP9207.

2Partially supported by NSERC grant OGP9207.

3Partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

structure and v, u be elements of V. Then ∑⟨v, u⟩ = v + u.

2. Ideals

Let L be a non empty loop structure and let F be a subset of L. We say that F is add closed if and only if:

(Def. 1) For all elements x, y of the carrier of L such that x ∈ F and y ∈ F holds

x + y ∈ F. Let L be a non empty groupoid and let F be a subset of L. We say that F

is left ideal if and only if: (Def. 2) For all elements p, x of the carrier of L such that x ∈ F holds p · x ∈ F.

We say that F is right ideal if and only if: (Def. 3) For all elements p, x of the carrier of L such that x ∈ F holds x · p ∈ F.

Let L be a non empty loop structure. Observe that there exists a non empty subset of L which is add closed.

Let L be a non empty groupoid. One can verify that there exists a non empty subset of L which is left ideal and there exists a non empty subset of L which is right ideal.

Let L be a non empty double loop structure. One can verify the following observations:

∗ there exists a non empty subset of L which is add closed, left ideal, and right ideal,

∗ there exists a non empty subset of L which is add closed and right ideal, and

∗ there exists a non empty subset of L which is add closed and left ideal. Let R be a commutative non empty groupoid. Observe that every non empty

subset of R which is left ideal is also right ideal and every non empty subset of R which is right ideal is also left ideal.

Let L be a non empty double loop structure. An ideal of L is an add closed left ideal right ideal non empty subset of L.

Let L be a non empty double loop structure. A right ideal of L is an add closed right ideal non empty subset of L.

Let L be a non empty double loop structure. A left ideal of L is an add closed left ideal non empty subset of L.

The following propositions are true:

(2) Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and I be a left ideal non empty subset of R. Then 0_{R} ∈ I.

(3) Let R be a left zeroed add-right-cancelable right distributive non empty double loop structure and I be a right ideal non empty subset of R. Then 0_{R} ∈ I.

(4) For every right zeroed non empty double loop structure L holds {0_{L}} is add closed.

(5) Let L be a left zeroed add-right-cancelable right distributive non empty double loop structure. Then {0_{L}} is left ideal.

(6) Let L be a right zeroed add-left-cancelable left distributive non empty double loop structure. Then {0_{L}} is right ideal.

(7) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure. Then {0_{L}} is an ideal of L.

(8) Let L be an add-associative right zeroed right complementable right distributive non empty double loop structure. Then {0_{L}} is a left ideal of L.

(9) Let L be an add-associative right zeroed right complementable left distributive non empty double loop structure. Then {0_{L}} is a right ideal of L.

(10) For every non empty double loop structure L holds the carrier of L is an ideal of L.

(11) For every non empty double loop structure L holds the carrier of L is a left ideal of L.

(12) For every non empty double loop structure L holds the carrier of L is a

right ideal of L. Let R be a left zeroed right zeroed add-cancelable distributive non empty

double loop structure and let I be an ideal of R. Let us observe that I is trivial if and only if:

(Def. 4) I = {0_{R}}.

Let S be a 1-sorted structure and let T be a subset of S. We say that T is proper if and only if:

(Def. 5) T ≠ the carrier of S.

Let S be a non empty 1-sorted structure. Note that there exists a subset of S which is proper.

Let R be a non trivial left zeroed right zeroed add-cancelable distributive non empty double loop structure. One can check that there exists an ideal of R which is proper.

The following propositions are true:

(13) Let L be an add-associative right zeroed right complementable left distributive left unital non empty double loop structure, I be a left ideal non empty subset of L, and x be an element of the carrier of L. If x ∈ I, then − x ∈ I.

(14) Let L be an add-associative right zeroed right complementable right distributive right unital non empty double loop structure, I be a right ideal non empty subset of L, and x be an element of the carrier of L. If x ∈ I, then − x ∈ I.

(15) Let L be an add-associative right zeroed right complementable left distributive left unital non empty double loop structure, I be a left ideal of L, and x, y be elements of the carrier of L. If x ∈ I and y ∈ I, then x − y ∈ I.

(16) Let L be an add-associative right zeroed right complementable right distributive right unital non empty double loop structure, I be a right ideal of L, and x, y be elements of the carrier of L. If x ∈ I and y ∈ I, then x − y ∈ I.

(17) Let R be a left zeroed right zeroed add-cancelable add-associative distributive non empty double loop structure, I be an add closed right ideal non empty subset of R, a be an element of I, and n be a natural number. Then n · a ∈ I.

(18) Let R be a unital left zeroed right zeroed add-cancelable associative distributive non empty double loop structure, I be a right ideal non empty subset of R, a be an element of I, and n be a natural number. If n ≠ 0, then a^{n} ∈ I.

Let R be a non empty loop structure and let I be an add closed non empty subset of R. The functor add|(I, R) yielding a binary operation on I is defined as follows:

(Def. 6) add|(I, R) = (the addition of R)↾[: I, I:].

Let R be a non empty groupoid and let I be a right ideal non empty subset of R. The functor mult|(I, R) yielding a binary operation on I is defined as follows:

(Def. 7) mult|(I, R) = (the multiplication of R)↾[: I, I:].

Let R be a non empty loop structure and let I be an add closed non empty subset of R. The functor Gr(I, R) yields a non empty loop structure and is defined by:

(Def. 8) Gr(I, R) = ⟨I, add|(I, R), 0_{R}(∈ I)⟩.

Let R be a left zeroed right zeroed add-cancelable add-associative distributive non empty double loop structure and let I be an add closed non empty subset of R. Note that Gr(I, R) is add-associative.

Let R be a left zeroed right zeroed add-cancelable add-associative distributive non empty double loop structure and let I be an add closed right ideal non empty subset of R. Observe that Gr(I, R) is right zeroed.

Let R be an Abelian non empty double loop structure and let I be an add closed non empty subset of R. Observe that Gr(I, R) is Abelian.

Let R be an Abelian right unital left zeroed right zeroed right complementable add-associative distributive non empty double loop structure and let I be an add closed right ideal non empty subset of R. Note that Gr(I, R) is right complementable.

We now state four propositions:

(19) Let R be a right unital non empty double loop structure and I be a left ideal non empty subset of R. Then I is proper if and only if 1_{R} ∉ I.

(20) Let R be a left unital right unital non empty double loop structure and I be a right ideal non empty subset of R. Then I is proper if and only if for every element u of R such that u is unital holds u ∉ I.

(21) Let R be a right unital non empty double loop structure and I be a left ideal right ideal non empty subset of R. Then I is proper if and only if for every element u of R such that u is unital holds u ∉ I.

(22) Let R be a non degenerated commutative ring. Then R is a field if and only if for every ideal I of R holds I = {0_{R}} or I = the carrier of R.

3. Linear Combinations

Let R be a non empty multiplicative loop structure and let A be a non empty subset of the carrier of R. A finite sequence of elements of the carrier of R is said to be a linear combination of A if:

(Def. 9) For every set i such that i ∈ domit there exist elements u, v of R and

there exists an element a of A such that it_{i} = u · a · v. A finite sequence of elements of the carrier of R is said to be a left linear combination of A if:

(Def. 10) For every set i such that i ∈ domit there exists an element u of R and

there exists an element a of A such that it_{i} = u · a. A finite sequence of elements of the carrier of R is said to be a right linear combination of A if:

(Def. 11) For every set i such that i ∈ domit there exists an element u of R and

there exists an element a of A such that it_{i} = a · u. Let R be a non empty multiplicative loop structure and let A be a non empty

subset of the carrier of R. One can verify the following observations: ∗ there exists a linear combination of A which is non empty,

∗ there exists a left linear combination of A which is non empty, and

∗ there exists a right linear combination of A which is non empty.

Let R be a non empty multiplicative loop structure, let A, B be non empty subsets of the carrier of R, let F be a linear combination of A, and let G be a linear combination of B. Then F ^{⌢} G is a linear combination of A ∪ B.

One can prove the following three propositions:

(23) Let R be an associative non empty multiplicative loop structure, A be a non empty subset of the carrier of R, a be an element of the carrier of R, and F be a linear combination of A. Then a · F is a linear combination of A.

(24) Let R be an associative non empty multiplicative loop structure, A be a non empty subset of the carrier of R, a be an element of the carrier of R, and F be a linear combination of A. Then F · a is a linear combination of A.

(25) Let R be a right unital non empty multiplicative loop structure and A be a non empty subset of the carrier of R. Then every left linear combination of A is a linear combination of A.

Let R be a non empty multiplicative loop structure, let A, B be non empty subsets of the carrier of R, let F be a left linear combination of A, and let G be a left linear combination of B. Then F ^{⌢} G is a left linear combination of A ∪ B.

One can prove the following three propositions:

(26) Let R be an associative non empty multiplicative loop structure, A be a non empty subset of the carrier of R, a be an element of the carrier of R, and F be a left linear combination of A. Then a · F is a left linear combination of A.

(27) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, a be an element of the carrier of R, and F be a left linear combination of A. Then F · a is a linear combination of A.

(28) Let R be a left unital non empty multiplicative loop structure and A be a non empty subset of the carrier of R. Then every right linear combination of A is a linear combination of A.

Let R be a non empty multiplicative loop structure, let A, B be non empty subsets of the carrier of R, let F be a right linear combination of A, and let G be a right linear combination of B. Then F ^{⌢} G is a right linear combination of A ∪ B.

Next we state several propositions:

(29) Let R be an associative non empty multiplicative loop structure, A be a non empty subset of the carrier of R, a be an element of the carrier of R, and F be a right linear combination of A. Then F · a is a right linear combination of A.

(30) Let R be an associative non empty multiplicative loop structure, A be a non empty subset of the carrier of R, a be an element of the carrier of R, and F be a right linear combination of A. Then a · F is a linear combination of A.

(31) Let R be a commutative associative non empty multiplicative loop structure and A be a non empty subset of the carrier of R. Then every linear combination of A is a left linear combination of A and a right linear combination of A.

(32) Let S be a non empty double loop structure, F be a non empty subset of the carrier of S, and l_{1} be a non empty linear combination of F. Then there exists a linear combination p of F and there exists an element e of the carrier of S such that l_{1} = p^{⌢} ⟨e⟩ and ⟨e⟩ is a linear combination of F.

(33) Let S be a non empty double loop structure, F be a non empty subset of the carrier of S, and l_{1} be a non empty left linear combination of F. Then there exists a left linear combination p of F and there exists an element e of the carrier of S such that l_{1} = p^{⌢}⟨e⟩ and ⟨e⟩ is a left linear combination of F.

(34) Let S be a non empty double loop structure, F be a non empty subset of the carrier of S, and l_{1} be a non empty right linear combination of F. Then there exists a right linear combination p of F and there exists an element e of the carrier of S such that l_{1} = p^{⌢} ⟨e⟩ and ⟨e⟩ is a right linear combination of F.

Let R be a non empty multiplicative loop structure, let A be a non empty subset of the carrier of R, let L be a linear combination of A, and let E be a finite sequence of elements of [: the carrier of R, the carrier of R, the carrier of R:]. We say that E represents L if and only if:

(Def. 12) lenE = lenL and for every set i such that i ∈ domL holds L(i) =

(E_{i})_{1} · (E_{i})_{2} · (E_{i})_{3} and (E_{i})_{2} ∈ A. The following propositions are true:

(35) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, and L be a linear combination of A. Then there exists a finite sequence E of elements of [: the carrier of R, the carrier of R, the carrier of R:] such that E represents L.

(36) Let R, S be non empty multiplicative loop structures, F be a non empty subset of the carrier of R, l_{1} be a linear combination of F, G be a non empty subset of the carrier of S, P be a function from the carrier of R into the carrier of S, and E be a finite sequence of elements of [: the carrier of R, the carrier of R, the carrier of R:]. Suppose P^{∘}F ⊆ G and E represents l_{1}. Then there exists a linear combination L_{1} of G such that lenl_{1} = lenL_{1} and for every set i such that i ∈ domL_{1} holds L_{1}(i) = P ((E_{i})_{1}) · P ((E_{i})_{2}) · P ((E_{i})_{3}).

Let R be a non empty multiplicative loop structure, let A be a non empty subset of the carrier of R, let L be a left linear combination of A, and let E be a finite sequence of elements of [: the carrier of R, the carrier of R:]. We say that

E represents L if and only if: (Def. 13) lenE = lenL and for every set i such that i ∈ domL holds L(i) =

(E_{i})_{1} · (E_{i})_{2} and (E_{i})_{2} ∈ A. One can prove the following two propositions:

(37) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, and L be a left linear combination of A. Then there exists a finite sequence E of elements of [: the carrier of R, the carrier of R:] such that E represents L.

(38) Let R, S be non empty multiplicative loop structures, F be a non empty subset of the carrier of R, l_{1} be a left linear combination of F, G be a non empty subset of the carrier of S, P be a function from the carrier of R into the carrier of S, and E be a finite sequence of elements of [: the carrier of R, the carrier of R:]. Suppose P^{∘}F ⊆ G and E represents l_{1}. Then there exists a left linear combination L_{1} of G such that lenl_{1} = lenL_{1} and for every set i such that i ∈ domL_{1} holds L_{1}(i) = P ((E_{i})_{1}) · P ((E_{i})_{2}).

Let R be a non empty multiplicative loop structure, let A be a non empty subset of the carrier of R, let L be a right linear combination of A, and let E be a finite sequence of elements of [: the carrier of R, the carrier of R:]. We say that E represents L if and only if:

(Def. 14) lenE = lenL and for every set i such that i ∈ domL holds L(i) =

(E_{i})_{1} · (E_{i})_{2} and (E_{i})_{1} ∈ A. One can prove the following propositions:

(39) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, and L be a right linear combination of A. Then there exists a finite sequence E of elements of [: the carrier of R, the carrier of R:] such that E represents L.

(40) Let R, S be non empty multiplicative loop structures, F be a non empty subset of the carrier of R, l_{1} be a right linear combination of F, G be a non empty subset of the carrier of S, P be a function from the carrier of R into the carrier of S, and E be a finite sequence of elements of [: the carrier of R, the carrier of R:]. Suppose P^{∘}F ⊆ G and E represents l_{1}. Then there exists a right linear combination L_{1} of G such that lenl_{1} = lenL_{1} and for every set i such that i ∈ domL_{1} holds L_{1}(i) = P ((E_{i})_{1}) · P ((E_{i})_{2}).

(41) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, l be a linear combination of A, and n be a natural number. Then l↾ Segn is a linear combination of A.

(42) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, l be a left linear combination of A, and n be a natural number. Then l↾ Segn is a left linear combination of A.

(43) Let R be a non empty multiplicative loop structure, A be a non empty subset of the carrier of R, l be a right linear combination of A, and n be a natural number. Then l↾ Segn is a right linear combination of A.

4. Generated Ideals

Let L be a non empty double loop structure and let F be a subset of the carrier of L. Let us assume that F is non empty. The functor F –ideal yielding an ideal of L is defined by:

(Def. 15) F ⊆ F –ideal and for every ideal I of L such that F ⊆ I holds F –ideal ⊆

I. The functor F –left-ideal yields a left ideal of L and is defined by:

(Def. 16) F ⊆ F –left-ideal and for every left ideal I of L such that F ⊆ I holds

F –left-ideal ⊆ I. The functor F –right-ideal yields a right ideal of L and is defined as follows:

(Def. 17) F ⊆ F –right-ideal and for every right ideal I of L such that F ⊆ I holds

F –right-ideal ⊆ I. One can prove the following three propositions:

(44) For every non empty double loop structure L and for every ideal I of L holds I–ideal = I.

(45) For every non empty double loop structure L and for every left ideal I of L holds I–left-ideal = I.

(46) For every non empty double loop structure L and for every right ideal I

of L holds I–right-ideal = I. Let L be a non empty double loop structure and let I be an ideal of L. A

non empty subset of L is said to be a basis of I if: (Def. 18) It–ideal = I.

We now state a number of propositions:

(47) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure. Then {0_{L}}–ideal = {0_{L}}.

(48) For every left zeroed right zeroed add-cancelable distributive non empty double loop structure L holds {0_{L}}–ideal = {0_{L}}.

(49) Let L be a left zeroed right zeroed add-right-cancelable right distributive non empty double loop structure. Then {0_{L}}–left-ideal = {0_{L}}.

(50) For every right zeroed add-left-cancelable left distributive non empty double loop structure L holds {0_{L}}–right-ideal = {0_{L}}.

(51) For every well unital non empty double loop structure L holds {1_{L}}–ideal = the carrier of L.

(52) For every right unital non empty double loop structure L holds {1_{L}}–left-ideal = the carrier of L.

(53) For every left unital non empty double loop structure L holds {1_{L}}–right-ideal = the carrier of L.

(54) For every non empty double loop structure L holds Ω_{L}–ideal = the carrier of L.

(55) For every non empty double loop structure L holds Ω_{L}–left-ideal = the carrier of L.

(56) For every non empty double loop structure L holds Ω_{L}–right-ideal = the carrier of L.

(57) Let L be a non empty double loop structure and A, B be non empty subsets of the carrier of L. If A ⊆ B, then A–ideal ⊆ B–ideal.

(58) Let L be a non empty double loop structure and A, B be non empty subsets of the carrier of L. If A ⊆ B, then A–left-ideal ⊆ B–left-ideal.

(59) Let L be a non empty double loop structure and A, B be non empty subsets of the carrier of L. If A ⊆ B, then A–right-ideal ⊆ B–right-ideal.

(60) Let L be an add-associative left zeroed right zeroed add-cancelable associative distributive well unital non empty double loop structure, F be a non empty subset of the carrier of L, and x be a set. Then x ∈ F –ideal if and only if there exists a linear combination f of F such that x = ∑ f.

(61) Let L be an add-associative left zeroed right zeroed add-cancelable associative distributive well unital non empty double loop structure, F be a non empty subset of the carrier of L, and x be a set. Then x ∈ F –left-ideal if and only if there exists a left linear combination f of F such that x = ∑ f.

(62) Let L be an add-associative left zeroed right zeroed add-cancelable associative distributive well unital non empty double loop structure, F be a non empty subset of the carrier of L, and x be a set. Then x ∈ F –right-ideal if and only if there exists a right linear combination f of F such that x = ∑ f.

(63) Let R be an add-associative left zeroed right zeroed add-cancelable well unital associative commutative distributive non empty double loop structure and F be a non empty subset of the carrier of R. Then F –ideal = F –left-ideal and F –ideal = F –right-ideal.

(64) Let R be an add-associative left zeroed right zeroed add-cancelable well unital associative commutative distributive non empty double loop structure and a be an element of R. Then {a}–ideal = {a · r : r ranges over elements of R}.

(65) Let R be an Abelian left zeroed right zeroed add-cancelable well unital add-associative associative commutative distributive non empty double loop structure and a, b be elements of R. Then {a, b}–ideal = {a·r+b·s : r

ranges over elements of R, s ranges over elements of R}.

(66) For every non empty double loop structure R and for every element a of R holds a ∈ {a}–ideal.

(67) Let R be an Abelian left zeroed right zeroed right complementable addassociative associative commutative distributive well unital non empty double loop structure, A be a non empty subset of R, and a be an element of R. If a ∈ A–ideal, then {a}–ideal ⊆ A–ideal.

(68) For every non empty double loop structure R and for all elements a, b of R holds a ∈ {a, b}–ideal and b ∈ {a, b}–ideal.

(69) For every non empty double loop structure R and for all elements a, b of R holds {a}–ideal ⊆ {a, b}–ideal and {b}–ideal ⊆ {a, b}–ideal.

5. Some Operations on Ideals

Let R be a non empty groupoid, let I be a subset of R, and let a be an element of R. The functor a · I yielding a subset of R is defined by:

(Def. 19) a · I = {a · i; i ranges over elements of R: i ∈ I}.

Let R be a non empty multiplicative loop structure, let I be a non empty subset of R, and let a be an element of R. Observe that a · I is non empty.

Let R be a distributive non empty double loop structure, let I be an add closed subset of R, and let a be an element of R. Observe that a·I is add closed.

Let R be an associative non empty double loop structure, let I be a right ideal subset of R, and let a be an element of R. One can check that a · I is right ideal.

