FORMALIZED MATHEMATICS Vol.1, No.3, May–August 1990 Université Catholique de Louvain

Zermelo’s Theorem ^{1}

Bogdan Nowak Łódź University Sławomir Białecki Łódź University

Summary. The article contains direct proof of Zermelo’s theorem about the existence of a well ordering for any set and the lemma the proof depends on.

MML Identifier: WELLSET1.

The articles [4], [3], [5], [2], and [1] provide the notation and terminology for this paper. For simplicity we follow the rules: a, x, y will be arbitrary, B, D, N, X, Y will denote sets, R, S, T will denote relations, F will denote a function, and W will denote a relation. We now state several propositions:

(1) x ∈ field R if and only if there exists y such that ⟨⟨x, y⟩⟩ ∈ R or ⟨⟨y, x⟩⟩ ∈ R.

(2) R ∪ S is a relation.

(3) If X ≠ ∅ and Y ≠ ∅ and W = [: X, Y :], then field W = X ∪ Y.

(4) If y = R, then y is a relation.

(5) For all a, T holds x ∈ T −Seg(a) if and only if x ≠ a and ⟨⟨x, a⟩⟩ ∈ T. In the article we present several logical schemes. The scheme R Separation

deals with a set A, and a unary predicate P, and states that:

there exists B such that for every relation R holds R ∈ B if and only if R ∈ A and P[R] for all values of the parameters.

The scheme S Separation deals with a set A, and a unary predicate P, and states that:

there exists B such that for every set X holds X ∈ B if and only if X ∈ A and P[X] for all values of the parameters.

The following four propositions are true:

(6) For all x, y, W such that x ∈ field W and y ∈ field W and W is well ordering relation holds if x ∉ W −Seg(y), then ⟨⟨y, x⟩⟩ ∈ W.

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(7) For all x, y, W such that x ∈ field W and y ∈ field W and W is well ordering relation holds if x ∈ W −Seg(y), then ⟨⟨y, x⟩⟩ ∉ W.

(8) Given F, D. Suppose for every X such that X ∈ D holds F (X) ∉ X and F (X) ∈ ⋃ D. Then there exists R such that field R ⊆ ⋃ D and R is well ordering relation and field R ∉ D and for every y such that y ∈ field R holds R−Seg(y) ∈ D and F (R−Seg(y)) = y.

(9) For every N there exists R such that R is well ordering relation and field R = N.

References

[1] Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.

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

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

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

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

Group and Field Definitions

Józef Białas^{1} Łódź University

Summary. The article contains exactly the same definitions of group and field as those in [3]. These definitions were prepared without the help of the definitions and properties of Nat and Real modes icluded in the MML. This is the first of a series of articles in which we are going to introduce the concept of the set of real numbers in a elementary axiomatic way.

MML Identifier: REALSET1.

The terminology and notation used here are introduced in the following papers: [4], [1], and [2]. Let x be arbitrary. The functor single(x) yields a set and is defined as follows:

single(x) = {x}. One can prove the following proposition

(1) For arbitrary x holds single(x) = {x}. Let X, Y be sets. The functor X#Y yields a set and is defined by: X#Y = [: X, Y :]. We now state several propositions:

(2) For all sets X, Y holds X#Y = [: X, Y :].

(3) For arbitrary z and for every set A holds z ∈ A#A if and only if there exist arbitrary x, y such that x ∈ A and y ∈ A and z = ⟨⟨x, y⟩⟩.

(4) For every set X and for every subset A of X holds A#A ⊆ X#X.

(5) For every set X such that X = ∅ holds X#X = ∅.

(6) For every set X such that X#X = ∅ holds X = ∅.

(7) For every set X holds X#X = ∅ if and only if X = ∅. Let X be a set. A binary operation of X is a function from X#X into X. The following propositions are true:

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(8) For every set X and for every function F from X#X into X holds F is a binary operation of X.

(9) For every set X and for every function F holds F is a function from X#X into X if and only if F is a binary operation of X.

(10) For every set X and for every function F from X#X into X and for arbitrary x such that x ∈ X#X holds F (x) ∈ X.

(11) For every set X and for every binary operation F of X there exists a subset A of X such that for arbitrary x such that x ∈ A#A holds F (x) ∈ A.

Let X be a set, and let F be a binary operation of X, and let A be a subset of X. We say that F is in A if and only if:

for arbitrary x such that x ∈ A#A holds F (x) ∈ A. Next we state a proposition

(12) For every set X and for every binary operation F of X and for every subset A of X holds F is in A if and only if for arbitrary x such that x ∈ A#A holds F (x) ∈ A.

Let X be a set, and let F be a binary operation of X. A subset of X is said to be a set closed w.r.t. F if:

for arbitrary x such that x ∈ it#it holds F (x) ∈ it. The following propositions are true:

(13) For every set X and for every binary operation F of X and for every subset A of X holds A is a set closed w.r.t. F if and only if for arbitrary x such that x ∈ A#A holds F (x) ∈ A.

(14) For every set X and for every binary operation F of X and for every

set A closed w.r.t. F holds F (A#A) is a binary operation of A. Let X be a set, and let F be a binary operation of X, and let A be a set

closed w.r.t. F. The functor F A yielding a binary operation of A, is defined by:

F A = F (A#A). The following propositions are true:

(15) For every set X and for every binary operation F of X and for every set A closed w.r.t. F holds F A = F (A#A).

(16) For every set X and for every binary operation F of X and for every subset A of X such that A is a set closed w.r.t. F holds F (A#A) is a binary operation of A.

(17) For every set X and for every binary operation F of X and for every

set A closed w.r.t. F holds F A is a binary operation of A. We consider group structures which are systems ⟨ a carrier, an addition, a zero ⟩ where the carrier is a non-empty set, the addition is a binary operation of

Group and Field Definitions 435

the carrier, and the zero is an element of the carrier. Let A be a non-empty set, and let og be a binary operation of A, and let ng be an element of A. The functor group(A, og, ng) yielding a group structure, is defined as follows:

A = the carrier of group(A, og, ng) and og = the addition of group(A, og, ng) and ng = the zero of group(A, og, ng).

The following propositions are true:

(18) For every non-empty set A and for every binary operation og of A and for every element ng of A and for every GR being a group structure holds GR = group(A, og, ng) if and only if A = the carrier of GR and og = the addition of GR and ng = the zero of GR.

(19) For every non-empty set A and for every binary operation og of A and for every element ng of A holds group(A, og, ng) is a group structure and A = the carrier of group(A, og, ng) and og = the addition of group(A, og, ng) and ng = the zero of group(A, og, ng).

A group structure is called a group if: there exists a non-empty set A and there exists a binary operation og of A

and there exists an element ng of A such that it = group(A, og, ng) and for all elements a, b, c of A holds og(⟨⟨og(⟨⟨a, b⟩⟩), c⟩⟩) = og(⟨⟨a, og(⟨⟨b, c⟩⟩)⟩⟩) and for every element a of A holds og(⟨⟨a, ng⟩⟩) = a and og(⟨⟨ng, a⟩⟩) = a and for every element a of A there exists an element b of A such that og(⟨⟨a, b⟩⟩) = ng and og(⟨⟨b, a⟩⟩) = ng and for all elements a, b of A holds og(⟨⟨a, b⟩⟩) = og(⟨⟨b, a⟩⟩).

Let D be a group. The carrier of D yields a non-empty set and is defined as follows:

there exists a binary operation od of the carrier of D and there exists an element nd of the carrier of D such that D = group(the carrier of D, od, nd).

The following two propositions are true:

(20) For every group D and for every non-empty set A holds A = the carrier of D if and only if there exists a binary operation od of A and there exists an element nd of A such that D = group(A, od, nd).

(21) For every group D holds the carrier of D is a non-empty set and there exists a binary operation od of the carrier of D and there exists an element nd of the carrier of D such that D = group(the carrier of D, od, nd).

Let D be a group. The functor +_{D} yielding a binary operation of the carrier of D, is defined as follows:

there exists an element nd of the carrier of D such that D = group(the carrier of D, +_{D}, nd) . The following propositions are true:

(22) For every group D and for every binary operation od of the carrier of D holds od = +_{D} if and only if there exists an element nd of the carrier of D such that D = group(the carrier of D, od, nd).

(23) For every group D holds +_{D} is a binary operation of the carrier of D and there exists an element nd of the carrier of D such that D = group(the carrier of D, +_{D}, nd) .

Let D be a group. The functor 0_{D} yielding an element of the carrier of D, is defined by:

D = group(the carrier of D, +_{D}, 0_{D}).

Next we state a number of propositions:

(24) For every group D and for every element ng of the carrier of D holds ng = 0_{D} if and only if D = group(the carrier of D, +_{D}, ng).

(25) For every group D holds 0_{D} is an element of the carrier of D and D = group(the carrier of D, +_{D}, 0_{D}).

(26) For every group D holds D = group(the carrier of D, +_{D}, 0_{D}).

(27) For every group D and for every non-empty set A and for every binary operation og of A and for every element ng of A such that D = group(A, og, ng) holds the carrier of D = A and +_{D} = og and 0_{D} = ng.

(28) For every group D and for all elements a, b, c of the carrier of D holds +_{D}(⟨⟨+_{D}(⟨⟨a, b⟩⟩), c⟩⟩) = +_{D}(⟨⟨a, +_{D}(⟨⟨b, c⟩⟩)⟩⟩).

(29) For every group D and for every element a of the carrier of D holds +_{D}(⟨⟨a, 0_{D}⟩⟩) = a and +_{D}(⟨⟨0_{D}, a⟩⟩) = a.

(30) For every group D and for every element a of the carrier of D there exists an element b of the carrier of D such that +_{D}(⟨⟨a, b⟩⟩) = 0_{D} and +_{D}(⟨⟨b, a⟩⟩) = 0_{D}.

(31) For every group D and for all elements a, b of the carrier of D holds +_{D}(⟨⟨a, b⟩⟩) = +_{D}(⟨⟨b, a⟩⟩).

(32) There exist arbitrary x, y such that x ≠ y.

(33) There exists a non-empty set A such that for every element z of A holds

A \ single(z) is a non-empty set. A non-empty set is said to be an at least 2-elements set if: for every element x of it holds it \ single(x) is a non-empty set. We now state two propositions:

(34) For every non-empty set A holds A is an at least 2-elements set if and only if for every element x of A holds A \ single(x) is a non-empty set.

(35) For every non-empty set A such that for every element x of A holds

A \ single(x) is a non-empty set holds A is an at least 2-elements set. We consider field structures which are systems ⟨ a carrier, an addition, a multiplication, a zero, a unit ⟩ where the carrier is an at least 2-elements set, the addition is a binary oper-

ation of the carrier, the multiplication is a binary operation of the carrier, the zero is an element of the carrier, and the unit is an element of the carrier. Let A be an at least 2-elements set, and let od, om be binary operations of A, and let nd be an element of A, and let nm be an element of A \ single(nd). The functor field(A, od, om, nd, nm) yielding a field structure, is defined as follows:

A = the carrier of field(A, od, om, nd, nm) and od = the addition of field(A, od, om, nd, nm) and om = the multiplication of field(A, od, om, nd, nm) and nd = the zero of field(A, od, om, nd, nm) and nm = the unit of field(A, od, om, nd, nm).

We now state two propositions:

(36) Let A be an at least 2-elements set. Let od, om be binary operations of A. Then for every element nd of A and for every element nm of A \ single(nd) and for every F being a field structure holds F = field(A, od, om, nd, nm) if and only if A = the carrier of F and od = the addition of F and om = the multiplication of F and nd = the zero of F and nm = the unit of F.

(37) Let A be an at least 2-elements set. Let od, om be binary operations of A. Let nd be an element of A. Let nm be an element of A \ single(nd). Then

(i) field(A, od, om, nd, nm) is a field structure, (ii) A = the carrier of field(A, od, om, nd, nm), (iii) od = the addition of field(A, od, om, nd, nm), (iv) om = the multiplication of field(A, od, om, nd, nm),

(v) nd = the zero of field(A, od, om, nd, nm), (vi) nm = the unit of field(A, od, om, nd, nm). Let X be an at least 2-elements set, and let F be a binary operation of X,

and let x be an element of X. We say that F is binary operation preserving x if and only if:

X \single(x) is a set closed w.r.t. F and F ((X \single(x))#(X \single(x))) is a binary operation of X \ single(x).

Next we state two propositions:

(38) For every at least 2-elements set X and for every binary operation F of X and for every element x of X holds F is binary operation preserving x if and only if X \ single(x) is a set closed w.r.t. F and F ((X \ single(x))#(X \ single(x))) is a binary operation of X \ single(x).

(39) For every set X and for every subset A of X there exists a binary operation F of X such that for arbitrary x such that x ∈ A#A holds F (x) ∈ A.

Let X be a set, and let A be a subset of X. A binary operation of X is said to be a binary operation of X preserving A if:

for arbitrary x such that x ∈ A#A holds it(x) ∈ A. One can prove the following two propositions:

(40) For every set X and for every subset A of X and for every binary operation F of X holds F is a binary operation of X preserving A if and only if for arbitrary x such that x ∈ A#A holds F (x) ∈ A.

(41) For every set X and for every subset A of X and for every binary operation F of X preserving A holds F (A#A) is a binary operation of A.

Let X be a set, and let A be a subset of X, and let F be a binary operation of X preserving A. The functor F A yielding a binary operation of A, is defined as follows:

F A = F (A#A). We now state two propositions:

(42) For every set X and for every subset A of X and for every binary operation F of X preserving A holds F A = F (A#A).

(43) For every at least 2-elements set A and for every element x of A there exists a binary operation F of A such that for arbitrary y such that y ∈ (A \ single(x))#(A \ single(x)) holds F (y) ∈ A \ single(x).

Let A be an at least 2-elements set, and let x be an element of A. A binary operation of A is called a binary operation of A preserving A \ {x} if:

for arbitrary y such that y ∈ (A \ single(x))#(A \ single(x)) holds it(y) ∈ A \ single(x).

One can prove the following two propositions:

(44) For every at least 2-elements set A and for every element x of A and for every binary operation F of A holds F is a binary operation of A preserving A \ {x} if and only if for arbitrary y such that y ∈ (A \ single(x))#(A \ single(x)) holds F (y) ∈ A \ single(x).

(45) For every at least 2-elements set A and for every element x of A and for every binary operation F of A preserving A \ {x} holds F ((A \ single(x))#(A \ single(x))) is a binary operation of A \ single(x).

Let A be an at least 2-elements set, and let x be an element of A, and let F be a binary operation of A preserving A \ {x}. The functor F _{x} A yields a binary operation of A \ single(x) and is defined as follows:

F _{x} A = F ((A \ single(x))#(A \ single(x))). One can prove the following proposition

(46) For every at least 2-elements set A and for every element x of A and for every binary operation F of A preserving A \ {x} holds F _{x} A = F ((A \ single(x))#(A \ single(x))).

A field structure is said to be a field if: there exists an at least 2-elements set A and there exists a binary operation od

of A and there exists an element nd of A and there exists a binary operation om of A preserving A \ {nd} and there exists an element nm of A \ single(nd) such that it = field(A, od, om, nd, nm) and group(A, od, nd) is a group and for every non-empty set B and for every binary operation P of B and for every element e of B such that B = A \ single(nd) and e = nm and P = om _{nd} A holds group(B, P, e) is a group and for all elements x, y, z of A holds om(⟨⟨x, od(⟨⟨y, z⟩⟩)⟩⟩) = od(⟨⟨om(⟨⟨x, y⟩⟩), om(⟨⟨x, z⟩⟩)⟩⟩).

We now state two propositions:

(47) Let F be a group structure. Then F is a group if and only if there exists a non-empty set A and there exists a binary operation og of A and there exists an element ng of A such that F = group(A, og, ng) and for all elements a, b, c of A holds og(⟨⟨og(⟨⟨a, b⟩⟩), c⟩⟩) = og(⟨⟨a, og(⟨⟨b, c⟩⟩)⟩⟩) and for every element a of A holds og(⟨⟨a, ng⟩⟩) = a and og(⟨⟨ng, a⟩⟩) = a and for every element a of A there exists an element b of A such that og(⟨⟨a, b⟩⟩) = ng and og(⟨⟨b, a⟩⟩) = ng and for all elements a, b of A holds og(⟨⟨a, b⟩⟩) = og(⟨⟨b, a⟩⟩).

(48) Let F be a field structure. Then F is a field if and only if there exists an at least 2-elements set A and there exists a binary operation od of A and there exists an element nd of A and there exists a binary operation om of A preserving A \ {nd} and there exists an element nm of A \ single(nd) such that F = field(A, od, om, nd, nm) and group(A, od, nd) is a group and for every non-empty set B and for every binary operation P of B and for every element e of B such that B = A \ single(nd) and e = nm and P = om _{nd} A holds group(B, P, e) is a group and for all elements x, y, z of A holds om(⟨⟨x, od(⟨⟨y, z⟩⟩)⟩⟩) = od(⟨⟨om(⟨⟨x, y⟩⟩), om(⟨⟨x, z⟩⟩)⟩⟩).

References

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

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

[3] Jean Dieudonné. Foundations of Modern Analises. Academic Press, New York and London, 1960.

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

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Equivalence Relations and Classes of Abstraction ^{1}

Konrad Raczkowski Warsaw University Białystok Paweł Sadowski Warsaw University Białystok

Summary. In this article we deal with the notion of equivalence relation. The main properties of equivalence relations are proved. Then we define the classes of abstraction determined by an equivalence relation. Finally, the connections between a partition of a set and an equivalence relation are presented. We introduce the following notation of modes: Equivalence Relation, a partition.

MML Identifier: EQREL 1.

The notation and terminology used in this paper are introduced in the following articles: [6], [7], [9], [8], [5], [3], [2], [4], and [1]. For simplicity we adopt the following rules: x, y, z are arbitrary, i, j are natural numbers, X, Y are sets, A, B are subsets of X, R, R_{1}, R_{2} are relations on X, and SF XX is a family of subsets of [: X, X:]. The following two propositions are true:

(1) If i < j, then j − i is a natural number.

(2) For every Y such that Y ⊆ [: X, X:] holds Y is a relation on X.

Let us consider X. The functor ∇_{X} yielding a relation on X, is defined as follows:

∇_{X} = [: X, X:].

We now state a proposition

(3) ∇_{X} = [: X, X:].

Let us consider X, R_{1}, R_{2}. Then R_{1} ∩ R_{2} is a relation on X. Then R_{1} ∪ R_{2} is a relation on X.

Next we state a proposition

(4) △_{X} is reflexive in X and △_{X} is symmetric in X and △_{X} is transitive in X.

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Let us consider X. A relation on X is called an equivalence relation of X if: it is reflexive in X and it is symmetric in X and it is transitive in X. The following three propositions are true:

(5) R is an equivalence relation of X if and only if R is reflexive in X and R is symmetric in X and R is transitive in X.

(6) △_{X} is an equivalence relation of X.

(7) ∇_{X} is an equivalence relation of X.

Let us consider X. Then △_{X} is an equivalence relation of X. Then ∇_{X} is an equivalence relation of X.

In the sequel EqR, EqR_{1}, EqR_{2} will be equivalence relations of X. We now state several propositions:

(8) EqR is reflexive in X.

(9) EqR is symmetric in X.

(10) EqR is transitive in X.

(11) If x ∈ X, then ⟨⟨x, x⟩⟩ ∈ EqR.

(12) If ⟨⟨x, y⟩⟩ ∈ EqR, then ⟨⟨y, x⟩⟩ ∈ EqR.

(13) If ⟨⟨x, y⟩⟩ ∈ EqR and ⟨⟨y, z⟩⟩ ∈ EqR, then ⟨⟨x, z⟩⟩ ∈ EqR.

(14) If there exists x such that x ∈ X, then EqR ≠ .

(15) field EqR = X.

(16) R is an equivalence relation of X if and only if R is pseudo reflexive

and R is symmetric and R is transitive and field R = X. Let us consider X, EqR_{1}, EqR_{2}. Then EqR_{1} ∩ EqR_{2} is an equivalence

relation of X.

We now state four propositions:

(17) △_{X} ∩ EqR = △_{X}.

(18) (∇_{X}) ∩ R = R.

(19) For every SF XX such that SF XX ≠ ∅ and for every Y such that Y ∈ SF XX holds Y is an equivalence relation of X holds ⋂ SF XX is an equivalence relation of X.

(20) For every R there exists EqR such that R ⊆ EqR and for every EqR_{2}

such that R ⊆ EqR_{2} holds EqR ⊆ EqR_{2}. Let us consider X, EqR_{1}, EqR_{2}. The functor EqR_{1} ⊔ EqR_{2} yielding an

equivalence relation of X, is defined by:

EqR_{1} ∪ EqR_{2} ⊆ EqR_{1} ⊔ EqR_{2} and for every EqR such that EqR_{1} ∪ EqR_{2} ⊆ EqR holds EqR_{1} ⊔ EqR_{2} ⊆ EqR.

Next we state several propositions:

(21) For every equivalence relation R of X holds R = EqR_{1} ⊔ EqR_{2} if and only if EqR_{1} ∪ EqR_{2} ⊆ R and for every EqR such that EqR_{1} ∪ EqR_{2} ⊆ EqR holds R ⊆ EqR.

(22) EqR ⊔ EqR = EqR.

(23) EqR_{1} ⊔ EqR_{2} = EqR_{2} ⊔ EqR_{1}.

Equivalence Relations and Classes of … 443

(24) EqR_{1} ∩ (EqR_{1} ⊔ EqR_{2}) = EqR_{1}.

(25) EqR_{1} ⊔ (EqR_{1} ∩ EqR_{2}) = EqR_{1}.

The scheme Ex Eq Rel concerns a set A, and a binary predicate P, and states that:

there exists an equivalence relation EqR of A such that for all x, y holds ⟨⟨x, y⟩⟩ ∈ EqR if and only if x ∈ A and y ∈ A and P[x, y] provided the parameters satisfy the following conditions:

• for every x such that x ∈ A holds P[x, x],

• for all x, y such that P[x, y] holds P[y, x],

• for all x, y, z such that P[x, y] and P[y, z] holds P[x, z].

Let us consider X, EqR, x. The functor [x]_{EqR} yielding a subset of X, is defined by:

[x]_{EqR} = EqR^{∘} {x}.

We now state a number of propositions:

(26) [x]_{EqR} = EqR^{∘} {x}.

(27) y ∈ [x]_{EqR} if and only if ⟨⟨y, x⟩⟩ ∈ EqR.

(28) For every x such that x ∈ X holds x ∈ [x]_{EqR}.

(29) For every x such that x ∈ X there exists y such that x ∈ [y]_{EqR}.

(30) If y ∈ [x]_{EqR} and z ∈ [x]_{EqR}, then ⟨⟨y, z⟩⟩ ∈ EqR.

(31) For every x such that x ∈ X holds y ∈ [x]_{EqR} if and only if [x]_{EqR} = [y]_{EqR}.

(32) For all x, y such that x ∈ X and y ∈ X holds [x]_{EqR} = [y]_{EqR} or [x]_{EqR} misses [y]_{EqR}.

(33) For every x such that x ∈ X holds [x]_{△_{X}} = {x}.

(34) For every x such that x ∈ X holds [x]_{∇_{X}} = X.

(35) If there exists x such that [x]_{EqR} = X, then EqR = ∇_{X}.

(36) Suppose x ∈ X. Then ⟨⟨x, y⟩⟩ ∈ EqR_{1} ⊔EqR_{2} if and only if there exists a finite sequence f such that 1 ≤ len f and x = f(1) and y = f(len f) and for every i such that 1 ≤ i and i < len f holds ⟨⟨f(i), f(i+1)⟩⟩ ∈ EqR_{1} ∪EqR_{2}.

(37) For every equivalence relation E of X such that E = EqR_{1} ∪ EqR_{2} for every x such that x ∈ X holds [x]_{E} = [x]_{EqR_{1}} or [x]_{E} = [x]_{EqR_{2}}.

(38) If EqR_{1} ∪ EqR_{2} = ∇_{X}, then EqR_{1} = ∇_{X} or EqR_{2} = ∇_{X}.

Let us consider X, EqR. The functor Classes EqR yields a family of subsets of X and is defined as follows:

A ∈ Classes EqR if and only if there exists x such that x ∈ X and A = [x]_{EqR}. The following two propositions are true:

(39) A ∈ Classes EqR if and only if there exists x such that x ∈ X and A = [x]_{EqR}.

(40) If X = ∅, then Classes EqR = ∅.

Let us consider X. A family of subsets of X is said to be a partition of X if:

⋃ it = X and for every A such that A ∈ it holds A ≠ ∅ and for every B such that B ∈ it holds A = B or A misses B if X ≠ ∅, it = ∅, otherwise.

We now state several propositions:

(41) If X ≠ ∅, then for every family F of subsets of X holds F is a partition of X if and only if ⋃ F = X and for every A such that A ∈ F holds A ≠ ∅ and for every B such that B ∈ F holds A = B or A misses B.

(42) Classes EqR is a partition of X.

(43) For every partition P of X there exists EqR such that P = Classes EqR.

(44) For every x such that x ∈ X holds ⟨⟨x, y⟩⟩ ∈ EqR if and only if [x]_{EqR} = [y]_{EqR}.

(45) If x ∈ Classes EqR, then there exists an element y of X such that x = [y]_{EqR}.

(46) For every x such that x ∈ X holds [x]_{EqR} ∈ Classes EqR.

References

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

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

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

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

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

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

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

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

[9] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

Some Properties of Real Numbers ^{1}

Andrzej Trybulec Warsaw University Białystok Czesław Byliński Warsaw University Białystok

Summary. We define the following operations on real numbers: max(x, y), min(x, y), x^{2}, \sqrt{x}. We prove basic properties of introduced operations. A number of auxiliary theorems absent in [1] and [2] is proved.

MML Identifier: SQUARE 1.

The terminology and notation used here are introduced in the papers [1] and [2]. In the sequel a, b, x, y, z will be real numbers. Next we state a number of propositions:

(1) 1 < 2.

(2) If 1 < x, then \frac{1}{x} < 1.

(3) \frac{1}{2} < 1.

(4) 2^{−1} < 1.

(5) 2 · a = a + a.

(6) a = (a − x) + x.

(7) a = (a + x) − x.

(8) If x − y = 0, then x = y.

(9) x ≤ y if and only if z + x ≤ z + y.

(10) a ≤ a + 1.

(11) If x < y, then 0 < y − x.

(12) If x ≤ y, then 0 ≤ y − x.

(13) 1^{−1} = 1.

(14) \frac{x}{1} = x.

(15) \frac{x+x}{2} = x.

(16) If x ≠ 0, then _{\frac{1}{\frac{1}{x}}} = x.

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(17) If y ≠ 0 and z ≠ 0, then \frac{x}{y·z} = \frac{\frac{x}{y}}{z}.

(18) If z ≠ 0, then x · \frac{y}{z} = \frac{x·y}{z}.

(19) If 0 ≤ x and 0 ≤ y, then 0 ≤ x · y.

(20) If x ≤ 0 and y ≤ 0, then 0 ≤ x · y.

(21) If 0 < x and 0 < y, then 0 < x · y.

(22) If x < 0 and y < 0, then 0 < x · y.

(23) If 0 ≤ x and y ≤ 0, then x · y ≤ 0 and y · x ≤ 0.

(24) If 0 < x and y < 0, then x · y < 0 and y · x < 0.

(25) If 0 ≤ x · y, then 0 ≤ x and 0 ≤ y or x ≤ 0 and y ≤ 0.

(26) If 0 < x · y, then 0 < x and 0 < y or x < 0 and y < 0.

(27) If 0 ≤ a and 0 < b, then 0 ≤ \frac{a}{b}.

(28) If 0 ≤ x, then y − x ≤ y.

(29) If 0 < x, then y − x < y.

(30) If x ≤ y, then z − y ≤ z − x.

The scheme RealContinuity deals with two unary predicates P and Q, and states that:

there exists z such that for all x, y such that P[x] and Q[y] holds x ≤ z and z ≤ y provided the following requirements are met:

• there exists x such that P[x],

• there exists x such that Q[x],

• for all x, y such that P[x] and Q[y] holds x ≤ y.

We now define two new functors. Let us consider x, y. The functor min(x, y) yields a real number and is defined by:

min(x, y) = x if x ≤ y, min(x, y) = y, otherwise. The functor max(x, y) yielding a real number, is defined as follows:

max(x, y) = x if y ≤ x, max(x, y) = y, otherwise.

We now state a number of propositions:

(31) If x ≤ y, then z = x if and only if z = min(x, y) but x ≤ y or z = y if and only if z = min(x, y).

(32) If y ≤ x, then min(x, y) = y.

(33) If y ≰ x, then min(x, y) = x.

(34) min(x, y) = \frac{(x+y)−|x−y|}{2}.

(35) min(x, y) ≤ x and min(y, x) ≤ x.

(36) min(x, x) = x.

(37) min(x, y) = min(y, x).

(38) min(x, y) = x or min(x, y) = y.

(39) x ≤ y and x ≤ z if and only if x ≤ min(y, z).

(40) min(x, min(y, z)) = min(min(x, y), z).

(41) If z < x and z < y, then z < min(x, y).

Some Properties of Real Numbers 447

(42) If y ≤ x, then z = x if and only if z = max(x, y) but y ≤ x or z = y if and only if z = max(x, y).

(43) If x ≤ y, then max(x, y) = y.

(44) If x ≰ y, then max(x, y) = x.

(45) max(x, y) = \frac{(x+y)+|x−y|}{2}.

(46) x ≤ max(x, y) and x ≤ max(y, x).

(47) max(x, x) = x.

(48) max(x, y) = max(y, x).

(49) max(x, y) = x or max(x, y) = y.

(50) y ≤ x and z ≤ x if and only if max(y, z) ≤ x.

(51) max(x, max(y, z)) = max(max(x, y), z).

(52) If 0 < x and 0 < y, then 0 < max(x, y).

(53) min(x, y) + max(x, y) = x + y.

(54) 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.

(55) 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.

(56) 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)).

(57) 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)). Let us consider x. The functor x^{2} yields an element of and is defined by: x^{2} = x · x. The following proposition is true

(58) x^{2} = x · x.

Let us consider a. Then a^{2} is a real number.

The following propositions are true:

(59) 1^{2} = 1.

(60) 0^{2} = 0.