One can prove the following propositions:

(70) Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and I be a non empty subset of R. Then 0_{R} · I = {0_{R}}.

(71) For every left unital non empty double loop structure R and for every

subset I of R holds 1_{R} · I = I. Let R be a non empty loop structure and let I, J be subsets of R. The

functor I + J yielding a subset of R is defined by: (Def. 20) I + J = {a + b; a ranges over elements of R, b ranges over elements of R:

a ∈ I ∧ b ∈ J}. Let R be a non empty loop structure and let I, J be non empty subsets of

R. One can check that I + J is non empty.

Let R be an Abelian non empty loop structure and let I, J be subsets of R. Let us observe that the functor I + J is commutative.

Let R be an Abelian add-associative non empty loop structure and let I, J be add closed subsets of R. Note that I + J is add closed.

Let R be a left distributive non empty double loop structure and let I, J be right ideal subsets of R. Observe that I + J is right ideal.

Let R be a right distributive non empty double loop structure and let I, J be left ideal subsets of R. One can check that I + J is left ideal.

One can prove the following propositions:

(72) For every add-associative non empty loop structure R and for all subsets I, J, K of R holds I + (J + K) = (I + J) + K.

(73) Let R be a left zeroed right zeroed add-right-cancelable right distributive non empty double loop structure and I, J be right ideal non empty subsets of R. Then I ⊆ I + J.

(74) Let R be a left zeroed add-right-cancelable right distributive non empty double loop structure and I, J be right ideal non empty subsets of R. Then J ⊆ I + J.

(75) Let R be a non empty loop structure, I, J be subsets of R, and K be an add closed subset of R. If I ⊆ K and J ⊆ K, then I + J ⊆ K.

(76) Let R be an Abelian left zeroed right zeroed add-cancelable well unital add-associative associative commutative distributive non empty double loop structure and a, b be elements of R. Then {a, b}–ideal = {a}–ideal+ {b}–ideal.

Let R be a non empty 1-sorted structure and let I, J be subsets of R. The functor I ∩ J yielding a subset of R is defined as follows:

(Def. 21) I ∩ J = {x; x ranges over elements of R: x ∈ I ∧ x ∈ J}.

Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and let I, J be left ideal non empty subsets of R. Note that I ∩ J is non empty.

Let R be a non empty loop structure and let I, J be add closed subsets of R. Note that I ∩ J is add closed.

Let R be a non empty multiplicative loop structure and let I, J be left ideal subsets of R. Observe that I ∩ J is left ideal.

Let R be a non empty multiplicative loop structure and let I, J be right ideal subsets of R. Note that I ∩ J is right ideal.

One can prove the following four propositions:

(77) For every non empty 1-sorted structure R and for all subsets I, J of R holds I ∩ J ⊆ I and I ∩ J ⊆ J.

(78) For every non empty 1-sorted structure R and for all subsets I, J, K of R holds I ∩ (J ∩ K) = (I ∩ J) ∩ K.

(79) For every non empty 1-sorted structure R and for all subsets I, J, K of R such that K ⊆ I and K ⊆ J holds K ⊆ I ∩ J.

(80) Let R be an Abelian left zeroed right zeroed right complementable left unital add-associative left distributive non empty double loop structure, I be an add closed left ideal non empty subset of R, J be a subset of R, and K be a non empty subset of R. If J ⊆ I, then I ∩ (J + K) = J + I ∩ K.

Let R be a non empty double loop structure and let I, J be subsets of R. The functor I ∗ J yields a subset of R and is defined by the condition (Def. 22).

(Def. 22) I ∗ J = {∑ s; s ranges over finite sequences of elements of the carrier of R: ⋀_{i: natural number} (1 i ∧ i lens ⇒ ⋁_{a,b: element of R} (s(i) = a · b ∧ a ∈ I ∧ b ∈ J))}.

Let R be a non empty double loop structure and let I, J be subsets of R. Note that I ∗ J is non empty.

Let R be a commutative non empty double loop structure and let I, J be subsets of R. Let us observe that the functor I ∗ J is commutative.

Let R be a right zeroed add-associative non empty double loop structure and let I, J be subsets of R. Note that I ∗ J is add closed.

Let R be a right zeroed add-left-cancelable associative left distributive non empty double loop structure and let I, J be right ideal subsets of R. One can check that I ∗ J is right ideal.

Let R be a left zeroed add-right-cancelable associative right distributive non empty double loop structure and let I, J be left ideal subsets of R. Note that I ∗ J is left ideal.

We now state several propositions:

(81) Let R be a left zeroed right zeroed add-left-cancelable left distributive non empty double loop structure and I be a non empty subset of R. Then {0_{R}} ∗ I = {0_{R}}.

(82) Let R be a left zeroed right zeroed add-cancelable distributive non empty double loop structure, I be an add closed right ideal non empty subset of R, and J be an add closed left ideal non empty subset of R. Then I ∗ J ⊆ I ∩ J.

(83) Let R be an Abelian left zeroed right zeroed add-cancelable addassociative associative distributive non empty double loop structure and I, J, K be right ideal non empty subsets of R. Then I ∗(J +K) = I ∗J +I ∗K.

(84) Let R be an Abelian left zeroed right zeroed add-cancelable addassociative commutative associative distributive non empty double loop structure and I, J be right ideal non empty subsets of R. Then (I + J) ∗ (I ∩ J) ⊆ I ∗ J.

(85) Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and I, J be add closed left ideal non empty subsets of R. Then (I + J) ∗ (I ∩ J) ⊆ I ∩ J.

Let R be a non empty loop structure and let I, J be subsets of R. We say

that I, J are co-prime if and only if: (Def. 23) I + J = the carrier of R.

We now state two propositions:

(86) Let R be a left zeroed left unital non empty double loop structure and I, J be non empty subsets of R. If I, J are co-prime, then I ∩ J ⊆ (I + J) ∗ (I ∩ J).

(87) Let R be an Abelian left zeroed right zeroed add-cancelable addassociative left unital commutative associative distributive non empty double loop structure, I be an add closed left ideal right ideal non empty subset of R, and J be an add closed left ideal non empty subset of R. If I, J are co-prime, then I ∗ J = I ∩ J.

Let R be a non empty groupoid and let I, J be subsets of R. The functor I % J yields a subset of R and is defined by:

(Def. 24) I % J = {a; a ranges over elements of R: a · J ⊆ I}.

Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and let I, J be left ideal non empty subsets of R. One can check that I % J is non empty.

Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure and let I, J be add closed left ideal non empty subsets of R. One can check that I % J is add closed.

Let R be a right zeroed add-left-cancelable left distributive associative commutative non empty double loop structure and let I, J be left ideal non empty subsets of R. Note that I % J is left ideal and I % J is right ideal.

We now state several propositions:

(88) Let R be a non empty multiplicative loop structure, I be a right ideal non empty subset of R, and J be a subset of R. Then I ⊆ I % J.

(89) Let R be a right zeroed add-left-cancelable left distributive non empty double loop structure, I be an add closed left ideal non empty subset of R, and J be a subset of R. Then (I % J) ∗ J ⊆ I.

(90) Let R be a left zeroed add-right-cancelable right distributive non empty double loop structure, I be an add closed right ideal non empty subset of R, and J be a subset of R. Then (I % J) ∗ J ⊆ I.

(91) Let R be a left zeroed add-right-cancelable right distributive commutative associative non empty double loop structure, I be an add closed right ideal non empty subset of R, and J, K be subsets of R. Then (I % J) % K = I % (J ∗ K).

(92) For every non empty multiplicative loop structure R and for all subsets I, J, K of R holds (J ∩ K) % I = (J % I) ∩ (K % I).

(93) Let R be a left zeroed right zeroed add-right-cancelable right distributive non empty double loop structure, I be an add closed subset of R, and J, K

be right ideal non empty subsets of R. Then I%(J +K) = (I%J)∩(I%K). Let R be a unital non empty double loop structure and let I be a subset of

R. The functor \sqrt{I} yielding a subset of R is defined as follows: (Def. 25) \sqrt{I} = \{a; a ranges over elements of R: ⋁_{n: natural number} a^{n} ∈ I\}.

Let R be a unital non empty double loop structure and let I be a non empty subset of R. One can verify that \sqrt{I} is non empty.

Let R be an Abelian add-associative left zeroed right zeroed commutative associative add-cancelable distributive unital non empty double loop structure and let I be an add closed right ideal non empty subset of R. Observe that \sqrt{I} is add closed.

Let R be a unital commutative associative non empty double loop structure and let I be a left ideal non empty subset of R. Observe that \sqrt{I} is left ideal and \sqrt{I} is right ideal.

One can prove the following propositions:

(94) Let R be a unital associative non empty double loop structure, I be a non empty subset of R, and a be an element of R. Then a ∈ \sqrt{I} if and only if there exists a natural number n such that a^{n} ∈ \sqrt{I}.

(95) Let R be a left zeroed right zeroed add-cancelable distributive unital associative non empty double loop structure, I be an add closed right ideal non empty subset of R, and J be an add closed left ideal non empty subset of R. Then \sqrt{I ∗ J} = \sqrt{I ∩ J} .

6. Noetherian Rings and PIDs

Let L be a non empty double loop structure and let I be an ideal of L. We say that I is finitely generated if and only if:

(Def. 26) There exists a non empty finite subset F of the carrier of L such that

I = F –ideal. Let L be a non empty double loop structure. Observe that there exists an

ideal of L which is finitely generated.

Let L be a non empty double loop structure and let F be a non empty finite subset of L. Note that F –ideal is finitely generated.

Let L be a non empty double loop structure. We say that L is Noetherian if and only if:

(Def. 27) Every ideal of L is finitely generated.

Let us observe that there exists a non empty double loop structure which is Euclidian, Abelian, add-associative, right zeroed, right complementable, well unital, distributive, commutative, associative, and non degenerated.

Let L be a non empty double loop structure and let I be an ideal of L. We say that I is principal if and only if:

(Def. 28) There exists an element e of the carrier of L such that I = {e}–ideal.

Let L be a non empty double loop structure. We say that L is PID if and only if:

(Def. 29) Every ideal of L is principal.

One can prove the following three propositions:

(96) Let L be a non empty double loop structure and F be a non empty subset of the carrier of L. Suppose F ≠ {0_{L}}. Then there exists an element x of the carrier of L such that x ≠ 0_{L} and x ∈ F.

(97) Every add-associative left zeroed right zeroed right complementable distributive left unital Euclidian non empty double loop structure is PID.

(98) For every non empty double loop structure L such that L is PID holds

L is Noetherian. Let us note that INT.Ring is Noetherian. Let us observe that there exists a non empty double loop structure which is

Noetherian, Abelian, add-associative, right zeroed, right complementable, well unital, distributive, commutative, associative, and non degenerated.

Next we state two propositions:

(99) Let R be a Noetherian add-associative left zeroed right zeroed addcancelable associative distributive well unital non empty double loop structure and B be a non empty subset of the carrier of R. Then there exists a non empty finite subset C of the carrier of R such that C ⊆ B and C–ideal = B–ideal.

(100) Let R be a non empty double loop structure. Suppose that for every non empty subset B of the carrier of R there exists a non empty finite subset C of the carrier of R such that C ⊆ B and C–ideal = B–ideal. Let a be a sequence of R. Then there exists a natural number m such that a(m + 1) ∈ (rng(a↾ℤ_{m+1}))–ideal.

Let X, Y be non empty sets, let f be a function from X into Y, and let A be a non empty subset of X. One can check that f↾A is non empty.

The following two propositions are true:

(101) Let R be a non empty double loop structure. Suppose that for every sequence a of R there exists a natural number m such that a(m + 1) ∈ (rng(a↾ℤ_{m+1}))–ideal. Then there does not exist a function F from ℕ into 2^{the carrier of R} such that

(i) for every natural number i holds F (i) is an ideal of R, and (ii) for all natural numbers j, k such that j < k holds F (j) ⊆ F (k) and

F (j) ≠ F (k).

ring ideals 581

(102) Let R be a non empty double loop structure. Suppose that it is not true that there exists a function F from ℕ into 2^{the carrier of R} such that for every natural number i holds F (i) is an ideal of R and for all natural numbers j, k such that j < k holds F (j) ⊆ F (k) and F (j) ≠ F (k). Then R is Noetherian.

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Received November 20, 2000

Hilbert Basis Theorem^{1}

Jonathan Backer University of Alberta Edmonton

Piotr Rudnicki University of Alberta Edmonton

Summary. We prove the Hilbert basis theorem following [5], page 145. First we prove the theorem for the univariate case and then for the multivariate case. Our proof for the latter is slightly different than in [5]. As a base case we take the ring of polynomilas with no variables. We also prove that a polynomial ring with infinite number of variables is not Noetherian.

MML Identifier: HILBASIS.

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

1. Preliminaries

One can prove the following propositions:

(1) Let A, B be finite sequences and f be a function. Suppose rngA∪rngB ⊆ domf. Then there exist finite sequences f_{1}, f_{2} such that f_{1} = f · A and f_{2} = f · B and f · (A^{⌢} B) = f_{1} ^{⌢} f_{2}.

(2) For every bag b of 0 holds decompb = ⟨⟨∅, ∅⟩⟩.

(3) For all natural numbers i, j and for every bag b of j such that i j holds b↾i is an element of Bagsi.

(4) Let i, j be sets, b_{1}, b_{2} be bags of j, and b′_{1}, b′_{2} be bags of i. If b′_{1} = b_{1}↾i and b′_{2} = b_{2}↾i and b_{1} divides b_{2}, then b′_{1} divides b′_{2}.

1This work has been partially supported by NSERC grant OGP9207.

(5) Let i, j be sets, b_{1}, b_{2} be bags of j, and b′_{1}, b′_{2} be bags of i. If b′_{1} = b_{1}↾i and b′_{2} = b_{2}↾i, then (b_{1} − ′b_{2})↾i = b′_{1} − ′b′_{2} and (b_{1} + b_{2})↾i = b′_{1} + b′_{2}.

Let n, k be natural numbers and let b be a bag of n. The functor b extended by k yields an element of Bagsn + 1 and is defined as follows:

(Def. 1) (b extended by k)↾n = b and (b extended by k)(n) = k. We now state two propositions:

(6) For every natural number n holds EmptyBagn + 1 = EmptyBagn extended by 0.

(7) For every ordinal number n and for all bags b, b_{1} of n holds b_{1} ∈ rngdivisorsb iff b_{1} divides b.

Let X be a set and let x be an element of X. The functor UnitBagx yields an element of BagsX and is defined as follows:

(Def. 2) UnitBagx = EmptyBagX +·(x, 1).

Next we state four propositions:

(8) For every non empty set X and for every element x of X holds supportUnitBagx = {x}.

(9) Let X be a non empty set and x be an element of X. Then (UnitBagx)(x) = 1 and for every element y of X such that x ≠ y holds (UnitBagx)(y) = 0.

(10) For every non empty set X and for all elements x_{1}, x_{2} of X such that UnitBagx_{1} = UnitBagx_{2} holds x_{1} = x_{2}.

(11) Let X be a non empty ordinal number, x be an element of X, L be a unital non trivial non empty double loop structure, and e be a function from X into L. Then eval(UnitBagx, e) = e(x).

Let X be a set, let x be an element of X, and let L be a unital non empty multiplicative loop with zero structure. The functor 1 1(x, L) yielding a Series of X, L is defined by:

(Def. 3) 1 1(x, L) = 0 (X, L) +·(UnitBagx, 1_{L}).

One can prove the following propositions:

(12) Let X be a set, L be a unital non trivial non empty double loop structure, and x be an element of X. Then (1 1(x, L))(UnitBagx) = 1_{L} and for every bag b of X such that b ≠ UnitBagx holds (1 1(x, L))(b) = 0_{L}.

(13) Let X be a set, x be an element of X, and L be an add-associative right zeroed right complementable unital right distributive non trivial non empty double loop structure. Then Support1 1(x, L) = {UnitBagx}.

Let X be an ordinal number, let x be an element of X, and let L be an add-associative right zeroed right complementable unital right distributive non trivial non empty double loop structure. Observe that 1 1(x, L) is finite-Support.

One can prove the following three propositions:

(14) Let L be an add-associative right zeroed right complementable unital right distributive non trivial non empty double loop structure, X be a non empty set, and x_{1}, x_{2} be elements of X. If 1 1(x_{1}, L) = 1 1(x_{2}, L), then x_{1} = x_{2}.

(15) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure, x be an element of the carrier of Polynom-RingL, and p be a sequence of L. If x = p, then − x = − p.

(16) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure, x, y be elements of the carrier of Polynom-RingL, and p, q be sequences of L. If x = p and y = q, then x − y = p − q.

Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure and let I be a non empty subset of the carrier of Polynom-RingL. The functor minlenI yields a non empty subset of I and is defined by:

(Def. 4) minlenI = {x; x ranges over elements of I: ⋀_{x′,y′: Polynomial of L} (x′ =

x ∧ y′∈ I ⇒ lenx′ leny′)}. We now state the proposition

(17) Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure, I be a non empty subset of the carrier of Polynom-RingL, and i_{1}, i_{2} be Polynomials of L. If i_{1} ∈ minlenI and i_{2} ∈ I, then i_{1} ∈ I and leni_{1} leni_{2}.

Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure, let n be a natural number, and let a be an element of the carrier of L. The functor monomial(a, n) yields a Polynomial of L and is defined as follows:

(Def. 5) For every natural number x holds if x = n, then (monomial(a, n))(x) = a

and if x ≠ n, then (monomial(a, n))(x) = 0_{L}. The following four propositions are true:

(18) Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure, n be a natural number, and a be an element of the carrier of L. Then if a ≠ 0_{L}, then lenmonomial(a, n) = n + 1 and if a = 0_{L}, then lenmonomial(a, n) = 0 and lenmonomial(a, n) n + 1.

(19) Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure, n, x be natural numbers, a be an element of the carrier of L, and p be a Polynomial of L. Then (monomial(a, n) ∗ p)(x + n) = a · p(x).

(20) Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure, n, x be natural numbers, a be an element of the carrier of L, and p be a Polynomial of L. Then (p ∗ monomial(a, n))(x + n) = p(x) · a.

(21) Let L be a right zeroed add-associative right complementable unital distributive non empty double loop structure and p, q be Polynomials of L. Then len(p ∗ q) (lenp + lenq) − ′1.

2. On Ring Isomorphism

The following propositions are true:

(22) Let R, S be non empty double loop structures, I be an ideal of R, and P be a map from R into S. If P is a ring isomorphism, then P^{∘}I is an ideal of S.

(23) Let R, S be add-associative right zeroed right complementable non empty double loop structures and f be a map from R into S. If f is a ring homomorphism, then f(0_{R}) = 0_{S}.

(24) Let R, S be add-associative right zeroed right complementable non empty double loop structures, F be a non empty subset of the carrier of R, G be a non empty subset of the carrier of S, P be a map from R into S, l_{1} be a linear combination of F, L_{1} be a linear combination of G, and E be a finite sequence of elements of [: the carrier of R, the carrier of R, the carrier of R:]. Suppose that

(i) P is a ring homomorphism, (ii) lenl_{1} = lenL_{1}, (iii) E represents l_{1}, and (iv) for every set i such that i ∈ domL_{1} holds L_{1}(i) = P ((E_{i})_{1}) · P ((E_{i})_{2}) ·

P ((E_{i})_{3}). Then P (∑ l_{1}) = ∑ L_{1}.

(25) Let R, S be non empty double loop structures and P be a map from R into S. Suppose P is a ring isomorphism. Then there exists a map P_{1} from S into R such that P_{1} is a ring isomorphism and P_{1} = P^{−1} .

(26) Let R, S be Abelian add-associative right zeroed right complementable associative distributive well unital non empty double loop structures, F be a non empty subset of the carrier of R, and P be a map from R into S. If P is a ring isomorphism, then P^{∘}F –ideal = (P^{∘}F)–ideal.

(27) Let R, S be Abelian add-associative right zeroed right complementable associative distributive well unital non empty double loop structures and P be a map from R into S. If P is a ring isomorphism and R is Noetherian, then S is Noetherian.