(61) a^{2} = (−a)^{2}.

(62) |a|^{2} = a^{2}.

(63) (a + b)^{2} = (a^{2} + (2 · a) · b) + b^{2}.

(64) (a − b)^{2} = (a^{2} − (2 · a) · b) + b^{2}.

(65) (a + 1)^{2} = (a^{2} + 2 · a) + 1.

(66) (a − 1)^{2} = (a^{2} − 2 · a) + 1.

(67) (a − b) · (a + b) = a^{2} − b^{2} and (a + b) · (a − b) = a^{2} − b^{2}.

(68) (a · b)^{2} = a^{2} · b^{2}.

(69) If 0 ≠ b, then \frac{a}{b}^{2} = \frac{a2}{b2} .

(70) If a^{2} − b^{2} ≠ 0, then \frac{1}{a+b} = \frac{a−b}{a2−b2} .

(71) If a^{2} − b^{2} ≠ 0, then \frac{1}{a−b} = \frac{a+b}{a2−b2} .

(72) 0 ≤ a^{2}.

(73) If a^{2} = 0, then a = 0.

(74) If 0 ≠ a, then 0 < a^{2}.

(75) If 0 < a and a < 1, then a^{2} < a.

(76) If 1 < a, then a < a^{2}.

(77) If 0 ≤ x and x ≤ y, then x^{2} ≤ y^{2}.

(78) If 0 ≤ x and x < y, then x^{2} < y^{2}.

(79) If 0 ≤ x and 0 ≤ y and x^{2} ≤ y^{2}, then x ≤ y.

(80) If 0 ≤ x and 0 ≤ y and x^{2} < y^{2}, then x < y.

Let us consider a. Let us assume that 0 ≤ a. The functor \sqrt{a} yielding a real number, is defined by:

0 ≤ \sqrt{a} and \sqrt{a}^{2} = a.

We now state a number of propositions:

(81) If 0 ≤ a, then for every b holds b = \sqrt{a} if and only if 0 ≤ b and b^{2} = a.

(82) \sqrt{0} = 0.

(83) \sqrt{1} = 1.

(84) 1 < \sqrt{2}.

(85) \sqrt{4} = 2.

(86) \sqrt{2} < 2.

(87) If 0 ≤ a, then 0 ≤ \sqrt{a}.

(88) If 0 ≤ a, then \sqrt{a}^{2} = a.

(89) If 0 ≤ a, then \sqrt{a^{2}} = a.

(90) If a ≤ 0, then \sqrt{a^{2}} = −a.

(91) \sqrt{a^{2}} = |a|.

(92) If 0 ≤ a and \sqrt{a} = 0, then a = 0.

(93) If 0 < a, then 0 < \sqrt{a}.

(94) If 0 ≤ x and x ≤ y, then \sqrt{x} ≤ \sqrt{y}.

(95) If 0 ≤ x and x < y, then \sqrt{x} < \sqrt{y}.

(96) If 0 ≤ x and 0 ≤ y and \sqrt{x} = \sqrt{y}, then x = y.

(97) If 0 ≤ a and 0 ≤ b, then \sqrt{a · b} = \sqrt{a} · \sqrt{b}.

(98) If 0 ≤ a · b, then \sqrt{a · b} = \sqrt{|a|} · \sqrt{|b|}.

(99) If 0 ≤ a and 0 < b, then \sqrt{\frac{a}{b}} = \begin{matrix}√ & \overline{a} \\ √ & \overline{b} \end{matrix}.

(100) If 0 < \frac{a}{b} and b ≠ 0, then \sqrt{\frac{a}{b}} = ^{\begin{matrix}√ & \overline{|a|} \\ √ & \overline{|b|} \end{matrix}}.

Some Properties of Real Numbers 449

(101) If 0 < a, then \sqrt{\frac{1}{a}} = \frac{1}{\sqrt{a}}.

(102) If 0 < a, then \frac{\sqrt{a}}{a} = \frac{1}{\sqrt{a}}.

(103) If 0 < a, then \frac{a}{\sqrt{a}} = \sqrt{a}.

(104) If 0 ≤ a and 0 ≤ b, then (\sqrt{a} − \sqrt{b}) · (\sqrt{a} + \sqrt{b}) = a − b.

(105) If 0 ≤ a and 0 ≤ b and a ≠ b, then \frac{1}{\sqrt{a}+\sqrt{b}} = ^{\frac{\sqrt{a}−\sqrt{b}}{a−b}}.

(106) If 0 ≤ b and b < a, then \frac{1}{\sqrt{a}+\sqrt{b}} = ^{\frac{\sqrt{a}−\sqrt{b}}{a−b}}.

(107) If 0 ≤ a and 0 ≤ b and a ≠ b, then \frac{1}{\sqrt{a}−\sqrt{b}} = ^{\frac{\sqrt{a}+\sqrt{b}}{a−b}}.

(108) If 0 ≤ b and b < a, then \frac{1}{\sqrt{a}−\sqrt{b}} = ^{\frac{\sqrt{a}+\sqrt{b}}{a−b}}.

References

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

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

450

Connectives and Subformulae of the First Order Language

Grzegorz Bancerek^{1} Warsaw University Białystok

Summary. In the article the development of the first order language defined in [5] is continued. The following connectives are introduced: implication (⇒), disjunction (∨), and equivalence (⇔). We introduce also the existential quantifier (∃) and FALSUM. Some theorems on disjunctive, conditional, biconditional and existential formulae are proved and their selector functors are introduced. The second part of the article deals with notions of subformula, proper subformula and immediate constituent of a QC-formula.

MML Identifier: QC LANG2.

The papers [7], [6], [3], [4], [1], [2], and [5] provide the terminology and notation for this paper. We adopt the following convention: x, y, z will be bound variables and p, q, p_{1}, p_{2}, q_{1} will be elements of WFF. One can prove the following propositions:

(1) If ¬p = ¬q, then p = q.

(2) Arg(¬p) = p.

(3) If p ∧ q = p_{1} ∧ q_{1}, then p = p_{1} and q = q_{1}.

(4) If p is conjunctive, then p = LeftArg(p) ∧ RightArg(p).

(5) LeftArg(p ∧ q) = p and RightArg(p ∧ q) = q.

(6) If ∀_{x}p = ∀_{y}q, then x = y and p = q.

(7) If p is universal, then p = ∀_{Bound(p)} Scope(p).

(8) Bound(∀_{x}p) = x and Scope(∀_{x}p) = p. We now define three new functors. The formula FALSUM is defined as

follows:

FALSUM = ¬ VERUM.

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452 Grzegorz Bancerek

Let p, q be elements of WFF. The functor p ⇒ q yields a formula and is defined by:

p ⇒ q = ¬(p ∧ ¬q). The functor p ∨ q yields a formula and is defined as follows:

p ∨ q = ¬(¬p ∧ ¬q).

Let p, q be elements of WFF. The functor p ⇔ q yielding a formula, is defined as follows:

p ⇔ q = (p ⇒ q) ∧ (q ⇒ p).

Let x be a bound variable, and let p be an element of WFF. The functor ∃_{x}p yielding a formula, is defined as follows:

∃_{x}p = ¬(∀_{x}¬p).

The following propositions are true:

(9) FALSUM = ¬ VERUM.

(10) p ⇒ q = ¬(p ∧ ¬q).

(11) p ∨ q = ¬(¬p ∧ ¬q).

(12) p ⇔ q = (p ⇒ q) ∧ (q ⇒ p).

(13) FALSUM is negative and Arg(FALSUM) = VERUM.

(14) p ∨ q = ¬p ⇒ q.

(15) ∃_{x}p = ¬(∀_{x}¬p).

(16) If p ∨ q = p_{1} ∨ q_{1}, then p = p_{1} and q = q_{1}.

(17) If p ⇒ q = p_{1} ⇒ q_{1}, then p = p_{1} and q = q_{1}.

(18) If p ⇔ q = p_{1} ⇔ q_{1}, then p = p_{1} and q = q_{1}.

(19) If ∃_{x}p = ∃_{y}q, then x = y and p = q.

We now define two new functors. Let x, y be bound variables, and let p be an element of WFF. The functor ∀_{x,y}p yielding a formula, is defined by:

∀_{x,y}p = ∀_{x}(∀_{y}p). The functor ∃_{x,y}p yields a formula and is defined by:

∃_{x,y}p = ∃_{x}(∃_{y}p).

Next we state several propositions:

(20) ∀_{x,y}p = ∀_{x}(∀_{y}p) and ∃_{x,y}p = ∃_{x}(∃_{y}p).

(21) For all bound variables x_{1}, x_{2}, y_{1}, y_{2} such that ∀_{x_{1},y_{1}}p_{1} = ∀_{x_{2},y_{2}}p_{2} holds x_{1} = x_{2} and y_{1} = y_{2} and p_{1} = p_{2}.

(22) If ∀_{x,y}p = ∀_{z}q, then x = z and ∀_{y}p = q.

(23) For all bound variables x_{1}, x_{2}, y_{1}, y_{2} such that ∃_{x_{1},y_{1}}p_{1} = ∃_{x_{2},y_{2}}p_{2} holds x_{1} = x_{2} and y_{1} = y_{2} and p_{1} = p_{2}.

(24) If ∃_{x,y}p = ∃_{z}q, then x = z and ∃_{y}p = q.

(25) ∀_{x,y}p is universal and Bound(∀_{x,y}p) = x and Scope(∀_{x,y}p) = ∀_{y}p. We now define two new functors. Let x, y, z be bound variables, and let p

be an element of WFF. The functor ∀_{x,y,z}p yields a formula and is defined by:

∀_{x,y,z}p = ∀_{x}(∀_{y,z}p). The functor ∃_{x,y,z}p yields a formula and is defined by:

Connectives and Subformulae of the First … 453

∃_{x,y,z}p = ∃_{x}(∃_{y,z}p). The following propositions are true:

(26) ∀_{x,y,z}p = ∀_{x}(∀_{y,z}p) and ∃_{x,y,z}p = ∃_{x}(∃_{y,z}p).

(27) For all bound variables x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2} such that ∀_{x_{1},y_{1},z_{1}}p_{1} =

∀_{x_{2},y_{2},z_{2}}p_{2} holds x_{1} = x_{2} and y_{1} = y_{2} and z_{1} = z_{2} and p_{1} = p_{2}. In the sequel s, t will be bound variables. We now state several propositions:

(28) If ∀_{x,y,z}p = ∀_{t}q, then x = t and ∀_{y,z}p = q.

(29) If ∀_{x,y,z}p = ∀_{t,s}q, then x = t and y = s and ∀_{z}p = q.

(30) For all bound variables x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2} such that ∃_{x_{1},y_{1},z_{1}}p_{1} = ∃_{x_{2},y_{2},z_{2}}p_{2} holds x_{1} = x_{2} and y_{1} = y_{2} and z_{1} = z_{2} and p_{1} = p_{2}.

(31) If ∃_{x,y,z}p = ∃_{t}q, then x = t and ∃_{y,z}p = q.

(32) If ∃_{x,y,z}p = ∃_{t,s}q, then x = t and y = s and ∃_{z}p = q.

(33) ∀_{x,y,z}p is universal and Bound(∀_{x,y,z}p) = x and Scope(∀_{x,y,z}p) = ∀_{y,z}p. We now define four new predicates. Let H be an element of WFF. We say

that H is disjunctive if and only if:

there exist elements p, q of WFF such that H = p ∨ q. We say that H is conditional if and only if:

there exist elements p, q of WFF such that H = p ⇒ q. We say that H is biconditional if and only if:

there exist elements p, q of WFF such that H = p ⇔ q. We say that H is existential if and only if:

there exists a bound variable x and there exists an element p of WFF such that H = ∃_{x}p.

We now state several propositions:

(34) For every element H of WFF holds H is disjunctive if and only if there exist elements p, q of WFF such that H = p ∨ q.

(35) For every element H of WFF holds H is conditional if and only if there exist elements p, q of WFF such that H = p ⇒ q.

(36) For every element H of WFF holds H is biconditional if and only if there exist elements p, q of WFF such that H = p ⇔ q.

(37) For every element H of WFF holds H is existential if and only if there exists a bound variable x and there exists an element p of WFF such that H = ∃_{x}p.

(38) ∃_{x,y}p is existential and ∃_{x,y,z}p is existential.

We now define four new functors. Let H be an element of WFF. The functor LeftDisj(H) yields a formula and is defined by:

LeftDisj(H) = Arg(LeftArg(Arg(H))). The functor RightDisj(H) yielding a formula, is defined as follows:

RightDisj(H) = Arg(RightArg(Arg(H))). The functor Antecedent(H) yields a formula and is defined by:

Antecedent(H) = LeftArg(Arg(H)). The functor Consequent(H) yields a formula and is defined by:

Consequent(H) = Arg(RightArg(Arg(H))).

We now define two new functors. Let H be an element of WFF. The functor LeftSide(H) yields a formula and is defined by:

LeftSide(H) = Antecedent(LeftArg(H)). The functor RightSide(H) yielding a formula, is defined as follows:

RightSide(H) = Consequent(LeftArg(H)).

The following propositions are true:

(39) For every element H of WFF holds LeftDisj(H) = Arg(LeftArg(Arg(H))) .

(40) For every element H of WFF holds RightDisj(H) = Arg(RightArg(Arg(H))) .

(41) For every element H of WFF holds Antecedent(H) = LeftArg(Arg(H)).

(42) For every element H of WFF holds Consequent(H) = Arg(RightArg(Arg(H))) .

(43) For every element H of WFF holds LeftSide(H) = Antecedent(LeftArg(H)) .

(44) For every element H of WFF holds

RightSide(H) = Consequent(LeftArg(H)) . In the sequel F, G, H will be elements of WFF. We now state a number of

propositions:

(45) LeftDisj(F ∨ G) = F and RightDisj(F ∨ G) = G and Arg(F ∨ G) = ¬F ∧ ¬G.

(46) Antecedent(F ⇒ G) = F and Consequent(F ⇒ G) = G and Arg(F ⇒ G) = F ∧ ¬G.

(47) LeftSide(F ⇔ G) = F and RightSide(F ⇔ G) = G and LeftArg(F ⇔ G) = F ⇒ G and RightArg(F ⇔ G) = G ⇒ F.

(48) Arg(∃_{x}H) = ∀_{x}¬H.

(49) If H is disjunctive, then H is conditional and H is negative and Arg(H) is conjunctive and LeftArg(Arg(H)) is negative and RightArg(Arg(H)) is negative.

(50) If H is conditional, then H is negative and Arg(H) is conjunctive and RightArg(Arg(H)) is negative.

(51) If H is biconditional, then H is conjunctive and LeftArg(H) is conditional and RightArg(H) is conditional.

(52) If H is existential, then H is negative and Arg(H) is universal and Scope(Arg(H)) is negative.

(53) If H is disjunctive, then H = LeftDisj(H) ∨ RightDisj(H).

(54) If H is conditional, then H = Antecedent(H) ⇒ Consequent(H).

(55) If H is biconditional, then H = LeftSide(H) ⇔ RightSide(H).

(56) If H is existential, then H = ∃_{Bound(Arg(H))} Arg(Scope(Arg(H))).

Let G, H be elements of WFF. We say that G is an immediate constituent of H if and only if:

H = ¬G or there exists an element F of WFF such that H = G ∧ F or H = F ∧ G or there exists a bound variable x such that H = ∀_{x}G.

For simplicity we adopt the following convention: x is a bound variable, k, n are natural numbers, P is a k-ary predicate symbol, and V is a list of variables of the length k. One can prove the following propositions:

(57) G is an immediate constituent of H if and only if H = ¬G or there exists F such that H = G ∧ F or H = F ∧ G or there exists x such that H = ∀_{x}G.

(58) H is not an immediate constituent of VERUM.

(59) H is not an immediate constituent of P [V].

(60) F is an immediate constituent of ¬H if and only if F = H.

(61) H is an immediate constituent of FALSUM if and only if H = VERUM.

(62) F is an immediate constituent of G ∧ H if and only if F = G or F = H.

(63) F is an immediate constituent of ∀_{x}H if and only if F = H.

(64) If H is atomic, then F is not an immediate constituent of H.

(65) If H is negative, then F is an immediate constituent of H if and only if F = Arg(H).

(66) If H is conjunctive, then F is an immediate constituent of H if and only if F = LeftArg(H) or F = RightArg(H).

(67) If H is universal, then F is an immediate constituent of H if and only

if F = Scope(H). In the sequel L denotes a finite sequence. Let us consider G, H. We say that

G is a subformula of H if and only if:

there exist n, L such that 1 ≤ n and len L = n and L(1) = G and L(n) = H and for every k such that 1 ≤ k and k < n there exist elements G_{1}, H_{1} of WFF such that L(k) = G_{1} and L(k + 1) = H_{1} and G_{1} is an immediate constituent of H_{1}.

We now state two propositions:

(68) G is a subformula of H if and only if there exist n, L such that 1 ≤ n and len L = n and L(1) = G and L(n) = H and for every k such that 1 ≤ k and k < n there exist elements G_{1}, H_{1} of WFF such that L(k) = G_{1} and L(k + 1) = H_{1} and G_{1} is an immediate constituent of H_{1}.

(69) H is a subformula of H.

Let us consider H, F. We say that H is a proper subformula of F if and only if:

H is a subformula of F and H ≠ F.

One can prove the following propositions:

(70) H is a proper subformula of F if and only if H is a subformula of F and H ≠ F.

(71) If H is an immediate constituent of F, then len(@H) < len(@F).

(72) If H is an immediate constituent of F, then H is a subformula of F.

(73) If H is an immediate constituent of F, then H is a proper subformula of F.

(74) If H is a proper subformula of F, then len(@H) < len(@F).

(75) If H is a proper subformula of F, then there exists G such that G is an immediate constituent of F.

(76) If F is a proper subformula of G and G is a proper subformula of H, then F is a proper subformula of H.

(77) If F is a subformula of G and G is a subformula of H, then F is a subformula of H.

(78) If G is a subformula of H and H is a subformula of G, then G = H.

(79) It is not true that: G is a proper subformula of H and H is a subformula of G.

(80) It is not true that: G is a proper subformula of H and H is a proper subformula of G.

(81) It is not true that: G is a subformula of H and H is an immediate constituent of G.

(82) It is not true that: G is a proper subformula of H and H is an immediate constituent of G.

(83) Suppose F is a proper subformula of G and G is a subformula of H or F is a subformula of G and G is a proper subformula of H or F is a subformula of G and G is an immediate constituent of H or F is an immediate constituent of G and G is a subformula of H or F is a proper subformula of G and G is an immediate constituent of H or F is an immediate constituent of G and G is a proper subformula of H. Then F is a proper subformula of H.

(84) F is not a proper subformula of VERUM.

(85) F is not a proper subformula of P [V].

(86) F is a subformula of H if and only if F is a proper subformula of ¬H.

(87) If ¬F is a subformula of H, then F is a proper subformula of H.

(88) F is a proper subformula of FALSUM if and only if F is a subformula of VERUM.

(89) F is a subformula of G or F is a subformula of H if and only if F is a proper subformula of G ∧ H.

(90) If F ∧ G is a subformula of H, then F is a proper subformula of H and G is a proper subformula of H.

(91) F is a subformula of H if and only if F is a proper subformula of ∀_{x}H.

(92) If ∀_{x}H is a subformula of F, then H is a proper subformula of F.

(93) F ∧ ¬G is a proper subformula of F ⇒ G and F is a proper subformula of F ⇒ G and ¬G is a proper subformula of F ⇒ G and G is a proper subformula of F ⇒ G.

(94) ¬F ∧¬G is a proper subformula of F ∨G and ¬F is a proper subformula of F ∨ G and ¬G is a proper subformula of F ∨ G and F is a proper subformula of F ∨ G and G is a proper subformula of F ∨ G.

(95) If H is atomic, then F is not a proper subformula of H.

(96) If H is negative, then Arg(H) is a proper subformula of H.

(97) If H is conjunctive, then LeftArg(H) is a proper subformula of H and RightArg(H) is a proper subformula of H.

(98) If H is universal, then Scope(H) is a proper subformula of H.

(99) H is a subformula of VERUM if and only if H = VERUM.

(100) H is a subformula of P [V] if and only if H = P [V].

(101) H is a subformula of FALSUM if and only if H = FALSUM or H =

VERUM. Let us consider H. The functor Subformulae H yields a set and is defined

by:

for arbitrary a holds a ∈ Subformulae H if and only if there exists F such that F = a and F is a subformula of H.

Next we state a number of propositions:

(102) For arbitrary a holds a ∈ Subformulae H if and only if there exists F such that F = a and F is a subformula of H.

(103) If G ∈ Subformulae H, then G is a subformula of H.

(104) If F is a subformula of H, then Subformulae F ⊆ Subformulae H.

(105) If G ∈ Subformulae H, then Subformulae G ⊆ Subformulae H.

(106) H ∈ Subformulae H.

(107) Subformulae VERUM = {VERUM}.

(108) Subformulae(P [V]) = {P [V]}.

(109) Subformulae FALSUM = {VERUM, FALSUM}.

(110) Subformulae ¬H = Subformulae H ∪ {¬H}.

(111) Subformulae H ∧ F = (Subformulae H ∪ Subformulae F) ∪ {H ∧ F }.

(112) Subformulae ∀_{x}H = Subformulae H ∪ {∀_{x}H}.

(113) Subformulae F ⇒ G = (Subformulae F ∪ Subformulae G) ∪ {¬G, F ∧ ¬G, F ⇒ G}.

(114) Subformulae F ∨G = (Subformulae F ∪Subformulae G)∪{¬G, ¬F, ¬F ∧ ¬G, F ∨ G}.

(115) Subformulae F ⇔ G = (Subformulae F ∪ Subformulae G) ∪ {¬G, F ∧ ¬G, F ⇒ G, ¬F, G ∧ ¬F, G ⇒ F, F ⇔ G}.

(116) H = VERUM or H is atomic if and only if Subformulae H = {H}.

(117) If H is negative, then Subformulae H = Subformulae Arg(H) ∪ {H}.

(118) If H is conjunctive, then Subformulae H = (Subformulae LeftArg(H) ∪ Subformulae RightArg(H)) ∪ {H}.

(119) If H is universal, then Subformulae H = Subformulae Scope(H) ∪ {H}.

(120) If H is an immediate constituent of G or H is a proper subformula of G or H is a subformula of G but G ∈ Subformulae F, then H ∈ Subformulae F.

References

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

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

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

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

[5] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

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

Variables in Formulae of the First Order Language ^{1}

Czesław Byliński Warsaw University Białystok Grzegorz Bancerek Warsaw University Białystok

Summary. We develop the first order language defined in [5]. We continue the work done in the article [1]. We prove some schemes of defining by structural induction. We deal with notions of closed subformulae and of still not bound variables in a formula. We introduce the concept of the set of all free variables and the set of all fixed variables occurring in a formula.

MML Identifier: QC LANG3.

The notation and terminology used in this paper have been introduced in the following articles: [6], [3], [4], [2], [5], and [1]. For simplicity we follow the rules: i, j, k are natural numbers, x is a bound variable, a is a free variable, p, q are elements of WFF, l is a finite sequence of elements of Var, P is a predicate symbol, and V is a non-empty subset of Var. Let F be a function from WFF into WFF, and let us consider p. Then F (p) is an element of WFF.

In the article we present several logical schemes. The scheme QC Func Uniq deals with a non-empty set A, a function B from WFF into A, a function C from WFF into A, an element D of A, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A and states that:

B = C provided the following conditions are satisfied:

460 Czesław Byliński and Grzegorz Bancerek

• Given p. Let d_{1}, d_{2} be elements of A. Then (i) if p = VERUM, then B(p) = D, (ii) if p is atomic, then B(p) = F(p), (iii) if p is negative and d_{1} = B(Arg(p)), then B(p) = G(d_{1}), (iv) if p is conjunctive and d_{1} = B(LeftArg(p)) and d_{2} = B(RightArg(p)) , then B(p) = H(d_{1}, d_{2}),

1Partially supported by RPBP III.24.C1

(v) if p is universal and d_{1} = B(Scope(p)), then B(p) = I(p, d_{1}),

• Given p. Let d_{1}, d_{2} be elements of A. Then (i) if p = VERUM, then C(p) = D, (ii) if p is atomic, then C(p) = F(p), (iii) if p is negative and d_{1} = C(Arg(p)), then C(p) = G(d_{1}), (iv) if p is conjunctive and d_{1} = C(LeftArg(p)) and

d_{2} = C(RightArg(p)) , then C(p) = H(d_{1}, d_{2}),

(v) if p is universal and d_{1} = C(Scope(p)), then C(p) = I(p, d_{1}). The scheme QC Def D deals with a non-empty set A, an element B of A, an

element C of WFF, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A and states that:

(i) there exists an element d of A and there exists a function F from WFF into A such that d = F (C) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}), (ii) for all elements x_{1}, x_{2} of A such that there exists a function F from WFF into A such that x_{1} = F (C) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}) and there exists a function F from WFF into A such that x_{2} = F (C) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}) holds x_{1} = x_{2} for all values of the parameters.

The scheme QC D Result’VERU deals with a non-empty set A, a unary functor F yielding an element of A, an element B of A, a unary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(VERUM) = B provided the parameters fulfill the following condition:

Variables in Formulae of the First Order … 461

• Let p be a formula. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = G(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = H(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = I(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = J (p, d_{1}).

The scheme QC D Result’atom concerns a non-empty set A, an element B of A, a unary functor F yielding an element of A, a formula C, a unary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(C) = G(C) provided the following conditions are fulfilled:

• Let p be a formula. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = G(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = H(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = I(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = J (p, d_{1}),

• C is atomic. The scheme QC D Result’nega deals with a non-empty set A, an element B

of A, a formula C, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, a binary functor I yielding an element of A, and a unary functor J yielding an element of A and states that:

J (C) = G(J (Arg(C))) provided the following requirements are met:

• Let p be a formula. Let d be an element of A. Then d = J (p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}),

• C is negative. The scheme QC D Result’conj concerns a non-empty set A, an element B of

A, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, a binary functor I yielding an element of A, a unary functor J yielding an element of A, and a formula C and states that:

for all elements d_{1}, d_{2} of A such that d_{1} = J (LeftArg(C)) and d_{2} = J (RightArg(C)) holds J (C) = H(d_{1}, d_{2})

provided the parameters satisfy the following conditions:

• Let p be a formula. Let d be an element of A. Then d = J (p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}),

• C is conjunctive. The scheme QC D Result’univ deals with a non-empty set A, an element B

of A, a formula C, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, a binary functor I yielding an element of A, and a unary functor J yielding an element of A and states that:

J (C) = I(C, J (Scope(C))) provided the following requirements are fulfilled:

• Let p be a formula. Let d be an element of A. Then d = J (p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}),

• C is universal. Let us consider V. The functor ∅_{V} yields an element of 2^{V} qua a non-empty

set and is defined as follows:

∅_{V} = ∅. Next we state three propositions:

(1) ∅_{V} = ∅.

(2) For every k-ary predicate symbol P holds P is a predicate symbol.

(3) P is a Arity(P)-ary predicate symbol. Let us consider l, V. The functor variables_{V} (l) yielding an element of 2^{V}, is

defined by:

variables_{V} (l) = {l(k) : 1 ≤ k ∧ k ≤ len l ∧ l(k) ∈ V }. One can prove the following propositions:

(4) variables_{V} (l) = {l(k) : 1 ≤ k ∧ k ≤ len l ∧ l(k) ∈ V }.

(5) variables_{V} (l) ⊆ V.

(6) snb(l) = variables_{BoundVar}(l).

(7) snb(VERUM) = ∅.

(8) For every formula p such that p is atomic holds snb(p) = snb(Args(p)).

(9) For every k-ary predicate symbol P and for every list of variables l of the length k holds snb(P [l]) = snb(l).

(10) For every formula p such that p is negative holds snb(p) = snb(Arg(p)).

(11) For every formula p holds snb(¬p) = snb(p).

(12) snb(FALSUM) = ∅.

(13) For every formula p such that p is conjunctive holds snb(p) = snb(LeftArg(p)) ∪ snb(RightArg(p)) .

(14) For all formulae p, q holds snb(p ∧ q) = snb(p) ∪ snb(q).

(15) For every formula p such that p is universal holds snb(p) = snb(Scope(p)) \ {Bound(p)} .

(16) For every formula p holds snb(∀_{x}p) = snb(p) \ {x}.

(17) For every formula p such that p is disjunctive holds snb(p) = snb(LeftDisj(p)) ∪ snb(RightDisj(p)) .

(18) For all formulae p, q holds snb(p ∨ q) = snb(p) ∪ snb(q).

(19) For every formula p such that p is conditional holds snb(p) = snb(Antecedent(p)) ∪ snb(Consequent(p)) .

(20) For all formulae p, q holds snb(p ⇒ q) = snb(p) ∪ snb(q).

(21) For every formula p such that p is biconditional holds snb(p) = snb(LeftSide(p)) ∪ snb(RightSide(p)) .

(22) For all formulae p, q holds snb(p ⇔ q) = snb(p) ∪ snb(q).

(23) For every formula p holds snb(∃_{x}p) = snb(p) \ {x}.

(24) VERUM is closed and FALSUM is closed.

(25) For every formula p holds p is closed if and only if ¬p is closed.

(26) For all formulae p, q holds p is closed and q is closed if and only if p ∧ q is closed.

(27) For every formula p holds ∀_{x}p is closed if and only if snb(p) ⊆ {x}.

(28) For every formula p such that p is closed holds ∀_{x}p is closed.

(29) For all formulae p, q holds p is closed and q is closed if and only if p ∨ q is closed.

(30) For all formulae p, q holds p is closed and q is closed if and only if p ⇒ q is closed.

(31) For all formulae p, q holds p is closed and q is closed if and only if p ⇔ q is closed.

(32) For every formula p holds ∃_{x}p is closed if and only if snb(p) ⊆ {x}.

(33) For every formula p such that p is closed holds ∃_{x}p is closed. Let us consider V, and let F be a function from WFF into 2^{V}, and let us

consider p. Then F (p) is an element of 2^{V}.

Let us consider k. The functor x_{k} yielding a bound variable, is defined as follows:

x_{k} = ⟨⟨4, k⟩⟩.

One can prove the following propositions:

(34) x_{k} = ⟨⟨4, k⟩⟩.

(35) If x_{i} = x_{j}, then i = j.

(36) There exists i such that x_{i} = x.

Let us consider k. The functor a_{k} yields a free variable and is defined as follows:

a_{k} = ⟨⟨6, k⟩⟩.

One can prove the following propositions:

(37) a_{k} = ⟨⟨6, k⟩⟩.

(38) If a_{i} = a_{j}, then i = j.

(39) There exists i such that a_{i} = a.

(40) For every element c of FixedVar and for every element a of FreeVar holds c ≠ a.

(41) For every element c of FixedVar and for every element x of BoundVar holds c ≠ x.

(42) For every element a of FreeVar and for every element x of BoundVar

holds a ≠ x. Let us consider V, and let V_{1}, V_{2} be elements of 2^{V}. Then V_{1} ∪ V_{2} is an

element of 2^{V}.

Let D be a non-empty family of sets, and let d be an element of D. The functor @d yields an element of D qua a non-empty set and is defined as follows:

@d = d.

One can prove the following proposition

(43) For every non-empty family D of sets and for every element d of D

holds @d = d. Let D be a non-empty family of sets, and let d be an element of D qua a

non-empty set. The functor @d yielding an element of D, is defined as follows: @d = d. We now state a proposition

(44) For every non-empty family D of sets and for every element d of D

qua a non-empty set holds @d = d. Now we present several schemes. The scheme QC Def SETD deals with a

non-empty family A of sets, an element B of A, an element C of WFF, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A and states that:

(i) there exists an element d of A and there exists a function F from WFF into A such that d = F (C) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}), (ii) for all elements x_{1}, x_{2} of A such that there exists a function F from WFF into A such that x_{1} = F (C) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}) and there exists a function F from WFF into A such that x_{2} = F (C) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}) holds x_{1} = x_{2} for all values of the parameters.

The scheme QC SETD Result’V concerns a non-empty family A of sets, a unary functor F yielding an element of A, an element B of A, a unary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(VERUM) = B provided the parameters meet the following requirement:

• Let p be an element of WFF. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = G(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = H(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = I(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = J (p, d_{1}).

The scheme QC SETD Result’a concerns a non-empty family A of sets, an element B of A, a unary functor F yielding an element of A, an element C of WFF, a unary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(C) = G(C) provided the parameters fulfill the following requirements:

• Let p be an element of WFF. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = G(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = H(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = I(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = J (p, d_{1}),

• C is atomic.

The scheme QC SETD Result’n deals with a non-empty family A of sets, an element B of A, an element C of WFF, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, a binary functor I yielding an element of A, and a unary functor J yielding an element of A and states that:

J (C) = G(J (Arg(C))) provided the following requirements are met:

• Let p be an element of WFF. Let d be an element of A. Then d = J (p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}),

• C is negative. The scheme QC SETD Result’c deals with a non-empty family A of sets, an

element B of A, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, a binary functor I yielding an element of A, a unary functor J yielding an element of A, and an element C of WFF and states that:

for all elements d_{1}, d_{2} of A such that d_{1} = J (LeftArg(C)) and d_{2} = J (RightArg(C)) holds J (C) = H(d_{1}, d_{2})

provided the parameters fulfill the following conditions:

• Let p be an element of WFF. Let d be an element of A. Then d = J (p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}),

• C is conjunctive. The scheme QC SETD Result’u deals with a non-empty family A of sets, an

element B of A, an element C of WFF, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, a binary functor I yielding an element of A, and a unary functor J yielding an element of A and states that:

J (C) = I(C, J (Scope(C))) provided the parameters meet the following requirements:

• Let p be an element of WFF. Let d be an element of A. Then d = J (p) if and only if there exists a function F from WFF into A such that d = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of A holds if p = VERUM, then F (p) = B but if p is atomic, then F (p) = F(p) but if p is negative and d_{1} = F (Arg(p)), then F (p) = G(d_{1}) but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = H(d_{1}, d_{2}) but if p is universal and d_{1} = F (Scope(p)), then F (p) = I(p, d_{1}),

• C is universal.

Let us consider V, p. The functor Vars_{V} (p) yielding an element of 2^{V}, is defined as follows:

there exists a function F from WFF into 2^{V} such that Vars_{V} (p) = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of 2^{V} holds if p = VERUM, then F (p) = @(∅_{V}) but if p is atomic, then F (p) = variables_{V} (Args(p)) but if p is negative and d_{1} = F (Arg(p)), then F (p) = d_{1} but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = d_{1} ∪ d_{2} but if p is universal and d_{1} = F (Scope(p)), then F (p) = d_{1}.

We now state a number of propositions:

(45) Let X be an element of 2^{V}. Then X = Vars_{V} (p) if and only if there exists a function F from WFF into 2^{V} such that X = F (p) and for every element p of WFF and for all elements d_{1}, d_{2} of 2^{V} holds if p = VERUM, then F (p) = @(∅_{V}) but if p is atomic, then F (p) = variables_{V} (Args(p)) but if p is negative and d_{1} = F (Arg(p)), then F (p) = d_{1} but if p is conjunctive and d_{1} = F (LeftArg(p)) and d_{2} = F (RightArg(p)), then F (p) = d_{1} ∪d_{2} but if p is universal and d_{1} = F (Scope(p)), then F (p) = d_{1}.

(46) Vars_{V} (VERUM) = ∅.

(47) If p is atomic, then Vars_{V} (p) = variables_{V} (Args(p)) and Vars_{V} (p) = {Args(p)(k) : 1 ≤ k ∧ k ≤ len Args(p) ∧ Args(p)(k) ∈ V }.

(48) For every k-ary predicate symbol P and for every list of variables l of the length k holds Vars_{V} (P [l]) = variables_{V} (l) and Vars_{V} (P [l]) = {l(i) : 1 ≤ i ∧ i ≤ len l ∧ l(i) ∈ V }.

(49) If p is negative, then Vars_{V} (p) = Vars_{V} (Arg(p)).

(50) Vars_{V} (¬p) = Vars_{V} (p).

(51) Vars_{V} (FALSUM) = ∅.

(52) If p is conjunctive, then Vars_{V} (p) = Vars_{V} (LeftArg(p)) ∪ Vars_{V} (RightArg(p)) .

(53) Vars_{V} (p ∧ q) = Vars_{V} (p) ∪ Vars_{V} (q).

(54) If p is universal, then Vars_{V} (p) = Vars_{V} (Scope(p)).

(55) Vars_{V} (∀_{x}p) = Vars_{V} (p).

(56) If p is disjunctive, then Vars_{V} (p) = Vars_{V} (LeftDisj(p)) ∪ Vars_{V} (RightDisj(p)) .

(57) Vars_{V} (p ∨ q) = Vars_{V} (p) ∪ Vars_{V} (q).

(58) If p is conditional, then Vars_{V} (p) = Vars_{V} (Antecedent(p)) ∪ Vars_{V} (Consequent(p)) .

(59) Vars_{V} (p ⇒ q) = Vars_{V} (p) ∪ Vars_{V} (q).

(60) If p is biconditional, then Vars_{V} (p) = Vars_{V} (LeftSide(p)) ∪ Vars_{V} (RightSide(p)) .

(61) Vars_{V} (p ⇔ q) = Vars_{V} (p) ∪ Vars_{V} (q).

(62) If p is existential, then Vars_{V} (p) = Vars_{V} (Arg(Scope(Arg(p)))).

(63) Vars_{V} (∃_{x}p) = Vars_{V} (p).

Let us consider p. The functor Free p yielding an element of 2^{FreeVar}, is defined as follows:

Free p = Vars_{FreeVar}(p).

One can prove the following propositions:

(64) Free p = Vars_{FreeVar}(p).

(65) Free VERUM = ∅.

(66) For every k-ary predicate symbol P and for every list of variables l of the length k holds Free(P [l]) = {l(i) : 1 ≤ i ∧ i ≤ len l ∧ l(i) ∈ FreeVar}.

(67) Free ¬p = Free p.

(68) Free FALSUM = ∅.

(69) Free p ∧ q = Free p ∪ Free q.

(70) Free ∀_{x}p = Free p.

(71) Free p ∨ q = Free p ∪ Free q.

(72) Free p ⇒ q = Free p ∪ Free q.

(73) Free p ⇔ q = Free p ∪ Free q.

(74) Free ∃_{x}p = Free p.

Let us consider p. The functor Fixed p yielding an element of 2^{FixedVar}, is defined as follows:

Fixed p = Vars_{FixedVar}(p).

Next we state a number of propositions:

(75) Fixed p = Vars_{FixedVar}(p).

(76) Fixed VERUM = ∅.

(77) For every k-ary predicate symbol P and for every list of variables l of the length k holds Fixed(P [l]) = {l(i) : 1 ≤ i∧i ≤ len l∧l(i) ∈ FixedVar}.

(78) Fixed ¬p = Fixed p.

(79) Fixed FALSUM = ∅.

(80) Fixed p ∧ q = Fixed p ∪ Fixed q.

(81) Fixed(∀_{x}p) = Fixed p.

(82) Fixed p ∨ q = Fixed p ∪ Fixed q.

(83) Fixed p ⇒ q = Fixed p ∪ Fixed q.

(84) Fixed p ⇔ q = Fixed p ∪ Fixed q.

(85) Fixed(∃_{x}p) = Fixed p.

References

[1] Grzegorz Bancerek. Connectives and subformulae of the first order language. Formalized Mathematics, 1(3):451–458, 1990.

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

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

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

[5] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

470

Monotone Real Sequences. Subsequences

Jarosław Kotowicz^{1} Warsaw University, Białystok

Summary. The article contains definitions of constant, increasing, decreasing, non decreasing, non increasing sequences, the definition of a subsequence and their basic properties.

MML Identifier: SEQM 3.

The articles [2], [4], [3], [1], and [5] provide the terminology and notation for this paper. We adopt the following convention: n, m, k will be natural numbers, r will be a real number, and seq, seq_{1}, seq_{2} will be sequences of real numbers. We now define five new predicates. Let us consider seq. We say that seq is increasing if and only if:

for every n holds seq(n) < seq(n + 1). We say that seq is decreasing if and only if:

for every n holds seq(n + 1) < seq(n). We say that seq is non-decreasing if and only if:

for every n holds seq(n) ≤ seq(n + 1). We say that seq is non-increasing if and only if:

for every n holds seq(n + 1) ≤ seq(n). We say that seq is constant if and only if:

there exists r such that for every n holds seq(n) = r.

Let us consider seq. We say that seq is monotone if and only if:

seq is non-decreasing or seq is non-increasing.

We now state a number of propositions:

(1) seq is increasing if and only if for every n holds seq(n) < seq(n + 1).

(2) seq is decreasing if and only if for every n holds seq(n + 1) < seq(n).

(3) seq is non-decreasing if and only if for every n holds seq(n) ≤ seq(n+1).

(4) seq is non-increasing if and only if for every n holds seq(n+1) ≤ seq(n).

1Supported by RPBP III.24

472 Jarosław Kotowicz

(5) seq is constant if and only if there exists r such that for every n holds seq(n) = r.

(6) seq is monotone if and only if seq is non-decreasing or seq is nonincreasing.

(7) seq is increasing if and only if for all n, m such that n < m holds seq(n) < seq(m).

(8) seq is increasing if and only if for all n, k holds seq(n) < seq((n+1)+k).

(9) seq is decreasing if and only if for all n, k holds seq((n+1)+k) < seq(n).

(10) seq is decreasing if and only if for all n, m such that n < m holds seq(m) < seq(n).

(11) seq is non-decreasing if and only if for all n, k holds seq(n) ≤ seq(n+k).

(12) seq is non-decreasing if and only if for all n, m such that n ≤ m holds seq(n) ≤ seq(m).

(13) seq is non-increasing if and only if for all n, k holds seq(n+k) ≤ seq(n).

(14) seq is non-increasing if and only if for all n, m such that n ≤ m holds seq(m) ≤ seq(n).

(15) seq is constant if and only if there exists r such that rng seq = {r}.

(16) seq is constant if and only if for every n holds seq(n) = seq(n + 1).

(17) seq is constant if and only if for all n, k holds seq(n) = seq(n + k).

(18) seq is constant if and only if for all n, m holds seq(n) = seq(m).

(19) If seq is increasing, then for every n such that 0 < n holds seq(0) < seq(n).

(20) If seq is decreasing, then for every n such that 0 < n holds seq(n) < seq(0).

(21) If seq is non-decreasing, then for every n holds seq(0) ≤ seq(n).

(22) If seq is non-increasing, then for every n holds seq(n) ≤ seq(0).

(23) If seq is increasing, then seq is non-decreasing.

(24) If seq is decreasing, then seq is non-increasing.

(25) If seq is constant, then seq is non-decreasing.

(26) If seq is constant, then seq is non-increasing.

(27) If seq is non-decreasing and seq is non-increasing, then seq is constant. A sequence of real numbers is said to be an increasing sequence of naturals

if:

rng it ⊆ and for every n holds it(n) < it(n + 1).

Let us consider seq, k. The functor seq^{}k yielding a sequence of real numbers, is defined as follows:

for every n holds (seq^{}k)(n) = seq(n + k).

In the sequel Nseq, Nseq_{1} will be increasing sequences of naturals. Next we state four propositions:

(28) seq is an increasing sequence of naturals if and only if rng seq ⊆ and for every n holds seq(n) < seq(n + 1).

Monotone Real Sequences. … 473

(29) seq is an increasing sequence of naturals if and only if seq is increasing and for every n holds seq(n) is a natural number.

(30) seq_{1} = seq^{}k if and only if for every n holds seq_{1}(n) = seq(n + k).

(31) For every n holds (seq · Nseq)(n) = seq(Nseq(n)).

Let us consider Nseq, n. Then Nseq(n) is a natural number.

Let us consider Nseq, seq. Then seq · Nseq is a sequence of real numbers. Let us consider Nseq, Nseq_{1}. Then Nseq_{1} · Nseq is an increasing sequence

of naturals. Let us consider Nseq, k. Then Nseq^{}k is an increasing sequence of naturals. Let us consider seq, seq_{1}. We say that seq is a subsequence of seq_{1} if and

only if:

there exists Nseq such that seq = seq_{1} · Nseq.

Next we state a number of propositions:

(32) seq is a subsequence of seq_{1} if and only if there exists Nseq such that seq = seq_{1} · Nseq.

(33) For every n holds n ≤ Nseq(n).

(34) seq^{}0 = seq.

(35) (seq^{}k)^{}m = (seq^{}m)^{}k.

(36) (seq^{}k)^{}m = seq^{}(k + m).

(37) (seq + seq_{1})^{}k = seq^{}k + seq_{1} ^{}k.

(38) (−seq)^{}k = −seq^{}k.

(39) (seq − seq_{1})^{}k = seq^{}k − seq_{1} ^{}k.

(40) If seq is non-zero, then seq^{}k is non-zero.

(41) If seq is non-zero, then seq^{−1} ^{}k = (seq^{}k)^{−1}.

(42) (seq · seq_{1})^{}k = (seq^{}k) · (seq_{1} ^{}k).

(43) If seq_{1} is non-zero, then \frac{seq}{seq_{1}} ^{}k = \frac{seq^{}k}{seq_{1}^{}k}.

(44) (r · seq)^{}k = r · (seq^{}k).

(45) (seq · Nseq)^{}k = seq · (Nseq^{}k).

(46) seq is a subsequence of seq.

(47) seq^{}k is a subsequence of seq.

(48) If seq is a subsequence of seq_{1} and seq_{1} is a subsequence of seq_{2}, then seq is a subsequence of seq_{2}.

(49) If seq is increasing and seq_{1} is a subsequence of seq, then seq_{1} is increasing.

(50) If seq is decreasing and seq_{1} is a subsequence of seq, then seq_{1} is decreasing.

(51) If seq is non-decreasing and seq_{1} is a subsequence of seq, then seq_{1} is non-decreasing.

(52) If seq is non-increasing and seq_{1} is a subsequence of seq, then seq_{1} is non-increasing.

(53) If seq is monotone and seq_{1} is a subsequence of seq, then seq_{1} is monotone.

(54) If seq is constant and seq_{1} is a subsequence of seq, then seq_{1} is constant.

(55) If seq is constant and seq_{1} is a subsequence of seq, then seq = seq_{1}.

(56) If seq is upper bounded and seq_{1} is a subsequence of seq, then seq_{1} is upper bounded.

(57) If seq is lower bounded and seq_{1} is a subsequence of seq, then seq_{1} is lower bounded.

(58) If seq is bounded and seq_{1} is a subsequence of seq, then seq_{1} is bounded.

(59) If seq is increasing and 0 < r, then r · seq is increasing but if seq is increasing and 0 = r, then r · seq is constant but if seq is increasing and r < 0, then r · seq is decreasing.

(60) If seq is decreasing and 0 < r, then r · seq is decreasing but if seq is decreasing and 0 = r, then r · seq is constant but if seq is decreasing and r < 0, then r · seq is increasing.

(61) If seq is non-decreasing and 0 ≤ r, then r · seq is non-decreasing but if seq is non-decreasing and r ≤ 0, then r · seq is non-increasing.

(62) If seq is non-increasing and 0 ≤ r, then r · seq is non-increasing but if seq is non-increasing and r ≤ 0, then r · seq is non-decreasing.

(63) If seq is increasing and seq_{1} is increasing, then seq + seq_{1} is increasing but if seq is decreasing and seq_{1} is decreasing, then seq+seq_{1} is decreasing but if seq is non-decreasing and seq_{1} is non-decreasing, then seq + seq_{1} is non-decreasing but if seq is non-increasing and seq_{1} is non-increasing, then seq + seq_{1} is non-increasing.

(64) If seq is increasing and seq_{1} is constant, then seq + seq_{1} is increasing but if seq is decreasing and seq_{1} is constant, then seq + seq_{1} is decreasing but if seq is non-decreasing and seq_{1} is constant, then seq + seq_{1} is nondecreasing but if seq is non-increasing and seq_{1} is constant, then seq+seq_{1} is non-increasing.

(65) If seq is constant, then for every r holds r · seq is constant and −seq is constant and |seq| is constant.

(66) If seq is constant and seq_{1} is constant, then seq · seq_{1} is constant and seq + seq_{1} is constant.

(67) If seq is constant and seq_{1} is constant, then seq − seq_{1} is constant.

(68) If seq is upper bounded and 0 < r, then r · seq is upper bounded but if seq is upper bounded and 0 = r, then r · seq is bounded but if seq is upper bounded and r < 0, then r · seq is lower bounded.

(69) If seq is lower bounded and 0 < r, then r · seq is lower bounded but if seq is lower bounded and 0 = r, then r · seq is bounded but if seq is lower bounded and r < 0, then r · seq is upper bounded.

(70) If seq is bounded, then for every r holds r · seq is bounded and −seq is bounded and |seq| is bounded.

(71) If seq is upper bounded and seq_{1} is upper bounded, then seq + seq_{1} is upper bounded but if seq is lower bounded and seq_{1} is lower bounded, then seq+seq_{1} is lower bounded but if seq is bounded and seq_{1} is bounded, then seq + seq_{1} is bounded.

(72) If seq is bounded and seq_{1} is bounded, then seq · seq_{1} is bounded and seq − seq_{1} is bounded.

(73) If seq is constant, then seq is bounded.

(74) If seq is constant, then for every r holds r · seq is bounded and −seq is bounded and |seq| is bounded.

(75) If seq is upper bounded and seq_{1} is constant, then seq + seq_{1} is upper bounded but if seq is lower bounded and seq_{1} is constant, then seq + seq_{1} is lower bounded but if seq is bounded and seq_{1} is constant, then seq+seq_{1} is bounded.

(76) If seq is upper bounded and seq_{1} is constant, then seq − seq_{1} is upper bounded but if seq is lower bounded and seq_{1} is constant, then seq − seq_{1} is lower bounded but if seq is bounded and seq_{1} is constant, then seq−seq_{1} is bounded and seq_{1} − seq is bounded and seq · seq_{1} is bounded.

(77) If seq is upper bounded and seq_{1} is non-increasing, then seq + seq_{1} is upper bounded.

(78) If seq is lower bounded and seq_{1} is non-decreasing, then seq + seq_{1} is lower bounded.

References

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

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

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

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

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

476

Convergent Real Sequences. Upper and Lower Bound of Sets of Real

Numbers

Jarosław Kotowicz^{1} Warsaw University, Białystok

Summary. The article contains theorems about convergent sequences and the limit of sequences occurring in [3] such as Bolzano-Weirrstrass theorem, Cauchy theorem and others. Bounded sets of real numbers and lower and upper bound of subset of real numbers are defined.

MML Identifier: SEQ 4.

The papers [7], [2], [5], [3], [1], [4], [8], and [6] provide the notation and terminology for this paper. For simplicity we follow a convention: n, k, m will denote natural numbers, r, r_{1}, p, g, g_{1}, g_{2}, s will denote real numbers, seq, seq_{1} will denote sequences of real numbers, Nseq will denote an increasing sequence of naturals, and X, Y will denote subsets of . One can prove the following propositions:

(1) If 0 < r_{1} and r_{1} ≤ r and 0 < g, then \frac{g}{r} ≤ \frac{g}{r_{1}} .

(2) If r < p, then 0 < p − r.

(3) r − (r − s) = s and r + (s − r) = s and (r + s) − r = s.

(4) If 0 < s, then 0 < \frac{s}{3}.

(5) (\frac{s}{3} + \frac{s}{3}) + \frac{s}{3} = s.

(6) If 0 < g and 0 < r and g ≤ g_{1} and r < r_{1}, then g · r < g_{1} · r_{1} and r · g < r_{1} · g_{1}.

(7) If 0 < g and 0 < r and g ≤ g_{1} and r ≤ r_{1}, then g · r ≤ g_{1} · r_{1} and r · g ≤ r_{1} · g_{1}.

(8) Given X, Y. Then if there exists r such that r ∈ X and there exists r such that r ∈ Y and for all r, p such that r ∈ X and p ∈ Y holds r < p, then there exists g such that for all r, p such that r ∈ X and p ∈ Y holds r ≤ g and g ≤ p.

1Supported by RPBP III.24

(9) If 0 < p and there exists r such that r ∈ X and for every r such that r ∈ X holds r + p ∈ X, then for every g there exists r such that r ∈ X and g < r.

(10) For every r there exists n such that r < n.

We now define two new predicates. Let us consider X. Let us assume that there exists r such that r ∈ X. We say that X is upper bounded if and only if:

there exists p such that for every r such that r ∈ X holds r ≤ p. We say that X is lower bounded if and only if:

there exists p such that for every r such that r ∈ X holds p ≤ r. Let us consider X. Let us assume that there exists r such that r ∈ X. We

say that X is bounded if and only if:

X is lower bounded and X is upper bounded. We now state several propositions:

(11) If there exists r such that r ∈ X, then X is upper bounded if and only if there exists p such that for every r such that r ∈ X holds r ≤ p.

(12) If there exists r such that r ∈ X, then X is lower bounded if and only if there exists p such that for every r such that r ∈ X holds p ≤ r.

(13) If there exists r such that r ∈ X, then X is bounded if and only if X is upper bounded and X is lower bounded.

(14) If there exists r such that r ∈ X, then X is bounded if and only if there exists s such that 0 < s and for every r such that r ∈ X holds |r| < s.

(15) If X = {r}, then X is bounded.

(16) If there exists r such that r ∈ X and X is upper bounded, then there exists g such that for every r such that r ∈ X holds r ≤ g and for every s such that 0 < s there exists r such that r ∈ X and g − s < r.

(17) Suppose that (i) for every r such that r ∈ X holds r ≤ g_{1}, (ii) for every s such that 0 < s there exists r such that r ∈ X and g_{1}−s < r, (iii) for every r such that r ∈ X holds r ≤ g_{2}, (iv) for every s such that 0 < s there exists r such that r ∈ X and g_{2}−s < r.

Then g_{1} = g_{2}.

(18) If there exists r such that r ∈ X and X is lower bounded, then there exists g such that for every r such that r ∈ X holds g ≤ r and for every s such that 0 < s there exists r such that r ∈ X and r < g + s.

(19) Suppose that (i) for every r such that r ∈ X holds g_{1} ≤ r, (ii) for every s such that 0 < s there exists r such that r ∈ X and r < g_{1}+s, (iii) for every r such that r ∈ X holds g_{2} ≤ r, (iv) for every s such that 0 < s there exists r such that r ∈ X and r < g_{2}+s.

Then g_{1} = g_{2}. Let us consider X. Let us assume that there exists r such that r ∈ X and

X is upper bounded. The functor sup X yielding a real number, is defined as follows:

Convergent Real Sequences. … 479

for every r such that r ∈ X holds r ≤ sup X and for every s such that 0 < s there exists r such that r ∈ X and (sup X) − s < r.

Let us consider X. Let us assume that there exists r such that r ∈ X and X is lower bounded. The functor inf X yields a real number and is defined by:

for every r such that r ∈ X holds inf X ≤ r and for every s such that 0 < s there exists r such that r ∈ X and r < (inf X) + s.

One can prove the following propositions:

(20) If there exists r such that r ∈ X and X is upper bounded, then sup X = g if and only if for every r such that r ∈ X holds r ≤ g and for every s such that 0 < s there exists r such that r ∈ X and g − s < r.

(21) If there exists r such that r ∈ X and X is lower bounded, then inf X = g if and only if for every r such that r ∈ X holds g ≤ r and for every s such that 0 < s there exists r such that r ∈ X and r < g + s.

(22) If X = {r}, then inf X = r and sup X = r.

(23) If X = {r}, then inf X = sup X.

(24) If X is bounded and there exists r such that r ∈ X, then inf X ≤ sup X.

(25) If X is bounded and there exists r such that r ∈ X, then there exist r,

p such that r ∈ X and p ∈ X and p ≠ r if and only if inf X < sup X. The scheme SepNat concerns a unary predicate P, and states that: there exists a X being sets of natural numbers such that for every n holds

n ∈ X if and only if P[n] for all values of the parameter.

We now state a number of propositions:

(26) If seq is convergent, then |seq| is convergent.

(27) If seq is convergent, then lim |seq| = | lim seq|.

(28) If |seq| is convergent and lim |seq| = 0, then seq is convergent and lim seq = 0.

(29) If seq_{1} is a subsequence of seq and seq is convergent, then seq_{1} is convergent.

(30) If seq_{1} is a subsequence of seq and seq is convergent, then lim seq_{1} = lim seq.

(31) If seq is convergent and there exists k such that for every n such that k ≤ n holds seq_{1}(n) = seq(n), then seq_{1} is convergent.

(32) If seq is convergent and there exists k such that for every n such that k ≤ n holds seq_{1}(n) = seq(n), then lim seq = lim seq_{1}.

(33) If seq is convergent, then seq^{}k is convergent and lim(seq^{}k) = lim seq.

(34) If seq is convergent and there exists k such that seq_{1} = seq ^{}k, then seq_{1} is convergent and lim seq_{1} = lim seq.

(35) If seq is convergent and there exists k such that seq = seq_{1} ^{}k, then seq_{1} is convergent.

(36) If seq is convergent and there exists k such that seq = seq_{1} ^{}k, then lim seq_{1} = lim seq.

(37) If seq is convergent and lim seq ≠ 0, then there exists k such that seq^{}k is non-zero.

(38) If seq is convergent and lim seq ≠ 0, then there exists seq_{1} such that seq_{1} is a subsequence of seq and seq_{1} is non-zero.

(39) If seq is constant, then seq is convergent.

(40) If seq is constant and r ∈ rng seq or seq is constant and there exists n such that seq(n) = r, then lim seq = r.

(41) If seq is constant, then for every n holds lim seq = seq(n).

(42) If seq is convergent and lim seq ≠ 0, then for every seq_{1} such that seq_{1} is a subsequence of seq and seq_{1} is non-zero holds lim seq_{1}^{−1} = (lim seq)^{−1}.

(43) For all r, seq such that 0 < r and for every n holds seq(n) = \frac{1}{n+r} holds seq is convergent.

(44) For all r, seq such that 0 < r and for every n holds seq(n) = \frac{1}{n+r} holds lim seq = 0.

(45) If for every n holds seq(n) = \frac{1}{n+1}, then seq is convergent and lim seq = 0.

(46) If 0 < r and for every n holds seq(n) = \frac{g}{n+r}, then seq is convergent and lim seq = 0.

(47) For all r, seq such that 0 < r and for every n holds seq(n) = \frac{1}{n·n+r} holds seq is convergent.

(48) For all r, seq such that 0 < r and for every n holds seq(n) = \frac{1}{n·n+r} holds lim seq = 0.

(49) If for every n holds seq(n) = \frac{1}{n·n+1}, then seq is convergent and lim seq = 0.

(50) If 0 < r and for every n holds seq(n) = \frac{g}{n·n+r}, then seq is convergent and lim seq = 0.

(51) If seq is non-decreasing and seq is upper bounded, then seq is convergent.

(52) If seq is non-increasing and seq is lower bounded, then seq is convergent.

(53) If seq is monotone and seq is bounded, then seq is convergent.

(54) If seq is upper bounded and seq is non-decreasing, then for every n holds seq(n) ≤ lim seq.

(55) If seq is lower bounded and seq is non-increasing, then for every n holds lim seq ≤ seq(n).

(56) For every seq there exists Nseq such that seq · Nseq is monotone.

(57) If seq is bounded, then there exists seq_{1} such that seq_{1} is a subsequence of seq and seq_{1} is convergent.

(58) seq is convergent if and only if for every s such that 0 < s there exists n such that for every m such that n ≤ m holds |seq(m) − seq(n)| < s.

(59) Suppose seq is constant and seq_{1} is convergent. Then lim(seq +seq_{1}) = seq(0) + lim seq_{1} and lim(seq − seq_{1}) = seq(0) − lim seq_{1} and lim(seq_{1} −

seq) = lim seq_{1} − seq(0) and lim(seq · seq_{1}) = seq(0) · (lim seq_{1}).

References

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

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

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

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

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

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

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

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

482

Midpoint algebras

Michał Muzalewski^{1} Warsaw University, Białystok

Summary. In this article basic properties of midpoint algebras are proved. We define a congruence relation ≡ on bound vectors and free vectors as the equivalence classes of ≡.

MML Identifier: MIDSP 1.