(28) Let R be an add-associative right zeroed right complementable associative distributive well unital non trivial non empty double loop structure. Then there exists a map from R into Polynom-Ring(0, R) which is a ring isomorphism.

(29) Let R be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, n be a natural number, b be a bag of n, p_{1} be a Polynomial of n, R, and F be a finite sequence of elements of the carrier of Polynom-Ring(n, R). Suppose p_{1} = ∑ F. Then there exists a function g from the carrier of Polynom-Ring(n, R) into the carrier of R such that for every Polynomial p of n, R holds g(p) = p(b) and p_{1}(b) = ∑(g · F).

Let R be an Abelian add-associative right zeroed right complementable associative distributive well unital commutative non trivial non empty double loop structure and let n be a natural number. The functor upm(n, R) yielding a map from Polynom-RingPolynom-Ring(n, R) into Polynom-Ring(n+1, R) is defined by the condition (Def. 6).

(Def. 6) Let p_{1} be a Polynomial of Polynom-Ring(n, R), p_{2} be a Polynomial of n, R, p_{3} be a Polynomial of n + 1, R, and b be a bag of n + 1. If p_{3} = (upm(n, R))(p_{1}) and p_{2} = p_{1}(b(n)), then p_{3}(b) = p_{2}(b↾n).

Let R be an Abelian add-associative right zeroed right complementable associative distributive well unital commutative non trivial non empty double loop structure and let n be a natural number. One can verify the following observations:

∗ upm(n, R) is additive,

∗ upm(n, R) is multiplicative,

∗ upm(n, R) is unity-preserving, and

∗ upm(n, R) is one-to-one. Let R be an Abelian add-associative right zeroed right complementable as-

sociative distributive well unital commutative non trivial non empty double loop structure and let n be a natural number. The functor mpu(n, R) yields a map from Polynom-Ring(n + 1, R) into Polynom-RingPolynom-Ring(n, R) and is defined by the condition (Def. 7).

(Def. 7) Let p_{1} be a Polynomial of n + 1, R, p_{2} be a Polynomial of n, R, p_{3} be a Polynomial of Polynom-Ring(n, R), i be a natural number, and b be a bag of n. If p_{3} = (mpu(n, R))(p_{1}) and p_{2} = p_{3}(i), then p_{2}(b) = p_{1}(b extended by i).

Next we state two propositions:

(30) Let R be an Abelian add-associative right zeroed right complementable associative distributive well unital commutative non trivial non empty double loop structure, n be a natural number, and p be an element of the

carrier of Polynom-Ring(n+1, R). Then (upm(n, R))((mpu(n, R))(p)) = p.

(31) Let R be an Abelian add-associative right zeroed right complementable associative distributive well unital commutative non trivial non empty double loop structure and n be a natural number. Then there exists a map from Polynom-RingPolynom-Ring(n, R) into Polynom-Ring(n+1, R) which is a ring isomorphism.

3. Hilbert Basis Theorem

Let R be a Noetherian Abelian add-associative right zeroed right complementable associative distributive well unital commutative non empty double loop structure. Observe that Polynom-RingR is Noetherian.

One can prove the following propositions:

(32) Let R be a Noetherian Abelian add-associative right zeroed right complementable associative distributive well unital commutative non empty double loop structure. Then Polynom-RingR is Noetherian.

(33) Let R be an Abelian add-associative right zeroed right complementable associative distributive well unital non trivial commutative non empty double loop structure. Suppose R is Noetherian. Let n be a natural number. Then Polynom-Ring(n, R) is Noetherian.

(34) Every field is Noetherian.

(35) For every field F and for every natural number n holds Polynom-Ring(n, F) is Noetherian.

(36) Let R be an Abelian right zeroed add-associative right complementable well unital distributive associative commutative non trivial non empty double loop structure and X be an infinite ordinal number. Then Polynom-Ring(X, R) is non Noetherian.

References

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 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] Thomas Becker and Volker Weispfenning. Gröbner bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.

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

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

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

hilbert basis theorem 589

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

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

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

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

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

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

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

[16] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.

[17] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[18] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97–104, 1991.

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

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

[21] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[22] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[23] Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69–79, 1998.

[24] Christoph Schwarzweller and Andrzej Trybulec. The evaluation of multivariate polynomials. Formalized Mathematics, 9(2):331–338, 2001.

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

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

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

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

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

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

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

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

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

590 jonathan backer and piotr rudnicki

Dynkin’s Lemma in Measure Theory

Franz Merkl University of Bielefeld

Summary. This article formalizes the proof of Dynkin’s lemma in measure theory. Dynkin’s lemma is a useful tool in measure theory and probability theory: it helps frequently to generalize a statement about all elements of a intersectionstable set system to all elements of the sigma-field generated by that system.

MML Identifier: DYNKIN.

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

1. Preliminaries

For simplicity, we adopt the following rules: O_{1} denotes a non empty set, f denotes a sequence of subsets of O_{1}, X, A, B denote subsets of O_{1}, D denotes a non empty subset of 2^{O_{1}}, n, m denote natural numbers, F denotes a non empty set, and x, Y denote sets.

Next we state two propositions:

(1) For every sequence f of subsets of O_{1} and for every x holds x ∈ rngf iff there exists n such that f(n) = x.

(2) For every n holds PSegn is finite.

Let us consider n. One can verify that PSegn is finite.

Next we state the proposition

(3) For all sets x, y, z such that x ⊆ y holds x misses z \ y.

Let a, b, c be sets. The functor a, b followed by c is defined as follows: (Def. 1) a, b followed by c = (ℕ ⟼→ c)+·[0 ⟼→ a, 1 ⟼→ b].

Let a, b, c be sets. Observe that a, b followed by c is function-like and relationlike.

Let X be a non empty set and let a, b, c be elements of X. Then a, b followed by c is a function from ℕ into X.

Next we state the proposition

(4) For every non empty set X and for all elements a, b, c of X holds a, b followed by c is a function from ℕ into X.

Let O_{1} be a non empty set and let a, b, c be subsets of O_{1}. Then a, b followed by c is a sequence of subsets of O_{1}.

One can prove the following propositions:

(5) For all sets a, b, c holds (a, b followed by c)(0) = a and (a, b followed by c)(1) = b and for every n such that n ≠ 0 and n ≠ 1 holds (a, b followed by c)(n) = c.

(6) For all subsets a, b of O_{1} holds ⋃ rng(a, b followed by ∅) = a ∪ b. Let O_{1} be a non empty set, let f be a sequence of subsets of O_{1}, and let X

be a subset of O_{1}. The functor seqIntersection(X, f) yields a sequence of subsets of O_{1} and is defined by:

(Def. 2) For every n holds (seqIntersection(X, f))(n) = X ∩ f(n).

2. Disjoint-valued Functions and Intersection

Let us consider O_{1} and let us consider f. Let us observe that f is disjoint valued if and only if:

(Def. 3) If n < m, then f(n) misses f(m).

We now state the proposition

(7) For every non empty set Y and for every x holds x ⊆ ⋂ Y iff for every element y of Y holds x ⊆ y.

Let x be a set. We introduce x is intersection stable as a synonym of x is multiplicative.

Let O_{1} be a non empty set, let f be a sequence of subsets of O_{1}, and let n be an element of ℕ. The functor disjointify(f, n) yielding an element of 2^{O_{1}} is defined by:

(Def. 5)^{1} disjointify(f, n) = f(n) \ ⋃ rng(f↾ PSegn).

Let O_{1} be a non empty set and let g be a sequence of subsets of O_{1}. The functor disjointifyg yielding a sequence of subsets of O_{1} is defined as follows:

(Def. 6) For every n holds (disjointifyg)(n) = disjointify(g, n). The following propositions are true:

1The definition (Def. 4) has been removed.

(8) For every n holds (disjointifyf)(n) = f(n) \ ⋃ rng(f↾ PSegn).

(9) For every sequence f of subsets of O_{1} holds disjointifyf is disjoint valued.

(10) For every sequence f of subsets of O_{1} holds ⋃ rngdisjointifyf = ⋃ rngf.

(11) For all subsets x, y of O_{1} such that x misses y holds x, y followed by ∅_{(O_{1})} is disjoint valued.

(12) Let f be a sequence of subsets of O_{1}. Suppose f is disjoint valued. Let X be a subset of O_{1}. Then seqIntersection(X, f) is disjoint valued.

(13) For every sequence f of subsets of O_{1} and for every subset X of O_{1} holds X ∩ Unionf = UnionseqIntersection(X, f).

3. Dynkin Systems: Definition and Closure Properties

Let us consider O_{1}. A subset of 2^{O_{1}} is called a Dynkin system of O_{1} if: (Def. 7) For every f such that rngf ⊆ it and f is disjoint valued holds Unionf ∈

it and for every X such that X ∈ it holds X^{c} ∈ it and ∅ ∈ it. Let us consider O_{1}. One can check that every Dynkin system of O_{1} is non

empty.

The following propositions are true:

(14) 2^{O_{1}} is a Dynkin system of O_{1}.

(15) If for every Y such that Y ∈ F holds Y is a Dynkin system of O_{1}, then ⋂ F is a Dynkin system of O_{1}.

(16) If D is a Dynkin system of O_{1} and intersection stable, then if A ∈ D and B ∈ D, then A \ B ∈ D.

(17) If D is a Dynkin system of O_{1} and intersection stable, then if A ∈ D and B ∈ D, then A ∪ B ∈ D.

(18) Suppose D is a Dynkin system of O_{1} and intersection stable. Let x be a finite set. If x ⊆ D, then ⋃ x ∈ D.

(19) Suppose D is a Dynkin system of O_{1} and intersection stable. Let f be a sequence of subsets of O_{1}. If rngf ⊆ D, then rngdisjointifyf ⊆ D.

(20) Suppose D is a Dynkin system of O_{1} and intersection stable. Let f be a sequence of subsets of O_{1}. If rngf ⊆ D, then ⋃ rngf ∈ D.

(21) For every Dynkin system D of O_{1} and for all elements x, y of D such that x misses y holds x ∪ y ∈ D.

(22) For every Dynkin system D of O_{1} and for all elements x, y of D such that x ⊆ y holds y \ x ∈ D.

594 franz merkl

4. Main Steps for Dynkin’s Lemma

One can prove the following proposition

(23) If D is a Dynkin system of O_{1} and intersection stable, then D is a σ-field of subsets of O_{1}.

Let O_{1} be a non empty set and let E be a subset of 2^{O_{1}}. The functor GenDynSysE yielding a Dynkin system of O_{1} is defined by:

(Def. 8) E ⊆ GenDynSysE and for every Dynkin system D of O_{1} such that E ⊆ D holds GenDynSysE ⊆ D.

Let O_{1} be a non empty set, let G be a set, and let X be a subset of O_{1}. The functor DynSys(X, G) yields a subset of 2^{O_{1}} and is defined as follows:

(Def. 9) For every subset A of O_{1} holds A ∈ DynSys(X, G) iff A ∩ X ∈ G. Let O_{1} be a non empty set, let G be a Dynkin system of O_{1}, and let X be

an element of G. Then DynSys(X, G) is a Dynkin system of O_{1}.

Next we state four propositions:

(24) Let E be a subset of 2^{O_{1}} and X, Y be subsets of O_{1}. If X ∈ E and Y ∈ GenDynSysE and E is intersection stable, then X ∩ Y ∈ GenDynSysE.

(25) Let E be a subset of 2^{O_{1}} and X, Y be subsets of O_{1}. If X ∈ GenDynSysE and Y ∈ GenDynSysE and E is intersection stable, then X ∩ Y ∈ GenDynSysE.

(26) For every subset E of 2^{O_{1}} such that E is intersection stable holds GenDynSysE is intersection stable.

(27) Let E be a subset of 2^{O_{1}}. Suppose E is intersection stable. Let D be a Dynkin system of O_{1}. If E ⊆ D, then σ(E) ⊆ D.

References

[1] Grzegorz Bancerek. Continuous, stable, and linear maps of coherence spaces. Formalized Mathematics, 5(3):381–393, 1996.

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

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

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

[5] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97–104, 1991.

[6] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745–749, 1990.

[7] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

dynkin’s lemma in measure theory 595

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

1990.

Received November 27, 2000 596 franz merkl

Lower Tolerance. Preliminaries to Wroclaw Taxonomy^{1}

Mariusz Giero University of Białystok Roman Matuszewski University of Białystok

Summary. The paper introduces some preliminary notions concerning the Wroclaw taxonomy according to [16]. The classifications and tolerances are defined and considered w.r.t. sets and metric spaces. We prove theorems showing various classifications based on tolerances.

MML Identifier: TAXONOM1.

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

1. Preliminaries

In this paper A, X are non empty sets, f is a partial function from [: X, X:] to ℝ, and a is a real number.

Let us note that there exists a real number which is non negative.

We now state a number of propositions:

(1) For every finite sequence p and for every natural number k such that k + 1 ∈ domp and k ∉ domp holds k = 0.

(2) Let p be a finite sequence and i, j be natural numbers. Suppose i ∈ domp and j ∈ domp and for every natural number k such that k ∈ domp and k + 1 ∈ domp holds p(k) = p(k + 1). Then p(i) = p(j).

1This work has been partially supported by the European Community TYPES grant IST-1999-29001 and CALCULEMUS grant HPRN-CT-2000-00102.

(3) For every set X and for every binary relation R on X such that R is reflexive in X holds domR = X.

(4) For every set X and for every binary relation R on X such that R is reflexive in X holds rngR = X.

(5) For every set X and for every binary relation R on X such that R is reflexive in X holds R^{∗} is reflexive in X.

(6) Let X, x, y be sets and R be a binary relation on X. Suppose R is reflexive in X. If R reduces x to y and x ∈ X, then ⟨⟨x, y⟩⟩∈ R^{∗}.

(7) Let X be a set and R be a binary relation on X. If R is reflexive in X and symmetric in X, then R^{∗} is symmetric in X.

(8) For every set X and for every binary relation R on X such that R is reflexive in X holds R^{∗} is transitive in X.

(9) Let X be a non empty set and R be a binary relation on X. Suppose R is reflexive in X and symmetric in X. Then R^{∗} is an equivalence relation of X.

(10) For all binary relations R_{1}, R_{2} on X such that R_{1} ⊆ R_{2} holds R_{1}^{∗} ⊆ R_{2}^{∗}.

(11) SmallestPartition(A) is finer than {A}.

2. The Notion of Classification

Let A be a non empty set. A subset of PARTITIONS(A) is called a classification of A if:

(Def. 1) For all partitions X, Y of A such that X ∈ it and Y ∈ it holds X is finer

than Y or Y is finer than X. One can prove the following propositions:

(12) {{A}} is a classification of A.

(13) {SmallestPartition(A)} is a classification of A.

(14) For every subset S of PARTITIONS(A) such that S = {{A},

SmallestPartition(A)} holds S is a classification of A. Let A be a non empty set. A subset of PARTITIONS(A) is called a strong

classification of A if: (Def. 2) It is a classification of A and {A} ∈ it and SmallestPartition(A) ∈ it.

Next we state the proposition

(15) For every subset S of PARTITIONS(A) such that S = {{A}, SmallestPartition(A)} holds S is a strong classification of A.

3. The Tolerance on a Non Empty Set

Let X be a non empty set, let f be a partial function from [: X, X:] to ℝ, and let a be a real number. The functor T_{l}(f, a) yields a binary relation on X and is defined as follows:

(Def. 3) For all elements x, y of X holds ⟨⟨x, y⟩⟩∈ T_{l}(f, a) iff f(x, y) a.

The following four propositions are true:

(16) If f is Reflexive and a 0, then T_{l}(f, a) is reflexive in X.

(17) If f is symmetric, then T_{l}(f, a) is symmetric in X.

(18) If a 0 and f is Reflexive and symmetric, then T_{l}(f, a) is a tolerance of X.

(19) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ,

and a_{1}, a_{2} be real numbers. If a_{1} a_{2}, then T_{l}(f, a_{1}) ⊆ T_{l}(f, a_{2}). Let X be a set and let f be a partial function from [: X, X:] to ℝ. We say

that f is non-negative if and only if: (Def. 4) For all elements x, y of X holds f(x, y) 0.

We now state three propositions:

(20) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ, and x, y be sets. Suppose f is non-negative, Reflexive, and discernible. If ⟨⟨x, y⟩⟩∈ T_{l}(f, 0), then x = y.

(21) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ, and x be an element of X. If f is Reflexive and discernible, then ⟨⟨x, x⟩⟩∈ T_{l}(f, 0).

(22) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ, and a be a real number. Suppose T_{l}(f, a) is reflexive in X and f is symmetric. Then (T_{l}(f, a))^{∗} is an equivalence relation of X.

4. The Partitions Defined by Lower Tolerance

Next we state several propositions:

(23) Let X be a non empty set and f be a partial function from [: X, X:] to ℝ. Suppose f is non-negative, Reflexive, and discernible. Then (T_{l}(f, 0))^{∗} = T_{l}(f, 0).

(24) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ, and R be an equivalence relation of X. Suppose R = (T_{l}(f, 0))^{∗} and f is non-negative, Reflexive, and discernible. Then R = △_{X}.

(25) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ, and R be an equivalence relation of X. Suppose R = (T_{l}(f, 0))^{∗} and f is non-negative, Reflexive, and discernible. Then ClassesR = SmallestPartition(X).

(26) Let X be a finite non empty subset of ℝ, f be a function from [: X, X:] into ℝ, z be a finite non empty subset of ℝ, and A be a real number. If z = rngf and A maxz, then for all elements x, y of X holds f(x,

y) A.

(27) Let X be a finite non empty subset of ℝ, f be a function from [: X, X:] into ℝ, z be a finite non empty subset of ℝ, and A be a real number. Suppose z = rngf and A maxz. Let R be an equivalence relation of X. If R = (T_{l}(f, A))^{∗}, then ClassesR = {X}.

(28) Let X be a finite non empty subset of ℝ, f be a function from [: X, X:] into ℝ, z be a finite non empty subset of ℝ, and A be a real number. If z = rngf and A maxz, then (T_{l}(f, A))^{∗} = T_{l}(f, A).

5. The Classification on a Non Empty Set

Let X be a non empty set and let f be a partial function from [: X, X:] to ℝ. The functor FamClassf yielding a subset of PARTITIONS(X) is defined by the condition (Def. 5).

(Def. 5) Let x be a set. Then x ∈ FamClassf if and only if there exists a non negative real number a and there exists an equivalence relation R of X such that R = (T_{l}(f, a))^{∗} and ClassesR = x.

We now state four propositions:

(29) Let X be a non empty set, f be a partial function from [: X, X:] to ℝ, and a be a non negative real number. If T_{l}(f, a) is reflexive in X and f is symmetric, then FamClassf is a non empty set.

(30) Let X be a finite non empty subset of ℝ and f be a function from [: X, X:] into ℝ. If f is symmetric and non-negative, then {X} ∈ FamClassf.

(31) For every non empty set X and for every partial function f from [: X, X:] to ℝ holds FamClassf is a classification of X.

(32) Let X be a finite non empty subset of ℝ and f be a function from [: X, X:] into ℝ. Suppose SmallestPartition(X) ∈ FamClassf and f is symmetric and non-negative. Then FamClassf is a strong classification of X.

6. The Classification on a Metric Space

Let M be a metric structure, let a be a real number, and let x, y be elements of the carrier of M. We say that x, y are in tolerance w.r.t. a if and only if:

(Def. 6) ρ(x, y) a.

Let M be a non empty metric structure and let a be a real number. The functor T_{m}(M, a) yielding a binary relation on M is defined by:

(Def. 7) For all elements x, y of the carrier of M holds ⟨⟨x, y⟩⟩ ∈ T_{m}(M, a) iff x,

y are in tolerance w.r.t. a. Next we state two propositions:

(33) For every non empty metric structure M and for every real number a holds T_{m}(M, a) = T_{l}(the distance of M, a).

(34) Let M be a non empty Reflexive symmetric metric structure, a be a real number, and T be a relation between the carrier of M and the carrier of M. If T = T_{m}(M, a) and a 0, then T is a tolerance of the carrier of M.