The notation and terminology used in this paper are introduced in the following articles: [5], [1], [2], [3], [4], and [6]. We consider midpoint algebra structures which are systems

⟨ points, a midpoint operation ⟩ where the points is a non-empty set and the midpoint operation is a binary

operation on the points. In the sequel MS is a midpoint algebra structure and a, b are elements of the points of MS. Let us consider MS, a, b. The functor a ⊕ b yielding an element of the points of MS, is defined by:

a ⊕ b = (the midpoint operation of MS)(a, b). We now state a proposition

(1) a ⊕ b = (the midpoint operation of MS)(a, b). Let x be arbitrary. Then {x} is a non-empty set. zo is a binary operation on {0}. One can prove the following propositions:

(2) zo is a function from [: {0}, {0}:] into {0}.

(3) For all elements x, y of {0} holds zo(x, y) = 0. The midpoint algebra structure EX is defined by: EX = ⟨{0}, zo⟩. The following propositions are true:

(4) EX = ⟨{0}, zo⟩.

(5) The points of EX = {0}.

1Supported by RPBP.III-24.C6.

484 Michał Muzalewski

(6) The midpoint operation of EX = zo.

(7) For every element a of the points of EX holds a = 0.

(8) For all elements a, b of the points of EX holds a ⊕ b = zo(a, b).

(9) For all elements a, b of the points of EX holds a ⊕ b = 0.

(10) For all elements a, b, c, d of the points of EX holds a ⊕ a = a and a ⊕ b = b ⊕ a and (a ⊕ b) ⊕ (c ⊕ d) = (a ⊕ c) ⊕ (b ⊕ d) and there exists an element x of the points of EX such that x ⊕ a = b.

A midpoint algebra structure is called a midpoint algebra if:

for all elements a, b, c, d of the points of it holds a ⊕ a = a and a ⊕ b = b ⊕ a and (a ⊕ b) ⊕ (c ⊕ d) = (a ⊕ c) ⊕ (b ⊕ d) and there exists an element x of the points of it such that x ⊕ a = b.

We follow the rules: M denotes a midpoint algebra and a, b, c, d, a′, b′, c′, d′, x, y, x′ denote elements of the points of M. Next we state several propositions:

(11) a ⊕ a = a.

(12) a ⊕ b = b ⊕ a.

(13) (a ⊕ b) ⊕ (c ⊕ d) = (a ⊕ c) ⊕ (b ⊕ d).

(14) There exists x such that x ⊕ a = b.

(15) (a ⊕ b) ⊕ c = (a ⊕ c) ⊕ (b ⊕ c).

(16) a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ (a ⊕ c).

(17) If a ⊕ b = a, then a = b.

(18) If x ⊕ a = x′ ⊕ a, then x = x′.

(19) If a ⊕ x = a ⊕ x′, then x = x′.

Let us consider M, a, b, c, d. The predicate a, b ≡ c, d is defined by:

a ⊕ d = b ⊕ c.

The following propositions are true:

(20) a, b ≡ c, d if and only if a ⊕ d = b ⊕ c.

(21) a, a ≡ b, b.

(22) If a, b ≡ c, d, then c, d ≡ a, b.

(23) If a, a ≡ b, c, then b = c.

(24) If a, b ≡ c, c, then a = b.

(25) a, b ≡ a, b.

(26) There exists d such that a, b ≡ c, d.

(27) If a, b ≡ c, d and a, b ≡ c, d′, then d = d′.

(28) If x, y ≡ a, b and x, y ≡ c, d, then a, b ≡ c, d.

(29) If a, b ≡ a′, b′ and b, c ≡ b′, c′, then a, c ≡ a′, c′.

In the sequel p, q, r will denote elements of [: the points of M, the points of M :]. Let us consider M, p. Then p_{1} is an element of the points of M.

Let us consider M, p. Then p_{2} is an element of the points of M.

Let us consider M, p, q. The predicate p ≡ q is defined as follows:

p_{1}, p_{2} ≡ q_{1}, q_{2}.

Midpoint algebras 485

One can prove the following proposition

(30) p ≡ q if and only if p_{1}, p_{2} ≡ q_{1}, q_{2}.

Let us consider M, a, b. Then ⟨⟨a, b⟩⟩ is an element of [: the points of M, the points of M :].

One can prove the following propositions:

(31) If a, b ≡ c, d, then ⟨⟨a, b⟩⟩ ≡ ⟨⟨c, d⟩⟩.

(32) If ⟨⟨a, b⟩⟩ ≡ ⟨⟨c, d⟩⟩, then a, b ≡ c, d.

(33) p ≡ p.

(34) If p ≡ q, then q ≡ p.

(35) If p ≡ q and p ≡ r, then q ≡ r.

(36) If p ≡ r and q ≡ r, then p ≡ q.

(37) If p ≡ q and q ≡ r, then p ≡ r.

(38) If p ≡ q, then r ≡ p if and only if r ≡ q.

(39) For every p holds {q : q ≡ p}is a non-empty subset of [: the points of

M, the points of M :]. Let us consider M, p. The functor p^{}yields a non-empty subset of [: the

points of M, the points of M :] and is defined as follows: p^{}= {q : q ≡ p}. The following propositions are true:

(40) For every p holds p^{}= {q : q ≡ p} and p^{}is a non-empty subset of [: the points of M, the points of M :].

(41) For every p holds r ∈ p^{}if and only if r ≡ p.

(42) If p ≡ q, then p^{}= q^{}.

(43) If p^{}= q^{}, then p ≡ q.

(44) If ⟨⟨a, b⟩⟩^{}= ⟨⟨c, d⟩⟩^{}, then a ⊕ d = b ⊕ c.

(45) p ∈ p^{}.

Let us consider M. A non-empty subset of [: the points of M, the points of M :] is said to be a vector of M if:

there exists p such that it = p^{}.

The following proposition is true

(46) For every non-empty subset X of [: the points of M, the points of M :]

holds X is a vector of M if and only if there exists p such that X = p^{}. In the sequel u, v, w, w′ denote vectors of M. The following proposition is

true

(47) p^{}is a vector of M.

Let us consider M, p. Then p^{}is a vector of M.

We now state a proposition

(48) There exists u such that for every p holds p ∈ u if and only if p_{1} = p_{2}. Let us consider M. The functor I_{M} yielding a vector of M, is defined by: I_{M} = {p : p_{1} = p_{2}}.

Next we state four propositions:

(49) I_{M} = {p : p_{1} = p_{2}}.

(50) I_{M} = ⟨⟨b, b⟩⟩^{}.

(51) There exist w, p, q such that u = p^{}and v = q^{}and p_{2} = q_{1} and w = ⟨⟨p_{1}, q_{2}⟩⟩^{}.

(52) Suppose that

(i) there exist p, q such that u = p^{}and v = q^{}and p_{2} = q_{1} and w = ⟨⟨p_{1}, q_{2}⟩⟩^{},

(ii) there exist p, q such that u = p^{}and v = q^{}and p_{2} = q_{1} and w′ = ⟨⟨p_{1}, q_{2}⟩⟩^{}. Then w = w′.

Let us consider M, u, v. The functor u+v yields a vector of M and is defined by:

there exist p, q such that u = p^{}and v = q^{}and p_{2} = q_{1} and u + v = ⟨⟨p_{1}, q_{2}⟩⟩^{}.

We now state a proposition

(53) There exists b such that u = ⟨⟨a, b⟩⟩^{}.

Let us consider M, a, b. The functor \overrightarrow{[a, b]} yields a vector of M and is defined by:

\overrightarrow{[a, b]} = ⟨⟨a, b⟩⟩^{}.

Next we state a number of propositions:

(54) \overrightarrow{[a, b]} = ⟨⟨a, b⟩⟩^{}.

(55) There exists b such that u = \overrightarrow{[a, b]}.

(56) If ⟨⟨a, b⟩⟩ ≡ ⟨⟨c, d⟩⟩, then \overrightarrow{[a, b]} = \overrightarrow{[c, d]}.

(57) If \overrightarrow{[a, b]} = \overrightarrow{[c, d]}, then a ⊕ d = b ⊕ c.

(58) I_{M} = \overrightarrow{[b, b]}.

(59) If \overrightarrow{[a, b]} = \overrightarrow{[a, c]}, then b = c.

(60) \overrightarrow{[a, b]} + \overrightarrow{[b, c]} = \overrightarrow{[a, c]}.

(61) ⟨⟨a, a ⊕ b⟩⟩ ≡ ⟨⟨a ⊕ b, b⟩⟩.

(62) \overrightarrow{[a, a ⊕ b]} + \overrightarrow{[a, a ⊕ b]} = \overrightarrow{[a, b]}.

(63) (u + v) + w = u + (v + w).

(64) u + I_{M} = u.

(65) There exists v such that u + v = I_{M}.

(66) u + v = v + u.

(67) If u + v = u + w, then v = w.

Let us consider M, u. The functor −u yields a vector of M and is defined by: u + (−u) = I_{M}. We now state a proposition

(68) u + (−u) = I_{M}.

In the sequel X denotes an element of 2^{[: the points of M, the points of M :]}. Let us consider M. The functor setvect M yields a set and is defined as follows:

setvect M = {X : X is a vector of M}. Next we state a proposition

(69) setvect M = {X : X is a vector of M}.

In the sequel x is arbitrary. One can prove the following two propositions:

(70) u is an element of 2^{[: the points of M, the points of M :]}.

(71) x is a vector of M if and only if x ∈ setvect M.

Let us consider M. Then setvect M is a non-empty set. The following proposition is true

(72) x is a vector of M if and only if x is an element of setvect M. In the sequel u_{1}, v_{1}, w_{1}, W, W_{1}, W_{2}, T will denote elements of setvect M.

Let us consider M, u_{1}, v_{1}. The functor u_{1} + v_{1} yields an element of setvect M and is defined as follows:

for all u, v such that u_{1} = u and v_{1} = v holds u_{1} + v_{1} = u + v. One can prove the following propositions:

(73) If u_{1} = u and v_{1} = v, then u_{1} + v_{1} = u + v.

(74) u_{1} + v_{1} = v_{1} + u_{1}.

(75) (u_{1} + v_{1}) + w_{1} = u_{1} + (v_{1} + w_{1}).

Let us consider M. The functor addvect M yields a binary operation on setvect M and is defined as follows:

for all u_{1}, v_{1} holds (addvect M)(u_{1}, v_{1}) = u_{1} + v_{1}. The following three propositions are true:

(76) (addvect M)(u_{1}, v_{1}) = u_{1} + v_{1}.

(77) For every W there exists T such that W + T = I_{M}.

(78) For all W, W_{1}, W_{2} such that W + W_{1} = I_{M} and W + W_{2} = I_{M} holds

W_{1} = W_{2}. Let us consider M. The functor complvect M yielding a unary operation on

setvect M, is defined by:

for every W holds W + (complvect M)(W) = I_{M}. One can prove the following proposition

(79) W + (complvect M)(W) = I_{M}.

Let us consider M. The functor zerovect M yields an element of setvect M and is defined as follows:

zerovect M = I_{M}. The following proposition is true

(80) zerovect M = I_{M}.

Let us consider M. The functor vectgroup M yielding a group structure, is defined by:

vectgroup M = ⟨setvect M, addvect M, complvect M, zerovect M⟩.

Next we state several propositions:

(81) vectgroup M = ⟨setvect M, addvect M, complvect M, zerovect M⟩.

(82) The carrier of vectgroup M = setvect M.

(83) The addition of vectgroup M = addvect M.

(84) The reverse-map of vectgroup M = complvect M.

(85) The zero of vectgroup M = zerovect M.

(86) vectgroup M is an Abelian group.

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175– 180, 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] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

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

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

The Fundamental Logic Structure in Quantum Mechanics

Paweł Sadowski Warsaw University

Białystok Andrzej Trybulec Warsaw University Białystok Konrad Raczkowski Warsaw University Białystok

Summary. In this article we present the logical structure given by four axioms of Mackey [3] in the set of propositions of Quantum Mechanics. The equivalence relation (PropRel(Q)) in the set of propositions (Prop Q) for given Quantum Mechanics Q is considered. The main text for this article is [6] where the structure of quotient space and the properties of equivalence relations, classes and partitions are studied.

MML Identifier: QMAX 1.

The articles [10], [1], [4], [2], [9], [8], [7], [5], and [6] provide the notation and terminology for this paper. In the sequel x will be arbitrary, X will be a nonempty set, and X_{1} will be a set. Let us consider X, and let S be a σ-field of subsets of X. The functor probabilities S yields a non-empty set and is defined by:

x ∈ probabilities S if and only if x is a probability on S.

We now state a proposition

(1) For every σ-field S of subsets of X holds x ∈ probabilities S if and only if x is a probability on S.

We consider quantum mechanics structures which are systems

⟨ observables, states, a probability ⟩

where the observables, the states are non-empty sets and the probability is a function from [: the observables, the states :] into probabilities the Borel sets. In the sequel Q denotes a quantum mechanics structure. We now define two new functors. Let us consider Q. The functor Obs Q yields a non-empty set and is defined by:

Obs Q = the observables of Q. The functor Sts Q yields a non-empty set and is defined by:

Sts Q = the states of Q.

The following propositions are true:

490 Paweł Sadowski et al.

(2) Obs Q = the observables of Q.

(3) Sts Q = the states of Q.

We adopt the following convention: A_{1}, A_{2} will denote elements of Obs Q, s, s_{1}, s_{2} will denote elements of Sts Q, and E will denote an event of the Borel sets. Let us consider Q, A_{1}, s. The functor Meas(A_{1}, s) yielding a probability on the Borel sets, is defined as follows:

Meas(A_{1}, s) = (the probability of Q)(⟨⟨A_{1}, s⟩⟩).

One can prove the following proposition

(4) Meas(A_{1}, s) = (the probability of Q)(⟨⟨A_{1}, s⟩⟩).

A quantum mechanics structure is said to be a quantum mechanics if:

(i) for all elements A_{1}, A_{2} of Obs it such that for every element s of Sts it holds Meas(A_{1}, s) = Meas(A_{2}, s) holds A_{1} = A_{2}, (ii) for all elements s_{1}, s_{2} of Sts it such that for every element A of Obs it holds Meas(A, s_{1}) = Meas(A, s_{2}) holds s_{1} = s_{2}, (iii) for every elements s_{1}, s_{2} of Sts it there exists an element s of Sts it such that for every element A of Obs it and for every E there exists a real number t such that 0 ≤ t and t ≤ 1 and Meas(A, s)(E) = t · Meas(A, s_{1})(E) + (1 − t) · Meas(A, s_{2})(E).

Next we state a proposition

(5) Q is a quantum mechanics if and only if the following conditions are satisfied:

(i) for all A_{1}, A_{2} such that for every s holds Meas(A_{1}, s) = Meas(A_{2}, s) holds A_{1} = A_{2},

(ii) for all s_{1}, s_{2} such that for every A_{1} holds Meas(A_{1}, s_{1}) = Meas(A_{1}, s_{2})

holds s_{1} = s_{2}, (iii) for every s_{1}, s_{2} there exists s such that for every A_{1}, E there exists

a real number t such that 0 ≤ t and t ≤ 1 and Meas(A_{1}, s)(E) = t · Meas(A_{1}, s_{1})(E) + (1 − t) · Meas(A_{1}, s_{2})(E).

We follow the rules: Q denotes a quantum mechanics, A, A_{1}, A_{2} denote elements of Obs Q, and s, s_{1}, s_{2} denote elements of Sts Q. We now state three propositions:

(6) If for every s holds Meas(A_{1}, s) = Meas(A_{2}, s), then A_{1} = A_{2}.

(7) If for every A holds Meas(A, s_{1}) = Meas(A, s_{2}), then s_{1} = s_{2}.

(8) For every s_{1}, s_{2} there exists s such that for every A, E there exists a real number t such that 0 ≤ t and t ≤ 1 and Meas(A, s)(E) = t · Meas(A, s_{1})(E) + (1 − t) · Meas(A, s_{2})(E).

We consider POI structures which are systems

⟨ a carrier, an ordering, an involution ⟩

The Fundamental Logic Structure … 491

where the carrier is a set, the ordering is a relation on the carrier, and the involution is a function from the carrier into the carrier. In the sequel x_{1} will denote an element of X_{1}, Ord will denote a relation on X_{1}, and Inv will denote a function from X_{1} into X_{1}. Let us consider X_{1}. A POI structure is said to be a poset with involution over X_{1} if: the carrier of it = X_{1}. One can prove the following proposition

(9) For every poset W with involution over X_{1} holds the carrier of W = X_{1}. Let us consider X_{1}, Ord, Inv. The functor LOG(Ord, Inv) yielding a poset

with involution over X_{1}, is defined by:

LOG(Ord, Inv) = ⟨X_{1}, Ord, Inv⟩. Next we state a proposition

(10) LOG(Ord, Inv) = ⟨X_{1}, Ord, Inv⟩.

Let us consider X_{1}, Inv. We say that Inv is an involution in X_{1} if and only if:

Inv(Inv(x_{1})) = x_{1}. We now state a proposition

(11) Inv is an involution in X_{1} if and only if for every x_{1} holds

Inv(Inv(x_{1})) = x_{1} . Let us consider X_{1}, and let W be a poset with involution over X_{1}. We say

that W is a quantum logic on X_{1} if and only if:

there exists a relation Ord on X_{1} and there exists a function Inv from X_{1} into X_{1} such that W = LOG(Ord, Inv) and Ord partially orders X_{1} and Inv is an involution in X_{1} and for all elements x, y of X_{1} such that ⟨⟨x, y⟩⟩ ∈ Ord holds ⟨⟨Inv(y), Inv(x)⟩⟩ ∈ Ord.

Next we state a proposition

(12) Let W be a poset with involution over X_{1}. Then W is a quantum logic on X_{1} if and only if there exists a relation Ord on X_{1} and there exists a function Inv from X_{1} into X_{1} such that W = LOG(Ord, Inv) and Ord partially orders X_{1} and Inv is an involution in X_{1} and for all elements x, y of X_{1} such that ⟨⟨x, y⟩⟩ ∈ Ord holds ⟨⟨Inv(y), Inv(x)⟩⟩ ∈ Ord.

Let us consider Q. The functor Prop Q yielding a non-empty set, is defined by:

Prop Q = [: Obs Q, the Borel sets :]. The following proposition is true

(13) Prop Q = [: Obs Q, the Borel sets :].

In the sequel p, q, r, p_{1}, q_{1} are elements of Prop Q. Let us consider Q, p. Then p_{1} is an element of Obs Q. Then p_{2} is an event of the Borel sets.

The following propositions are true:

(14) p = ⟨⟨p_{1}, p_{2}⟩⟩.

(15) (E^{c})^{c} = E.

(16) For every E such that E = p_{2}^{c} holds

Meas(p_{1}, s)(p_{2}) = 1 − Meas(p_{1}, s)(E) . Let us consider Q, p. The functor ¬p yields an element of Prop Q and is

defined as follows:

¬p = ⟨⟨p_{1}, p_{2}^{c}⟩⟩.

492 Paweł Sadowski et al.

The following proposition is true

(17) ¬p = ⟨⟨p_{1}, p_{2}^{c}⟩⟩.

Let us consider Q, p, q. The predicate p ⊢ q is defined by: for every s holds Meas(p_{1}, s)(p_{2}) ≤ Meas(q_{1}, s)(q_{2}). We now state a proposition

(18) p ⊢ q if and only if for every s holds Meas(p_{1}, s)(p_{2}) ≤ Meas(q_{1}, s)(q_{2}). Let us consider Q, p, q. The predicate p ≡ q is defined as follows: p ⊢ q and q ⊢ p. One can prove the following propositions:

(19) p ≡ q if and only if p ⊢ q and q ⊢ p.

(20) p ≡ q if and only if for every s holds Meas(p_{1}, s)(p_{2}) = Meas(q_{1}, s)(q_{2}).

(21) p ⊢ p.

(22) If p ⊢ q and q ⊢ r, then p ⊢ r.

(23) p ≡ p.

(24) If p ≡ q, then q ≡ p.

(25) If p ≡ q and q ≡ r, then p ≡ r.

(26) (¬p)_{1} = p_{1} and (¬p)_{2} = p_{2}^{c}.

(27) ¬(¬p) = p.

(28) If p ⊢ q, then ¬q ⊢ ¬p.

Let us consider Q. The functor PropRel Q yields an equivalence relation of Prop Q and is defined as follows:

⟨⟨p, q⟩⟩ ∈ PropRel Q if and only if p ≡ q. We now state a proposition

(29) ⟨⟨p, q⟩⟩ ∈ PropRel Q if and only if p ≡ q.

In the sequel B, C will denote subsets of Prop Q. Next we state a proposition (30) For all B, C such that B ∈ Classes(PropRel Q) and

C ∈ Classes(PropRel Q)

for all elements a, b, c, d of Prop Q such that a ∈ B and b ∈ B and c ∈ C and d ∈ C and a ⊢ c holds b ⊢ d.

Let us consider Q. The functor OrdRel Q yielding a relation on Classes(PropRel Q) , is defined as follows: ⟨⟨B, C⟩⟩ ∈ OrdRel Q if and only if B ∈ Classes(PropRel Q) and C ∈ Classes(PropRel Q) and for all p, q such that p ∈ B and q ∈ C holds p ⊢ q. Next we state four propositions:

(31) ⟨⟨B, C⟩⟩ ∈ OrdRel Q if and only if B ∈ Classes(PropRel Q) and C ∈ Classes(PropRel Q) and for all p, q such that p ∈ B and q ∈ C holds p ⊢ q.

(32) p ⊢ q if and only if ⟨⟨[p]_{PropRel Q}, [q]_{PropRel Q}⟩⟩ ∈ OrdRel Q.

(33) For all B, C such that B ∈ Classes(PropRel Q) and C ∈ Classes(PropRel Q) for all p_{1}, q_{1} such that p_{1} ∈ B and q_{1} ∈ B and ¬p_{1} ∈ C holds ¬q_{1} ∈ C.

(34) For all B, C such that B ∈ Classes(PropRel Q) and C ∈ Classes(PropRel Q) for all p, q such that ¬p ∈ C and ¬q ∈ C and p ∈ B holds q ∈ B.

Let us consider Q. The functor InvRel Q yielding a function from Classes(PropRel Q) into Classes(PropRel Q), is defined by: (InvRel Q)([p]_{PropRel Q}) = [¬p]_{PropRel Q}. One can prove the following two propositions:

(35) (InvRel Q)([p]_{PropRel Q}) = [¬p]_{PropRel Q}.

(36) For every Q holds LOG(OrdRel Q, InvRel Q) is a quantum logic on Classes(PropRel Q).

References

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

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

[3] G.W.Mackey. The Mathematical Foundations of Quantum Mechanics. North Holland, New York,Amsterdam, 1963.

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

[5] Andrzej Ne‘dzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

[9] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.

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

494 Paweł Sadowski et al. Received December 18, 1989

Function Domains and Frænkel Operator

Andrzej Trybulec^{1} Warsaw University, Białystok

Summary. We deal with a non–empty set of functions and a non– empty set of functions from a set A to a non–empty set B. In the case when B is a non–empty set, B^{A} is redefined. It yields a non–empty set of functions from A to B. An element of such a set is redefined as a function from A to B. Some theorems concerning these concepts are proved, as well as a number of schemes dealing with infinity and the Axiom of Choice. The article contains a number of schemes allowing for simple logical transformations related to terms constructed with the Frænkel Operator.

MML Identifier: FRAENKEL.

The articles [5], [4], [6], [1], [2], and [3] provide the notation and terminology for this paper. In the sequel A, B will be non-empty sets. We now state a proposition

(1) For arbitrary x holds {x} is a non-empty set.

In the article we present several logical schemes. The scheme Fraenkel5’ deals with a non-empty set A, a unary functor F, and two unary predicates P and Q, and states that:

{F(v′) : P[v′]} ⊆ {F(u′) : Q[u′]} provided the parameters enjoy the following property:

• for every element v of A such that P[v] holds Q[v].

The scheme Fraenkel5” concerns a non-empty set A, a non-empty set B, a binary functor F, and two binary predicates P and Q, and states that:

{F(u_{1}, v_{1}) : P[u_{1}, v_{1}]} ⊆ {F(u_{2}, v_{2}) : Q[u_{2}, v_{2}]} provided the following condition is fulfilled:

• for every element u of A and for every element v of B such that P[u, v] holds Q[u, v].

The scheme Fraenkel6’ deals with a non-empty set A, a unary functor F, and two unary predicates P and Q, and states that:

1Supported by RPBP III.24 C1

496 Andrzej Trybulec

{F(v_{1}) : P[v_{1}]} = {F(v_{2}) : Q[v_{2}]} provided the following requirement is fulfilled:

• for every element v of A holds P[v] if and only if Q[v]. The scheme Fraenkel6” concerns a non-empty set A, a non-empty set B, a

binary functor F, and two binary predicates P and Q, and states that:

{F(u_{1}, v_{1}) : P[u_{1}, v_{1}]} = {F(u_{2}, v_{2}) : Q[u_{2}, v_{2}]} provided the parameters fulfill the following requirement:

• for every element u of A and for every element v of B holds P[u, v] if and only if Q[u, v].

The scheme FraenkelF’ concerns a non-empty set A, a unary functor F, a unary functor G, and a unary predicate P, and states that:

{F(v_{1}) : P[v_{1}]} = {G(v_{2}) : P[v_{2}]} provided the following requirement is met:

• for every element v of A holds F(v) = G(v). The scheme FraenkelF’R concerns a non-empty set A, a unary functor F, a

unary functor G, and a unary predicate P, and states that:

{F(v_{1}) : P[v_{1}]} = {G(v_{2}) : P[v_{2}]} provided the parameters fulfill the following condition:

• for every element v of A such that P[v] holds F(v) = G(v). The scheme FraenkelF” concerns a non-empty set A, a non-empty set B, a

binary functor F, a binary functor G, and a binary predicate P, and states that:

{F(u_{1}, v_{1}) : P[u_{1}, v_{1}]} = {G(u_{2}, v_{2}) : P[u_{2}, v_{2}]} provided the parameters meet the following requirement:

• for every element u of A and for every element v of B holds F(u, v) = G(u, v).

The scheme FraenkelF6”C deals with a non-empty set A, a non-empty set B, a binary functor F, and a binary predicate P, and states that:

{F(u_{1}, v_{1}) : P[u_{1}, v_{1}]} = {F(v_{2}, u_{2}) : P[u_{2}, v_{2}]} provided the following requirement is met:

• for every element u of A and for every element v of B holds F(u, v) = F(v, u).

The scheme FraenkelF6” deals with a non-empty set A, a non-empty set B, a binary functor F, and two binary predicates P and Q, and states that:

{F(u_{1}, v_{1}) : P[u_{1}, v_{1}]} = {F(v_{2}, u_{2}) : Q[u_{2}, v_{2}]} provided the parameters meet the following requirements:

• for every element u of A and for every element v of B holds P[u, v] if and only if Q[u, v],

• for every element u of A and for every element v of B holds F(u, v) = F(v, u).

The following propositions are true:

(2) For all non-empty sets A, B and for every function F from A into B and for every set X and for every element x of A such that x ∈ X holds (F X)(x) = F (x).

Function Domains and Frænkel Operator 497

(3) For all non-empty sets A, B and for all functions F, G from A into B and for every set X such that F X = G X for every element x of A such that x ∈ X holds F (x) = G(x).

(4) For every function f from A into B holds f ∈ B^{A}.

(5) For all sets A, B holds B^{A} ⊆ 2^{[: A, B:]}.

(6) For all sets X, Y such that Y^{X} ≠ ∅ and X ⊆ A and Y ⊆ B for every element f of Y^{X} holds f is a partial function from A to B.

Now we present a number of schemes. The scheme RelevantArgs deals with a non-empty set A, a non-empty set B, a set C, a function D from A into B, a function E from A into B, and two unary predicates P and Q, and states that:

{D(u′) : P[u′] ∧ u′ ∈ C} = {E(v′) : Q[v′] ∧ v′ ∈ C} provided the following requirements are met:

• D C = E C,

• for every element u of A such that u ∈ C holds P[u] if and only if Q[u].

The scheme Fr Set0 deals with a non-empty set A, and a unary predicate P, and states that:

{xx : P[xx]} ⊆ A for all values of the parameters.

The scheme Gen1” concerns a non-empty set A, a non-empty set B, a binary functor F, a unary predicate Q, and a binary predicate P, and states that:

for every element s of A and for every element t of B such that P[s, t] holds Q[F(s, t)] provided the parameters meet the following requirement:

• for arbitrary s_{t} such that s_{t} ∈ {F(s_{1}, t_{1}) : P[s_{1}, t_{1}]} holds Q[s_{t}].

The scheme Gen1”A deals with a non-empty set A, a non-empty set B, a binary functor F, a unary predicate Q, and a binary predicate P, and states that:

for arbitrary s_{t} such that s_{t} ∈ {F(s_{1}, t_{1}) : P[s_{1}, t_{1}]} holds Q[s_{t}] provided the following requirement is met:

• for every element s of A and for every element t of B such that P[s, t] holds Q[F(s, t)].

The scheme Gen2” deals with a non-empty set A, a non-empty set B, a nonempty set C, a binary functor F yielding an element of C, a unary predicate Q, and a binary predicate P, and states that:

{s_{t} : s_{t} ∈ {F(s_{1}, t_{1}) : P[s_{1}, t_{1}]}∧Q[s_{t}]} = {F(s_{2}, t_{2}) : P[s_{2}, t_{2}]∧Q[F(s_{2}, t_{2})]} for all values of the parameters.

The scheme Gen3’ concerns a non-empty set A, a unary functor F, and two unary predicates P and Q, and states that:

{F(s) : s ∈ {s_{1} : Q[s_{1}]} ∧ P[s]} = {F(s_{2}) : Q[s_{2}] ∧ P[s_{2}]} for all values of the parameters.

The scheme Gen3” concerns a non-empty set A, a non-empty set B, a binary functor F, a unary predicate Q, and a binary predicate P, and states that:

{F(s, t) : s ∈ {s_{1} : Q[s_{1}]} ∧ P[s, t]} = {F(s_{2}, t_{2}) : Q[s_{2}] ∧ P[s_{2}, t_{2}]} for all values of the parameters.

The scheme Gen4” deals with a non-empty set A, a non-empty set B, a binary functor F, and two binary predicates P and Q, and states that:

{F(s, t) : P[s, t]} ⊆ {F(s_{1}, t_{1}) : Q[s_{1}, t_{1}]} provided the following condition is satisfied:

• for every element s of A and for every element t of B such that P[s, t] there exists an element s′ of A such that Q[s′, t] and F(s, t) = F(s′, t).

The scheme FrSet 1 concerns a non-empty set A, a set B, a unary functor F, and a unary predicate P, and states that:

{F(y) : F(y) ∈ B ∧ P[y]} ⊆ B for all values of the parameters.

The scheme FrSet 2 deals with a non-empty set A, a set B, a unary functor F, and a unary predicate P, and states that:

{F(y) : P[y] ∧ F(y) ∉ B} misses B for all values of the parameters.

The scheme FrEqua1 deals with a non-empty set A, a non-empty set B, a binary functor F, an element C of B, and two binary predicates P and Q, and states that:

{F(s, t) : Q[s, t]} = {F(s′, C) : P[s′, C]} provided the parameters meet the following requirement:

• for every element s of A and for every element t of B holds Q[s, t] if and only if t = C and P[s, t].

The scheme FrEqua2 concerns a non-empty set A, a non-empty set B, a binary functor F, an element C of B, and a binary predicate P, and states that:

{F(s, t) : t = C ∧ P[s, t]} = {F(s′, C) : P[s′, C]} for all values of the parameters.

A non-empty set is said to be a non-empty set of functions if: for every element x of it holds x is a function. Next we state two propositions:

(7) A is a non-empty set of functions if and only if for every element x of A holds x is a function.

(8) For every function f holds {f} is a non-empty set of functions. Let A be a set, and let B be a non-empty set. A non-empty set of functions

is called a non-empty set of functions from A to B if:

for every element x of it holds x is a function from A into B. Next we state three propositions:

(9) For every set A and for every non-empty set B and for every non-empty set C of functions holds C is a non-empty set of functions from A to B if and only if for every element x of C holds x is a function from A into B.

(10) For every function f from A into B holds {f} is a non-empty set of functions from A to B.

(11) For every set A and for every non-empty set B holds B^{A} is a non-empty

set of functions from A to B. Let A be a set, and let B be a non-empty set. Then B^{A} is a non-empty set of

functions from A to B. Let F be a non-empty set of functions from A to B. We see that it makes sense to consider the following mode for restricted scopes of arguments. Then all the objests of the mode element of F are a function from A into B.

In the sequel phi will be an element of B^{A}. The following propositions are true:

(12) For every function f from A into B holds f is an element of B^{A}.

(13) For every element f of B^{A} holds dom f = A and rng f ⊆ B.

(14) For all sets X, Y such that Y^{X} ≠ ∅ and X ⊆ A and Y ⊆ B for every element f of Y^{X} there exists an element phi of B^{A} such that phi X = f.

(15) For every set X and for every phi holds phi X = phi (A ∩ X). Now we present four schemes. The scheme FraenkelFin deals with a non-

empty set A, a set B, and a unary functor F and states that:

{F(w) : w ∈ B} is finite provided the parameters meet the following requirement:

• B is finite.

The scheme CartFin deals with a non-empty set A, a set B, a set C, and a binary functor F and states that:

{F(u′, v′) : u′ ∈ B ∧ v′ ∈ C} is finite provided the parameters fulfill the following requirements:

• B is finite,

• C is finite.

The scheme Finiteness deals with a non-empty set A, an element B of Fin A, and a binary predicate P, and states that:

for every element x of A such that x ∈ B there exists an element y of A such that y ∈ B and P[y, x] and for every element z of A such that z ∈ B and P[z, y] holds P[y, z] provided the following requirements are fulfilled:

• for every element x of A holds P[x, x],

• for all elements x, y, z of A such that P[x, y] and P[y, z] holds P[x, z].

The scheme Fin Im deals with a non-empty set A, a non-empty set B, an element C of Fin B, a unary functor F yielding an element of A, and a binary predicate P, and states that:

there exists an element c_{1} of Fin A such that for every element t of A holds t ∈ c_{1} if and only if there exists an element t′ of B such that t′ ∈ C and t = F(t′) and P[t, t′] for all values of the parameters.

The following proposition is true

(16) For all sets A, B such that A is finite and B is finite holds B^{A} is finite.

Now we present three schemes. The scheme ImFin concerns a non-empty set A, a non-empty set B, a set C, a set D, and a unary functor F and states that:

{F(phi′) : phi′ ^{∘} C ⊆ D} is finite provided the parameters fulfill the following conditions:

• C is finite,

• D is finite,

• for all elements phi, psi of B^{A} such that phi C = psi C holds F(phi) = F(psi).

The scheme FunctChoice concerns a non-empty set A, a non-empty set B, an element C of Fin A, and a binary predicate P, and states that:

there exists a function ff from A into B such that for every element t of A such that t ∈ C holds P[t, ff(t)] provided the parameters fulfill the following condition:

• for every element t of A such that t ∈ C there exists an element ff of B such that P[t, ff].

The scheme FuncsChoice concerns a non-empty set A, a non-empty set B, an element C of Fin A, and a binary predicate P, and states that:

there exists an element ff of B^{A} such that for every element t of A such that t ∈ C holds P[t, ff(t)] provided the parameters meet the following requirement:

• for every element t of A such that t ∈ C there exists an element ff of B such that P[t, ff].

References

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

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

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

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

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

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

Integers

Michał J. Trybulec Białystok

Summary. In the article the following concepts were introduced: the set of integers ( ) and its elements (integers), congruences (i_{1} ≡ i_{2}(mod i_{3})), the ceiling and floor functors (⌈x⌉ and ⌊x⌋), also the fraction part of a real number (frac), the integer division (÷) and remainder of integer division (mod). The following schemes were also included: the separation scheme (SepInt), the schemes of integer induction (Int Ind Down, Int Ind Up, Int Ind Full), the minimum (Int Min) and maximum (Int Max) schemes (the existence of minimum and maximum integers enjoying a given property).

MML Identifier: INT 1.

The papers [2], and [1] provide the notation and terminology for this paper. For simplicity we follow a convention: x is arbitrary, k, n_{1}, n_{2} denote natural numbers, r, r_{1}, r_{2} denote real numbers, and D denotes a non-empty set. The following propositions are true:

(1) (r + r_{1}) − r_{2} = (r − r_{2}) + r_{1}.

(2) (−r_{1}) + r_{2} = r_{2} − r_{1}.

(3) r_{1} = ((−r_{2}) + r_{1}) + r_{2} and r_{1} = r_{2} + ((−r_{2}) + r_{1}) and r_{1} = r_{2} + (r_{1} − r_{2}) and r_{1} = (r_{2} + r_{1}) − r_{2}.

(4) (r_{1} − r_{2}) + r_{2} = r_{1} and (r_{1} + r_{2}) − r_{2} = r_{1}.

(5) r_{1} ≤ r_{2} if and only if r_{1} < r_{2} or r_{1} = r_{2}. The non-empty set is defined by: x ∈ if and only if there exists k such that x = k or x = −k. One can prove the following proposition

(6) For every x holds x ∈ D if and only if there exists k such that x = k or x = −k if and only if D = .

A real number is called an integer if: it is an element of . The following propositions are true:

502 Michał J. Trybulec

(7) r is an integer if and only if r is an element of .

(8) r is an integer if and only if there exists k such that r = k or r = −k.

(9) If x is a natural number, then x is an integer.

(10) 0 is an integer and 1 is an integer.

(11) If x ∈ , then x ∈ .

(12) x is an integer if and only if x ∈ .

(13) x is an integer if and only if x is an element of .

(14) ⊆ .

(15) ⊆ .

In the sequel i_{0}, i_{1}, i_{2}, i_{3}, i_{4}, i_{5} are integers. Let i_{1}, i_{2} be integers. Then i_{1} + i_{2} is an integer. Then i_{1} · i_{2} is an integer.

Let i_{0} be an integer. Then −i_{0} is an integer.

Let i_{1}, i_{2} be integers. Then i_{1} − i_{2} is an integer.

Let n be a natural number. Then −n is an integer. Let i_{1} be an integer. Then n + i_{1} is an integer. Then n · i_{1} is an integer. Then n − i_{1} is an integer.

Let i_{1} be an integer, and let n be a natural number. Then i_{1} +n is an integer. Then i_{1} · n is an integer. Then i_{1} − n is an integer.

Let us consider n_{1}, n_{2}. Then n_{1} − n_{2} is an integer.

We now state a number of propositions:

(16) If 0 ≤ i_{0}, then i_{0} is a natural number.

(17) If r is an integer, then r + 1 is an integer and r − 1 is an integer.

(18) If i_{2} ≤ i_{1}, then i_{1} − i_{2} is a natural number.

(19) If i_{1} + k = i_{2} or k + i_{1} = i_{2}, then i_{1} ≤ i_{2}.

(20) If i_{0} < i_{1}, then i_{0} + 1 ≤ i_{1} and 1 + i_{0} ≤ i_{1}.

(21) If i_{1} < 0, then i_{1} ≤ −1.

(22) i_{1} · i_{2} = 1 if and only if i_{1} = 1 and i_{2} = 1 or i_{1} = −1 and i_{2} = −1.

(23) i_{1} · i_{2} = −1 if and only if i_{1} = −1 and i_{2} = 1 or i_{1} = 1 and i_{2} = −1.

(24) If i_{0} ≠ 0, then i_{1} ≠ i_{1} + i_{0}.

(25) i_{1} < i_{1} + 1.

(26) i_{1} − 1 < i_{1}.

(27) For no i_{0} holds for every i_{1} holds i_{0} < i_{1}.

(28) For no i_{0} holds for every i_{1} holds i_{1} < i_{0}.

In the article we present several logical schemes. The scheme SepInt deals with a unary predicate P, and states that:

there exists a subset X of such that for every integer x holds x ∈ X if and only if P[x] for all values of the parameter.

The scheme Int Ind Up concerns an integer A, and a unary predicate P, and states that:

Integers 503

for every i_{0} such that A ≤ i_{0} holds P[i_{0}] provided the following conditions are fulfilled:

• P[A],

• for every i_{2} such that A ≤ i_{2} holds if P[i_{2}], then P[i_{2} + 1]. The scheme Int Ind Down deals with an integer A, and a unary predicate P,

and states that:

for every i_{0} such that i_{0} ≤ A holds P[i_{0}] provided the parameters fulfill the following conditions:

• P[A],

• for every i_{2} such that i_{2} ≤ A holds if P[i_{2}], then P[i_{2} − 1]. The scheme Int Ind Full deals with an integer A, and a unary predicate P,

and states that:

for every i_{0} holds P[i_{0}] provided the following requirements are fulfilled:

• P[A],

• for every i_{2} such that P[i_{2}] holds P[i_{2} − 1] and P[i_{2} + 1]. The scheme Int Min concerns an integer A, and a unary predicate P, and

states that:

there exists i_{0} such that P[i_{0}] and for every i_{1} such that P[i_{1}] holds i_{0} ≤ i_{1} provided the following conditions are satisfied:

• for every i_{1} such that P[i_{1}] holds A ≤ i_{1},

• there exists i_{1} such that P[i_{1}]. The scheme Int Max deals with an integer A, and a unary predicate P, and

states that:

there exists i_{0} such that P[i_{0}] and for every i_{1} such that P[i_{1}] holds i_{1} ≤ i_{0} provided the parameters satisfy the following conditions:

• for every i_{1} such that P[i_{1}] holds i_{1} ≤ A,

• there exists i_{1} such that P[i_{1}]. Let us consider r. Then sgn r is an integer. We now state two propositions:

(29) sgn r = 1 or sgn r = −1 or sgn r = 0.

(30) |r| = r or |r| = −r.

Let us consider i_{0}. Then |i_{0}| is an integer. Let i_{1}, i_{2}, i_{3} be integers. The predicate i_{1} ≡ i_{2}(mod i_{3}) is defined by: there exists i_{4} such that i_{3} · i_{4} = i_{1} − i_{2}. We now state a number of propositions:

(31) i_{1} ≡ i_{2}(mod i_{3}) if and only if there exists an integer i_{4} such that i_{3} ·i_{4} = i_{1} − i_{2}.

(32) i_{1} ≡ i_{1}(mod i_{2}).

(33) If i_{2} = 0, then i_{1} ≡ i_{2}(mod i_{1}) and i_{2} ≡ i_{1}(mod i_{1}).

(34) If i_{3} = 1, then i_{1} ≡ i_{2}(mod i_{3}).

(35) If i_{1} ≡ i_{2}(mod i_{3}), then i_{2} ≡ i_{1}(mod i_{3}).

(36) If i_{1} ≡ i_{2}(mod i_{5}) and i_{2} ≡ i_{3}(mod i_{5}), then i_{1} ≡ i_{3}(mod i_{5}).

(37) If i_{1} ≡ i_{2}(mod i_{5}) and i_{3} ≡ i_{4}(mod i_{5}), then i_{1} + i_{3} ≡ i_{2} + i_{4}(mod i_{5}).

(38) If i_{1} ≡ i_{2}(mod i_{5}) and i_{3} ≡ i_{4}(mod i_{5}), then i_{1} − i_{3} ≡ i_{2} − i_{4}(mod i_{5}).

(39) If i_{1} ≡ i_{2}(mod i_{5}) and i_{3} ≡ i_{4}(mod i_{5}), then i_{1} · i_{3} ≡ i_{2} · i_{4}(mod i_{5}).

(40) i_{1} + i_{2} ≡ i_{3}(mod i_{5}) if and only if i_{1} ≡ i_{3} − i_{2}(mod i_{5}).

(41) If i_{4} · i_{5} = i_{3}, then if i_{1} ≡ i_{2}(mod i_{3}), then i_{1} ≡ i_{2}(mod i_{4}).

(42) i_{1} ≡ i_{2}(mod i_{5}) if and only if i_{1} + i_{5} ≡ i_{2}(mod i_{5}).

(43) i_{1} ≡ i_{2}(mod i_{5}) if and only if i_{1} − i_{5} ≡ i_{2}(mod i_{5}).

(44) If i_{1} ≤ r and r − 1 < i_{1} and i_{2} ≤ r and r − 1 < i_{2}, then i_{1} = i_{2}.

(45) If r ≤ i_{1} and i_{1} < r + 1 and r ≤ i_{2} and i_{2} < r + 1, then i_{1} = i_{2}. Let us consider r. The functor ⌊r⌋ yielding an integer, is defined as follows: ⌊r⌋ ≤ r and r − 1 < ⌊r⌋. The following propositions are true:

(46) i_{0} ≤ r and r − 1 < i_{0} if and only if ⌊r⌋ = i_{0}.

(47) ⌊r⌋ = r if and only if r is an integer.

(48) ⌊r⌋ < r if and only if r is not an integer.

(49) ⌊r⌋ ≤ r.

(50) ⌊r⌋ − 1 < r and ⌊r⌋ < r + 1.

(51) ⌊r⌋ + i_{0} = ⌊r + i_{0}⌋.

(52) r ≤ ⌊r⌋ + 1.

Let us consider r. The functor ⌈r⌉ yields an integer and is defined as follows: r ≤ ⌈r⌉ and ⌈r⌉ < r + 1. We now state a number of propositions:

(53) r ≤ i_{0} and i_{0} < r + 1 if and only if ⌈r⌉ = i_{0}.

(54) ⌈r⌉ = r if and only if r is an integer.

(55) r < ⌈r⌉ if and only if r is not an integer.

(56) r ≤ ⌈r⌉.

(57) r − 1 < ⌈r⌉ and r < ⌈r⌉ + 1.

(58) ⌈r⌉ + i_{0} = ⌈r + i_{0}⌉.

(59) ⌊r⌋ = ⌈r⌉ if and only if r is an integer.

(60) ⌊r⌋ < ⌈r⌉ if and only if r is not an integer.

(61) ⌊r⌋ ≤ ⌈r⌉.

(62) ⌊⌈r⌉⌋ = ⌈r⌉.

(63) ⌊⌊r⌋⌋ = ⌊r⌋.

(64) ⌈⌈r⌉⌉ = ⌈r⌉.

(65) ⌈⌊r⌋⌉ = ⌊r⌋.

(66) ⌊r⌋ = ⌈r⌉ if and only if ⌊r⌋ + 1 ≠ ⌈r⌉.

Let us consider r. The functor frac r yielding a real number, is defined by: frac r = r − ⌊r⌋. One can prove the following propositions:

(67) frac r = r − ⌊r⌋.

(68) r = ⌊r⌋ + frac r.

(69) frac r < 1 and 0 ≤ frac r.

(70) ⌊frac r⌋ = 0.

(71) frac r = 0 if and only if r is an integer.

(72) 0 < frac r if and only if r is not an integer.

Let i_{1}, i_{2} be integers. The functor i_{1} ÷ i_{2} yields an integer and is defined by: i_{1} ÷ i_{2} = ⌊\frac{i_{1}}{i_{2}} ⌋. One can prove the following proposition

(73) i_{1} ÷ i_{2} = ⌊\frac{i_{1}}{i_{2}} ⌋.

Let i_{1}, i_{2} be integers. The functor i_{1} mod i_{2} yielding an integer, is defined as follows:

i_{1} mod i_{2} = i_{1} − (i_{1} ÷ i_{2}) · i_{2}. Next we state a proposition

(74) i_{1} mod i_{2} = i_{1} − (i_{1} ÷ i_{2}) · i_{2}.

Let i_{1}, i_{2} be integers. The predicate i_{1} | i_{2} is defined as follows: there exists i_{3} such that i_{2} = i_{1} · i_{3}. The following proposition is true

(75) i_{1} | i_{2} if and only if there exists i_{3} such that i_{1} · i_{3} = i_{2}.

References

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

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

506

The Complex Numbers

Czesław Byliński^{1} Warsaw University Białystok

Summary. We define the set of complex numbers as the set of all ordered pairs z = ⟨a, b⟩ where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for a field. 0_{}= 0 + 0i and 1_{}= 1 + 0i are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in . The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers as a subset of . From here on we do not abandon the ordered pair notation for complex numbers. For example: i^{2} = (0+1i)^{2} = −1 + 0i ≠ −1. We conclude this article by introducing two operations on which are not field operations. We define the absolute value of z denoted by |z| and the conjugate of z denoted by z^{∗}.

MML Identifier: COMPLEX1.

The articles [1], [3], [2], and [4] provide the notation and terminology for this paper. In the sequel a, b, a_{1}, b_{1}, a_{2}, b_{2} denote real numbers. The following two propositions are true:

(1) If a ≠ 0, then \frac{0}{a} = 0.

(2) a^{2} + b^{2} = 0 if and only if a = 0 and b = 0. The non-empty set is defined as follows: = [: , :]. One can prove the following proposition

(3) = [: , :]. In the sequel z, z_{1}, z_{2}, z_{3}, z_{4} will denote elements of . We now define two

new functors. Let us consider z. The functor ℜ(z) yielding a real number, is defined by:

ℜ(z) = z_{1}.

1Supported by RPBP.III-24.C1.

508 Czesław Byliński

The functor ℑ(z) yielding a real number, is defined as follows:

ℑ(z) = z_{2}. We now state two propositions:

(4) ℜ(z) = z_{1}.

(5) ℑ(z) = z_{2}. Let x, y be elements of . The functor x + yi yields an element of and is

defined as follows:

x + yi = ⟨⟨x, y⟩⟩. Next we state several propositions:

(6) For all elements x, y of holds x + yi = ⟨⟨x, y⟩⟩.

(7) ℜ(a + bi) = a and ℑ(a + bi) = b.

(8) ℜ(z) + ℑ(z)i = z.

(9) If ℜ(z_{1}) = ℜ(z_{2}) and ℑ(z_{1}) = ℑ(z_{2}), then z_{1} = z_{2}.

(10) If a_{1} + b_{1}i = a_{2} + b_{2}i, then a_{1} = a_{2} and b_{1} = b_{2}.

Let us consider z_{1}, z_{2}. Let us note that one can characterize the predicate z_{1} = z_{2} by the following (equivalent) condition: ℜ(z_{1}) = ℜ(z_{2}) and ℑ(z_{1}) = ℑ(z_{2}).

We now define three new functors. The element 0_{}of is defined as follows: 0_{}= 0 + 0i.

The element 1_{}of is defined by:

1_{}= 1 + 0i. The element i of is defined as follows:

i = 0 + 1i. The following propositions are true:

(11) 0_{}= 0 + 0i.

(12) ℜ(0_{}) = 0 and ℑ(0_{}) = 0.

(13) z = 0_{}if and only if ℜ(z)^{2} + ℑ(z)^{2} = 0.

(14) 1_{}= 1 + 0i.

(15) ℜ(1_{}) = 1 and ℑ(1_{}) = 0.

(16) i = 0 + 1i.

(17) ℜ(i) = 0 and ℑ(i) = 1.

Let us consider z_{1}, z_{2}. The functor z_{1} + z_{2} yields an element of and is defined as follows:

z_{1} + z_{2} = ℜ(z_{1}) + ℜ(z_{2}) + ℑ(z_{1}) + ℑ(z_{2})i. We now state several propositions:

(18) z_{1} + z_{2} = ℜ(z_{1}) + ℜ(z_{2}) + ℑ(z_{1}) + ℑ(z_{2})i.

(19) ℜ(z_{1} + z_{2}) = ℜ(z_{1}) + ℜ(z_{2}) and ℑ(z_{1} + z_{2}) = ℑ(z_{1}) + ℑ(z_{2}).

(20) z_{1} + z_{2} = z_{2} + z_{1}.

(21) z_{1} + (z_{2} + z_{3}) = (z_{1} + z_{2}) + z_{3}.

(22) 0_{}+ z = z and z + 0_{}= z.

The Complex Numbers 509

Let us consider z_{1}, z_{2}. The functor z_{1} · z_{2} yielding an element of , is defined as follows:

z_{1} · z_{2} = ℜ(z_{1}) · ℜ(z_{2}) − ℑ(z_{1}) · ℑ(z_{2}) + ℜ(z_{1}) · ℑ(z_{2}) + ℜ(z_{2}) · ℑ(z_{1})i.

Next we state a number of propositions:

(23) z_{1} · z_{2} = ℜ(z_{1}) · ℜ(z_{2}) − ℑ(z_{1}) · ℑ(z_{2}) + ℜ(z_{1}) · ℑ(z_{2}) + ℜ(z_{2}) · ℑ(z_{1})i.

(24) ℜ(z_{1} · z_{2}) = ℜ(z_{1}) · ℜ(z_{2}) − ℑ(z_{1}) · ℑ(z_{2}) and ℑ(z_{1} · z_{2}) = ℜ(z_{1}) · ℑ(z_{2}) + ℜ(z_{2}) · ℑ(z_{1}).

(25) z_{1} · z_{2} = z_{2} · z_{1}.

(26) z_{1} · (z_{2} · z_{3}) = (z_{1} · z_{2}) · z_{3}.

(27) z · (z_{1} + z_{2}) = z · z_{1} + z · z_{2} and (z_{1} + z_{2}) · z = z_{1} · z + z_{2} · z.

(28) 0_{}· z = 0_{}and z · 0_{}= 0_{}.

(29) 1_{}· z = z and z · 1_{}= z.

(30) If ℑ(z_{1}) = 0 and ℑ(z_{2}) = 0, then ℜ(z_{1}·z_{2}) = ℜ(z_{1})·ℜ(z_{2}) and ℑ(z_{1}·z_{2}) = 0.

(31) If ℜ(z_{1}) = 0 and ℜ(z_{2}) = 0, then ℜ(z_{1} · z_{2}) = −ℑ(z_{1}) · ℑ(z_{2}) and ℑ(z_{1} · z_{2}) = 0.

(32) ℜ(z · z) = ℜ(z)^{2} − ℑ(z)^{2} and ℑ(z · z) = 2 · (ℜ(z) · ℑ(z)).

Let us consider z. The functor −z yielding an element of , is defined by: −z = −ℜ(z) + −ℑ(z)i. One can prove the following propositions:

(33) −z = −ℜ(z) + −ℑ(z)i.

(34) ℜ(−z) = −ℜ(z) and ℑ(−z) = −ℑ(z).

(35) −0_{}= 0_{}.

(36) If −z = 0_{}, then z = 0_{}.

(37) i · i = −1_{}.

(38) z + (−z) = 0_{}and (−z) + z = 0_{}.

(39) If z_{1} + z_{2} = 0_{}, then z_{2} = −z_{1} and z_{1} = −z_{2}.

(40) −(−z) = z.

(41) If −z_{1} = −z_{2}, then z_{1} = z_{2}.

(42) If z_{1} + z = z_{2} + z or z_{1} + z = z + z_{2}, then z_{1} = z_{2}.

(43) −(z_{1} + z_{2}) = (−z_{1}) + (−z_{2}).

(44) (−z_{1}) · z_{2} = −z_{1} · z_{2} and z_{1} · (−z_{2}) = −z_{1} · z_{2}.

(45) (−z_{1}) · (−z_{2}) = z_{1} · z_{2}.

(46) −z = (−1_{}) · z.

Let us consider z_{1}, z_{2}. The functor z_{1} − z_{2} yields an element of and is defined by:

z_{1} − z_{2} = ℜ(z_{1}) − ℜ(z_{2}) + ℑ(z_{1}) − ℑ(z_{2})i.

We now state a number of propositions:

(47) z_{1} − z_{2} = ℜ(z_{1}) − ℜ(z_{2}) + ℑ(z_{1}) − ℑ(z_{2})i.

(48) ℜ(z_{1} − z_{2}) = ℜ(z_{1}) − ℜ(z_{2}) and ℑ(z_{1} − z_{2}) = ℑ(z_{1}) − ℑ(z_{2}).

(49) z_{1} − z_{2} = z_{1} + (−z_{2}).

(50) If z_{1} − z_{2} = 0_{}, then z_{1} = z_{2}.

(51) z − z = 0_{}.

(52) z − 0_{}= z.

(53) 0_{}− z = −z.

(54) z_{1} − (−z_{2}) = z_{1} + z_{2}.

(55) −(z_{1} − z_{2}) = (−z_{1}) + z_{2}.

(56) −(z_{1} − z_{2}) = z_{2} − z_{1}.

(57) z_{1} + (z_{2} − z_{3}) = (z_{1} + z_{2}) − z_{3}.

(58) z_{1} − (z_{2} − z_{3}) = (z_{1} − z_{2}) + z_{3}.

(59) (z_{1} − z_{2}) − z_{3} = z_{1} − (z_{2} + z_{3}).

(60) z_{1} = (z_{1} + z) − z.

(61) z_{1} = (z_{1} − z) + z.

(62) z · (z_{1} − z_{2}) = z · z_{1} − z · z_{2} and (z_{1} − z_{2}) · z = z_{1} · z − z_{2} · z. Let us consider z. The functor z^{−1} yields an element of and is defined by: z^{−1} = \frac{ℜ(z)}{ℜ(z)2+ℑ(z)2} + \frac{−ℑ(z)}{ℜ(z)2+ℑ(z)2} i.

Next we state a number of propositions:

(63) z^{−1} = \frac{ℜ(z)}{ℜ(z)2+ℑ(z)2} + \frac{−ℑ(z)}{ℜ(z)2+ℑ(z)2} i.

(64) ℜ(z^{−1}) = \frac{ℜ(z)}{ℜ(z)2+ℑ(z)2} and ℑ(z^{−1}) = \frac{−ℑ(z)}{ℜ(z)2+ℑ(z)2} .

(65) If z ≠ 0_{}, then z · z^{−1} = 1_{}and z^{−1} · z = 1_{}.

(66) If z_{1} · z_{2} = 0_{}, then z_{1} = 0_{}or z_{2} = 0_{}.

(67) If z ≠ 0_{}, then z^{−1} ≠ 0_{}.

(68) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}and z_{1}^{−1} = z_{2}^{−1}, then z_{1} = z_{2}.

(69) If z_{2} ≠ 0_{}but z_{1} · z_{2} = 1_{}or z_{2} · z_{1} = 1_{}, then z_{1} = z_{2}^{−1}.

(70) If z_{2} ≠ 0_{}but z_{1} · z_{2} = z_{3} or z_{2} · z_{1} = z_{3}, then z_{1} = z_{3} · z_{2}^{−1} and z_{1} = z_{2}^{−1} · z_{3}.

(71) 1_{}^{−1} = 1_{}.

(72) i^{−1} = −i.

(73) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then (z_{1} · z_{2})^{−1} = z_{1}^{−1} · z_{2}^{−1}.

(74) If z ≠ 0_{}, then (z^{−1})^{−1} = z.

(75) If z ≠ 0_{}, then (−z)^{−1} = −z^{−1}.

(76) If z ≠ 0_{}but z_{1} · z = z_{2} · z or z_{1} · z = z · z_{2}, then z_{1} = z_{2}.

(77) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then z_{1}^{−1} + z_{2}^{−1} = (z_{1} + z_{2}) · (z_{1} · z_{2})^{−1}.

(78) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then z_{1}^{−1} − z_{2}^{−1} = (z_{2} − z_{1}) · (z_{1} · z_{2})^{−1}.

(79) If ℜ(z) ≠ 0 and ℑ(z) = 0, then ℜ(z^{−1}) = ℜ(z)^{−1} and ℑ(z^{−1}) = 0.

(80) If ℜ(z) = 0 and ℑ(z) ≠ 0, then ℜ(z^{−1}) = 0 and ℑ(z^{−1}) = −ℑ(z)^{−1}. Let us consider z_{1}, z_{2}. The functor \frac{z_{1}}{z_{2}} yields an element of and is defined

by:

\frac{z_{1}}{z_{2}} = \frac{ℜ(z_{1})·ℜ(z_{2})+ℑ(z_{1})·ℑ(z_{2})}{ℜ(z_{2})2+ℑ(z_{2})2} + \frac{ℜ(z_{2})·ℑ(z_{1})−ℜ(z_{1})·ℑ(z_{2})}{ℜ(z_{2})2+ℑ(z_{2})2}i.

Next we state a number of propositions:

(81) \frac{z_{1}}{z_{2}} = \frac{ℜ(z_{1})·ℜ(z_{2})+ℑ(z_{1})·ℑ(z_{2})}{ℜ(z_{2})2+ℑ(z_{2})2} + \frac{ℜ(z_{2})·ℑ(z_{1})−ℜ(z_{1})·ℑ(z_{2})}{ℜ(z_{2})2+ℑ(z_{2})2}i.