Let M be a Reflexive symmetric non empty metric structure. The functor MetricFamClassM yielding a subset of PARTITIONS(the carrier of M) is defined by the condition (Def. 8).

(Def. 8) Let x be a set. Then x ∈ MetricFamClassM if and only if there exists a non negative real number a and there exists an equivalence relation R of M such that R = (T_{m}(M, a))^{∗} and ClassesR = x.

The following propositions are true:

(35) For every Reflexive symmetric non empty metric structure M holds MetricFamClassM = FamClassthe distance of M.

(36) Let M be a non empty metric space and R be an equivalence relation of M. If R = (T_{m}(M, 0))^{∗}, then ClassesR = SmallestPartition(the carrier of M).

(37) For every Reflexive symmetric bounded non empty metric structure M such that a Ø(Ω_{M}) holds T_{m}(M, a) = ∇_{the carrier of M}.

(38) For every Reflexive symmetric bounded non empty metric structure M such that a Ø(Ω_{M}) holds T_{m}(M, a) = (T_{m}(M, a))^{∗}.

(39) For every Reflexive symmetric bounded non empty metric structure M such that a Ø(Ω_{M}) holds (T_{m}(M, a))^{∗} = ∇_{the carrier of M}.

(40) Let M be a Reflexive symmetric bounded non empty metric structure, R be an equivalence relation of M, and a be a non negative real number. If a Ø(Ω_{M}) and R = (T_{m}(M, a))^{∗}, then ClassesR = {the carrier of M}.

Let M be a Reflexive symmetric triangle non empty metric structure and let C be a non empty bounded subset of M. Observe that ØC is non negative.

We now state three propositions:

602 mariusz giero and roman matuszewski

(41) For every bounded non empty metric space M holds {the carrier of M} ∈ MetricFamClassM.

(42) For every Reflexive symmetric non empty metric structure M holds MetricFamClassM is a classification of the carrier of M.

(43) For every bounded non empty metric space M holds MetricFamClassM is a strong classification of the carrier of M.

Acknowledgments

The authors thank Prof. Andrzej Trybulec for his introduction to this topic. We thank Dr. Artur Kornilowicz for his advice on this article. We also thank Robert Milewski and Adam Naumowicz for their helpful comments.

References

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[8] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part 1. Formalized Mathematics, 2(5):683–687, 1991.

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

[10] Alicia de la Cruz. Totally bounded metric spaces. Formalized Mathematics, 2(4):559–562, 1991.

[11] Krzysztof Hryniewiecki. Relations of tolerance. Formalized Mathematics, 2(1):105–109, 1991.

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

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

[14] Małgorzata Korolkiewicz. Homomorphisms of algebras. Quotient universal algebra. Formalized Mathematics, 4(1):109–113, 1993.

[15] Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part I: Theory. Formalized Mathematics, 9(1):135–142, 2001.

[16] Roman Matuszewski and Andrzej Trybulec. Certain algorithm of classification in metric spaces, volume V, Number 20 of Mathematical Papers. Warsaw University, Bialystok Campus, 1977.

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

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

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

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lower tolerance. preliminaries to … 603

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

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[25] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Received December 5, 2000 604 mariusz giero and roman matuszewski

Concrete Categories

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Summary. In the paper, we develop the notation of duality and equivalence of categories and concrete categories based on [9]. The development was motivated by the duality theory for continuous lattices (see [5, p. 189]), where we need to cope with concrete categories of lattices and maps preserving their properties. For example, the category UPS of complete lattices and directed suprema preserving maps; or the category INF of complete lattices and infima preserving maps. As the main result of this paper it is shown that every category is isomorphic to its concretization (the concrete category with the same objects). Some useful schemes to construct categories and functors are also presented.

MML Identifier: YELLOW18.

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

1. Definability of Categories and Functors

In this article we present several logical schemes. The scheme AltCatStr-Lambda deals with a non empty set A, a binary functor F yielding a set, and a 5-ary functor G yielding a set, and states that:

There exists a strict non empty transitive category structure C such that

(i) the carrier of C = A, (ii) for all objects a, b of C holds ⟨a, b⟩ = F(a, b), and (iii) for all objects a, b, c of C such that ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅ and for every morphism f from a to b and for every morphism g from b to c holds g · f = G(a, b, c, f, g) provided the following requirement is met:

• For all elements a, b, c of A and for all sets f, g such that f ∈ F(a, b) and g ∈ F(b, c) holds G(a, b, c, f, g) ∈ F(a, c).

The scheme CatAssocSch deals with a non empty transitive category structure A and a 5-ary functor F yielding a set, and states that:

A is associative provided the parameters meet the following requirements:

• Let a, b, c be objects of A. 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 = F(a, b, c, f, g), and

• Let a, b, c, d be objects of A and f, g, h be sets. If f ∈ ⟨a, b⟩ and g ∈ ⟨b, c⟩ and h ∈ ⟨c, d⟩, then F(a, c, d, F(a, b, c, f, g), h) = F(a, b, d, f, F(b, c, d, g, h)).

The scheme CatUnitsSch deals with a non empty transitive category structure A and a 5-ary functor F yielding a set, and states that:

A has units provided the parameters satisfy the following conditions:

• Let a, b, c be objects of A. 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 = F(a, b, c, f, g),

• Let a be an object of A. Then there exists a set f such that f ∈ ⟨a, a⟩ and for every object b of A and for every set g such that g ∈ ⟨a, b⟩ holds F(a, a, b, f, g) = g, and

• Let a be an object of A. Then there exists a set f such that f ∈ ⟨a, a⟩ and for every object b of A and for every set g such that g ∈ ⟨b, a⟩ holds F(b, a, a, g, f) = g.

The scheme CategoryLambda deals with a non empty set A, a binary functor F yielding a set, and a 5-ary functor G yielding a set, and states that:

There exists a strict category C such that

(i) the carrier of C = A, (ii) for all objects a, b of C holds ⟨a, b⟩ = F(a, b), and (iii) for all objects a, b, c of C such that ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅ and for every morphism f from a to b and for every morphism g from b to c holds g · f = G(a, b, c, f, g)

provided the parameters satisfy the following conditions:

• For all elements a, b, c of A and for all sets f, g such that f ∈ F(a, b) and g ∈ F(b, c) holds G(a, b, c, f, g) ∈ F(a, c),

• Let a, b, c, d be elements of A and f, g, h be sets. If f ∈ F(a, b) and g ∈ F(b, c) and h ∈ F(c, d), then G(a, c, d, G(a, b, c, f, g), h) = G(a, b, d, f, G(b, c, d, g, h)),

• Let a be an element of A. Then there exists a set f such that f ∈ F(a, a) and for every element b of A and for every set g such

that g ∈ F(a, b) holds G(a, a, b, f, g) = g, and

• Let a be an element of A. Then there exists a set f such that f ∈ F(a, a) and for every element b of A and for every set g such that g ∈ F(b, a) holds G(b, a, a, g, f) = g.

The scheme CategoryLambdaUniq deals with a non empty set A, a binary functor F yielding a set, and a 5-ary functor G yielding a set, and states that:

Let C_{1}, C_{2} be non empty transitive category structures. Suppose that

(i) the carrier of C_{1} = A, (ii) for all objects a, b of C_{1} holds ⟨a, b⟩ = F(a, b), (iii) for all objects a, b, c of C_{1} such that ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅ and for every morphism f from a to b and for every morphism g from b to c holds g · f = G(a, b, c, f, g), (iv) the carrier of C_{2} = A,

(v) for all objects a, b of C_{2} holds ⟨a, b⟩ = F(a, b), and (vi) for all objects a, b, c of C_{2} such that ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅ and for every morphism f from a to b and for every morphism g from b to c holds g · f = G(a, b, c, f, g).

Then the category structure of C_{1} = the category structure of C_{2}

for all values of the parameters.

The scheme CategoryQuasiLambda deals with a non empty set A, a binary functor F yielding a set, a 5-ary functor G yielding a set, and a ternary predicate P, and states that:

There exists a strict category C such that

(i) the carrier of C = A, (ii) for all objects a, b of C and for every set f holds f ∈ ⟨a, b⟩ iff f ∈ F(a, b) and P[a, b, f], and (iii) for all objects a, b, c of C such that ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅ and for every morphism f from a to b and for every morphism g from b to c holds g · f = G(a, b, c, f, g)

provided the following requirements are met:

• Let a, b, c be elements of A and f, g be sets. Suppose f ∈ F(a, b) and P[a, b, f] and g ∈ F(b, c) and P[b, c, g]. Then G(a, b, c, f, g) ∈ F(a, c) and P[a, c, G(a, b, c, f, g)],

• Let a, b, c, d be elements of A and f, g, h be sets. Suppose f ∈ F(a, b) and P[a, b, f] and g ∈ F(b, c) and P[b, c, g] and h ∈ F(c, d) and P[c, d, h]. Then G(a, c, d, G(a, b, c, f, g), h) = G(a, b, d, f, G(b, c, d, g, h)),

• Let a be an element of A. Then there exists a set f such that f ∈ F(a, a) and P[a, a, f] and for every element b of A and for every set g such that g ∈ F(a, b) and P[a, b, g] holds G(a, a, b, f, g) = g, and

• Let a be an element of A. Then there exists a set f such that f ∈ F(a, a) and P[a, a, f] and for every element b of A and for every set g such that g ∈ F(b, a) and P[b, a, g] holds G(b, a, a, g, f) = g.

Let f be a function yielding function and let a, b, c be sets. Note that f(a, b, c) is relation-like and function-like.

Now we present two schemes. The scheme SubcategoryEx deals with a category A, a unary predicate P, and a ternary predicate Q, and states that:

There exists a subcategory B of A such that

(i) for every object a of A holds a is an object of B iff P[a], and (ii) for all objects a, b of A and for all objects a′, b′ of B such that a′ = a and b′ = b and ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds f ∈ ⟨a′, b′⟩ iff Q[a, b, f]

provided the parameters meet the following requirements:

• There exists an object a of A such that P[a],

• Let a, b, c be objects of A. Suppose P[a] and P[b] and P[c] and ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅. Let f be a morphism from a to b and g be a morphism from b to c. If Q[a, b, f] and Q[b, c, g], then Q[a, c, g · f], and

• For every object a of A such that P[a] holds Q[a, a, id_{a}]. The scheme CovariantFunctorLambda deals with categories A, B, a unary

functor F yielding a set, and a ternary functor G yielding a set, 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(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)

provided the parameters have the following properties:

• For every object a of A holds F(a) is an object of B,

• Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then G(a, b, f) ∈ (the arrows of B)(F(a), F(b)),

• Let a, b, c be objects of A. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅. Let f be a morphism from a to b, g be a morphism from b to c, and a′, b′, c′ be objects of B. Suppose a′ = F(a) and b′ = F(b) and c′ = F(c). Let f′ be a morphism from a′ to b′ and g′ be a morphism from b′ to c′. If f′= G(a, b, f) and g′= G(b, c, g), then G(a, c, g · f) = g′· f′, and

• For every object a of A and for every object a′ of B such that a′= F(a) holds G(a, a, id_{a}) = id_{a′}.

The following proposition is true (1) Let A, B be categories and F, G be covariant functors from A to B.

Suppose that

(i) for every object a of A holds F (a) = G(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(f). Then the functor structure of F = the functor structure of G.

The scheme ContravariantFunctorLambda deals with categories A, B, a unary functor F yielding a set, and a ternary functor G yielding a set, 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(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)

provided the parameters meet the following requirements:

• For every object a of A holds F(a) is an object of B,

• Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then G(a, b, f) ∈ (the arrows of B)(F(b), F(a)),

• Let a, b, c be objects of A. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅. Let f be a morphism from a to b, g be a morphism from b to c, and a′, b′, c′ be objects of B. Suppose a′ = F(a) and b′ = F(b) and c′ = F(c). Let f′ be a morphism from b′ to a′ and g′ be a morphism from c′ to b′. If f′= G(a, b, f) and g′= G(b, c, g), then G(a, c, g · f) = f′· g′, and

• For every object a of A and for every object a′ of B such that a′= F(a) holds G(a, a, id_{a}) = id_{a′}.

One can prove the following proposition

(2) Let A, B be categories and F, G be contravariant functors from A to B. Suppose that

(i) for every object a of A holds F (a) = G(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(f). Then the functor structure of F = the functor structure of G.

2. Isomorphism and Equivalence of Categories

Let A, B, C be non empty sets and let f be a function from [: A, B:] into C. Let us observe that f is one-to-one if and only if:

(Def. 1) For all elements a_{1}, a_{2} of A and for all elements b_{1}, b_{2} of B such that f(a_{1}, b_{1}) = f(a_{2}, b_{2}) holds a_{1} = a_{2} and b_{1} = b_{2}.

Now we present four schemes. The scheme CoBijectiveSch deals with categories A, B, a covariant functor C from A to B, a unary functor F yielding a set, and a ternary functor C yielding a set, and states that:

C is bijective provided the parameters meet the following requirements:

• For every object a of A holds C(a) = F(a),

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds C(f) = C(a, b, f),

• For all objects a, b of A such that F(a) = F(b) holds a = b,

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for all morphisms f, g from a to b such that C(a, b, f) = C(a, b, g) holds f = g, and

• 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 = C(c, d, g).

The scheme CatIsomorphism deals with categories A, B, a unary functor F yielding a set, and a ternary functor G yielding a set, and states that:

A and B are isomorphic provided the parameters meet the following requirements:

• 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 objects a, b of A such that F(a) = F(b) holds a = b,

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for all morphisms f, g from a to b such that G(a, b, f) = G(a, b, g) holds f = g, and

• 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 = G(c, d, g).

The scheme ContraBijectiveSch deals with categories A, B, a contravariant functor C from A to B, a unary functor F yielding a set, and a ternary functor C yielding a set, and states that:

C is bijective provided the following conditions are met:

• For every object a of A holds C(a) = F(a),

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for every morphism f from a to b holds C(f) = C(a, b, f),

• For all objects a, b of A such that F(a) = F(b) holds a = b,

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for all morphisms f, g from a to b such that C(a, b, f) = C(a, b, g) holds f = g, and • 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 = C(c, d, g).

The scheme CatAntiIsomorphism deals with categories A, B, a unary functor F yielding a set, and a ternary functor G yielding a set, and states that:

A, B are anti-isomorphic provided the parameters meet the following conditions:

• 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 objects a, b of A such that F(a) = F(b) holds a = b,

• For all objects a, b of A such that ⟨a, b⟩ ≠ ∅ and for all morphisms f, g from a to b such that G(a, b, f) = G(a, b, g) holds f = g, and

• 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 = G(c, d, g).

Let A, B be categories. We say that A and B are equivalent if and only if the condition (Def. 2) is satisfied.

(Def. 2) There exists a covariant functor F from A to B and there exists a covariant functor G from B to A such that G · F and id_{A} are naturally equivalent and F · G and id_{B} are naturally equivalent.

Let us notice that the predicate A and B are equivalent is reflexive and symmetric.

The following propositions are true:

(3) Let A, B, C be non empty reflexive graphs, F_{1}, F_{2} be feasible functor structures from A to B, and G_{1}, G_{2} be functor structures from B to C. Suppose that

(i) the functor structure of F_{1} = the functor structure of F_{2}, and (ii) the functor structure of G_{1} = the functor structure of G_{2}.

Then G_{1} · F_{1} = G_{2} · F_{2}.

(4) Let A, B, C be categories. Suppose A and B are equivalent and B and C are equivalent. Then A and C are equivalent.

(5) For all categories A, B such that A and B are isomorphic holds A and B are equivalent.

Now we present two schemes. The scheme NatTransLambda deals with categories A, B, covariant functors C, D from A to B, and a unary functor F yielding a set, and states that:

There exists a natural transformation t from C to D such that C is naturally transformable to D and for every object a of A holds t[a] = F(a)

provided the parameters have the following properties:

• For every object a of A holds F(a) ∈ ⟨C(a), D(a)⟩, and

• Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b and g_{1} be a morphism from C(a) to D(a). Suppose g_{1} = F(a). Let g_{2} be a morphism from C(b) to D(b). If g_{2} = F(b), then g_{2} · C(f) = D(f) · g_{1}.

The scheme NatEquivalenceLambda deals with categories A, B, covariant functors C, D from A to B, and a unary functor F yielding a set, and states that:

There exists a natural equivalence t of C and D such that C and D are naturally equivalent and for every object a of A holds t[a] = F(a)

provided the following conditions are satisfied:

• Let a be an object of A. Then F(a) ∈ ⟨C(a), D(a)⟩ and ⟨D(a), C(a)⟩ ≠ ∅ and for every morphism f from C(a) to D(a) such that f = F(a) holds f is iso, and

• Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b and g_{1} be a morphism from C(a) to D(a). Suppose g_{1} = F(a). Let g_{2} be a morphism from C(b) to D(b). If g_{2} = F(b), then g_{2} · C(f) = D(f) · g_{1}.

3. Dual Categories

Let C_{1}, C_{2} be non empty category structures. We say that C_{1} and C_{2} are opposite if and only if the conditions (Def. 3) are satisfied.

(Def. 3)(i) The carrier of C_{2} = the carrier of C_{1}, (ii) the arrows of C_{2} = ↶(the arrows of C_{1}), and (iii) for all objects a, b, c of C_{1} and for all objects a′, b′, c′ of C_{2} such

that a′ = a and b′ = b and c′ = c holds (the composition of C_{2})(a′, b′, c′) = ↶(the composition of C_{1})(c, b, a).

Let us note that the predicate C_{1} and C_{2} are opposite is symmetric. Next we state several propositions:

(6) For all non empty category structures A, B such that A and B are opposite holds every object of A is an object of B.

(7) Let A, B be non empty category structures. Suppose A and B are opposite. Let a, b be objects of A and a′, b′ be objects of B. If a′ = a and b′= b, then ⟨a, b⟩ = ⟨b′, a′⟩.

(8) Let A, B be non empty category structures. Suppose A and B are opposite. Let a, b be objects of A and a′, b′ be objects of B. If a′ = a and b′= b, then every morphism from a to b is a morphism from b′ to a′.

(9) Let C_{1}, C_{2} be non empty transitive category structures. Then C_{1} and C_{2} are opposite if and only if the following conditions are satisfied:

(i) the carrier of C_{2} = the carrier of C_{1}, and (ii) for all objects a, b, c of C_{1} and for all objects a′, b′, c′ of C_{2} such that

a′ = a and b′ = b and c′ = c holds ⟨a, b⟩ = ⟨b′, a′⟩ and if ⟨a, b⟩ ≠ ∅ and ⟨b, c⟩ ≠ ∅, then for every morphism f from a to b and for every morphism g from b to c and for every morphism f′from b′to a′and for every morphism g′ from c′ to b′ such that f′= f and g′= g holds f′· g′= g · f.

(10) Let A, B be categories. Suppose A and B are opposite. Let a be an object of A and b be an object of B. If a = b, then id_{a} = id_{b} .

(11) Let C be a transitive non empty category structure. Then there exists a strict transitive non empty category structure C′ such that C and C′ are opposite.

(12) Let A, B be transitive non empty category structures. Suppose A and B are opposite. If A is associative, then B is associative.

(13) For all transitive non empty category structures A, B such that A and B are opposite holds if A has units, then B has units.

(14) Let C, C_{1}, C_{2} be non empty category structures. Suppose C and C_{1} are opposite. Then C and C_{2} are opposite if and only if the category structure of C_{1} = the category structure of C_{2}.

Let C be a transitive non empty category structure. The functor C^{op} yields a strict transitive non empty category structure and is defined as follows:

(Def. 4) C and C^{op} are opposite.

Let C be an associative transitive non empty category structure. One can check that C^{op} is associative.

Let C be a transitive non empty category structure with units. One can verify that C^{op} has units.

Let A, B be categories. Let us assume that A and B are opposite. The dualizing functor from A into B is a contravariant strict functor from A to B and is defined by the conditions (Def. 5).

(Def. 5)(i) For every object a of A holds (the dualizing functor from A into

B)(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 (the dualizing functor from A into B)(f) = f. Next we state two propositions:

(15) Let A, B be categories. Suppose A and B are opposite. Then (the dualizing functor from A into B) · (the dualizing functor from B into A) = id_{B}.