(82) ℜ(\frac{z_{1}}{z_{2}} ) = \frac{ℜ(z_{1})·ℜ(z_{2})+ℑ(z_{1})·ℑ(z_{2})}{ℜ(z_{2})2+ℑ(z_{2})2} and ℑ(\frac{z_{1}}{z_{2}} ) = \frac{ℜ(z_{2})·ℑ(z_{1})−ℜ(z_{1})·ℑ(z_{2})}{ℜ(z_{2})2+ℑ(z_{2})2}.

(83) If z_{2} ≠ 0_{}, then \frac{z_{1}}{z_{2}} = z_{1} · z_{2}^{−1}.

(84) If z ≠ 0_{}, then z^{−1} = \frac{1_{}}{z} .

(85) \frac{z}{1_{}}= z.

(86) If z ≠ 0_{}, then \frac{z}{z} = 1_{}.

(87) If z ≠ 0_{}, then \frac{0_{}}{z} = 0_{}.

(88) If z_{2} ≠ 0_{}and \frac{z_{1}}{z_{2}} = 0_{}, then z_{1} = 0_{}.

(89) If z_{2} ≠ 0_{}and z_{4} ≠ 0_{}, then \frac{z_{1}}{z_{2}} · \frac{z_{3}}{z_{4}} = \frac{z_{1}·z_{3}}{z_{2}·z_{4}} .

(90) If z_{2} ≠ 0_{}, then z · \frac{z_{1}}{z_{2}} = \frac{z·z_{1}}{z_{2}} .

(91) If z_{2} ≠ 0_{}and \frac{z_{1}}{z_{2}} = 1_{}, then z_{1} = z_{2}.

(92) If z ≠ 0_{}, then z_{1} = \frac{z_{1}·z}{z}.

(93) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then \frac{z_{1}}{z_{2}} ^{−1} = \frac{z_{2}}{z_{1}} .

(94) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then \frac{z_{1}^{−1}}{z_{2}^{−1}} = \frac{z_{2}}{z_{1}} .

(95) If z_{2} ≠ 0_{}, then \frac{z_{1}}{z_{2}^{−1}} = z_{1} · z_{2}.

(96) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then \frac{z_{1}^{−1}}{z_{2}} = (z_{1} · z_{2})^{−1}.

(97) If z_{1} ≠ 0_{}and z_{2} ≠ 0_{}, then z_{1}^{−1} · \frac{z}{z_{2}} = \frac{z}{z_{1}·z_{2}} .

(98) If z ≠ 0_{}and z_{2} ≠ 0_{}, then \frac{z_{1}}{z_{2}} = \frac{z_{1}·z}{z_{2}·z} and \frac{z_{1}}{z_{2}} = \frac{z·z_{1}}{z·z_{2}} .

(99) If z_{2} ≠ 0_{}and z_{3} ≠ 0_{}, then \frac{z_{1}}{z_{2}·z_{3}} = ^{\frac{\frac{z_{1}}{z_{2}}}{z_{3}}} .

(100) If z_{2} ≠ 0_{}and z_{3} ≠ 0_{}, then \frac{z_{1}·z_{3}}{z_{2}} = _{\frac{z_{1}}{\frac{z_{2}}{z_{3}}}} .

(101) If z_{2} ≠ 0_{}and z_{3} ≠ 0_{}and z_{4} ≠ 0_{}, then \frac{\frac{z_{1}}{z_{2}}}{\frac{z_{3}}{z_{4}}} = \frac{z_{1}·z_{4}}{z_{2}·z_{3}} .

(102) If z_{2} ≠ 0_{}and z_{4} ≠ 0_{}, then \frac{z_{1}}{z_{2}} + \frac{z_{3}}{z_{4}} = \frac{z_{1}·z_{4}+z_{3}·z_{2}}{z_{2}·z_{4}} .

(103) If z ≠ 0_{}, then \frac{z_{1}}{z} + \frac{z_{2}}{z} = \frac{z_{1}+z_{2}}{z}.

(104) If z_{2} ≠ 0_{}, then −\frac{z_{1}}{z_{2}} = \frac{−z_{1}}{z_{2}} and −\frac{z_{1}}{z_{2}} = \frac{z_{1}}{−z_{2}} .

(105) If z_{2} ≠ 0_{}, then \frac{z_{1}}{z_{2}} = \frac{−z_{1}}{−z_{2}} .

(106) If z_{2} ≠ 0_{}and z_{4} ≠ 0_{}, then \frac{z_{1}}{z_{2}} − \frac{z_{3}}{z_{4}} = \frac{z_{1}·z_{4}−z_{3}·z_{2}}{z_{2}·z_{4}} .

(107) If z ≠ 0_{}, then \frac{z_{1}}{z} − \frac{z_{2}}{z} = \frac{z_{1}−z_{2}}{z}.

(108) If z_{2} ≠ 0_{}but z_{1} · z_{2} = z_{3} or z_{2} · z_{1} = z_{3}, then z_{1} = \frac{z_{3}}{z_{2}} .

(109) If ℑ(z_{1}) = 0 and ℑ(z_{2}) = 0 and ℜ(z_{2}) ≠ 0, then ℜ(\frac{z_{1}}{z_{2}} ) = \frac{ℜ(z_{1})}{ℜ(z_{2})} and ℑ(\frac{z_{1}}{z_{2}} ) = 0.

(110) If ℜ(z_{1}) = 0 and ℜ(z_{2}) = 0 and ℑ(z_{2}) ≠ 0, then ℜ(\frac{z_{1}}{z_{2}} ) = \frac{ℑ(z_{1})}{ℑ(z_{2})} and

ℑ(\frac{z_{1}}{z_{2}} ) = 0. Let us consider z. The functor z^{∗} yielding an element of , is defined as

follows:

z^{∗} = ℜ(z) + −ℑ(z)i.

The following propositions are true:

(111) z^{∗} = ℜ(z) + −ℑ(z)i.

(112) ℜ(z^{∗}) = ℜ(z) and ℑ(z^{∗}) = −ℑ(z).

(113) 0_{}^{∗} = 0_{}.

(114) If z^{∗} = 0_{}, then z = 0_{}.

(115) 1_{}^{∗} = 1_{}.

(116) i^{∗} = −i.

(117) z^{∗}^{∗} = z.

(118) (z_{1} + z_{2})^{∗} = z_{1}^{∗} + z_{2}^{∗}.

(119) (−z)^{∗} = −z^{∗}.

(120) (z_{1} − z_{2})^{∗} = z_{1}^{∗} − z_{2}^{∗}.

(121) (z_{1} · z_{2})^{∗} = z_{1}^{∗} · z_{2}^{∗}.

(122) If z ≠ 0_{}, then (z^{−1})^{∗} = z^{∗}^{−1}.

(123) If z_{2} ≠ 0_{}, then \frac{z_{1}}{z_{2}} ^{∗} = \begin{matrix}z_{1} & ^{∗} \\ z_{2} & ^{∗} \end{matrix}.

(124) If ℑ(z) = 0, then z^{∗} = z.

(125) If ℜ(z) = 0, then z^{∗} = −z.

(126) ℜ(z · z^{∗}) = ℜ(z)^{2} + ℑ(z)^{2} and ℑ(z · z^{∗}) = 0.

(127) ℜ(z + z^{∗}) = 2 · ℜ(z) and ℑ(z + z^{∗}) = 0.

(128) ℜ(z − z^{∗}) = 0 and ℑ(z − z^{∗}) = 2 · ℑ(z).

Let us consider z. The functor |z| yielding a real number, is defined as follows: |z| = \sqrt{ℜ(z)^{2} + ℑ(z)^{2}}. One can prove the following propositions:

(129) |z| = \sqrt{ℜ(z)^{2} + ℑ(z)^{2}}.

(130) |0_{}| = 0.

(131) If |z| = 0, then z = 0_{}.

(132) 0 ≤ |z|.

(133) z ≠ 0_{}if and only if 0 < |z|.

(134) |1_{}| = 1.

(135) |i| = 1.

(136) If ℑ(z) = 0, then |z| = |ℜ(z)|.

(137) If ℜ(z) = 0, then |z| = |ℑ(z)|.

(138) | − z| = |z|.

(139) |z^{∗}| = |z|.

(140) ℜ(z) ≤ |z|.

(141) ℑ(z) ≤ |z|.

(142) |z_{1} + z_{2}| ≤ |z_{1}| + |z_{2}|.

(143) |z_{1} − z_{2}| ≤ |z_{1}| + |z_{2}|.

(144) |z_{1}| − |z_{2}| ≤ |z_{1} + z_{2}|.

(145) |z_{1}| − |z_{2}| ≤ |z_{1} − z_{2}|.

(146) |z_{1} − z_{2}| = |z_{2} − z_{1}|.

(147) |z_{1} − z_{2}| = 0 if and only if z_{1} = z_{2}.

(148) z_{1} ≠ z_{2} if and only if 0 < |z_{1} − z_{2}|.

(149) |z_{1} − z_{2}| ≤ |z_{1} − z| + |z − z_{2}|.

(150) ||z_{1}| − |z_{2}|| ≤ |z_{1} − z_{2}|.

(151) |z_{1} · z_{2}| = |z_{1}| · |z_{2}|.

(152) If z ≠ 0_{}, then |z^{−1}| = |z|^{−1}.

(153) If z_{2} ≠ 0_{}, then \frac{|z_{1}|}{|z_{2}|} = |\frac{z_{1}}{z_{2}} |.

(154) |z · z| = ℜ(z)^{2} + ℑ(z)^{2}.

(155) |z · z| = |z · z^{∗}|.

References

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

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

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

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

514

Ordinal Arithmetics

Grzegorz Bancerek Warsaw University Białystok

Summary. At the beginning the article contains some auxiliary theorems concerning the constructors defined in papers [1] and [2]. Next simple properties of addition and multiplication of ordinals are shown, e.g. associativity of addition. Addition and multiplication of a transfinite sequence of ordinals and a ordinal are also introduced here. The goal of the article is the proof that the distributivity of multiplication wrt addition and the associativity of multiplication hold. Additionally new binary functors of ordinals are introduced: subtraction, exact division, and remainder and some of their basic properties are presented.

MML Identifier: ORDINAL3.

The notation and terminology used here are introduced in the following papers: [5], [3], [1], [4], and [2]. For simplicity we adopt the following convention: fi, psi denote sequences of ordinal numbers, A, B, C, D denote ordinal numbers, X, Y denote sets, and x is arbitrary. We now state a number of propositions:

(1) X ⊆ succ X.

(2) If succ X ⊆ Y, then X ⊆ Y.

(3) If succ A ⊆ B, then A ∈ B.

(4) A ⊆ B if and only if succ A ⊆ succ B.

(5) A ∈ B if and only if succ A ∈ succ B.

(6) If X ⊆ A, then ⋃ X is an ordinal number.

(7) ⋃(On X) is an ordinal number.

(8) If X ⊆ A, then On X = X.

(9) On{A} = {A}.

(10) If A ≠ 0, then 0 ∈ A.

(11) inf A = 0.

(12) inf{A} = A.

(13) If X ⊆ A, then ⋂ X is an ordinal number.

Let us consider x. Let us assume that x is an ordinal number. The functor x (as an ordinal) yielding an ordinal number, is defined as follows:

x (as an ordinal) = x.

The following proposition is true

(14) If x is an ordinal number, then x (as an ordinal) = x.

Let us consider A, B. Then A ∪ B is an ordinal number. Then A ∩ B is an ordinal number.

We now state a number of propositions:

(15) A ∪ B = A or A ∪ B = B.

(16) A ∩ B = A or A ∩ B = B.

(17) If A ∈ 1, then A = 0.

(18) 1 = {0}.

(19) If A ⊆ 1, then A = 0 or A = 1.

(20) If A ⊆ B or A ∈ B but C ∈ D, then A + C ∈ B + D.

(21) If A ⊆ B and C ⊆ D, then A + C ⊆ B + D.

(22) If A ∈ B but C ⊆ D and D ≠ 0 or C ∈ D, then A · C ∈ B · D.

(23) If A ⊆ B and C ⊆ D, then A · C ⊆ B · D.

(24) If B + C = B + D, then C = D.

(25) If B + C ∈ B + D, then C ∈ D.

(26) If B + C ⊆ B + D, then C ⊆ D.

(27) A ⊆ A + B and B ⊆ A + B.

(28) If A ∈ B, then A ∈ B + C and A ∈ C + B.

(29) If A + B = 0, then A = 0 and B = 0.

(30) If A ⊆ B, then there exists C such that B = A + C.

(31) If A ∈ B, then there exists C such that B = A + C and C ≠ 0.

(32) If A ≠ 0 and A is a limit ordinal number, then B + A is a limit ordinal number.

(33) (A + B) + C = A + (B + C).

(34) If A · B = 0, then A = 0 or B = 0.

(35) If A ∈ B and C ≠ 0, then A ∈ B · C and A ∈ C · B.

(36) If B · A = C · A and A ≠ 0, then B = C.

(37) If B · A ∈ C · A, then B ∈ C.

(38) If B · A ⊆ C · A and A ≠ 0, then B ⊆ C.

(39) If B ≠ 0, then A ⊆ A · B and A ⊆ B · A.

(40) If A ∈ B and C ≠ 0, then A ∈ B · C and A ∈ C · B.

(41) If A · B = 1, then A = 1 and B = 1.

(42) If A ∈ B + C, then A ∈ B or there exists D such that D ∈ C and

A = B + D. We now define four new functors. Let us consider C, fi. The functor C + fi

yields a sequence of ordinal numbers and is defined by:

Ordinal Arithmetics 517

dom(C + fi) = dom fi and for every A such that A ∈ dom fi holds (C + fi)(A) = C + (fi(A)) (as an ordinal). The functor fi + C yields a sequence of ordinal numbers and is defined by:

dom(fi + C) = dom fi and for every A such that A ∈ dom fi holds (fi + C)(A) = (fi(A)) (as an ordinal) + C. The functor C ·fi yields a sequence of ordinal numbers and is defined as follows:

dom(C · fi) = dom fi and for every A such that A ∈ dom fi holds (C · fi)(A) = C · (fi(A)) (as an ordinal). The functor fi · C yields a sequence of ordinal numbers and is defined by:

dom(fi · C) = dom fi and for every A such that A ∈ dom fi holds (fi · C)(A) = (fi(A)) (as an ordinal) · C.

The following propositions are true:

(43) psi = C + fi if and only if dom psi = dom fi and for every A such that A ∈ dom fi holds psi(A) = C + (fi(A)) (as an ordinal).

(44) psi = fi + C if and only if dom psi = dom fi and for every A such that A ∈ dom fi holds psi(A) = (fi(A)) (as an ordinal) + C.

(45) psi = C · fi if and only if dom psi = dom fi and for every A such that A ∈ dom fi holds psi(A) = C · (fi(A)) (as an ordinal).

(46) psi = fi · C if and only if dom psi = dom fi and for every A such that A ∈ dom fi holds psi(A) = (fi(A)) (as an ordinal) · C.

(47) If 0 ≠ dom fi and dom fi = dom psi and for all A, B such that A ∈ dom fi and B = fi(A) holds psi(A) = C + B, then sup psi = C + sup fi.

(48) If A is a limit ordinal number, then A · B is a limit ordinal number.

(49) If A ∈ B · C and B is a limit ordinal number, then there exists D such that D ∈ B and A ∈ D · C.

(50) If 0 ≠ dom fi and dom fi = dom psi and C ≠ 0 and sup fi is a limit ordinal number and for all A, B such that A ∈ dom fi and B = fi(A) holds psi(A) = B · C, then sup psi = sup fi · C.

(51) If 0 ≠ dom fi, then sup(C + fi) = C + sup fi.

(52) If 0 ≠ dom fi and C ≠ 0 and sup fi is a limit ordinal number, then sup(fi · C) = sup fi · C.

(53) If B ≠ 0, then ⋃(A + B) = A + ⋃ B.

(54) (A + B) · C = A · C + B · C.

(55) If A ≠ 0, then there exist C, D such that B = C · A + D and D ∈ A.

(56) For all ordinal numbers C_{1}, D_{1}, C_{2}, D_{2} such that C_{1}·A+D_{1} = C_{2}·A+D_{2} and D_{1} ∈ A and D_{2} ∈ A holds C_{1} = C_{2} and D_{1} = D_{2}.

(57) If 1 ∈ B and A ≠ 0 and A is a limit ordinal number, then for every fi such that dom fi = A and for every C such that C ∈ A holds fi(C) = C·B holds A · B = sup fi.

(58) (A · B) · C = A · (B · C).

We now define two new functors. Let us consider A, B. The functor A − B yields an ordinal number and is defined as follows:

A = B + (A − B) if B ⊆ A, A − B = 0, otherwise. The functor A ÷ B yielding an ordinal number, is defined by:

there exists C such that A = (A÷B)·B +C and C ∈ B if B ≠ 0, A÷B = 0, otherwise.

Let us consider A, B. The functor A mod B yielding an ordinal number, is defined by:

A mod B = A − (A ÷ B) · B.

The following propositions are true:

(59) If A ⊆ B, then B = A + (B − A).

(60) If A ∈ B, then B = A + (B − A).

(61) If A ⊈ B, then B − A = 0.

(62) If B ≠ 0, then there exists C such that A = (A÷B)·B +C and C ∈ B.

(63) A ÷ 0 = 0.

(64) A mod B = A − (A ÷ B) · B.

(65) (A + B) − A = B.

(66) If A ∈ B but C ⊆ A or C ∈ A, then A − C ∈ B − C.

(67) A − A = 0.

(68) If A ∈ B, then B − A ≠ 0 and 0 ∈ B − A.

(69) A − 0 = A and 0 − A = 0.

(70) A − (B + C) = (A − B) − C.

(71) If A ⊆ B, then C − B ⊆ C − A.

(72) If A ⊆ B, then A − C ⊆ B − C.

(73) If C ≠ 0 and A ∈ B + C, then A − B ∈ C.

(74) If A + B ∈ C, then B ∈ C − A.

(75) A ⊆ B + (A − B).

(76) A · C − B · C = (A − B) · C.

(77) (A ÷ B) · B ⊆ A.

(78) A = (A ÷ B) · B + (A mod B).

(79) If A = B · C + D and D ∈ C, then B = A ÷ C and D = A mod C.

(80) If A ∈ B · C, then A ÷ C ∈ B and A mod C ∈ C.

(81) If B ≠ 0, then A · B ÷ B = A.

(82) A · B mod B = 0.

(83) 0 ÷ A = 0 and 0 mod A = 0 and A mod 0 = A.

(84) A ÷ 1 = A and A mod 1 = 0.

References

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

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

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

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

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

520

The Modification of a Function by a Function

and the Iteration of the Composition of a Function

Czesław Byliński^{1} Warsaw University Białystok

Summary. In the article we introduce some operation on functions. We define the natural ordering relation on functions. The fact that a function f is less than a function g we denote by f ≤ g and we define by graphf ⊆ graphf. In the sequel we define the modifications of a function f by a function g denoted f+·g and the n-th iteration of the composition of a function f denoted by f^{n}. We prove some propositions related to the introduced notions.

MML Identifier: FUNCT 4.

The papers [7], [1], [2], [3], [4], [5], and [6] provide the terminology and notation for this paper. For simplicity we adopt the following rules: a, b, x, x′, y, y′, z will be arbitrary, X, X′, Y, Y′, Z, Z′ will be sets, D, D′ will be non-empty sets, and f, g, h will be functions. We now state several propositions:

(1) If for every z such that z ∈ Z there exist x, y such that z = ⟨⟨x, y⟩⟩, then there exist X, Y such that Z ⊆ [: X, Y :].

(2) If rng f ∩ dom g = ∅, then g · f = .

(3) g · f = g rng f · f.

(4) = ∅ ⟼ a.

(5) graph(id_{X}) ⊆ graph(id_{Y}) if and only if X ⊆ Y.

(6) If X ⊆ Y, then graph(X ⟼ a) ⊆ graph(Y ⟼ a).

(7) If graph(X ⟼ a) ⊆ graph(Y ⟼ b), then X ⊆ Y.

(8) If X ≠ ∅ and graph(X ⟼ a) ⊆ graph(Y ⟼ b), then a = b.

1Supported by RPBP.III-24.C1.

(9) If x ∈ dom f, then graph({x} ⟼ f(x)) ⊆ graph f.

Let us consider f, g. The predicate f ≤ g is defined as follows:

graph f ⊆ graph g.

We now state a number of propositions:

(10) For all f, g holds f ≤ g if and only if graph f ⊆ graph g.

(11) f ≤ g if and only if dom f ⊆ dom g and for every x such that x ∈ dom f holds f(x) = g(x).

(12) If f ≤ g, then f ≈ g.

(13) If f ≤ g, then dom f ⊆ dom g and rng f ⊆ rng g.

(14) If f ≤ g and dom f = dom g, then f = g.

(15) ≤ f.

(16) f ≤ f.

(17) If f ≤ g and g ≤ h, then f ≤ h.

(18) f ≤ g and g ≤ f if and only if f = g.

(19) id_{X} ≤ id_{Y} if and only if X ⊆ Y.

(20) If X ⊆ Y, then X ⟼ a ≤ Y ⟼ a.

(21) If X ⟼ a ≤ Y ⟼ b, then X ⊆ Y.

(22) If X ≠ ∅ and X ⟼ a ≤ Y ⟼ b, then a = b.

(23) If x ∈ dom f, then {x} ⟼ f(x) ≤ f.

(24) If f ≤ g and g is one-to-one, then f is one-to-one.

(25) If f ≤ g, then g dom f = f.

(26) If f ≤ g and g is one-to-one, then rng f g = f.

(27) f X ≤ f.

(28) If X ⊆ Y, then f X ≤ f Y.

(29) If X ⊆ Y, then X f ≤ Y f.

(30) Y f ≤ f.

(31) (Y f) X ≤ f.

(32) f_{X ˙→Y} ≤ f.

(33) If f ≤ g, then f · h ≤ g · h.

(34) If f ≤ g, then h · f ≤ h · g.

(35) For all functions f_{1}, f_{2}, g_{1}, g_{2} such that f_{1} ≤ g_{1} and f_{2} ≤ g_{2} holds f_{1} · f_{2} ≤ g_{1} · g_{2}.

(36) If f ≤ g, then f X ≤ g X.

(37) If f ≤ g, then Y f ≤ Y g.

(38) If f ≤ g, then (Y f) X ≤ (Y g) X.

(39) If f ≤ g, then f_{X ˙→Y} ≤ g_{X ˙→Y} .

(40) If f ≤ h and g ≤ h, then f ≈ g.

Let us consider f, g. The functor f +· g yields a function and is defined by:

The Modification of a Function by a Function … 523

dom(f +· g) = dom f ∪ dom g and for every x such that x ∈ dom f ∪ dom g holds if x ∈ dom g, then (f +· g)(x) = g(x) but if x ∉ dom g, then (f +· g)(x) = f(x).

We now state a number of propositions:

(41) Let f, g, h be functions. Then h = f +· g if and only if the following

conditions are satisfied:

(i) dom h = dom f ∪ dom g, (ii) for every x such that x ∈ dom f ∪ dom g holds if x ∈ dom g, then

h(x) = g(x) but if x ∉ dom g, then h(x) = f(x).

(42) If x ∈ dom(f +· g) and x ∉ dom g, then (f +· g)(x) = f(x).

(43) x ∈ dom(f +· g) if and only if x ∈ dom f or x ∈ dom g.

(44) If x ∈ dom g, then (f +· g)(x) = g(x).

(45) If x ∈ dom f \ dom g, then (f +· g)(x) = f(x).

(46) If f ≈ g and x ∈ dom f, then (f +· g)(x) = f(x).

(47) If dom f ∩ dom g = ∅ and x ∈ dom f, then (f +· g)(x) = f(x).

(48) rng(f +· g) ⊆ rng f ∪ rng g.

(49) rng g ⊆ rng(f +· g).

(50) If dom f ⊆ dom g, then f +· g = g.

(51) If dom f = dom g, then f +· g = g.

(52) f +· f = f.

(53) +· f = f.

(54) f +· = f.

(55) id_{X} +· id_{Y} = id_{X∪Y}.

(56) (f +· g) dom g = g.

(57) graph((f +· g) (dom f \ dom g)) ⊆ graph f.

(58) (f +· g) (dom f \ dom g) ≤ f.

(59) graph g ⊆ graph(f +· g).

(60) g ≤ f +· g.

(61) If f ≈ g +· h, then f (dom f \ dom h) ≈ g.

(62) If f ≈ g +· h, then f ≈ h.

(63) f ≈ g if and only if graph f ⊆ graph(f +· g).

(64) f ≈ g if and only if f ≤ f +· g.

(65) graph(f +· g) ⊆ graph f ∪ graph g.

(66) f ≈ g if and only if graph f ∪ graph g = graph(f +· g).

(67) If dom f ∩ dom g = ∅, then graph f ∪ graph g = graph(f +· g).

(68) If dom f ∩ dom g = ∅, then graph f ⊆ graph(f +· g).

(69) If dom f ∩ dom g = ∅, then f ≤ f +· g.

(70) If dom f ∩ dom g = ∅, then (f +· g) dom f = f.

(71) f ≈ g if and only if f +· g = g +· f.

(72) If dom f ∩ dom g = ∅, then f +· g = g +· f.

(73) For all partial functions f, g from X to Y such that g is total holds f +· g = g.

(74) For all functions f, g from X into Y such that if Y = ∅, then X = ∅ holds f +· g = g.

(75) For all functions f, g from X into X holds f +· g = g.

(76) For all functions f, g from X into D holds f +· g = g.

(77) For all partial functions f, g from X to Y holds f +· g is a partial

function from X to Y. Let us consider f. The functor f yields a function and is defined by: for every x holds x ∈ dom( f) if and only if there exist y, z such that

x = ⟨⟨z, y⟩⟩ and ⟨⟨y, z⟩⟩ ∈ dom f and for all y, z such that ⟨⟨y, z⟩⟩ ∈ dom f holds ( f)(⟨⟨z, y⟩⟩) = f(⟨⟨y, z⟩⟩).

We now state a number of propositions:

(78) Let f, h be functions. Then h = f if and only if for every z holds z ∈ dom h if and only if there exist x, y such that z = ⟨⟨y, x⟩⟩ and ⟨⟨x, y⟩⟩ ∈ dom f and for all x, y such that ⟨⟨x, y⟩⟩ ∈ dom f holds h(⟨⟨y, x⟩⟩) = f(⟨⟨x, y⟩⟩).

(79) rng( f) ⊆ rng f.

(80) ⟨⟨x, y⟩⟩ ∈ dom f if and only if ⟨⟨y, x⟩⟩ ∈ dom( f).

(81) If ⟨⟨y, x⟩⟩ ∈ dom( f), then f(⟨⟨y, x⟩⟩) = f(⟨⟨x, y⟩⟩).

(82) There exist X, Y such that dom( f) ⊆ [: X, Y :].

(83) If dom f ⊆ [: X, Y :], then dom( f) ⊆ [: Y, X:].

(84) If dom f = [: X, Y :], then dom( f) = [: Y, X:].

(85) If dom f ⊆ [: X, Y :], then rng( f) = rng f.

(86) If dom f = [: X, Y :], then rng( f) = rng f.

(87) For every partial function f from [: X, Y :] to Z holds f is a partial function from [: Y, X:] to Z.

(88) For every function f from [: X, Y :] into Z such that Z ≠ ∅ holds f is a function from [: Y, X:] into Z.

(89) For every function f from [: X, Y :] into D holds f is a function from [: Y, X:] into D.

(90) graph( ( f)) ⊆ graph f.

(91) If dom f ⊆ [: X, Y :], then ( f) = f.

(92) If dom f = [: X, Y :], then ( f) = f.

(93) For every partial function f from [: X, Y :] to Z holds ( f) = f.

(94) For every function f from [: X, Y :] into Z such that Z ≠ ∅ holds ( f) = f.

(95) For every function f from [: X, Y :] into D holds ( f) = f. Let us consider f, g. The functor |:f, g:| yielding a function, is defined as

follows:

(i) for every z holds z ∈ dom|:f, g:| if and only if there exist x, y, x′, y′ such that z = ⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩ and ⟨⟨x, y⟩⟩ ∈ dom f and ⟨⟨x′, y′⟩⟩ ∈ dom g, (ii) for all x, y, x′, y′ such that ⟨⟨x, y⟩⟩ ∈ dom f and ⟨⟨x′, y′⟩⟩ ∈ dom g holds |:f, g:|(⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩) = ⟨⟨f(⟨⟨x, y⟩⟩), g(⟨⟨x′, y′⟩⟩)⟩⟩.

The following propositions are true:

(96) Given f, g, h. Then h = |:f, g:| if and only if the following conditions

are satisfied:

(i) for every z holds z ∈ dom h if and only if there exist x, y, x′, y′ such that z = ⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩ and ⟨⟨x, y⟩⟩ ∈ dom f and ⟨⟨x′, y′⟩⟩ ∈ dom g,

(ii) for all x, y, x′, y′ such that ⟨⟨x, y⟩⟩ ∈ dom f and ⟨⟨x′, y′⟩⟩ ∈ dom g holds

h(⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩) = ⟨⟨f(⟨⟨x, y⟩⟩), g(⟨⟨x′, y′⟩⟩)⟩⟩.

(97) ⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩ ∈ dom|:f, g:| if and only if ⟨⟨x, y⟩⟩ ∈ dom f and ⟨⟨x′, y′⟩⟩ ∈ dom g.

(98) If ⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩ ∈ dom|:f, g:|, then |:f, g:|(⟨⟨⟨⟨x, x′⟩⟩, ⟨⟨y, y′⟩⟩⟩⟩) = ⟨⟨f(⟨⟨x, y⟩⟩), g(⟨⟨x′, y′⟩⟩)⟩⟩ .

(99) rng|:f, g:| ⊆ [: rng f, rng g:].

(100) If dom f ⊆ [: X, Y :] and dom g ⊆ [: X′, Y′:], then dom|:f, g:| ⊆ [: [: X, X′:], [: Y, Y′:] :].

(101) If dom f = [: X, Y :] and dom g = [: X′, Y′:], then dom|:f, g:| = [: [: X, X′:], [: Y, Y′:] :].

(102) For every partial function f from [: X, Y :] to Z and for every partial function g from [: X′, Y′:] to Z′ holds |:f, g:| is a partial function from [: [: X, X′:], [: Y, Y′:] :] to [: Z, Z′:].