(16) Let A, B be categories. Suppose A and B are opposite. Then the duali-

zing functor from A into B is bijective. Let A be a category. One can verify that the dualizing functor from A into

A^{op} is bijective and the dualizing functor from A^{op} into A is bijective.

Next we state a number of propositions:

(17) For all categories A, B such that A and B are opposite holds A, B are anti-isomorphic.

(18) Let A, B, C be categories. Suppose A and B are opposite. Then A and C are isomorphic if and only if B, C are anti-isomorphic.

(19) Let A, B, C, D be categories. Suppose A and B are opposite and C and D are opposite. If A and C are isomorphic, then B and D are isomorphic.

(20) Let A, B, C, D be categories. Suppose A and B are opposite and C and D are opposite. If A, C are anti-isomorphic, then B, D are anti-isomorphic.

(21) Let A, B be categories. Suppose A and B are opposite. Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, a⟩ ≠ ∅. Let a′, b′be objects of B. Suppose a′= a and b′= b. Let f be a morphism from a to b and f′be a morphism from b′ to a′. If f′= f, then f is retraction iff f′ is coretraction.

(22) Let A, B be categories. Suppose A and B are opposite. Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, a⟩ ≠ ∅. Let a′, b′be objects of B. Suppose a′= a and b′= b. Let f be a morphism from a to b and f′be a morphism from b′ to a′. If f′= f, then f is coretraction iff f′ is retraction.

(23) Let A, B be categories. Suppose A and B are opposite. Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, a⟩ ≠ ∅. Let a′, b′ be objects of B. Suppose a′ = a and b′ = b. Let f be a morphism from a to b and f′ be a morphism from b′ to a′. If f′ = f and f is retraction and coretraction, then f′^{−1} = f^{−1} .

(24) Let A, B be categories. Suppose A and B are opposite. Let a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅ and ⟨b, a⟩ ≠ ∅. Let a′, b′ be objects of B. Suppose a′= a and b′= b. Let f be a morphism from a to b and f′ be a morphism from b′ to a′. If f′= f, then f is iso iff f′ is iso.

(25) Let A, B, C, D be categories. Suppose A and B are opposite and C and D are opposite. Let F, G be covariant functors from B to C. Suppose F and G are naturally equivalent. Then (the dualizing functor from C into D) · G · the dualizing functor from A into B and (the dualizing functor from C into D) · F · the dualizing functor from A into B are naturally equivalent.

(26) Let A, B, C, D be categories. Suppose A and B are opposite and C and

D are opposite. If A and C are equivalent, then B and D are equivalent. Let A, B be categories. We say that A and B are dual if and only if:

(Def. 6) A and B^{op} are equivalent.

Let us note that the predicate A and B are dual is symmetric.

We now state four propositions:

(27) For all categories A, B such that A, B are anti-isomorphic holds A and B are dual.

(28) Let A, B, C be categories. Suppose A and B are opposite. Then A and C are equivalent if and only if B and C are dual.

(29) For all categories A, B, C such that A and B are dual and B and C are equivalent holds A and C are dual.

(30) For all categories A, B, C such that A and B are dual and B and C are dual holds A and C are equivalent.

4. Concrete Categories

The following proposition is true

(31) For all sets X, Y, x holds x ∈ Y^{X} iff x is a function and π_{1}(x) = X and

π_{2}(x) ⊆ Y. Let C be a 1-sorted structure. A many sorted set indexed by C is a many

sorted set indexed by the carrier of C.

Let C be a category. We say that C is para-functional if and only if: (Def. 7) There exists a many sorted set F indexed by C such that for all objects

a_{1}, a_{2} of C holds ⟨a_{1}, a_{2}⟩ ⊆ F (a_{2})^{F (a_{1})}. Let us note that every category which is quasi-functional is also para-functional. Let C be a category and let a be a set. C-carrier of a is defined as follows:

(Def. 8)(i) There exists an object b of C such that b = a and C-carrier of a =

π_{1}(id_{b}) if a is an object of C, (ii) C-carrier of a = ∅, otherwise. Let C be a category and let a be an object of C. Then C-carrier of a can be

characterized by the condition: (Def. 9) C-carrier of a = π_{1}(id_{a}).

We introduce the carrier of a as a synonym of C-carrier of a.

We now state two propositions:

(32) For every non empty set A and for every object a of Ens_{A} holds the identity morphism of a = the identity function on a.

(33) For every non empty set A and for every object a of Ens_{A} holds the

carrier of a = a. Let C be a category. We say that C is set-id-inheriting if and only if:

(Def. 10) For every object a of C holds id_{a} = id_{the carrier of a}.

Let A be a non empty set. Observe that Ens_{A} is set-id-inheriting. Let C be a category. We say that C is concrete if and only if:

(Def. 11) C is para-functional, semi-functional, and set-id-inheriting.

One can verify that every category which is concrete is also para-functional, semi-functional, and set-id-inheriting and every category which is para-functional, semi-functional, and set-id-inheriting is also concrete.

Let us mention that there exists a category which is concrete, quasi-functional, and strict.

The following propositions are true:

(34) Let C be a category. Then C is para-functional if and only if for all objects a_{1}, a_{2} of C holds ⟨a_{1}, a_{2}⟩ ⊆ (the carrier of a_{2})^{the carrier of a_{1}}.

(35) Let C be a para-functional category and a, b be objects of C. Suppose ⟨a, b⟩ ≠ ∅. Then every morphism from a to b is a function from the carrier of a into the carrier of b.

Let A be a para-functional category and let a, b be objects of A. One can verify that every morphism from a to b is function-like and relation-like.

We now state four propositions:

(36) Let C be a para-functional category and a, b be objects of C. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then domf = the carrier of a and rngf ⊆ the carrier of b.

(37) For every para-functional semi-functional category C and for every object a of C holds the carrier of a = dom(id_{a}).

(38) Let C be a para-functional semi-functional 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 qua function) · (f qua function).

(39) Let C be a para-functional semi-functional category and a be an object

of C. If id_{the carrier of a} ∈ ⟨a, a⟩, then id_{a} = id_{the carrier of a}. Now we present several schemes. The scheme ConcreteCategoryLambda deals

with a non empty set A, a binary functor F yielding a set, and a unary functor G yielding a set, and states that:

There exists a concrete strict category C such that

(i) the carrier of C = A, (ii) for every object a of C holds the carrier of a = G(a), and (iii) for all objects a, b of C holds ⟨a, b⟩ = F(a, b)

provided the following requirements are met:

• For all elements a, b, c of A and for all functions f, g such that f ∈ F(a, b) and g ∈ F(b, c) holds g · f ∈ F(a, c),

• For all elements a, b of A holds F(a, b) ⊆ G(b)^{G(a)}, and

• For every element a of A holds id_{G(a)} ∈ F(a, a).

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

There exists a concrete strict category C such that

(i) the carrier of C = A, (ii) for every object a of C holds the carrier of a = F(a), and (iii) for all elements a, b of A and for every function f holds f ∈ (the arrows of C)(a, b) iff f ∈ F(b)^{F(a)} and P[a, b, f]

provided the parameters satisfy the following conditions:

• For all elements a, b, c of A and for all functions f, g such that P[a, b, f] and P[b, c, g] holds P[a, c, g · f], and

• For every element a of A holds P[a, a, id_{F(a)}]. The scheme ConcreteCategoryEx deals with a non empty set A, a unary

functor F yielding a set, a binary predicate P, and a ternary predicate Q, and states that:

There exists a concrete strict category C such that

(i) the carrier of C = A, (ii) for every object a of C and for every set x holds x ∈ the carrier of a iff x ∈ F(a) and P[a, x], and (iii) for all elements a, b of A and for every function f holds f ∈ (the arrows of C)(a, b) iff f ∈ (C-carrier of b)^{C-carrier of a} and Q[a, b, f]

provided the following requirements are met:

• For all elements a, b, c of A and for all functions f, g such that Q[a, b, f] and Q[b, c, g] holds Q[a, c, g · f], and

• Let a be an element of A and X be a set. If for every set x holds x ∈ X iff x ∈ F(a) and P[a, x], then Q[a, a, id_{X}].

The scheme ConcreteCategoryUniq1 deals with a non empty set A and a binary functor F yielding a set, and states that:

Let C_{1}, C_{2} be para-functional semi-functional categories. Suppose that

(i) the carrier of C_{1} = A, (ii) for all objects a, b of C_{1} holds ⟨a, b⟩ = F(a, b), (iii) the carrier of C_{2} = A, and (iv) for all objects a, b of C_{2} holds ⟨a, b⟩ = F(a, b).

Then the category structure of C_{1} = the category structure of C_{2}

for all values of the parameters.

The scheme ConcreteCategoryUniq2 deals with a non empty set A, a unary functor F yielding a set, and a ternary predicate P, and states that:

Let C_{1}, C_{2} be para-functional semi-functional categories. Suppose that

(i) the carrier of C_{1} = A, (ii) for all elements a, b of A and for every function f holds f ∈ (the arrows of C_{1})(a, b) iff f ∈ F(b)^{F(a)} and P[a, b, f], (iii) the carrier of C_{2} = A, and (iv) for all elements a, b of A and for every function f holds f ∈ (the arrows of C_{2})(a, b) iff f ∈ F(b)^{F(a)} and 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 ConcreteCategoryUniq3 deals with a non empty set A, a unary functor F yielding a set, a binary predicate P, and a ternary predicate Q, and states that:

Let C_{1}, C_{2} be para-functional semi-functional categories. Suppose that

(i) the carrier of C_{1} = A, (ii) for every object a of C_{1} and for every set x holds x ∈ the carrier of a iff x ∈ F(a) and P[a, x], (iii) for all elements a, b of A and for every function f holds f ∈ (the arrows of C_{1})(a, b) iff f ∈ (C_{1}-carrier of b)^{C_{1}-carrier of a} and Q[a, b, f], (iv) the carrier of C_{2} = A,

(v) for every object a of C_{2} and for every set x holds x ∈ the carrier of a iff x ∈ F(a) and P[a, x], and (vi) for all elements a, b of A and for every function f holds f ∈ (the arrows of C_{2})(a, b) iff f ∈ (C_{2}-carrier of b)^{C_{2}-carrier of a} and Q[a, b, f].

Then the category structure of C_{1} = the category structure of C_{2}

for all values of the parameters.

5. Equivalence Between Concrete Categories

One can prove the following propositions:

(40) Let C be a concrete 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 retraction, then rngf = the carrier of b.

(41) Let C be a concrete 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 coretraction, then f is one-to-one.

(42) Let C be a concrete 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 oneto-one and rngf = the carrier of b.

(43) Let C be a para-functional semi-functional category and a, b be objects of C. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. If f is one-to-one and (f qua function) ^{−1} ∈ ⟨b, a⟩, then f is iso.

(44) Let C be a concrete 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^{−1} = (f qua function) ^{−1} .

The scheme ConcreteCatEquivalence deals with para-functional semi-functional categories A, B, two unary functors F and G yielding sets, two ternary functors H and I yielding functions, and two unary functors A and B yielding functions, and states that:

A and B are equivalent provided the following conditions are met:

• 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),

• For all objects a, b of A such that a = G(F(b)) holds A(b) ∈ ⟨a, b⟩ and A(b)^{−1} ∈ ⟨b, a⟩ and A(b) is one-to-one,

• For all objects a, b of B such that b = F(G(a)) holds B(a) ∈ ⟨a, b⟩ and B(a)^{−1} ∈ ⟨b, a⟩ and B(a) is one-to-one,

• 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.

6. Concretization of Categories

Let C be a category. The concretized C is a concrete strict category and is defined by the conditions (Def. 12).

(Def. 12)(i) The carrier of the concretized C = the carrier of C,

(ii) for every object a of the concretized C and for every set x holds x ∈ the carrier of a iff x ∈ Uniondisjoint(the arrows of C) and a = x_{2,2} , and (iii) for all elements a, b of the carrier of C and for every function f holds f ∈

(the arrows of the concretized C)(a, b) iff f ∈ ((the concretized C)-carrier of b)^{(the concretized C)-carrier of a} and there exist objects f_{1}, f_{2} of C and there exists a morphism g from f_{1} to f_{2} such that f_{1} = a and f_{2} = b and ⟨f_{1}, f_{2}⟩ ≠ ∅ and for every object o of C such that ⟨o, f_{1}⟩ ≠ ∅ and for every morphism h from o to f_{1} holds f(⟨⟨h, ⟨⟨o, f_{1}⟩⟩⟩⟩) = ⟨⟨g · h, ⟨⟨o, f_{2}⟩⟩⟩⟩.

One can prove the following proposition

(45) Let A be a category, a be an object of A, and x be a set. Then x ∈ (the concretized A)-carrier of a if and only if there exists an object b of A and there exists a morphism f from b to a such that ⟨b, a⟩ ≠ ∅ and x = ⟨⟨f, ⟨⟨b, a⟩⟩⟩⟩.

Let A be a category and let a be an object of A. Observe that (the concretized A)-carrier of a is non empty.

One can prove the following two propositions:

(46) Let A be a category and a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let f be a morphism from a to b. Then there exists a function F from (the concretized A)-carrier of a into (the concretized A)-carrier of b such that

(i) F ∈ (the arrows of the concretized A)(a, b), and (ii) for every object c of A and for every morphism g from c to a such that

⟨c, a⟩ ≠ ∅ holds F (⟨⟨g, ⟨⟨c, a⟩⟩⟩⟩) = ⟨⟨f · g, ⟨⟨c, b⟩⟩⟩⟩.

(47) Let A be a category and a, b be objects of A. Suppose ⟨a, b⟩ ≠ ∅. Let

F_{1}, F_{2} be functions. Suppose that

(i) F_{1} ∈ (the arrows of the concretized A)(a, b), (ii) F_{2} ∈ (the arrows of the concretized A)(a, b), and (iii) F_{1}(⟨⟨id_{a}, ⟨⟨a, a⟩⟩⟩⟩) = F_{2}(⟨⟨id_{a}, ⟨⟨a, a⟩⟩⟩⟩).

Then F_{1} = F_{2}. The scheme NonUniqMSFunctionEx deals with a set A, many sorted sets B,

C indexed by A, and a ternary predicate P, and states that:

There exists a many sorted function F from B into C such that for all sets i, x if i ∈ A and x ∈ B(i), then F (i)(x) ∈ C(i) and P[i, x, F (i)(x)]

provided the following condition is met:

• For all sets i, x such that i ∈ A and x ∈ B(i) there exists a set y such that y ∈ C(i) and P[i, x, y].

Let A be a category. The concretization of A is a covariant strict functor from A to the concretized A and is defined by the conditions (Def. 13).

(Def. 13)(i) For every object a of A holds (the concretization of A)(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 (the concretization of A)(f)(⟨⟨id_{a}, ⟨⟨a, a⟩⟩⟩⟩) = ⟨⟨f, ⟨⟨a, b⟩⟩⟩⟩. Let A be a category. One can check that the concretization of A is bijective. The following proposition is true

concrete categories 621

(48) For every category A holds A and the concretized A are isomorphic.

References

[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537– 541, 1990.

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

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

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

[5] 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.

[6] Artur Korniłowicz. The composition of functors and transformations in alternative categories. Formalized Mathematics, 7(1):1–7, 1998.

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

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

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

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

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

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

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

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

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

Received January 12, 2001 622 grzegorz bancerek

Classes of Independent Partitions

Andrzej Trybulec University of Białystok

Summary. The paper includes proofs of few theorems proved earlier by Shunichi Kobayashi in many different settings.

MML Identifier: PARTIT 2.

The terminology and notation used in this paper have been introduced in the following articles: [1], [3], [4], [5], [9], [2], [10], [12], [11], [7], [6], and [8].

1. Preliminaries

Let X, Y be sets and let R, S be relations between X and Y. Let us observe that R ⊆ S if and only if:

(Def. 1) For every element x of X and for every element y of Y such that ⟨⟨x,

y⟩⟩∈ R holds ⟨⟨x, y⟩⟩∈ S. For simplicity, we adopt the following rules: Y is a non empty set, a is an

element of Boolean^{Y}, G is a subset of PARTITIONS(Y), and P, Q are partitions of Y.

Let Y be a non empty set and let G be a non empty subset of PARTITIONS(Y). We see that the element of G is a partition of Y.

One can prove the following propositions:

(1) ⋀ ∅_{PARTITIONS (Y)} = O(Y).

(2) For all equivalence relations R, S of Y holds R ∪ S ⊆ R · S.

(3) For every binary relation R on Y holds R ⊆ ∇_{Y} .

(4) For every equivalence relation R of Y holds ∇_{Y} · R = ∇_{Y} and R · ∇_{Y} = ∇_{Y} .

(5) For every partition P of Y and for all elements x, y of Y holds ⟨⟨x, y⟩⟩∈ ≡_{P} iff x ∈ EqClass(y, P).

(6) Let P, Q, R be partitions of Y. Suppose ≡_{R} = ≡_{P} · ≡_{Q} . Let x, y be elements of Y. Then x ∈ EqClass(y, R) if and only if there exists an element z of Y such that x ∈ EqClass(z, P) and z ∈ EqClass(y, Q).

(7) Let R, S be binary relations and Y be a set. If R is reflexive in Y and S is reflexive in Y, then R · S is reflexive in Y.

(8) For every binary relation R and for every set Y such that R is reflexive in Y holds Y ⊆ fieldR.

(9) For every set Y and for every binary relation R on Y such that R is reflexive in Y holds Y = fieldR.

(10) For all equivalence relations R, S of Y such that R · S = S · R holds R · S is an equivalence relation of Y.

2. Boolean-Valued Functions

The following propositions are true:

(11) For all elements a, b of Boolean^{Y} such that a ⋐ b holds ¬b ⋐ ¬a.

(12) For every element a of Boolean^{Y} and for every subset G of PARTITIONS(Y) and for every partition A of Y holds ∀_{a,A}G ⋐ a.

(13) Let a, b be elements of Boolean^{Y}, G be a subset of PARTITIONS(Y), and P be a partition of Y. If a ⋐ b, then ∀_{a,P} G ⋐ ∀_{b,P} G.

(14) For every element a of Boolean^{Y} and for every subset G of PARTITIONS(Y) and for every partition A of Y holds a ⋐ ∃_{a,A}G.

(15) Let a, b be elements of Boolean^{Y}, G be a subset of PARTITIONS(Y), and P be a partition of Y. If a ⋐ b, then ∃_{a,P} G ⋐ ∃_{b,P} G.

3. Independent Classes of Partitions

One can prove the following four propositions:

(16) If G is independent, then for all subsets P, Q of PARTITIONS(Y) such that P ⊆ G and Q ⊆ G holds ≡_{⋀ P} · ≡_{⋀ Q} = ≡_{⋀ Q} · ≡_{⋀ P} .

(17) If G is independent, then ∀_{∀_{a,P} G,Q}G = ∀_{∀_{a,Q}G,P} G.

(18) If G is independent, then ∃_{∃_{a,P} G,Q}G = ∃_{∃_{a,Q}G,P} G.

(19) Let a be an element of Boolean^{Y}, G be a subset of PARTITIONS(Y), and P, Q be partitions of Y. If G is independent, then ∃_{∀_{a,P} G,Q}G ⋐ ∀_{∃_{a,Q}G,P} G. classes of independent partitions 625

References

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

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

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

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

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

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

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

[8] Edmund Woronowicz. Interpretation and satisfiability in the first order logic. Formalized Mathematics, 1(4):739–743, 1990.

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

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

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

[12] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

626 andrzej trybulec

Some Properties of Dyadic Numbers and Intervals

Józef Białas Łódź University Yatsuka Nakamura Shinshu University Nagano

Summary. The article is the second part of a paper proving the fundamental Urysohn Theorem concerning the existence of a real valued continuous function on a normal topological space. The paper is divided into two parts. In the first part, we introduce some definitions and theorems concerning properties of intervals; in the second we prove some of properties of dyadic numbers used in proving Urysohn Lemma.

MML Identifier: URYSOHN2.

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

The following proposition is true

(1) For every interval A such that A ≠ ∅ holds if inf A < supA, then vol(A) = supA − inf A and if supA = inf A, then vol(A) = 0\overline{ℝ}.

Let A be a subset of ℝ and let x be a real number. The functor x·A yielding a subset of ℝ is defined as follows:

(Def. 1) For every real number y holds y ∈ x · A iff there exists a real number z

such that z ∈ A and y = x · z. Next we state a number of propositions:

(2) For every subset A of ℝ and for every real number x such that x ≠ 0 holds x^{−1} · (x · A) = A.

(3) For every real number x such that x ≠ 0 and for every subset A of ℝ such that A = ℝ holds x · A = A.

(4) For every subset A of ℝ such that A ≠ ∅ holds 0 · A = {0}.

(5) For every subset A of ℝ such that A ≠ ∅ holds 0 · A = {0}.

(6) For every real number x holds x · ∅ = ∅.

(7) For every real number y holds y < 0 or y = 0 or 0 < y.

(8) Let a, b be extended real numbers. Suppose a b. Then a = −∞ and b = −∞ or a = −∞ and b ∈ ℝ or a = −∞ and b = +∞ or a ∈ ℝ and b ∈ ℝ or a ∈ ℝ and b = +∞ or a = +∞ and b = +∞.

(9) For every extended real number x holds [x, x] is an interval.

(10) For every interval A holds 0 · A is an interval.

(11) For all real numbers q, x such that x ≠ 0 holds q = x · \frac{q}{x}.

(12) For all real numbers p, q, x such that 0 < x and x · p < x · q holds p < q.

(13) For all real numbers p, q, x such that x < 0 and x · p < x · q holds q < p.

(14) For all real numbers p, q, x such that 0 < x and x · p x · q holds p q.

(15) For all real numbers p, q, x such that x < 0 and x · p x · q holds q p.

(16) Let A be an interval and x be a real number. If x ≠ 0, then if A is open interval, then x · A is open interval.

(17) Let A be an interval and x be a real number. If x ≠ 0, then if A is closed interval, then x · A is closed interval.

(18) Let A be an interval and x be a real number. Suppose 0 < x. If A is right open interval, then x · A is right open interval.

(19) Let A be an interval and x be a real number. Suppose x < 0. If A is right open interval, then x · A is left open interval.

(20) Let A be an interval and x be a real number. Suppose 0 < x. If A is left open interval, then x · A is left open interval.

(21) Let A be an interval and x be a real number. Suppose x < 0. If A is left open interval, then x · A is right open interval.

(22) Let A be an interval. Suppose A ≠ ∅. Let x be a real number. Suppose 0 < x. Let B be an interval. Suppose B = x·A. Suppose A = [inf A, supA]. Then B = [inf B, supB] and for all real numbers s, t such that s = inf A and t = supA holds inf B = x · s and supB = x · t.

(23) Let A be an interval. Suppose A ≠ ∅. Let x be a real number. Suppose 0 < x. Let B be an interval. Suppose B = x·A. Suppose A = ]inf A, supA]. Then B = ]inf B, supB] and for all real numbers s, t such that s = inf A and t = supA holds inf B = x · s and supB = x · t.

(24) Let A be an interval. Suppose A ≠ ∅. Let x be a real number. Suppose 0 < x. Let B be an interval. Suppose B = x·A. Suppose A = ]inf A, supA[. Then B = ]inf B, supB[ and for all real numbers s, t such that s = inf A and t = supA holds inf B = x · s and supB = x · t.

(25) Let A be an interval. Suppose A ≠ ∅. Let x be a real number. Suppose 0 < x. Let B be an interval. Suppose B = x·A. Suppose A = [inf A, supA[.

Then B = [inf B, supB[ and for all real numbers s, t such that s = inf A and t = supA holds inf B = x · s and supB = x · t.

(26) For every interval A and for every real number x holds x·A is an interval. Let A be an interval and let x be a real number. Observe that x·A is interval. The following propositions are true:

(27) Let A be an interval and x be a real number. If 0 x, then for every real number y such that y = vol(A) holds x · y = vol(x · A).

(28) For all real numbers x, y, z such that x < y and y z or x y and y < z holds x < z.

(29) For every natural number n holds n < 2^{n}.

(30) For every integer n such that 0 n holds n is a natural number.

(31) For all natural numbers n, m such that n < m holds 2^{n} < 2^{m}.

(32) For every real number e_{1} such that 0 < e_{1} there exists a natural number n such that 1 < 2^{n} · e_{1}.

(33) For all real numbers a, b such that 0 a and 1 < b − a there exists a natural number n such that a < n and n < b.

(34) For every integer n such that 0 < n holds n is a natural number.

(35) For every rational number n such that 0 n holds 0 numn.

(36) For every rational number n such that 0 < n holds 0 < numn.

(37) For all real numbers a, b, c, d such that 0 < b and 0 < d or b < 0 and d < 0 holds if \frac{a}{b} < \frac{c}{d}, then a · d < c · b.

(38) For every natural number n holds dyadic(n) ⊆ DYADIC.

(39) For all real numbers a, b such that a < b and 0 a and b 1 there exists a real number c such that c ∈ DYADIC and a < c and c < b.

(40) For all real numbers a, b such that a < b there exists a real number c such that c ∈ DOM and a < c and c < b.

(41) For every non empty subset A of \overline{ℝ} and for all extended real numbers a, b such that A ⊆ [a, b] holds a inf A and supA b.

(42) 0 ∈ DYADIC and 1 ∈ DYADIC.

(43) For all extended real numbers a, b such that a = 0 and b = 1 holds DYADIC ⊆ [a, b].

(44) For all natural numbers n, k such that n k holds dyadic(n) ⊆ dyadic(k).

(45) For all real numbers a, b, c, d such that a < c and c < b and a < d and d < b holds |d − c| < b − a.

(46) Let e_{1} be a real number. Suppose 0 < e_{1}. Let d be a real number. Suppose 0 < d and d 1. Then there exist real numbers r_{1}, r_{2} such that r_{1} ∈ DYADIC∪ℝ_{>1} and r_{2} ∈ DYADIC∪ℝ_{>1} and 0 < r_{1} and r_{1} < d and d < r_{2} and r_{2} − r_{1} < e_{1}.

630 józef białas and yatsuka nakamura

References

[1] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[2] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[3] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2):263–269, 1992.

[4] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21–26, 1996.

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

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

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

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

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

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

[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.

The Urysohn Lemma

Józef Białas Łódź University Yatsuka Nakamura Shinshu University Nagano

Summary. This article is the third part of a paper proving the fundamental Urysohn Theorem concerning the existence of a real valued continuous function on a normal topological space. The paper is divided into two parts. In the first part, we describe the construction of the function solving thesis of the Urysohn Lemma. The second part contains the proof of the Urysohn Lemma in normal space and the proof of the same theorem for compact space.

MML Identifier: URYSOHN3.

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

Let D be a non empty subset of ℝ. One can check that every element of D is real.

One can prove the following proposition

(1) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A∩B = ∅. Let n be a natural number. Then there exists a function G from dyadic(n) into 2^{the carrier of T} such that for all elements r_{1}, r_{2} of dyadic(n) if r_{1} < r_{2}, then G(r_{1}) is open and G(r_{2}) is open and \overline{G(r_{1})} ⊆ G(r_{2}) and A ⊆ G(0) and B = Ω_{T} \ G(1).

Let T be a non empty topological space, let A, B be subsets of T, and let n be a natural number. Let us assume that T is a T_{4} space and A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. A function from dyadic(n) into 2^{the carrier of T} is said to be a drizzle of A, B, n if it satisfies the condition (Def. 1).

(Def. 1) Let r_{1}, r_{2} be elements of dyadic(n). Suppose r_{1} < r_{2}. Then it(r_{1}) is open and it(r_{2}) is open and \overline{it(r_{1})} ⊆ it(r_{2}) and A ⊆ it(0) and B = Ω_{T} \ it(1).

One can prove the following propositions:

(2) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let n be a natural number and D be a drizzle of A, B, n. Then A ⊆ D(0) and B = Ω_{T} \ D(1).

(3) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let n be a natural number and G be a drizzle of A, B, n. Then there exists a drizzle F of A, B, n + 1 such that for every element r of dyadic(n + 1) if r ∈ dyadic(n), then F (r) = G(r).

Let A, B be non empty sets, let F be a function from ℕ into A ˙→ B, and let n be a natural number. Then F (n) is a partial function from A to B.

Next we state the proposition

(4) Let T be a non empty topological space, A, B be subsets of T, and n be a natural number. Then every drizzle of A, B, n is an element of DYADIC ˙→ 2^{the carrier of T} .

Let A, B be non empty sets, let F be a function from ℕ into A ˙→ B, and let n be a natural number. Then F (n) is an element of A ˙→ B.

One can prove the following proposition

(5) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Then there exists a sequence F of partial functions from DYADIC into 2^{the carrier of T} such that for every natural number n holds F (n) is a drizzle of A, B, n and for every element r of domF (n) holds F (n)(r) = F (n + 1)(r).

Let T be a non empty topological space and let A, B be subsets of T. Let us assume that T is a T_{4} space and A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. A sequence of partial functions from DYADIC into 2^{the carrier of T} is said to be a rain of A, B if it satisfies the condition (Def. 2).

(Def. 2) Let n be a natural number. Then it(n) is a drizzle of A, B, n and for

every element r of domit(n) holds it(n)(r) = it(n + 1)(r). Let x be a real number. Let us assume that x ∈ DYADIC. The functor

InfDyadicx yields a natural number and is defined by: (Def. 3) x ∈ dyadic(0) iff InfDyadicx = 0 and for every natural number n such

that x ∈ dyadic(n + 1) and x ∉ dyadic(n) holds InfDyadicx = n + 1. The following propositions are true:

(6) For every real number x such that x ∈ DYADIC holds x ∈ dyadic(InfDyadicx).

(7) For every real number x such that x ∈ DYADIC and for every natural number n such that InfDyadicx n holds x ∈ dyadic(n).

(8) For every real number x such that x ∈ DYADIC and for every natural number n such that x ∈ dyadic(n) holds InfDyadicx n.

(9) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B and x be a real number. If x ∈ DYADIC, then for every natural number n holds G(InfDyadicx)(x) = G(InfDyadicx + n)(x).

(10) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B and x be a real number. Suppose x ∈ DYADIC. Then there exists an element y of 2^{the carrier of T} such that for every natural number n if x ∈ dyadic(n), then y = G(n)(x).

(11) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B. Then there exists a function F from DOM into 2^{the carrier of T} such that for every real number x holds

(i) if x ∈ ℝ_{<0}, then F (x) = ∅, (ii) if x ∈ ℝ_{>1}, then F (x) = the carrier of T, and (iii) if x ∈ DYADIC, then for every natural number n such that x ∈

dyadic(n) holds F (x) = G(n)(x). Let T be a non empty topological space and let A, B be subsets of T. Let

us assume that T is a T_{4} space and A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let R be a rain of A, B. The functor TempestR yielding a function from DOM into 2^{the carrier of T} is defined by the condition (Def. 4).

(Def. 4) Let x be a real number such that x ∈ DOM. Then (i) if x ∈ ℝ_{<0}, then (TempestR)(x) = ∅, (ii) if x ∈ ℝ_{>1}, then (TempestR)(x) = the carrier of T, and (iii) if x ∈ DYADIC, then for every natural number n such that x ∈

dyadic(n) holds (TempestR)(x) = R(n)(x). Let X be a non empty set, let T be a topological space, let F be a function

from X into 2^{the carrier of T}, and let x be an element of X. Then F (x) is a subset of T.

One can prove the following three propositions:

(12) Let T be a non empty topological space and A, B be subsets of T. Suppose T is a T_{4} space and A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B and r be a real number. If r ∈ DOM, then for every subset C of T such that C = (TempestG)(r) holds C is open.

(13) Let T be a non empty topological space and A, B be subsets of T. Suppose T is a T_{4} space and A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B and r_{1}, r_{2} be real numbers. Suppose r_{1} ∈ DOM and r_{2} ∈ DOM and r_{1} < r_{2}. Let C be a subset of T. If C = (TempestG)(r_{1}), then \overline{C} ⊆ (TempestG)(r_{2}).

(14) Let T be a non empty topological space, A, B be subsets of T, G be a rain of A, B, and p be a point of T. Then there exists a subset R of \overline{ℝ} such that for every set x holds x ∈ R if and only if the following conditions are satisfied:

(i) x ∈ DYADIC, and (ii) for every real number s such that s = x holds p ∉ (TempestG)(s). Let T be a non empty topological space, let A, B be subsets of T, let R be

a rain of A, B, and let p be a point of T. The functor Rainbow(p, R) yielding a subset of \overline{ℝ} is defined by:

(Def. 5) For every set x holds x ∈ Rainbow(p, R) iff x ∈ DYADIC and for every

real number s such that s = x holds p ∉ (TempestR)(s). Let T, S be non empty topological spaces, let F be a function from the

carrier of T into the carrier of S, and let p be a point of T. Then F (p) is a point of S.

One can prove the following propositions:

(15) Let T be a non empty topological space, A, B be subsets of T, G be a rain of A, B, and p be a point of T. Then Rainbow(p, G) ⊆ DYADIC.

(16) Let T be a non empty topological space, A, B be subsets of T, and R be a rain of A, B. Then there exists a map F from T into ℝ^{1} such that for every point p of T holds if Rainbow(p, R) = ∅, then F (p) = 0 and for every non empty subset S of \overline{ℝ} such that S = Rainbow(p, R) holds F (p) = supS.

Let T be a non empty topological space, let A, B be subsets of T, and let R be a rain of A, B. The functor ThunderR yielding a map from T into ℝ^{1} is defined by the condition (Def. 6).

(Def. 6) Let p be a point of T. Then if Rainbow(p, R) = ∅, then (ThunderR)(p) = 0 and for every non empty subset S of \overline{ℝ} such that S = Rainbow(p, R) holds (ThunderR)(p) = supS.

Let T be a non empty topological space, let F be a map from T into ℝ^{1}, and let p be a point of T. Then F (p) is a real number.

One can prove the following propositions:

(17) Let T be a non empty topological space, A, B be subsets of T, G be a rain of A, B, p be a point of T, and S be a non empty subset of \overline{ℝ}. Suppose S = Rainbow(p, G). Let ℓ_{1} be an extended real number. If ℓ_{1} = 1, then 0\overline{ℝ} supS and supS ℓ_{1}.

(18) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B, r be an element of DOM, and p be a

point of T. If (ThunderG)(p) < r, then p ∈ (TempestG)(r).

(19) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B and r be a real number. Suppose r ∈ DYADIC∪ℝ_{>1} and 0 < r. Let p be a point of T. If p ∈ (TempestG)(r), then (ThunderG)(p) r.

(20) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B, n be a natural number, and r_{1} be an element of DOM. If 0 < r_{1}, then for every point p of T such that r_{1} < (ThunderG)(p) holds p ∉ (TempestG)(r_{1}).

(21) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Let G be a rain of A, B. Then

(i) ThunderG is continuous, and (ii) for every point x of T holds 0 (ThunderG)(x) and (ThunderG)(x)

1 and if x ∈ A, then (ThunderG)(x) = 0 and if x ∈ B, then (ThunderG)(x) = 1.

(22) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Then there exists a map F from T into ℝ^{1} such that

(i) F is continuous, and (ii) for every point x of T holds 0 F (x) and F (x) 1 and if x ∈ A, then

F (x) = 0 and if x ∈ B, then F (x) = 1.

(23) Let T be a non empty topological space. Suppose T is a T_{4} space. Let A, B be subsets of T. Suppose A is closed and B is closed and A ∩ B = ∅. Then there exists a map F from T into ℝ^{1} such that

(i) F is continuous, and (ii) for every point x of T holds 0 F (x) and F (x) 1 and if x ∈ A, then

F (x) = 0 and if x ∈ B, then F (x) = 1.

(24) Let T be a non empty topological space. Suppose T is a T_{2} space and compact. Let A, B be subsets of T. Suppose A is closed and B is closed and A ∩ B = ∅. Then there exists a map F from T into ℝ^{1} such that

(i) F is continuous, and (ii) for every point x of T holds 0 F (x) and F (x) 1 and if x ∈ A, then

F (x) = 0 and if x ∈ B, then F (x) = 1.

References

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

636 józef białas and yatsuka nakamura

[2] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[3] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[4] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2):263–269, 1992.

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

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

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

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

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

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

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

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

[13] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17–21, 1992.

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

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

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

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

The Algebra of Polynomials^{1}

Ewa Grądzka University of Białystok

Summary. In this paper we define the algebra of formal power series and the algebra of polynomials over an arbitrary field and prove some properties of these structures. We also formulate and prove theorems showing some general properties of sequences. These preliminaries will be used for defining and considering linear functionals on the algebra of polynomials.

MML Identifier: POLYALG1.

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

1. Preliminaries

Let F be a 1-sorted structure. We introduce algebra structures over F which are extensions of double loop structure and vector space structure over F and are systems

⟨ a carrier, an addition, a multiplication, a reverse-map, a zero, a unity, a left multiplication ⟩, where the carrier is a set, the addition and the multiplication are binary operations on the carrier, the reverse-map is a unary operation on the carrier, the zero and the unity are elements of the carrier, and the left multiplication is a function from [: the carrier of F, the carrier :] into the carrier.

Let L be a non empty double loop structure. Note that there exists an algebra structure over L which is strict and non empty.

Let L be a non empty double loop structure and let A be a non empty algebra structure over L. We say that A is mix-associative if and only if:

1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

(Def. 1) For every element a of L and for all elements x, y of A holds a · (x · y) =

(a · x) · y. Let L be a non empty double loop structure. Note that there exists a non

empty algebra structure over L which is well unital, distributive, vector spacelike, and mix-associative.

Let L be a non empty double loop structure. An algebra of L is a well unital distributive vector space-like mix-associative non empty algebra structure over L.

Next we state two propositions:

(1) For all sets X, Y and for every function f from [: X, Y :] into X holds domf = [: X, Y :].

(2) For all sets X, Y and for every function f from [: X, Y :] into Y holds domf = [: X, Y :].

2. The Algebra of Formal Power Series

Let L be a non empty double loop structure. The functor Formal-SeriesL yields a strict non empty algebra structure over L and is defined by the conditions (Def. 2).

(Def. 2) For every set x holds x ∈ the carrier of Formal-SeriesL iff x is a sequence of L and for all elements x, y of the carrier of Formal-SeriesL and for all sequences p, q of L such that x = p and y = q holds x + y = p + q and for all elements x, y of the carrier of Formal-SeriesL and for all sequences p, q of L such that x = p and y = q holds x · y = p ∗ q and for every element x of the carrier of Formal-SeriesL and for every sequence p of L such that x = p holds − x = − p and for every element a of L and for every element x of the carrier of Formal-SeriesL and for every sequence p of L such that x = p holds a·x = a·p and 0_{Formal-Series L} = 0. L and 1_{Formal-Series L} = 1. L.

Let L be an Abelian non empty double loop structure. Note that Formal-SeriesL is Abelian.

Let L be an add-associative non empty double loop structure. Note that Formal-SeriesL is add-associative.

Let L be a right zeroed non empty double loop structure. Note that Formal-SeriesL is right zeroed.

Let L be an add-associative right zeroed right complementable non empty double loop structure. Note that Formal-SeriesL is right complementable.

Let L be an Abelian add-associative right zeroed commutative non empty double loop structure. Observe that Formal-SeriesL is commutative.

Let L be an Abelian add-associative right zeroed right complementable unital associative distributive non empty double loop structure. Note that Formal-SeriesL is associative.

Let L be an add-associative right zeroed right complementable right unital right distributive non empty double loop structure. Note that Formal-SeriesL is right unital.

One can verify that there exists a non empty double loop structure which is add-associative, associative, right zeroed, left zeroed, right unital, left unital, right complementable, and distributive.

We now state three propositions:

(3) For every non empty set D and for every non empty finite sequence f of elements of D holds f_{⇂1} = f_{↾1}.

(4) For every non empty set D and for every non empty finite sequence f of elements of D holds f = ⟨f(1)⟩^{⌢} (f_{↾1}).