(103) For every function f from [: X, Y :] into Z and for every function g from [: X′, Y′:] into Z′ such that Z ≠ ∅ and Z′ ≠ ∅ holds |:f, g:| is a function from [: [: X, X′:], [: Y, Y′:] :] into [: Z, Z′:].

(104) For every function f from [: X, Y :] into D and for every function g from [: X′, Y′:] into D′ holds |:f, g:| is a function from [: [: X, X′:], [: Y, Y′:] :] into [: D, D′:].

Let f be a function, and let n be an element of . The functor f^{n} yields a function and is defined as follows:

there exists a function p from into (dom f ∪ rng f) ˙→(dom f ∪ rng f) such that f^{n} = p(n) and p(0) = id_{dom f∪rng f} and for every element k of there exists a function g such that g = p(k) and p(k + 1) = g · f.

One can prove the following proposition

(105) Let f be a function. Let n be an element of . Suppose rng f ⊆ dom f. Let h be a function. Then h = f^{n} if and only if there exists a function p from into (dom f ∪ rng f) ˙→(dom f ∪ rng f) such that h = p(n) and p(0) = id_{dom f∪rng f} and for every element k of there exists a function g such that g = p(k) and p(k + 1) = g · f.

In the sequel m, n will be natural numbers. Next we state a number of propositions:

(106) f^{0} = id_{dom f∪rng f}.

(107) f^{n+1} = (f^{n}) · f.

(108) f^{1} = f.

(109) f^{n+1} = f · (f^{n}).

(110) dom(f^{n}) ⊆ dom f ∪ rng f and rng(f^{n}) ⊆ dom f ∪ rng f.

(111) If n ≠ 0, then dom(f^{n}) ⊆ dom f and rng(f^{n}) ⊆ rng f.

(112) If rng f ⊆ dom f, then dom(f^{n}) = dom f and rng(f^{n}) ⊆ dom f.

(113) (f^{n}) · id_{dom f∪rng f} = f^{n}.

(114) id_{dom f∪rng f} ·(f^{n}) = f^{n}.

(115) (f^{n}) · (f^{m}) = f^{n+m}.

(116) If n ≠ 0, then (f^{m})^{n} = f^{m·n}.

(117) If rng f ⊆ dom f, then (f^{m})^{n} = f^{m·n}.

(118) ^{n} = .

(119) id_{X}^{n} = id_{X}.

(120) If rng f ∩ dom f = ∅, then f^{2} = .

(121) For every function f from X into X holds f^{n} is a function from X into X.

(122) For every function f from X into X holds f^{0} = id_{X}.

(123) For every function f from X into X holds (f^{m})^{n} = f^{m·n}.

(124) For every partial function f from X to X holds f^{n} is a partial function from X to X.

(125) If n ≠ 0 and a ∈ X and f = X ⟼ a, then f^{n} = f.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 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. Graphs of functions. Formalized Mathematics, 1(1):169– 173, 1990.

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

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

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

The Modification of a Function by a Function … 527 Received March 1, 1990 528

Finite Sequences and

Tuples of Elements of a Non-empty Sets

Czesław Byliński^{1} Warsaw University Białystok

Summary. The first part of the article is a continuation of [2]. Next, we define the identity sequence of natural numbers and the constant sequences. The main part of this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a non-empty set D and it is denoted by element of D^{n}. Also some basic facts about tuples of a non-empty set are proved.

MML Identifier: FINSEQ 2.

The notation and terminology used here have been introduced in the following articles: [9], [8], [6], [1], [10], [4], [5], [2], [3], and [7]. For simplicity we adopt the following rules: i, j, l denote natural numbers, a, b, x_{1}, x_{2}, x_{3} are arbitrary, D, D′, E denote non-empty sets, d, d_{1}, d_{2}, d_{3} denote elements of D, d′, d′_{1}, d′_{2}, d′_{3} denote elements of D′, and p, q, r denote finite sequences. Next we state a number of propositions:

(1) min(i, j) is a natural number and max(i, j) is a natural number.

(2) If l = min(i, j), then Seg i ∩ Seg j = Seg l.

(3) If i ≤ j, then max(0, i − j) = 0.

(4) If j ≤ i, then max(0, i − j) = i − j.

(5) max(0, i − j) is a natural number.

(6) min(0, i) = 0 and min(i, 0) = 0 and max(0, i) = i and max(i, 0) = i.

(7) If i ≠ 0, then Seg i is a non-empty subset of .

(8) If i ∈ Seg(l + 1), then i ∈ Seg l or i = l + 1.

(9) If i ∈ Seg l, then i ∈ Seg(l + j).

(10) If len p = i and len q = i and for every j such that j ∈ Seg i holds p(j) = q(j), then p = q.

1Supported by RPBP.III-24.C1.

(11) If b ∈ rng p, then there exists i such that i ∈ Seg(len p) and p(i) = b.

(12) If i ∈ Seg(len p), then p(i) ∈ rng p.

(13) For every finite sequence p of elements of D such that i ∈ Seg(len p) holds p(i) ∈ D.

(14) If for every i such that i ∈ Seg(len p) holds p(i) ∈ D, then p is a finite sequence of elements of D.

(15) ⟨d_{1}, d_{2}⟩ is a finite sequence of elements of D.

(16) ⟨d_{1}, d_{2}, d_{3}⟩ is a finite sequence of elements of D.

(17) If i ∈ Seg(len p), then (p^{}q)(i) = p(i).

(18) If i ∈ Seg(len p), then i ∈ Seg(len(p^{}q)).

(19) len(p^{}⟨a⟩) = len p + 1.

(20) If p^{}⟨a⟩ = q^{}⟨b⟩, then p = q and a = b.

(21) If len p = i + 1, then there exist q, a such that p = q^{}⟨a⟩.

(22) For every finite sequence p of elements of D such that len p ≠ 0 there exists a finite sequence q of elements of D and there exists d such that p = q^{}⟨d⟩.

(23) If q = p Seg i and len p ≤ i, then p = q.

(24) If q = p Seg i, then len q = min(i, len p).

(25) If len r = i + j, then there exist p, q such that len p = i and len q = j and r = p^{}q.

(26) For every finite sequence r of elements of D such that len r = i + j there exist finite sequences p, q of elements of D such that len p = i and len q = j and r = p^{}q.

In the article we present several logical schemes. The scheme SeqLambdaD concerns a natural number A, a non-empty set B, and a unary functor F yielding an element of B and states that:

there exists a finite sequence z of elements of B such that len z = A and for every j such that j ∈ Seg A holds z(j) = F(j) for all values of the parameters.

The scheme IndSeqD deals with a non-empty set A, and a unary predicate P, and states that:

for every finite sequence p of elements of A holds P[p] provided the parameters meet the following requirements:

• P[ε_{A}],

• for every finite sequence p of elements of A and for every element x of A such that P[p] holds P[p^{}⟨x⟩].

We now state a number of propositions:

(27) For every non-empty subset D′ of D and for every finite sequence p of elements of D′ holds p is a finite sequence of elements of D.

(28) For every function f from Seg i into D holds f is a finite sequence of elements of D.

(29) p is a function from Seg(len p) into rng p.

Finite Sequences and … 531

(30) For every finite sequence p of elements of D holds p is a function from Seg(len p) into D.

(31) For every function f from into D holds f Seg i is a finite sequence of elements of D.

(32) For every function f from into D such that q = f Seg i holds len q = i.

(33) For every function f such that rng p ⊆ dom f and q = f · p holds len q = len p.

(34) If D = Seg i, then for every finite sequence p and for every finite sequence q of elements of D such that i ≤ len p holds p·q is a finite sequence.

(35) If D = Seg i, then for every finite sequence p of elements of D′ and for every finite sequence q of elements of D such that i ≤ len p holds p · q is a finite sequence of elements of D′.

(36) For every finite sequence p of elements of D and for every function f from D into D′ holds f · p is a finite sequence of elements of D′.

(37) For every finite sequence p of elements of D and for every function f from D into D′ such that q = f · p holds len q = len p.

(38) For every function f from D into D′ holds f · ε_{D} = ε_{D′}.

(39) For every finite sequence p of elements of D and for every function f from D into D′ such that p = ⟨x_{1}⟩ holds f · p = ⟨f(x_{1})⟩.

(40) For every finite sequence p of elements of D and for every function f from D into D′ such that p = ⟨x_{1}, x_{2}⟩ holds f · p = ⟨f(x_{1}), f(x_{2})⟩.

(41) For every finite sequence p of elements of D and for every function f from D into D′ such that p = ⟨x_{1}, x_{2}, x_{3}⟩ holds f · p = ⟨f(x_{1}), f(x_{2}), f(x_{3})⟩.

(42) For every function f from Seg i into Seg j such that if j = 0, then i = 0 but j ≤ len p holds p · f is a finite sequence.

(43) For every function f from Seg i into Seg i such that i ≤ len p holds p · f is a finite sequence.

(44) For every function f from Seg(len p) into Seg(len p) holds p·f is a finite sequence.

(45) For every function f from Seg i into Seg i such that rng f = Seg i and i ≤ len p and q = p · f holds len q = i.

(46) For every function f from Seg(len p) into Seg(len p) such that rng f = Seg(len p) and q = p · f holds len q = len p.

(47) For every permutation f of Seg i such that i ≤ len p and q = p · f holds len q = i.

(48) For every permutation f of Seg(len p) such that q = p · f holds len q = len p.

(49) For every finite sequence p of elements of D and for every function f from Seg i into Seg j such that if j = 0, then i = 0 but j ≤ len p holds p · f is a finite sequence of elements of D.

(50) For every finite sequence p of elements of D and for every function f from Seg i into Seg i such that i ≤ len p holds p · f is a finite sequence of elements of D.

(51) For every finite sequence p of elements of D and for every function f from Seg(len p) into Seg(len p) holds p · f is a finite sequence of elements of D.

(52) id_{Seg i} is a finite sequence of elements of .

Let us consider i. The functor id_{i} yielding a finite sequence, is defined as follows:

id_{i} = id_{Seg i}.

One can prove the following propositions:

(53) id_{i} = id_{Seg i}.

(54) dom(id_{i}) = Seg i.

(55) len(id_{i}) = i.

(56) If j ∈ Seg i, then id_{i}(j) = j.

(57) If i ≠ 0, then for every element k of Seg i holds id_{i}(k) = k.

(58) id_{0} = ε.

(59) id_{1} = ⟨1⟩.

(60) id_{i+1} = id_{i} ^{}⟨i + 1⟩.

(61) id_{2} = ⟨1, 2⟩.

(62) id_{3} = ⟨1, 2, 3⟩.

(63) p · id_{i} = p Seg i.

(64) If len p ≤ i, then p · id_{i} = p.

(65) id_{i} is a permutation of Seg i.

(66) Seg i ⟼ a is a finite sequence.

Let us consider i, a. The functor i ⟼ a yielding a finite sequence, is defined as follows:

i ⟼ a = Seg i ⟼ a.

We now state a number of propositions:

(67) i ⟼ a = Seg i ⟼ a.

(68) dom(i ⟼ a) = Seg i.

(69) len(i ⟼ a) = i.

(70) If j ∈ Seg i, then (i ⟼ a)(j) = a.

(71) If i ≠ 0, then for every element k of Seg i holds (i ⟼ d)(k) = d.

(72) 0 ⟼ a = ε.

(73) 1 ⟼ a = ⟨a⟩.

(74) i + 1 ⟼ a = (i ⟼ a)^{}⟨a⟩.

(75) 2 ⟼ a = ⟨a, a⟩.

(76) 3 ⟼ a = ⟨a, a, a⟩.

(77) i ⟼ d is a finite sequence of elements of D.

(78) For every function F such that [: rng p, rng q:] ⊆ dom F holds F^{∘}(p, q) is a finite sequence.

(79) For every function F such that [: rng p, rng q:] ⊆ dom F and r = F^{∘}(p, q) holds len r = min(len p, len q).

(80) For every function F such that [: {a}, rng p:] ⊆ dom F holds F^{∘}(a, p) is a finite sequence.

(81) For every function F such that [: {a}, rng p:] ⊆ dom F and r = F^{∘}(a, p) holds len r = len p.

(82) For every function F such that [: rng p, {a}:] ⊆ dom F holds F^{∘}(p, a) is a finite sequence.

(83) For every function F such that [: rng p, {a}:] ⊆ dom F and r = F^{∘}(p, a) holds len r = len p.

(84) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence q of elements of D′ holds F^{∘}(p, q) is a finite sequence of elements of E.

(85) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence q of elements of D′ such that r = F^{∘}(p, q) holds len r = min(len p, len q).

(86) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence q of elements of D′ such that len p = len q and r = F^{∘}(p, q) holds len r = len p and len r = len q.

(87) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence p′ of elements of D′ holds F^{∘}(ε_{D}, p′) = ε_{E} and F^{∘}(p, ε_{D′}) = ε_{E}.

(88) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence q of elements of D′ such that p = ⟨d_{1}⟩ and q = ⟨d′_{1}⟩ holds F^{∘}(p, q) = ⟨F (d_{1}, d′_{1})⟩.

(89) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence q of elements of D′ such that p = ⟨d_{1}, d_{2}⟩ and q = ⟨d′_{1}, d′_{2}⟩ holds F^{∘}(p, q) = ⟨F (d_{1}, d′_{1}), F (d_{2}, d′_{2})⟩.

(90) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D and for every finite sequence q of elements of D′ such that p = ⟨d_{1}, d_{2}, d_{3}⟩ and q = ⟨d′_{1}, d′_{2}, d′_{3}⟩ holds F^{∘}(p, q) = ⟨F (d_{1}, d′_{1}), F (d_{2}, d′_{2}), F (d_{3}, d′_{3})⟩.

(91) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D′ holds F^{∘}(d, p) is a finite sequence of elements of E.

(92) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D′ such that r = F^{∘}(d, p) holds len r = len p.

(93) For every function F from [: D, D′:] into E holds F^{∘}(d, ε_{D′}) = ε_{E}.

(94) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D′ such that p = ⟨d′_{1}⟩ holds F^{∘}(d, p) = ⟨F (d, d′_{1})⟩.

(95) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D′ such that p = ⟨d′_{1}, d′_{2}⟩ holds F^{∘}(d, p) = ⟨F (d, d′_{1}), F (d, d′_{2})⟩.

(96) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D′ such that p = ⟨d′_{1}, d′_{2}, d′_{3}⟩ holds F^{∘}(d, p) = ⟨F (d, d′_{1}), F (d, d′_{2}), F (d, d′_{3})⟩.

(97) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D holds F^{∘}(p, d′) is a finite sequence of elements of E.

(98) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D such that r = F^{∘}(p, d′) holds len r = len p.

(99) For every function F from [: D, D′:] into E holds F^{∘}(ε_{D}, d′) = ε_{E}.

(100) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D such that p = ⟨d_{1}⟩ holds F^{∘}(p, d′) = ⟨F (d_{1}, d′)⟩.

(101) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D such that p = ⟨d_{1}, d_{2}⟩ holds F^{∘}(p, d′) = ⟨F (d_{1}, d′), F (d_{2}, d′)⟩.

(102) For every function F from [: D, D′:] into E and for every finite sequence p of elements of D such that p = ⟨d_{1}, d_{2}, d_{3}⟩ holds F^{∘}(p, d′) = ⟨F (d_{1}, d′), F (d_{2}, d′), F (d_{3}, d′)⟩.

Let us consider D. A non-empty set is said to be a non-empty set of finite sequences of D if:

if a ∈ it, then a is a finite sequence of elements of D. We now state two propositions:

(103) For all D, D′ holds D′ is a non-empty set of finite sequences of D if and only if for every a such that a ∈ D′ holds a is a finite sequence of elements of D.

(104) D^{∗} is a non-empty set of finite sequences of D.

Let us consider D. Then D^{∗} is a non-empty set of finite sequences of D. Next we state two propositions:

(105) For every non-empty set D′ of finite sequences of D holds D′ ⊆ D^{∗}.

(106) For every non-empty set S of finite sequences of D and for every element

s of S holds s is a finite sequence of elements of D. Let us consider D, and let S be a non-empty set of finite sequences of D. We

see that it makes sense to consider the following mode for restricted scopes of arguments. Then all the objests of the mode element of S are a finite sequence of elements of D.

One can prove the following proposition

(107) For every non-empty subset D′ of D and for every non-empty set S of

finite sequences of D′ holds S is a non-empty set of finite sequences of D. In the sequel s is an element of D^{∗}. Let us consider i, D. The functor D^{i}

yielding a non-empty set of finite sequences of D, is defined as follows: D^{i} = {s : len s = i}. Next we state a number of propositions:

(108) D^{i} = {s : len s = i}.

(109) For every element z of D^{i} holds len z = i.

(110) For every finite sequence z of elements of D holds z is an element of D^{len z}.

(111) D^{i} = D^{Seg i}.

(112) D^{0} = {ε_{D}}.

(113) For every element z of D^{0} holds z = ε_{D}.

(114) ε_{D} is an element of D^{0}.

(115) For every element z of D^{0} and for every element t of D^{i} holds z^{}t = t and t^{}z = t.

(116) D^{1} = {⟨d⟩}.

(117) For every element z of D^{1} there exists d such that z = ⟨d⟩.

(118) ⟨d⟩ is an element of D^{1}.

(119) D^{2} = {⟨d_{1}, d_{2}⟩}.

(120) For every element z of D^{2} there exist d_{1}, d_{2} such that z = ⟨d_{1}, d_{2}⟩.

(121) ⟨d_{1}, d_{2}⟩ is an element of D^{2}.

(122) D^{3} = {⟨d_{1}, d_{2}, d_{3}⟩}.

(123) For every element z of D^{3} there exist d_{1}, d_{2}, d_{3} such that z = ⟨d_{1}, d_{2}, d_{3}⟩.

(124) ⟨d_{1}, d_{2}, d_{3}⟩ is an element of D^{3}.

(125) D^{i+j} = {z ^{}t}.

(126) For every element s of D^{i+j} there exists an element z of D^{i} and there exists an element t of D^{j} such that s = z ^{}t.

(127) For every element z of D^{i} and for every element t of D^{j} holds z ^{}t is an element of D^{i+j}.

(128) D^{∗} = ⋃{D^{i}}.

(129) For every non-empty subset D′ of D and for every element z of D′^{i} holds z is an element of D^{i}.

(130) If D^{i} = D^{j}, then i = j.

(131) id_{i} is an element of ^{i}.

(132) i ⟼ d is an element of D^{i}.

(133) For every element z of D^{i} and for every function f from D into D′ holds f · z is an element of D′^{i}.

(134) For every element z of D^{i} and for every function f from Seg i into Seg i such that rng f = Seg i holds z · f is an element of D^{i}.

(135) For every element z of D^{i} and for every permutation f of Seg i holds z · f is an element of D^{i}.

(136) For every element z of D^{i} and for every d holds (z ^{}⟨d⟩)(i + 1) = d.

(137) For every element z of D^{i+1} there exists an element t of D^{i} and there exists d such that z = t^{}⟨d⟩.

(138) For every element z of D^{i} holds z · id_{i} = z.

(139) For all elements z_{1}, z_{2} of D^{i} such that for every j such that j ∈ Seg i holds z_{1}(j) = z_{2}(j) holds z_{1} = z_{2}.

(140) For every function F from [: D, D′:] into E and for every element z_{1} of D^{i} and for every element z_{2} of D′^{i} holds F^{∘}(z_{1}, z_{2}) is an element of E^{i}.

(141) For every function F from [: D, D′:] into E and for every element z of D′^{i} holds F^{∘}(d, z) is an element of E^{i}.

(142) For every function F from [: D, D′:] into E and for every element z of D^{i} holds F^{∘}(z, d′) is an element of E^{i}.

References

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

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

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

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

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

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

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

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

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

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

Curried and Uncurried Functions

Grzegorz Bancerek Warsaw University Białystok

Summary. In the article following functors are introduced: the projections of subsets of the Cartesian product, the functor which for every function f : X ×Y → Z gives some curried function (X → (Y → Z)), and the functor which from curried functions makes uncurried functions. Some of their properties and some properties of the set of all functions from a set into a set are also shown.

MML Identifier: FUNCT 5.

The papers [8], [3], [2], [4], [9], [1], [6], [7], and [5] provide the terminology and notation for this paper. We follow a convention: X, Y, Z, X_{1}, X_{2}, Y_{1}, Y_{2} are sets, f, g, f_{1}, f_{2} are functions, and x, y, z, t are arbitrary. The scheme LambdaFS deals with a set A and a unary functor F and states that:

there exists f such that dom f = A and for every g such that g ∈ A holds f(g) = F(g) for all values of the parameters.

We now state a proposition

(1) = .

We now define two new functors. Let us consider X. The functor π_{1}(X) yields a set and is defined as follows:

x ∈ π_{1}(X) if and only if there exists y such that ⟨⟨x, y⟩⟩ ∈ X. The functor π_{2}(X) yields a set and is defined as follows:

y ∈ π_{2}(X) if and only if there exists x such that ⟨⟨x, y⟩⟩ ∈ X.

The following propositions are true:

(2) Z = π_{1}(X) if and only if for every x holds x ∈ Z if and only if there exists y such that ⟨⟨x, y⟩⟩ ∈ X.

(3) Z = π_{2}(X) if and only if for every y holds y ∈ Z if and only if there exists x such that ⟨⟨x, y⟩⟩ ∈ X.

(4) If ⟨⟨x, y⟩⟩ ∈ X, then x ∈ π_{1}(X) and y ∈ π_{2}(X).

(5) If X ⊆ Y, then π_{1}(X) ⊆ π_{1}(Y) and π_{2}(X) ⊆ π_{2}(Y).

(6) π_{1}(X ∪ Y) = π_{1}(X) ∪ π_{1}(Y) and π_{2}(X ∪ Y) = π_{2}(X) ∪ π_{2}(Y).

(7) π_{1}(X ∩ Y) ⊆ π_{1}(X) ∩ π_{1}(Y) and π_{2}(X ∩ Y) ⊆ π_{2}(X) ∩ π_{2}(Y).

(8) π_{1}(X) \ π_{1}(Y) ⊆ π_{1}(X \ Y) and π_{2}(X) \ π_{2}(Y) ⊆ π_{2}(X \ Y).

(9) π_{1}(X)− . π_{1}(Y) ⊆ π_{1}(X− . Y) and π_{2}(X)− . π_{2}(Y) ⊆ π_{2}(X− . Y).

(10) π_{1}(∅) = ∅ and π_{2}(∅) = ∅.

(11) If Y ≠ ∅ or [: X, Y :] ≠ ∅ or [: Y, X:] ≠ ∅, then π_{1}([: X, Y :]) = X and π_{2}([: Y, X:]) = X.

(12) π_{1}([: X, Y :]) ⊆ X and π_{2}([: X, Y :]) ⊆ Y.

(13) If Z ⊆ [: X, Y :], then π_{1}(Z) ⊆ X and π_{2}(Z) ⊆ Y.

(14) π_{1}([: X, {x}:]) = X and π_{2}([: {x}, X:]) = X and π_{1}([: X, {x, y}:]) = X and π_{2}([: {x, y}, X:]) = X.

(15) π_{1}({⟨⟨x, y⟩⟩}) = {x} and π_{2}({⟨⟨x, y⟩⟩}) = {y}.

(16) π_{1}({⟨⟨x, y⟩⟩, ⟨⟨z, t⟩⟩}) = {x, z} and π_{2}({⟨⟨x, y⟩⟩, ⟨⟨z, t⟩⟩}) = {y, t}.

(17) If for no x, y holds ⟨⟨x, y⟩⟩ ∈ X, then π_{1}(X) = ∅ and π_{2}(X) = ∅.

(18) If π_{1}(X) = ∅ or π_{2}(X) = ∅, then for no x, y holds ⟨⟨x, y⟩⟩ ∈ X.

(19) π_{1}(X) = ∅ if and only if π_{2}(X) = ∅.

(20) π_{1}(dom f) = π_{2}(dom( f)) and π_{2}(dom f) = π_{1}(dom( f)).

(21) π_{1}(graph f) = dom f and π_{2}(graph f) = rng f.

We now define two new functors. Let us consider f. The functor curry f yielding a function, is defined by:

(i) dom(curry f) = π_{1}(dom f), (ii) for every x such that x ∈ π_{1}(dom f) there exists g such that (curry f)(x) = g and dom g = π_{2}(dom f ∩ [: {x}, π_{2}(dom f) :]) and for every y such that y ∈ dom g holds g(y) = f(⟨⟨x, y⟩⟩). The functor uncurry f yields a function and is defined as follows:

(i) for every t holds t ∈ dom(uncurry f) if and only if there exist x, g, y such that t = ⟨⟨x, y⟩⟩ and x ∈ dom f and g = f(x) and y ∈ dom g, (ii) for all x, g such that x ∈ dom(uncurry f) and g = f(x_{1}) holds

(uncurry f)(x) = g(x_{2}) .

We now define two new functors. Let us consider f. The functor curry′ f yields a function and is defined as follows:

curry′ f = curry( f). The functor uncurry′ f yielding a function, is defined by:

uncurry′ f = (uncurry f).

The following propositions are true:

(22) Let F be a function. Then F = curry f if and only if the following

conditions are satisfied:

(i) dom F = π_{1}(dom f), (ii) for every x such that x ∈ π_{1}(dom f) there exists g such that F (x) = g

and dom g = π_{2}(dom f ∩ [: {x}, π_{2}(dom f) :]) and for every y such that y ∈ dom g holds g(y) = f(⟨⟨x, y⟩⟩).

Curried and Uncurried Functions 539

(23) curry′ f = curry( f).

(24) Let F be a function. Then F = uncurry f if and only if the following

conditions are satisfied:

(i) for every t holds t ∈ dom F if and only if there exist x, g, y such that t = ⟨⟨x, y⟩⟩ and x ∈ dom f and g = f(x) and y ∈ dom g,

(ii) for all x, g such that x ∈ dom F and g = f(x_{1}) holds F (x) = g(x_{2}).

(25) uncurry′ f = (uncurry f).

(26) If ⟨⟨x, y⟩⟩ ∈ dom f, then x ∈ dom(curry f) and curry f(x) is a function.

(27) If ⟨⟨x, y⟩⟩ ∈ dom f and g = curry f(x), then y ∈ dom g and g(y) = f(⟨⟨x, y⟩⟩).

(28) If ⟨⟨x, y⟩⟩ ∈ dom f, then y ∈ dom(curry′ f) and curry′ f(y) is a function.

(29) If ⟨⟨x, y⟩⟩ ∈ dom f and g = curry′ f(y), then x ∈ dom g and g(x) = f(⟨⟨x, y⟩⟩).

(30) dom(curry′ f) = π_{2}(dom f).

(31) If [: X, Y :] ≠ ∅ and dom f = [: X, Y :], then dom(curry f) = X and dom(curry′ f) = Y.

(32) If dom f ⊆ [: X, Y :], then dom(curry f) ⊆ X and dom(curry′ f) ⊆ Y.

(33) If rng f ⊆ Y^{X}, then dom(uncurry f) = [: dom f, X:] and dom(uncurry′ f) = [: X, dom f :] .

(34) If for no x, y holds ⟨⟨x, y⟩⟩ ∈ dom f, then curry f = and curry′ f = .

(35) If for no x holds x ∈ dom f and f(x) is a function, then uncurry f = and uncurry′ f = .

(36) Suppose [: X, Y :] ≠ ∅ and dom f = [: X, Y :] and x ∈ X. Then there exists g such that curry f(x) = g and dom g = Y and rng g ⊆ rng f and for every y such that y ∈ Y holds g(y) = f(⟨⟨x, y⟩⟩).

(37) If x ∈ dom(curry f), then curry f(x) is a function.

(38) Suppose x ∈ dom(curry f) and g = curry f(x). Then (i) dom g = π_{2}(dom f ∩ [: {x}, π_{2}(dom f) :]), (ii) dom g ⊆ π_{2}(dom f), (iii) rng g ⊆ rng f, (iv) for every y such that y ∈ dom g holds g(y) = f(⟨⟨x, y⟩⟩) and ⟨⟨x, y⟩⟩ ∈

dom f.

(39) Suppose [: X, Y :] ≠ ∅ and dom f = [: X, Y :] and y ∈ Y. Then there exists g such that curry′ f(y) = g and dom g = X and rng g ⊆ rng f and for every x such that x ∈ X holds g(x) = f(⟨⟨x, y⟩⟩).

(40) If x ∈ dom(curry′ f), then curry′ f(x) is a function.

(41) Suppose x ∈ dom(curry′ f) and g = curry′ f(x). Then (i) dom g = π_{1}(dom f ∩ [: π_{1}(dom f), {x}:]), (ii) dom g ⊆ π_{1}(dom f), (iii) rng g ⊆ rng f, (iv) for every y such that y ∈ dom g holds g(y) = f(⟨⟨y, x⟩⟩) and ⟨⟨y, x⟩⟩ ∈

dom f.

(42) If dom f = [: X, Y :], then rng(curry f) ⊆ (rng f)^{Y} and rng(curry′ f) ⊆ (rng f)^{X}.

(43) rng(curry f) ⊆ π_{2}(dom f) ˙→(rng f) and rng(curry′ f) ⊆ π_{1}(dom f) ˙→(rng f) .

(44) If rng f ⊆ X ˙→Y, then dom(uncurry f) ⊆ [: dom f, X:] and dom(uncurry′ f) ⊆ [: X, dom f :] .

(45) If x ∈ dom f and g = f(x) and y ∈ dom g, then ⟨⟨x, y⟩⟩ ∈ dom(uncurry f) and uncurry f(⟨⟨x, y⟩⟩) = g(y) and g(y) ∈ rng(uncurry f).

(46) If x ∈ dom f and g = f(x) and y ∈ dom g, then ⟨⟨y, x⟩⟩ ∈ dom(uncurry′ f) and uncurry′ f(⟨⟨y, x⟩⟩) = g(y) and g(y) ∈ rng(uncurry′ f).

(47) If rng f ⊆ X ˙→Y, then rng(uncurry f) ⊆ Y and rng(uncurry′ f) ⊆ Y.

(48) If rng f ⊆ Y^{X}, then rng(uncurry f) ⊆ Y and rng(uncurry′ f) ⊆ Y.

(49) curry = and curry′ = .

(50) uncurry = and uncurry′ = .