(5) Let L be an add-associative right zeroed left unital right complementable left distributive non empty double loop structure and p be a sequence of L. Then 1. L ∗ p = p.

Let L be an add-associative right zeroed right complementable left unital left distributive non empty double loop structure. One can verify that Formal-SeriesL is left unital.

Let L be an Abelian add-associative right zeroed right complementable distributive non empty double loop structure. One can check that Formal-SeriesL is right distributive and Formal-SeriesL is left distributive.

We now state four propositions:

(6) Let L be an Abelian add-associative right zeroed right complementable distributive non empty double loop structure, a be an element of L, and p, q be sequences of L. Then a · (p + q) = a · p + a · q.

(7) Let L be an Abelian add-associative right zeroed right complementable distributive non empty double loop structure, a, b be elements of L, and p be a sequence of L. Then (a + b) · p = a · p + b · p.

(8) Let L be an associative non empty double loop structure, a, b be elements of L, and p be a sequence of L. Then (a · b) · p = a · (b · p).

(9) Let L be an associative left unital non empty double loop structure and p be a sequence of L. Then (the unity of L) · p = p.

Let L be an Abelian add-associative associative right zeroed right complementable left unital distributive non empty double loop structure. One can check that Formal-SeriesL is vector space-like.

We now state the proposition

(10) Let L be an Abelian left zeroed add-associative associative right zeroed right complementable distributive non empty double loop structure, a be

an element of L, and p, q be sequences of L. Then a · (p ∗ q) = (a · p) ∗ q. Let L be an Abelian left zeroed add-associative associative right zeroed right

complementable distributive non empty double loop structure. One can verify that Formal-SeriesL is mix-associative.

Let L be a left zeroed right zeroed add-associative left unital right unital right complementable distributive non empty double loop structure. Observe that Formal-SeriesL is well unital.

Let L be a 1-sorted structure and let A be an algebra structure over L. An algebra structure over L is said to be a subalgebra of A if it satisfies the conditions (Def. 3).

(Def. 3) The carrier of it ⊆ the carrier of A and 1_{it} = 1_{A} and 0_{it} = 0_{A} and the addition of it = (the addition of A)↾[: the carrier of it, the carrier of it :] and the multiplication of it = (the multiplication of A)↾[: the carrier of it, the carrier of it :] and the reverse-map of it = (the reverse-map of A)↾(the carrier of it) and the left multiplication of it = (the left multiplication of A)↾[: the carrier of L, the carrier of it :].

We now state four propositions:

(11) For every 1-sorted structure L holds every algebra structure A over L is a subalgebra of A.

(12) Let L be a 1-sorted structure and A, B, C be algebra structures over L. Suppose A is a subalgebra of B and B is a subalgebra of C. Then A is a subalgebra of C.

(13) Let L be a 1-sorted structure and A, B be algebra structures over L. Suppose A is a subalgebra of B and B is a subalgebra of A. Then the algebra structure of A = the algebra structure of B.

(14) Let L be a 1-sorted structure and A, B be algebra structures over L. Suppose the algebra structure of A = the algebra structure of B. Then A is a subalgebra of B and B is a subalgebra of A.

Let L be a non empty 1-sorted structure. Observe that there exists an algebra structure over L which is non empty and strict.

Let L be a 1-sorted structure and let B be an algebra structure over L. Observe that there exists a subalgebra of B which is strict.

Let L be a non empty 1-sorted structure and let B be a non empty algebra structure over L. Note that there exists a subalgebra of B which is strict and non empty.

Let L be a non empty groupoid, let B be a non empty algebra structure over L, and let A be a subset of B. We say that A is operations closed if and only if the conditions (Def. 4) are satisfied.

(Def. 4)(i) A is linearly closed, (ii) for all elements x, y of B such that x ∈ A and y ∈ A holds x · y ∈ A, (iii) for every element x of B such that x ∈ A holds − x ∈ A, (iv) 1_{B} ∈ A, and

(v) 0_{B} ∈ A. The following propositions are true:

(15) Let L be a non empty groupoid, B be a non empty algebra structure over L, A be a non empty subalgebra of B, x, y be elements of the carrier of B, and x′, y′ be elements of the carrier of A. If x = x′ and y = y′, then x + y = x′+ y′.

(16) Let L be a non empty groupoid, B be a non empty algebra structure over L, A be a non empty subalgebra of B, x, y be elements of the carrier of B, and x′, y′ be elements of the carrier of A. If x = x′ and y = y′, then x · y = x′· y′.

(17) Let L be a non empty groupoid, B be a non empty algebra structure over L, A be a non empty subalgebra of B, a be an element of the carrier of L, x be an element of the carrier of B, and x′ be an element of the carrier of A. If x = x′, then a · x = a · x′.

(18) Let L be a non empty groupoid, B be a non empty algebra structure over L, A be a non empty subalgebra of B, x be an element of the carrier of B, and x′ be an element of the carrier of A. If x = x′, then − x = − x′.

(19) Let L be a non empty groupoid, B be a non empty algebra structure over L, and A be a non empty subalgebra of B. Then there exists a subset C of B such that the carrier of A = C and C is operations closed.

(20) Let L be a non empty groupoid, B be a non empty algebra structure over L, and A be a subset of B. Suppose A is operations closed. Then there exists a strict subalgebra C of B such that the carrier of C = A.

(21) Let L be a non empty groupoid, B be a non empty algebra structure over L, A be a non empty subset of B, and X be a family of subsets of the carrier of B. Suppose that for every set Y holds Y ∈ X iff Y ∈ 2^{the carrier of B} and there exists a subalgebra C of B such that Y = the carrier of C and A ⊆ Y. Then ⋂ X is operations closed.

Let L be a non empty groupoid, let B be a non empty algebra structure over L, and let A be a non empty subset of B. The functor GenAlgA yielding a strict non empty subalgebra of B is defined by the conditions (Def. 5).

(Def. 5)(i) A ⊆ the carrier of GenAlgA, and

(ii) for every subalgebra C of B such that A ⊆ the carrier of C holds the

carrier of GenAlgA ⊆ the carrier of C. We now state the proposition

(22) Let L be a non empty groupoid, B be a non empty algebra structure over L, and A be a non empty subset of B. If A is operations closed, then the carrier of GenAlgA = A.

642 ewa grądzka

3. The Algebra of Polynomials

Let L be an add-associative right zeroed right complementable distributive non empty double loop structure. The functor Polynom-AlgebraL yields a strict non empty algebra structure over L and is defined as follows:

(Def. 6) There exists a non empty subset A of Formal-SeriesL such that A = the carrier of Polynom-RingL and Polynom-AlgebraL = GenAlgA.

Let L be an add-associative right zeroed right complementable distributive non empty double loop structure. One can verify that Polynom-RingL is looplike.

The following propositions are true:

(23) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure and A be a non empty subset of Formal-SeriesL. If A = the carrier of Polynom-RingL, then A is operations closed.

(24) Let L be an add-associative right zeroed right complementable distributive non empty double loop structure. Then the double loop structure of Polynom-AlgebraL = Polynom-RingL.

Acknowledgments

I would like to thank Dr Andrzej Trybulec for his helpful advices on this paper.

References

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

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

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

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

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

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

[7] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461– 470, 2001.

[8] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339–346, 2001.

[9] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

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

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

[12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[13] Wojciech Skaba and Michał Muzalewski. From double loops to fields. Formalized Mathematics, 2(1):185–191, 1991.

the algebra of polynomials 643

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

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

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

644 ewa grądzka

Circuit Generated by Terms and Circuit Calculating Terms

Grzegorz Bancerek University of Białystok Shinshu University, Nagano

Summary. In the paper we investigate the dependence between the structure of circuits and sets of terms. Circuits in our terminology (see [19]) are treated as locally-finite many sorted algebras over special signatures. Such approach enables to formalize every real circuit. The goal of this investigation is to specify circuits by terms and, enentualy, to have methods of formal verification of real circuits. The following notation is introduced in this paper:

• structural equivalence of circuits, i.e. equivalence of many sorted signatures,

• embedding of a circuit into another one,

• similarity of circuits (a concept narrower than isomorphism of many sorted algebras over equivalent signatures),

• calculation of terms by a circuit according to an algebra,

• specification of circuits by terms and an algebra.

MML Identifier: CIRCTRM1.

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

1. Circuit Structure Generated by Terms

One can prove the following proposition

(1) Let S be a non empty non void many sorted signature, A be a non-empty algebra over S, V be a variables family of A, t be a term of S over V, and T be a term of A over V. If T = t, then the sort of T = the sort of t.

Let D be a non empty set and let X be a subset of D. Then id_{X} is a function from X into D.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a non empty subset of S -Terms(V). The functor X-CircuitStr yields a non empty strict many sorted signature and is defined by the condition (Def. 1).

(Def. 1) X-CircuitStr = ⟨Subtrees(X), [: the operation symbols of S, {the carrier of S}:] -Subtrees(X), [: the operation symbols of S, {the carrier of S}:] -ImmediateSubtrees(X), id_{[: the operation symbols of S, {the carrier of S}:]-Subtrees (X)}⟩.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a non empty subset of S -Terms(V). Observe that X-CircuitStr is unsplit.

In the sequel S denotes a non empty non void many sorted signature, V denotes a non-empty many sorted set indexed by the carrier of S, A denotes a non-empty algebra over S, and X denotes a non empty subset of S -Terms(V).

The following propositions are true:

(2) X-CircuitStr is void if and only if for every element t of X holds t is root and t(∅) ∉ [: the operation symbols of S, {the carrier of S}:].

(3) X is a set with a compound term of S over V iff X-CircuitStr is non void.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a set with a compound term of S over V. One can check that X-CircuitStr is non void.

The following three propositions are true:

(4)(i) Every vertex of X-CircuitStr is a term of S over V, and

(ii) for every set s such that s ∈ the operation symbols of X-CircuitStr

holds s is a compound term of S over V.

(5) Let t be a vertex of X-CircuitStr. Then t ∈ the operation symbols of X-CircuitStr if and only if t is a compound term of S over V.

(6) Let X be a set with a compound term of S over V and g be a gate of X-CircuitStr. Then (the result sort of X-CircuitStr)(g) = g and the result sort of g = g.

Let us consider S, V, let X be a set with a compound term of S over V, and let g be a gate of X-CircuitStr. Note that Arity(g) is decorated tree yielding.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a non empty subset of S -Terms(V). Note that every vertex of X-CircuitStr is finite, function-like, and relation-like.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a non empty subset of S -Terms(V). One can verify that every vertex of X-CircuitStr is decorated tree-like.

Let S be a non empty non void many sorted signature, let V be a nonempty many sorted set indexed by the carrier of S, and let X be a set with a compound term of S over V. One can check that every gate of X-CircuitStr is finite, function-like, and relation-like.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a set with a compound term of S over V. Note that every gate of X-CircuitStr is decorated tree-like.

Next we state the proposition

(7) Let X_{1}, X_{2} be non empty subsets of S -Terms(V). Then the arity of X_{1}-CircuitStr ≈ the arity of X_{2}-CircuitStr and the result sort of X_{1}-CircuitStr ≈ the result sort of X_{2}-CircuitStr.

Let X, Y be constituted of decorated trees sets. Note that X ∪ Y is constituted of decorated trees.

One can prove the following propositions:

(8) For all constituted of decorated trees non empty sets X_{1}, X_{2} holds Subtrees(X_{1} ∪ X_{2}) = Subtrees(X_{1}) ∪ Subtrees(X_{2}).

(9) For all constituted of decorated trees non empty sets X_{1}, X_{2} and for every set C holds C -Subtrees(X_{1}∪X_{2}) = (C -Subtrees(X_{1}))∪(C -Subtrees(X_{2})).

(10) Let X_{1}, X_{2} be constituted of decorated trees non empty sets. If every element of X_{1} is finite and every element of X_{2} is finite, then for every set C holds C -ImmediateSubtrees(X_{1} ∪ X_{2}) = (C -ImmediateSubtrees(X_{1}))+·(C -ImmediateSubtrees(X_{2})).

(11) For all non empty subsets X_{1}, X_{2} of S -Terms(V) holds (X_{1} ∪ X_{2})-CircuitStr = (X_{1}-CircuitStr)+·(X_{2}-CircuitStr).

(12) Let x be a set. Then x ∈ InputVertices(X-CircuitStr) if and only if the

following conditions are satisfied:

(i) x ∈ Subtrees(X), and (ii) there exists a sort symbol s of S and there exists an element v of V (s)

such that x = the root tree of ⟨⟨v, s⟩⟩.

(13) For every set X with a compound term of S over V and for every gate g of X-CircuitStr holds g = g(∅)-tree(Arity(g)).

2. Circuit Generated by Terms

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), let v be a vertex of X-CircuitStr, and let A be an algebra over S. The sort of v w.r.t. A is defined as follows:

(Def. 2) For every term u of S over V such that u = v holds the sort of v w.r.t.

A = (the sorts of A)(the sort of u). Let S be a non empty non void many sorted signature, let V be a non-empty

many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), let v be a vertex of X-CircuitStr, and let A be a non-empty algebra over S. Note that the sort of v w.r.t. A is non empty.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let X be a non empty subset of S -Terms(V). Let us assume that X is a set with a compound term of S over V. Let o be a gate of X-CircuitStr and let A be an algebra over S. The action of o w.r.t A is a function and is defined by the condition (Def. 3).

(Def. 3) Let X′ be a set with a compound term of S over V. Suppose X′ = X. Let o′ be a gate of X′-CircuitStr. Suppose o′ = o. Then the action of o w.r.t A = (the characteristics of A)(o′(∅)_{1}).

The scheme MSFuncEx deals with a non empty set A, non-empty many sorted sets B, C indexed by A, and a ternary predicate P, and states that:

There exists a many sorted function f from B into C such that for every element i of A and for every element a of B(i) holds P[i, a, f(i)(a)]

provided the following requirement is met:

• For every element i of A and for every element a of B(i) there exists an element b of C(i) such that P[i, a, b].

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), and let A be an algebra over S. The functor X-CircuitSorts(A) yielding a many sorted set indexed by the carrier of X-CircuitStr is defined by:

(Def. 4) For every vertex v of X-CircuitStr holds (X-CircuitSorts(A))(v) = the

sort of v w.r.t. A. Let S be a non empty non void many sorted signature, let V be a non-

empty many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), and let A be a non-empty algebra over S. Note that X-CircuitSorts(A) is non-empty.

We now state the proposition

(14) Let X be a set with a compound term of S over V, g be a gate of X-CircuitStr, and o be an operation symbol of S. If g(∅) = ⟨⟨o, the carrier of S⟩⟩, then (X-CircuitSorts(A)) · Arity(g) = (the sorts of A) · Arity(o).

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), and let A be a non-empty algebra over S. The functor X-CircuitCharact(A) yields a many sorted function from (X-CircuitSorts(A))^{#} · the arity of X-CircuitStr into (X-CircuitSorts(A)) · the result sort of X-CircuitStr and is defined by:

(Def. 5) For every gate g of X-CircuitStr such that g ∈ the operation symbols of

X-CircuitStr holds (X-CircuitCharact(A))(g) = the action of g w.r.t A. Let S be a non empty non void many sorted signature, let V be a non-

empty many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), and let A be a non-empty algebra over S. The functor X-Circuit(A) yielding a non-empty strict algebra over X-CircuitStr is defined by:

(Def. 6) X-Circuit(A) = ⟨X-CircuitSorts(A), X-CircuitCharact(A)⟩.

Next we state four propositions:

(15) For every vertex v of X-CircuitStr holds (the sorts of X-Circuit(A))(v) = the sort of v w.r.t. A.

(16) Let A be a locally-finite non-empty algebra over S, X be a set with a compound term of S over V, and g be an operation symbol of X-CircuitStr. Then Den(g, X-Circuit(A)) = the action of g w.r.t A.

(17) Let A be a locally-finite non-empty algebra over S, X be a set with a compound term of S over V, g be an operation symbol of X-CircuitStr, and o be an operation symbol of S. If g(∅) = ⟨⟨o, the carrier of S⟩⟩, then Den(g, X-Circuit(A)) = Den(o, A).

(18) Let A be a locally-finite non-empty algebra over S and X be a non empty

subset of S -Terms(V). Then X-Circuit(A) is locally-finite. Let S be a non empty non void many sorted signature, let V be a non-empty

many sorted set indexed by the carrier of S, let X be a set with a compound term of S over V, and let A be a locally-finite non-empty algebra over S. Note that X-Circuit(A) is locally-finite.

The following two propositions are true:

(19) Let S be a non empty non void many sorted signature, V be a nonempty many sorted set indexed by the carrier of S, X_{1}, X_{2} be sets with compound terms of S over V, and A be a non-empty algebra over S. Then X_{1}-Circuit(A) ≈ X_{2}-Circuit(A).

(20) Let S be a non empty non void many sorted signature, V be a nonempty many sorted set indexed by the carrier of S, X_{1}, X_{2} be sets with compound terms of S over V, and A be a non-empty algebra over S. Then (X_{1} ∪ X_{2})-Circuit(A) = (X_{1}-Circuit(A))+·(X_{2}-Circuit(A)).

3. Correctness of a Term Circuit

In the sequel S is a non empty non void many sorted signature, A is a nonempty locally-finite algebra over S, V is a variables family of A, and X is a set with a compound term of S over V.

Let S be a non empty non void many sorted signature, let A be a non-empty algebra over S, let V be a variables family of A, and let t be a decorated tree. Let us assume that t is a term of S over V. Let f be a many sorted function from V into the sorts of A. The functor [[t]]_{A}(f) is defined by:

(Def. 7) There exists a term t′ of A over V such that t′= t and [[t]]_{A}(f) = t′^{@} f.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, let X be a set with a compound term of S over V, let A be a non-empty locally-finite algebra over S, and let s be a state of X-Circuit(A). A many sorted function from V into the sorts of A is said to be a valuation compatible with s if it satisfies the condition (Def. 8).

(Def. 8) Let x be a vertex of S and v be an element of V (x). If the root tree of

⟨⟨v, x⟩⟩∈ Subtrees(X), then it(x)(v) = s(the root tree of ⟨⟨v, x⟩⟩). Next we state the proposition

(21) Let s be a state of X-Circuit(A), f be a valuation compatible with s, and n be a natural number. Then f is a valuation compatible with Following(s, n).

Let x be a set, let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let p be a finite sequence of elements of S -Terms(V). One can verify that x-tree(p) is finite.

The following propositions are true:

(22) Let s be a state of X-Circuit(A), f be a valuation compatible with s, and t be a term of S over V. If t ∈ Subtrees(X), then Following(s, 1 + heightdomt) is stable at t and (Following(s, 1 + heightdomt))(t) = [[t]]_{A}(f).

(23) Suppose that it is not true that there exists a term t of S over V and there exists an operation symbol o of S such that t ∈ Subtrees(X) and t(∅) = ⟨⟨o, the carrier of S⟩⟩ and Arity(o) = ∅. Let s be a state of X-Circuit(A), f be a valuation compatible with s, and t be a term of S over V. If t ∈ Subtrees(X), then Following(s, heightdomt) is stable at t and (Following(s, heightdomt))(t) = [[t]]_{A}(f).

4. Circuit Similarity

Let X be a set. One can verify that id_{X} is one-to-one. Let f be an one-to-one function. One can verify that f^{−1} is one-to-one. Let

g be an one-to-one function. Note that g · f is one-to-one.

Let S_{1}, S_{2} be non empty many sorted signatures and let f, g be functions. We say that S_{1} and S_{2} are equivalent w.r.t. f and g if and only if the conditions (Def. 9) are satisfied.

(Def. 9)(i) f is one-to-one, (ii) g is one-to-one, (iii) f and g form morphism between S_{1} and S_{2}, and (iv) f^{−1} and g^{−1} form morphism between S_{2} and S_{1}. One can prove the following propositions:

(24) Let S_{1}, S_{2} be non empty many sorted signatures and f, g be functions. Suppose S_{1} and S_{2} are equivalent w.r.t. f and g. Then the carrier of S_{2} = f^{∘}(the carrier of S_{1}) and the operation symbols of S_{2} = g^{∘}(the operation symbols of S_{1}).

(25) Let S_{1}, S_{2} be non empty many sorted signatures and f, g be functions. Suppose S_{1} and S_{2} are equivalent w.r.t. f and g. Then rngf = the carrier of S_{2} and rngg = the operation symbols of S_{2}.

(26) Let S be a non empty many sorted signature. Then S and S are equivalent w.r.t. id_{the carrier of S} and id_{the operation symbols of S}.

(27) Let S_{1}, S_{2} be non empty many sorted signatures and f, g be functions. Suppose S_{1} and S_{2} are equivalent w.r.t. f and g. Then S_{2} and S_{1} are equivalent w.r.t. f^{−1} and g^{−1} .

(28) Let S_{1}, S_{2}, S_{3} be non empty many sorted signatures and f_{1}, g_{1}, f_{2}, g_{2} be functions. Suppose S_{1} and S_{2} are equivalent w.r.t. f_{1} and g_{1} and S_{2} and S_{3} are equivalent w.r.t. f_{2} and g_{2}. Then S_{1} and S_{3} are equivalent w.r.t. f_{2} · f_{1} and g_{2} · g_{1}.

(29) Let S_{1}, S_{2} be non empty many sorted signatures and f, g be functions. Suppose S_{1} and S_{2} are equivalent w.r.t. f and g. Then f^{∘} InputVertices(S_{1}) = InputVertices(S_{2}) and f^{∘} InnerVertices(S_{1}) = InnerVertices(S_{2}).

Let S_{1}, S_{2} be non empty many sorted signatures. We say that S_{1} and S_{2} are equivalent if and only if:

(Def. 10) There exist one-to-one functions f, g such that S_{1} and S_{2} are equivalent

w.r.t. f and g. Let us notice that the predicate S_{1} and S_{2} are equivalent is reflexive and symmetric.

One can prove the following proposition

(30) Let S_{1}, S_{2}, S_{3} be non empty many sorted signatures. Suppose S_{1} and S_{2}

are equivalent and S_{2} and S_{3} are equivalent. Then S_{1} and S_{3} are equivalent. Let S_{1}, S_{2} be non empty many sorted signatures and let f be a function.

We say that f preserves inputs of S_{1} in S_{2} if and only if: (Def. 11) f^{∘} InputVertices(S_{1}) ⊆ InputVertices(S_{2}).

Next we state four propositions:

(31) Let S_{1}, S_{2} be non empty many sorted signatures and f, g be functions. Suppose f and g form morphism between S_{1} and S_{2}. Let v be a vertex of S_{1}. Then f(v) is a vertex of S_{2}.

(32) Let S_{1}, S_{2} be non empty non void many sorted signatures and f, g be functions. Suppose f and g form morphism between S_{1} and S_{2}. Let v be a gate of S_{1}. Then g(v) is a gate of S_{2}.

(33) Let S_{1}, S_{2} be non empty many sorted signatures and f, g be functions. If f and g form morphism between S_{1} and S_{2}, then f^{∘} InnerVertices(S_{1}) ⊆ InnerVertices(S_{2}).

(34) Let S_{1}, S_{2} be circuit-like non void non empty many sorted signatures and f, g be functions. Suppose f and g form morphism between S_{1} and S_{2}. Let v_{1} be a vertex of S_{1}. Suppose v_{1} ∈ InnerVertices(S_{1}). Let v_{2} be a vertex of S_{2}. If v_{2} = f(v_{1}), then the action at v_{2} = g(the action at v_{1}).

Let S_{1}, S_{2} be non empty many sorted signatures, let f, g be functions, let C_{1} be a non-empty algebra over S_{1}, and let C_{2} be a non-empty algebra over S_{2}. We say that f and g form embedding of C_{1} into C_{2} if and only if the conditions (Def. 12) are satisfied.

(Def. 12)(i) f is one-to-one, (ii) g is one-to-one, (iii) f and g form morphism between S_{1} and S_{2}, (iv) the sorts of C_{1} = (the sorts of C_{2}) · f, and

(v) the characteristics of C_{1} = (the characteristics of C_{2}) · g. The following propositions are true:

(35) Let S be a non empty many sorted signature and C be a non-empty algebra over S. Then id_{the carrier of S} and id_{the operation symbols of S} form embedding of C into C.

(36) Let S_{1}, S_{2}, S_{3} be non empty many sorted signatures, f_{1}, g_{1}, f_{2}, g_{2} be functions, C_{1} be a non-empty algebra over S_{1}, C_{2} be a non-empty algebra over S_{2}, and C_{3} be a non-empty algebra over S_{3}. Suppose f_{1} and g_{1} form embedding of C_{1} into C_{2} and f_{2} and g_{2} form embedding of C_{2} into C_{3}. Then f_{2} · f_{1} and g_{2} · g_{1} form embedding of C_{1} into C_{3}.

Let S_{1}, S_{2} be non empty many sorted signatures, let f, g be functions, let C_{1} be a non-empty algebra over S_{1}, and let C_{2} be a non-empty algebra over S_{2}.

We say that C_{1} and C_{2} are similar w.r.t. f and g if and only if: (Def. 13) f and g form embedding of C_{1} into C_{2} and f^{−1} and g^{−1} form embedding

of C_{2} into C_{1}. The following propositions are true:

(37) Let S_{1}, S_{2} be non empty many sorted signatures, f, g be functions, C_{1} be a non-empty algebra over S_{1}, and C_{2} be a non-empty algebra over S_{2}. Suppose C_{1} and C_{2} are similar w.r.t. f and g. Then S_{1} and S_{2} are equivalent w.r.t. f and g.

(38) Let S_{1}, S_{2} be non empty many sorted signatures, f, g be functions, C_{1} be a non-empty algebra over S_{1}, and C_{2} be a non-empty algebra over S_{2}. Then C_{1} and C_{2} are similar w.r.t. f and g if and only if the following conditions are satisfied:

(i) S_{1} and S_{2} are equivalent w.r.t. f and g, (ii) the sorts of C_{1} = (the sorts of C_{2}) · f, and (iii) the characteristics of C_{1} = (the characteristics of C_{2}) · g.

(39) Let S be a non empty many sorted signature and C be a non-empty algebra over S. Then C and C are similar w.r.t. id_{the carrier of S} and id_{the operation symbols of S}.

(40) Let S_{1}, S_{2} be non empty many sorted signatures, f, g be functions, C_{1} be a non-empty algebra over S_{1}, and C_{2} be a non-empty algebra over S_{2}. Suppose C_{1} and C_{2} are similar w.r.t. f and g. Then C_{2} and C_{1} are similar w.r.t. f^{−1} and g^{−1} .

(41) Let S_{1}, S_{2}, S_{3} be non empty many sorted signatures, f_{1}, g_{1}, f_{2}, g_{2} be functions, C_{1} be a non-empty algebra over S_{1}, C_{2} be a non-empty algebra over S_{2}, and C_{3} be a non-empty algebra over S_{3}. Suppose C_{1} and C_{2} are similar w.r.t. f_{1} and g_{1} and C_{2} and C_{3} are similar w.r.t. f_{2} and g_{2}. Then C_{1} and C_{3} are similar w.r.t. f_{2} · f_{1} and g_{2} · g_{1}.

Let S_{1}, S_{2} be non empty many sorted signatures, let C_{1} be a non-empty algebra over S_{1}, and let C_{2} be a non-empty algebra over S_{2}. We say that C_{1} and C_{2} are similar if and only if:

(Def. 14) There exist functions f, g such that C_{1} and C_{2} are similar w.r.t. f and

g. For simplicity, we use the following convention: G_{1}, G_{2} denote circuit-like

non void non empty many sorted signatures, f, g denote functions, C_{1} denotes a non-empty circuit of G_{1}, and C_{2} denotes a non-empty circuit of G_{2}.

Next we state a number of propositions:

(42) Suppose f and g form embedding of C_{1} into C_{2}. Then (i) domf = the carrier of G_{1}, (ii) rngf ⊆ the carrier of G_{2}, (iii) domg = the operation symbols of G_{1}, and (iv) rngg ⊆ the operation symbols of G_{2}.

(43) Suppose f and g form embedding of C_{1} into C_{2}. Let o_{1} be a gate of G_{1} and o_{2} be a gate of G_{2}. If o_{2} = g(o_{1}), then Den(o_{2}, C_{2}) = Den(o_{1}, C_{1}).

(44) Suppose f and g form embedding of C_{1} into C_{2}. Let o_{1} be a gate of G_{1} and o_{2} be a gate of G_{2}. Suppose o_{2} = g(o_{1}). Let s_{1} be a state of C_{1} and s_{2} be a state of C_{2}. If s_{1} = s_{2} · f, then o_{2} depends-on-ins_{2} = o_{1} depends-on-ins_{1}.

(45) If f and g form embedding of C_{1} into C_{2}, then for every state s of C_{2} holds s · f is a state of C_{1}.

(46) Suppose f and g form embedding of C_{1} into C_{2}. Let s_{2} be a state of C_{2} and s_{1} be a state of C_{1}. Suppose s_{1} = s_{2} · f and for every vertex v of G_{1} such that v ∈ InputVertices(G_{1}) holds s_{2} is stable at f(v). Then Following(s_{1}) = Following(s_{2}) · f.

(47) Suppose f and g form embedding of C_{1} into C_{2} and f preserves inputs of G_{1} in G_{2}. Let s_{2} be a state of C_{2} and s_{1} be a state of C_{1}. If s_{1} = s_{2} · f, then Following(s_{1}) = Following(s_{2}) · f.

(48) Suppose f and g form embedding of C_{1} into C_{2} and f preserves inputs of G_{1} in G_{2}. Let s_{2} be a state of C_{2} and s_{1} be a state of C_{1}. If s_{1} = s_{2} ·f, then for every natural number n holds Following(s_{1}, n) = Following(s_{2}, n) · f.

(49) Suppose f and g form embedding of C_{1} into C_{2} and f preserves inputs of G_{1} in G_{2}. Let s_{2} be a state of C_{2} and s_{1} be a state of C_{1}. If s_{1} = s_{2} · f, then if s_{2} is stable, then s_{1} is stable.

(50) Suppose f and g form embedding of C_{1} into C_{2} and f preserves inputs of G_{1} in G_{2}. Let s_{2} be a state of C_{2} and s_{1} be a state of C_{1}. Suppose s_{1} = s_{2} · f. Let v_{1} be a vertex of G_{1}. Then s_{1} is stable at v_{1} if and only if s_{2} is stable at f(v_{1}).

(51) If C_{1} and C_{2} are similar w.r.t. f and g, then for every state s of C_{2} holds s · f is a state of C_{1}.

(52) Suppose C_{1} and C_{2} are similar w.r.t. f and g. Let s_{1} be a state of C_{1} and s_{2} be a state of C_{2}. Then s_{1} = s_{2} · f if and only if s_{2} = s_{1} · f^{−1} .

(53) If C_{1} and C_{2} are similar w.r.t. f and g, then f^{∘} InputVertices(G_{1}) = InputVertices(G_{2}) and f^{∘} InnerVertices(G_{1}) = InnerVertices(G_{2}).

(54) If C_{1} and C_{2} are similar w.r.t. f and g, then f preserves inputs of G_{1} in G_{2}.

(55) Suppose C_{1} and C_{2} are similar w.r.t. f and g. Let s_{1} be a state of C_{1} and s_{2} be a state of C_{2}. If s_{1} = s_{2} · f, then Following(s_{1}) = Following(s_{2}) · f.

(56) Suppose C_{1} and C_{2} are similar w.r.t. f and g. Let s_{1} be a state of C_{1} and s_{2} be a state of C_{2}. If s_{1} = s_{2} · f, then for every natural number n holds Following(s_{1}, n) = Following(s_{2}, n) · f.

(57) Suppose C_{1} and C_{2} are similar w.r.t. f and g. Let s_{1} be a state of C_{1} and s_{2} be a state of C_{2}. If s_{1} = s_{2} · f, then s_{1} is stable iff s_{2} is stable.

(58) Suppose C_{1} and C_{2} are similar w.r.t. f and g. Let s_{1} be a state of C_{1} and s_{2} be a state of C_{2}. Suppose s_{1} = s_{2} · f. Let v_{1} be a vertex of G_{1}. Then s_{1} is stable at v_{1} if and only if s_{2} is stable at f(v_{1}).

5. Term Specification

Let S be a non empty non void many sorted signature, let A be a non-empty algebra over S, let V be a non-empty many sorted set indexed by the carrier of S, let X be a non empty subset of S -Terms(V), let G be a circuit-like non void non empty many sorted signature, and let C be a non-empty circuit of G. We say that C calculates X in A if and only if:

(Def. 15) There exist f, g such that f and g form embedding of X-Circuit(A) into

C and f preserves inputs of X-CircuitStr in G. We say that X and A specify C if and only if:

(Def. 16) C and X-Circuit(A) are similar.

Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, let A be a non-empty algebra over S, let X be a non empty subset of S -Terms(V), let G be a circuit-like non void non empty many sorted signature, and let C be a non-empty circuit of G. Let us assume that C calculates X in A. An one-to-one function is said to be a sort map from X and A into C if:

(Def. 17) It preserves inputs of X-CircuitStr in G and there exists g such that it

and g form embedding of X-Circuit(A) into C. Let S be a non empty non void many sorted signature, let V be a non-empty

many sorted set indexed by the carrier of S, let A be a non-empty algebra over S, let X be a non empty subset of S -Terms(V), let G be a circuit-like non void non empty many sorted signature, and let C be a non-empty circuit of G. Let us assume that C calculates X in A. Let f be a sort map from X and A into C. An one-to-one function is said to be an operation map from X and A into C obeying f if:

(Def. 18) f and it form embedding of X-Circuit(A) into C.

The following propositions are true:

(59) Let G be a circuit-like non void non empty many sorted signature and C be a non-empty circuit of G. If X and A specify C, then C calculates X in A.

(60) Let G be a circuit-like non void non empty many sorted signature and C be a non-empty circuit of G. Suppose C calculates X in A. Let f be a sort map from X and A into C and t be a term of S over V. Suppose t ∈ Subtrees(X). Let s be a state of C. Then

(i) Following(s, 1 + heightdomt) is stable at f(t), and (ii) for every state s′ of X-Circuit(A) such that s′= s · f and for every va-

luation h compatible with s′holds (Following(s, 1+heightdomt))(f(t)) = [[t]]_{A}(h).

(61) Let G be a circuit-like non void non empty many sorted signature and C be a non-empty circuit of G. Suppose C calculates X in A. Let t be a term of S over V. Suppose t ∈ Subtrees(X). Then there exists a vertex v of G such that for every state s of C holds

(i) Following(s, 1 + heightdomt) is stable at v, and (ii) there exists a sort map f from X and A into C such that for every

state s′ of X-Circuit(A) such that s′ = s · f and for every valuation h compatible with s′ holds (Following(s, 1 + heightdomt))(v) = [[t]]_{A}(h).

(62) Let G be a circuit-like non void non empty many sorted signature and C be a non-empty circuit of G. Suppose X and A specify C. Let f be a sort map from X and A into C, s be a state of C, and t be a term of S over V. Suppose t ∈ Subtrees(X). Then

(i) Following(s, 1 + heightdomt) is stable at f(t), and (ii) for every state s′ of X-Circuit(A) such that s′= s · f and for every va-

luation h compatible with s′holds (Following(s, 1+heightdomt))(f(t)) = [[t]]_{A}(h).

(63) Let G be a circuit-like non void non empty many sorted signature and C be a non-empty circuit of G. Suppose X and A specify C. Let t be a term of S over V. Suppose t ∈ Subtrees(X). Then there exists a vertex v of G such that for every state s of C holds

(i) Following(s, 1 + heightdomt) is stable at v, and (ii) there exists a sort map f from X and A into C such that for every

state s′ of X-Circuit(A) such that s′ = s · f and for every valuation h compatible with s′ holds (Following(s, 1 + heightdomt))(v) = [[t]]_{A}(h).

References

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

[2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

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

[4] Grzegorz Bancerek. The reflection theorem. Formalized Mathematics, 1(5):973–977, 1990.

[5] Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195–204, 1992.

[6] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77–82, 1993.

[7] Grzegorz Bancerek. Minimal signature for partial algebra. Formalized Mathematics, 5(3):405–414, 1996.

[8] Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185–190, 1996.

circuit generated by terms and circuit … 657

[9] Grzegorz Bancerek. Terms over many sorted universal algebra. Formalized Mathematics, 5(2):191–198, 1996.

[10] Grzegorz Bancerek and Yatsuka Nakamura. Full adder circuit. Part I. Formalized Mathematics, 5(3):367–380, 1996.

[11] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91–101, 1993.

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

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

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

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

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

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

[18] Yatsuka Nakamura and Grzegorz Bancerek. Combining of circuits. Formalized Mathematics, 5(2):283–295, 1996.

[19] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Introduction to circuits, I. Formalized Mathematics, 5(2):227–232, 1996.

[20] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Introduction to circuits, II. Formalized Mathematics, 5(2):273–278, 1996.

[21] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, II. Formalized Mathematics, 5(2):215–220, 1996.

[22] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67– 74, 1996.

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

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

[25] Andrzej Trybulec. 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.

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

Received April 10, 2001

Index of MML Identifiers

BINOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 CARD LAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .485 CIRCTRM1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 COMPTRIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 DYNKIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591 EXTREAL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 EXTREAL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 FINSEQ 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .471 FINTOPO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 FUZZY 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 GATE 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 HEYTING3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 HILBASIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 HILBERT3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 IDEAL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 JCT MISC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .481 JORDAN1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 JORDAN1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 JORDAN1C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 JORDAN1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 MESFUNC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 MESFUNC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 PARTIT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .623 POLYALG1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 POLYNOM5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 SPRECT 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .549 TAXONOM1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 URYSOHN2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 URYSOHN3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 YELLOW18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Contents Formaliz. Math. 9 (3)

The Canonical Formulae

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

The Incompleteness of the Lattice of Substitutions

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Trigonometric Form of Complex Numbers

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Fundamental Theorem of Algebra

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

On Replace Function and Swap Function for Finite Sequences

By Hiroshi Yamazaki et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

The Correctness of the High Speed Array Multiplier Circuits

By Hiroshi Yamazaki and Katsumi Wasaki  . . . . . . . . . . . . . . . . 475

Some Lemmas for the Jordan Curve Theorem

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Mahlo and Inaccessible Cardinals

By Josef Urban  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Basic Properties of Extended Real Numbers

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Definitions and Basic Properties of Measurable Functions

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Gauges and Cages. Part I

By Artur Korniłowicz et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Some Properties of Extended Real Numbers Operations: abs, min

and max

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

The Concept of Fuzzy Relation and Basic Properties of its Ope-

ration

By Takashi Mitsuishi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Continued on inside back cover

The Measurability of Extended Real Valued Functions

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Some Properties of Cells and Arcs

By Robert Milewski et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Formal Topological Spaces

By Gang Liu et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Some Properties of Cells and Gauges

By Adam Grabowski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

Again on the Order on a Special Polygon

By Andrzej Trybulec and Yatsuka Nakamura  . . . . . . . . . . . 549

Gauges and Cages. Part II

By Artur Korniłowicz and Robert Milewski  . . . . . . . . . . . . 555

The Binomial Theorem for Algebraic Structures

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

Ring Ideals

By Jonathan Backer et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

Hilbert Basis Theorem

By Jonathan Backer and Piotr Rudnicki  . . . . . . . . . . . . . . . . . 583

Dynkin’s Lemma in Measure Theory

By Franz Merkl  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Lower Tolerance. Preliminaries to Wroclaw Taxonomy

By Mariusz Giero and Roman Matuszewski  . . . . . . . . . . . . . . 597

Concrete Categories

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Classes of Independent Partitions

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

Some Properties of Dyadic Numbers and Intervals

By Józef Białas and Yatsuka Nakamura  . . . . . . . . . . . . . . . . . 627

The Urysohn Lemma

By Józef Białas and Yatsuka Nakamura  . . . . . . . . . . . . . . . . . 631

The Algebra of Polynomials

By Ewa Grądzka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

Circuit Generated by Terms and Circuit Calculating Terms

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658