(51) If dom f_{1} = [: X, Y :] and dom f_{2} = [: X, Y :] and curry f_{1} = curry f_{2}, then f_{1} = f_{2}.

(52) If dom f_{1} = [: X, Y :] and dom f_{2} = [: X, Y :] and curry′ f_{1} = curry′ f_{2}, then f_{1} = f_{2}.

(53) If rng f_{1} ⊆ Y^{X} and rng f_{2} ⊆ Y^{X} and X ≠ ∅ and uncurry f_{1} = uncurry f_{2}, then f_{1} = f_{2}.

(54) If rng f_{1} ⊆ Y^{X} and rng f_{2} ⊆ Y^{X} and X ≠ ∅ and uncurry′ f_{1} = uncurry′ f_{2}, then f_{1} = f_{2}.

(55) If rng f ⊆ Y^{X} and X ≠ ∅, then curry(uncurry f) = f and curry′(uncurry′ f) = f .

(56) If dom f = [: X, Y :], then uncurry(curry f) = f and uncurry′(curry′ f) = f .

(57) If dom f ⊆ [: X, Y :], then uncurry(curry f) = f and uncurry′(curry′ f) = f .

(58) If rng f ⊆ X ˙→Y and ∉ rng f, then curry(uncurry f) = f and curry′(uncurry′ f) = f.

(59) If dom f_{1} ⊆ [: X, Y :] and dom f_{2} ⊆ [: X, Y :] and curry f_{1} = curry f_{2}, then f_{1} = f_{2}.

(60) If dom f_{1} ⊆ [: X, Y :] and dom f_{2} ⊆ [: X, Y :] and curry′ f_{1} = curry′ f_{2}, then f_{1} = f_{2}.

(61) If rng f_{1} ⊆ X ˙→Y and rng f_{2} ⊆ X ˙→Y and ∉ rng f_{1} and ∉ rng f_{2} and uncurry f_{1} = uncurry f_{2}, then f_{1} = f_{2}.

(62) If rng f_{1} ⊆ X ˙→Y and rng f_{2} ⊆ X ˙→Y and ∉ rng f_{1} and ∉ rng f_{2} and uncurry′ f_{1} = uncurry′ f_{2}, then f_{1} = f_{2}.

(63) If X ⊆ Y, then X^{Z} ⊆ Y^{Z}.

(64) X^{∅} = { }.

(65) X ≈ X^{\{x\}} and \overline{\overline{X}} = \overline{\overline{X^{\{x\}}}}.

(66) {x}^{X} = {X ⟼ x}.

(67) If X_{1} ≈ Y_{1} and X_{2} ≈ Y_{2}, then X_{2}^{X_{1}} ≈ Y_{2}^{Y_{1}} and \overline{\overline{X_{2}^{X_{1}}}} = \overline{\overline{Y_{2}^{Y_{1}}}} .

(68) If \overline{\overline{X_{1}}} = \overline{\overline{Y_{1}}} and \overline{\overline{X_{2}}} = \overline{\overline{Y_{2}}} , then \overline{\overline{X_{2}^{X_{1}}}} = \overline{\overline{Y_{2}^{Y_{1}}}} .

(69) If X_{1} ∩ X_{2} = ∅, then X^{X_{1}∪X_{2}} ≈ [: X^{X_{1}}, X^{X_{2}} :] and \overline{\overline{X^{X_{1}∪X_{2}}}} = \overline{\overline{[: X^{X_{1}}, X^{X_{2}} :]}} .

(70) Z^{[: X, Y :]} ≈ (Z^{Y})^{X} and \overline{\overline{Z^{[: X, Y :]}}} = \overline{\overline{(Z^{Y})^{X}}}.

(71) [: X, Y :]^{Z} ≈ [: X^{Z}, Y^{Z} :] and \overline{\overline{[: X, Y :]^{Z}}} = \overline{\overline{[: X^{Z}, Y^{Z} :]}}.

(72) If x ≠ y, then \{x, y\}^{X} ≈ 2^{X} and \overline{\overline{\{x, y\}^{X}}} = \overline{\overline{2^{X}}}.

(73) If x ≠ y, then X^{\{x,y\}} ≈ [: X, X:] and \overline{\overline{X^{\{x,y\}}}} = \overline{\overline{[: X, X:]}}.

References

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

[2] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

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

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

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

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

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

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

542

Cardinal Arithmetics

Grzegorz Bancerek Warsaw University Białystok

Summary. In the article addition, multiplication and power operation of cardinals are introduced. Presented are some properties of equipotence of Cartesian products, basic cardinal arithmetics laws (transformativity, associativity, distributivity), and some facts about finite sets.

MML Identifier: CARD 2.

The articles [12], [11], [7], [8], [3], [4], [5], [10], [2], [6], [9], and [1] provide the terminology and notation for this paper. For simplicity we follow a convention: A, B denote ordinal numbers, K, M, N denote cardinal numbers, x, x_{1}, x_{2}, y, y_{1}, y_{2} are arbitrary, X, Y, Z, X_{1}, X_{2}, Y_{1}, Y_{2} denote sets, and f denotes a function. Let us consider x. The functor [x] yielding a set, is defined by:

[x] = x.

Next we state several propositions:

(1) [x] = x.

(2) \overline{\overline{X}} ≤ \overline{\overline{Y}} if and only if there exists f such that X = f ^{∘} Y or X ⊆ f ^{∘} Y.

(3) \overline{\overline{f ^{∘} X}} ≤ \overline{\overline{X}}.

(4) If \overline{\overline{X}} < \overline{\overline{Y}} , then Y \ X ≠ ∅.

(5) If x ∈ X and X ≈ Y, then Y ≠ ∅ and there exists x such that x ∈ Y.

(6) 2^{X} ≈ 2\overline{\overline{X}} and \overline{\overline{2^{X}}} = \overline{\overline{2\overline{\overline{X}}}} .

(7) If Z ∈ Y^{X}, then Z ≈ X and \overline{\overline{Z}} = \overline{\overline{X}}.

We now define three new functors. Let us consider M, N. The functor M + N yielding a cardinal number, is defined as follows:

M + N = \overline{\overline{ord(M) + ord(N)}}. The functor M · N yielding a cardinal number, is defined by:

M · N = \overline{\overline{[: M, N :]}}. The functor M^{N} yielding a cardinal number, is defined by:

M^{N} = \overline{\overline{M^{N}}}.

Next we state a number of propositions:

(8) M + N = \overline{\overline{ord(M) + ord(N)}}.

(9) M · N = \overline{\overline{[: M, N :]}}.

(10) M^{N} = \overline{\overline{M^{N}}}.

(11) [: X, Y :] ≈ [: Y, X:] and \overline{\overline{[: X, Y :]}} = \overline{\overline{[: Y, X:]}}.

(12) [: [: X, Y :], Z:] ≈ [: X, [: Y, Z:] :] and \overline{\overline{[: [: X, Y :], Z:]}} = \overline{\overline{[: X, [: Y, Z:] :]}}.

(13) X ≈ [: X, \{x\}:] and \overline{\overline{X}} = \overline{\overline{[: X, \{x\}:]}}.

(14) (i) [: X, Y :] ≈ [: \overline{\overline{X}} , Y :], (ii) [: X, Y :] ≈ [: X, \overline{\overline{Y}} :], (iii) [: X, Y :] ≈ [: \overline{\overline{X}} , \overline{\overline{Y}} :], (iv) \overline{\overline{[: X, Y :]}} = \overline{\overline{[: \overline{\overline{X}} , Y :]}},

(v) \overline{\overline{[: X, Y :]}} = \overline{\overline{[: X, \overline{\overline{Y}} :]}}, (vi) \overline{\overline{[: X, Y :]}} = \overline{\overline{[: \overline{\overline{X}} , \overline{\overline{Y}} :]}}.

(15) If X_{1} ≈ Y_{1} and X_{2} ≈ Y_{2}, then [: X_{1}, X_{2} :] ≈ [: Y_{1}, Y_{2} :] and \overline{\overline{[: X_{1}, X_{2} :]}} = \overline{\overline{[: Y_{1}, Y_{2} :]}}.

(16) If x_{1} ≠ x_{2}, then A + B ≈ [: A, \{x_{1}\}:] ∪ [: B, \{x_{2}\}:] and \overline{\overline{A + B}} = \overline{\overline{[: A, \{x_{1}\}:] ∪ [: B, \{x_{2}\}:]}}.

(17) If x_{1} ≠ x_{2}, then K + M ≈ [: K, {x_{1}}:] ∪ [: M, {x_{2}}:] and K + M = \overline{\overline{[: K, \{x_{1}\}:] ∪ [: M, \{x_{2}\}:]}}.

(18) A · B ≈ [: A, B:] and \overline{\overline{A · B}} = \overline{\overline{[: A, B:]}}.

We now define three new functors. The cardinal number \overline{0} is defined by: \overline{0} = \overline{\overline{0}}.

The cardinal number \overline{1} is defined as follows:

\overline{1} = \overline{\overline{1}}. The cardinal number \overline{2} is defined as follows:

\overline{2} = \overline{\overline{succ 1}}.

The following propositions are true:

(19) \overline{0} = \overline{\overline{0}} and \overline{1} = \overline{\overline{1}} and \overline{2} = \overline{\overline{succ 1}}.

(20) \overline{0} = 0 and \overline{0} = ∅ and \overline{1} = 1.

(21) \overline{0} = \overline{\overline{0}} and \overline{1} = \overline{\overline{1}} and \overline{2} = \overline{\overline{2}}.

(22) \overline{2} = \{0, 1\} and \overline{2} = succ 1.

(23) Suppose X_{1} ≈ Y_{1} and X_{2} ≈ Y_{2} and x_{1} ≠ x_{2} and y_{1} ≠ y_{2}. Then [: X_{1}, {x_{1}}:] ∪ [: X_{2}, {x_{2}}:] ≈ [: Y_{1}, {y_{1}}:] ∪ [: Y_{2}, {y_{2}}:] and \overline{\overline{[: X_{1}, \{x_{1}\}:] ∪ [: X_{2}, \{x_{2}\}:]}} = \overline{\overline{[: Y_{1}, \{y_{1}\}:] ∪ [: Y_{2}, \{y_{2}\}:]}} .

(24) \overline{\overline{A + B}} = \overline{\overline{A}} + \overline{\overline{B}}.

Cardinal Arithmetics 545

(25) \overline{\overline{A · B}} = \overline{\overline{A}} · \overline{\overline{B}}.

(26) [: X, \{0\}:]∪[: Y, \{1\}:] ≈ [: Y, \{0\}:]∪[: X, \{1\}:] and \overline{\overline{[: X, \{0\}:] ∪ [: Y, \{1\}:]}} = \overline{\overline{[: Y, \{0\}:] ∪ [: X, \{1\}:]}}.

(27) [: X_{1}, X_{2} :] ∪ [: Y_{1}, Y_{2} :] ≈ [: X_{2}, X_{1} :] ∪ [: Y_{2}, Y_{1} :] and \overline{\overline{[: X_{1}, X_{2} :] ∪ [: Y_{1}, Y_{2} :]}} = \overline{\overline{[: X_{2}, X_{1} :] ∪ [: Y_{2}, Y_{1} :]}} .

(28) If x ≠ y, then \overline{\overline{X}} + \overline{\overline{Y}} = \overline{\overline{[: X, \{x\}:] ∪ [: Y, \{y\}:]}}.

(29) M + \overline{0} = M and \overline{0} + M = M.

(30) M + N = N + M.

(31) (K + M) + N = K + (M + N).

(32) K · \overline{0} = \overline{0} and \overline{0} · K = \overline{0}.

(33) K · \overline{1} = K and \overline{1} · K = K.

(34) K · M = M · K.

(35) (K · M) · N = K · (M · N).

(36) \overline{2} · K = K + K and K · \overline{2} = K + K.

(37) K · (M + N) = K · M + K · N and (M + N) · K = M · K + N · K.

(38) K\overline{0} = \overline{1}.

(39) If K ≠ \overline{0}, then \overline{0}^{K} = \overline{0}.

(40) K\overline{1} = K and \overline{1}^{K} = \overline{1}.

(41) K^{M+N} = (K^{M}) · (K^{N}).

(42) (K · M)^{N} = (K^{N}) · (M^{N}).

(43) K^{M·N} = (K^{M})^{N}.

(44) \overline{2}\overline{\overline{X}} = \overline{\overline{2^{X}}}.

(45) K\overline{2} = K · K.

(46) (K + M)\overline{2} = (K · K + (\overline{2} · K) · M) + M · M.

(47) \overline{\overline{X ∪ Y}} ≤ \overline{\overline{X}} + \overline{\overline{Y}} .

(48) If X ∩ Y = ∅, then \overline{\overline{X ∪ Y}} = \overline{\overline{X}} + \overline{\overline{Y}} .

In the sequel m, n will denote natural numbers. Next we state a number of propositions:

(49) ord(n + m) = ord(n) + ord(m).

(50) ord(n · m) = ord(n) · ord(m).

(51) \overline{\overline{n + m}} = \overline{\overline{n}} + \overline{\overline{m}}.

(52) \overline{\overline{n · m}} = \overline{\overline{n}} · \overline{\overline{m}}.

(53) If X is finite and Y is finite and X ∩ Y = ∅, then card(X ∪ Y) = card X + card Y.

(54) If X is finite and x ∉ X, then card(X ∪ {x}) = card X + 1.

(55) If X is finite and Y is finite, then card X = card Y if and only if X ≈ Y.

(56) If X is finite and Y is finite, then \overline{\overline{X}} = \overline{\overline{Y}} if and only if card X = card Y.

(57) If X is finite and Y is finite, then \overline{\overline{X}} ≤ \overline{\overline{Y}} if and only if card X ≤ card Y.

(58) If X is finite and Y is finite, then \overline{\overline{X}} < \overline{\overline{Y}} if and only if card X < card Y.

(59) If X is finite, then X = ∅ if and only if card X = 0.

(60) If X is finite, then card X = 1 if and only if there exists x such that X = {x}.

(61) If X is finite, then X ≈ ord(card X) and X ≈ \overline{\overline{card X}} and X ≈ Seg(card X).

(62) If X is finite and Y is finite, then card(X ∪ Y) ≤ card X + card Y.

(63) If Y ⊆ X and X is finite, then card(X \ Y) = card X − card Y.

(64) If X is finite and Y is finite, then card(X ∪ Y) = (card X + card Y) − card(X ∩ Y).

(65) If X is finite and Y is finite, then card[: X, Y :] = card X · card Y.

(66) If X ⊆ Y and Y is finite, then card X ≤ card Y.

(67) If X ⊆ Y and X ≠ Y and Y is finite, then card X < card Y and \overline{\overline{X}} < \overline{\overline{Y}} .

(68) If \overline{\overline{X}} ≤ \overline{\overline{Y}} or \overline{\overline{X}} < \overline{\overline{Y}} but Y is finite, then X is finite.

In the sequel x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8} are arbitrary. One can prove the following propositions:

(69) card{x_{1}, x_{2}} ≤ 2.

(70) card{x_{1}, x_{2}, x_{3}} ≤ 3.

(71) card{x_{1}, x_{2}, x_{3}, x_{4}} ≤ 4.

(72) card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} ≤ 5.

(73) card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} ≤ 6.

(74) card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}} ≤ 7.

(75) card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}} ≤ 8.

(76) If x_{1} ≠ x_{2}, then card{x_{1}, x_{2}} = 2.

(77) If x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{2} ≠ x_{3}, then card{x_{1}, x_{2}, x_{3}} = 3.

(78) If x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{2} ≠ x_{3} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{3} ≠ x_{4}, then card{x_{1}, x_{2}, x_{3}, x_{4}} = 4.

References

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

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

[3] Grzegorz Bancerek. 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. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

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

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

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

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

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

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

548

Fano-Desargues Parallelity Spaces ^{1}

Eugeniusz Kusak Warsaw University Białystok Wojciech Leończuk Warsaw University Białystok

Summary. This article is the second part of Parallelity Space. It contain definition of a Fano-Desargues space, axioms of a Fano-Desargues parallelity space, definition of the relations: collinearity, parallelogram and directed congruence and some basic facts concerned with them.

MML Identifier: PARSP 2.

The papers [2], and [1] provide the notation and terminology for this paper. In the sequel F will denote a field. We now state a proposition

(1) Aff_{F^{3}} is a parallelity space.

We follow the rules: a, b, c, d, p, q, r will denote elements of the universum of Aff_{F^{3}}, e, f, g, h will denote elements of [: the carrier of F, the carrier of F, the carrier of F :], and K, L will denote elements of the carrier of F. One can prove the following propositions:

(2) a, b ∥ c, d if and only if there exist e, f, g, h such that ⟨⟨a, b, c, d⟩⟩ = ⟨⟨e, f, g, h⟩⟩ but there exists K such that K · (e_{1} − f_{1}) = g_{1} − h_{1} and K · (e_{2} − f_{2}) = g_{2} − h_{2} and K · (e_{3} − f_{3}) = g_{3} − h_{3} or e_{1} − f_{1} = 0_{F} and e_{2} − f_{2} = 0_{F} and e_{3} − f_{3} = 0_{F}.

(3) If a, b a, c and ⟨⟨a, b, a, c⟩⟩ = ⟨⟨e, f, e, g⟩⟩, then e ≠ f and e ≠ g and f ≠ g.

(4) Suppose that

(i) a, b a, c, (ii) ⟨⟨a, b, a, c⟩⟩ = ⟨⟨e, f, e, g⟩⟩, (iii) K · (e_{1} − f_{1}) = L · (e_{1} − g_{1}), (iv) K · (e_{2} − f_{2}) = L · (e_{2} − g_{2}),

(v) K · (e_{3} − f_{3}) = L · (e_{3} − g_{3}). Then K = 0_{F} and L = 0_{F}.

1Supported by RPBP III.24

550 Eugeniusz Kusak and Wojciech Leończuk

(5) Suppose a, b a, c and a, b ∥ c, d and a, c ∥ b, d and ⟨⟨a, b, c, d⟩⟩ = ⟨⟨e, f, g, h⟩⟩. Then h_{1} = (f_{1} + g_{1}) − e_{1} and h_{2} = (f_{2} + g_{2}) − e_{2} and h_{3} = (f_{3} + g_{3}) − e_{3}.

(6) There exist a, b, c such that a, b a, c.

(7) If 1_{F} + 1_{F} ≠ 0_{F} and b, c ∥ a, d and a, b ∥ c, d and a, c ∥ b, d, then a, b ∥ a, c.

(8) If a, p a, b and a, p a, c and a, p ∥ b, q and a, p ∥ c, r and a, b ∥ p, q and a, c ∥ p, r, then b, c ∥ q, r.

A parallelity space is called a Fano-Desarques space if:

(i) there exist elements a, b, c of the universum of it such that a, b a, c, (ii) for all elements a, b, c, d of the universum of it such that b, c ∥ a, d and a, b ∥ c, d and a, c ∥ b, d holds a, b ∥ a, c, (iii) for all elements a, b, c, p, q, r of the universum of it such that a, p a, b and a, p a, c and a, p ∥ b, q and a, p ∥ c, r and a, b ∥ p, q and a, c ∥ p, r holds b, c ∥ q, r.

We now state a proposition

(9) Let F d be a parallelity space. Then the following conditions are equivalent:

(i) there exist elements a, b, c of the universum of F d such that a, b a, c and for all elements a, b, c, d of the universum of F d such that b, c ∥ a, d and a, b ∥ c, d and a, c ∥ b, d holds a, b ∥ a, c and for all elements a, b, c, p, q, r of the universum of F d such that a, p a, b and a, p a, c and a, p ∥ b, q and a, p ∥ c, r and a, b ∥ p, q and a, c ∥ p, r holds b, c ∥ q, r,

(ii) F d is a Fano-Desarques space. We adopt the following convention: F dSp is a Fano-Desarques space and a,

b, c, d, p, q, r, s, o, x, y are elements of the universum of F dSp. The following propositions are true:

(10) There exist a, b, c such that a, b a, c.

(11) If b, c ∥ a, d and a, b ∥ c, d and a, c ∥ b, d, then a, b ∥ a, c.

(12) If a, p a, b and a, p a, c and a, p ∥ b, q and a, p ∥ c, r and a, b ∥ p, q and a, c ∥ p, r, then b, c ∥ q, r.

(13) If p ≠ q, then there exists r such that p, q p, r.

Let us consider F dSp, a, b, c. The predicate L(a, b, c) is defined as follows: a, b ∥ a, c. The following propositions are true:

(14) L(a, b, c) if and only if a, b ∥ a, c.

(15) If L(a, b, c), then L(a, c, b) and L(c, b, a) and L(b, a, c) and L(b, c, a) and L(c, a, b).

(16) If not L(a, b, c), then not L(a, c, b) and not L(c, b, a) and not L(b, a, c) and not L(b, c, a) and not L(c, a, b).

(17) If not L(a, b, c) and a, b ∥ p, q and a, c ∥ p, r and p ≠ q and p ≠ r, then not L(p, q, r).

Fano-Desargues Parallelity Spaces 551

(18) If a = b or b = c or c = a, then L(a, b, c).

(19) If a ≠ b and L(a, b, p) and L(a, b, q) and L(a, b, r), then L(p, q, r).

(20) If p ≠ q, then there exists r such that not L(p, q, r).

(21) If L(a, b, c) and L(a, b, d), then a, b ∥ c, d.

(22) If not L(a, b, c) and a, b ∥ c, d, then not L(a, b, d).

(23) If not L(a, b, c) and a, b ∥ c, d and c ≠ d, then not L(a, b, x) or not L(c, d, x) .

(24) If not L(o, a, b), then not L(o, a, x) or not L(o, b, x) or o = x.

(25) If o ≠ a and o ≠ b and L(o, a, b) and L(o, a, p) and L(o, b, q), then a, b ∥ p, q.

(26) If a, b c, d and L(a, b, p) and L(a, b, q) and L(c, d, p) and L(c, d, q), then p = q.

(27) If a ≠ b and L(a, b, c) and a, b ∥ c, d, then a, c ∥ b, d.

(28) If a ≠ b and L(a, b, c) and a, b ∥ c, d, then c, b ∥ c, d.

(29) If not L(o, a, c) and L(o, a, b) and L(o, c, p) and L(o, c, q) and a, c ∥ b, p and a, c ∥ b, q, then p = q.

(30) If a ≠ b and L(a, b, c) and L(a, b, d), then L(a, c, d).

(31) If L(a, b, c) and L(a, c, d) and a ≠ c, then L(b, c, d).

(32) L(a, b, c) if and only if a, b ∥ a, c.

Let us consider F dSp, a, b, c, d. The predicate P(a, b, c, d) is defined by: not L(a, b, c) and a, b ∥ c, d and a, c ∥ b, d. Next we state a number of propositions:

(33) P(a, b, c, d) if and only if not L(a, b, c) and a, b ∥ c, d and a, c ∥ b, d.

(34) If P(a, b, c, d), then a ≠ b and b ≠ c and c ≠ a and a ≠ d and b ≠ d and c ≠ d.

(35) If P(a, b, c, d), then not L(a, b, c) and not L(b, a, d) and not L(c, d, a) and not L(d, c, b).

(36) Suppose P(a, b, c, d). Then not L(a, b, c) and not L(b, a, d) and not L(c, d, a) and not L(d, c, b) and not L(a, c, b) and not L(b, a, c) and not L(b, c, a) and not L(c, a, b) and not L(c, b, a) and not L(b, d, a) and not L(a, b, d) and not L(a, d, b) and not L(d, a, b) and not L(d, b, a) and not L(c, a, d) and not L(a, c, d) and not L(a, d, c) and not L(d, a, c) and not L(d, c, a) and not L(d, b, c) and not L(b, c, d) and not L(b, d, c) and not L(c, b, d) and not L(c, d, b).

(37) If P(a, b, c, d), then not L(a, b, x) or not L(c, d, x).

(38) If P(a, b, c, d), then P(a, c, b, d).

(39) If P(a, b, c, d), then P(c, d, a, b).

(40) If P(a, b, c, d), then P(b, a, d, c).

(41) If P(a, b, c, d), then P(a, c, b, d) and P(c, d, a, b) and P(b, a, d, c) and P(c, a, d, b) and P(d, b, c, a) and P(b, d, a, c) and P(d, c, b, a).

(42) If not L(a, b, c), then there exists d such that P(a, b, c, d).

(43) If P(a, b, c, p) and P(a, b, c, q), then p = q.

(44) If P(a, b, c, d), then a, d b, c.

(45) If P(a, b, c, d), then not P(a, b, d, c).

(46) If a ≠ b, then there exists c such that L(a, b, c) and c ≠ a and c ≠ b.

(47) If P(a, p, b, q) and P(a, p, c, r), then b, c ∥ q, r.

(48) If not L(b, q, c) and P(a, p, b, q) and P(a, p, c, r), then P(b, q, c, r).

(49) If L(a, b, c) and b ≠ c and P(a, p, b, q) and P(a, p, c, r), then P(b, q, c, r).

(50) If P(a, p, b, q) and P(a, p, c, r) and P(b, q, d, s), then c, d ∥ r, s.

(51) If a ≠ b, then there exist c, d such that P(a, b, c, d).

(52) If a ≠ d, then there exist b, c such that P(a, b, c, d).

(53) P(a, b, c, d) if and only if not L(a, b, c) and a, b ∥ c, d and a, c ∥ b, d. Let us consider F dSp, a, b, r, s. The predicate a, b r, s is defined as

follows:

a = b and r = s or there exist p, q such that P(p, q, a, b) and P(p, q, r, s). One can prove the following propositions:

(54) a, b r, s if and only if a = b and r = s or there exist p, q such that P(p, q, a, b) and P(p, q, r, s).

(55) If a, a b, c, then b = c.

(56) If a, b c, c, then a = b.

(57) If a, b b, a, then a = b.

(58) If a, b c, d, then a, b ∥ c, d.

(59) If a, b c, d, then a, c ∥ b, d.

(60) If a, b c, d and not L(a, b, c), then P(a, b, c, d).

(61) If P(a, b, c, d), then a, b c, d.

(62) If a, b c, d and L(a, b, c) and P(r, s, a, b), then P(r, s, c, d).

(63) If a, b c, x and a, b c, y, then x = y.

(64) There exists d such that a, b c, d.

(65) a, a b, b.

(66) a, b a, b.

(67) If r, s a, b and r, s c, d, then a, b c, d.

(68) If a, b c, d, then c, d a, b.

(69) If a, b c, d, then b, a d, c.

References

[1] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Parallelity spaces. Formalized Mathematics, 1(2):343–348, 1990.

Fano-Desargues Parallelity Spaces 553

[2] Andrzej Trybulec. Domains and their Cartesian products. Formalized

Mathematics, 1(1):115–122, 1990. Received March 23, 1990 554

Real Functions Spaces ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. This abstract contains a construction of the domain of functions defined in an arbitrary nonempty set, with values being real numbers. In every such set of functions we introduce several algebraic operations, which yield in this set the structures of a real linear space, of a ring, and of a real algebra. Formal definitions of such concepts are given.

MML Identifier: FUNCSDOM.

The notation and terminology used in this paper are introduced in the following papers: [3], [9], [11], [2], [7], [12], [6], [1], [10], [4], [5], and [8]. We adopt the following convention: x_{1}, x_{2}, z are arbitrary and A, B denote non-empty sets. Let us consider A, B, and let F be a binary operation on B^{A}, and let f, g be elements of B^{A}. Then F (f, g) is an element of B^{A}.

Let A, B, C, D be non-empty sets, and let F be a function from [: C, D:] into B^{A}, and let cd be an element of [: C, D:]. Then F (cd) is an element of B^{A}.

Let A, B be non-empty sets, and let f be a function from A into B. The functor @f yields an element of B^{A} and is defined by:

@f = f.

We now state a proposition

(1) For all functions f, g from A into B holds @f = g if and only if f = g. In the sequel f, g, h denote elements of ^{A}. Let A, B be non-empty sets,

and let x be an element of B^{A}. The functor x yields an element of B^{A} qua a non-empty set and is defined as follows:

x = x.

We now state a proposition

(2) For all elements f, g of B^{A} holds f = g if and only if f = g.

1Supported by RPBP.III-24.C2

556 Henryk Oryszczyszyn et al.

Let us consider A, B, and let f be an element of B^{A} qua a non-empty set. The functor f yielding an element of B^{A}, is defined by:

f = f. We now state two propositions:

(3) For all elements f, g of B^{A} qua non-empty sets holds f = g if and only if f = g.

(4) f = ( f) . Let X, Z be non-empty sets, and let F be a binary operation on X, and let

f, g be functions from Z into X. Then F^{∘}(f, g) is an element of X^{Z}.

Let X, Z be non-empty sets, and let F be a binary operation on X, and let a be an element of X, and let f be a function from Z into X. Then F^{∘}(a, f) is an element of X^{Z}.

Let us consider A. The functor +_{^{A}} yields a binary operation on ^{A} and is defined by:

for all elements f, g of ^{A} holds +_{^{A}} (f, g) = +_{}^{∘}(f, g). We now state a proposition

(5) For every binary operation F on ^{A} holds F = +_{^{A}} if and only if for all elements f, g of ^{A} holds F (f, g) = +_{}^{∘}(f, g).

Let us consider A. The functor ·_{^{A}} yields a binary operation on ^{A} and is defined as follows:

for all elements f, g of ^{A} holds ·_{^{A}} (f, g) = ·_{}^{∘}(f, g). Next we state a proposition

(6) For every binary operation F on ^{A} holds F = ·_{^{A}} if and only if for all elements f, g of ^{A} holds F (f, g) = ·_{}^{∘}(f, g).

Let us consider A, and let a be a real number, and let f be an element of A. Then ⟨⟨a, f⟩⟩ is an element of [: , ^{A}:].

Let us consider A. The functor ·_{^{A}}^{}yielding a function from [: , ^{A}:] into A, is defined as follows:

for every real number a and for every element f of ^{A} and for every element x of A holds (·_{^{A}}^{}(⟨⟨a, f⟩⟩))(x) = a · f(x).

The following proposition is true

(7) For every function F from [: , ^{A}:] into ^{A} holds F = ·_{^{A}}^{}if and only if for every real number a and for every element f of ^{A} and for every element x of A holds (F (⟨⟨a, f⟩⟩))(x) = a · f(x).

Let us consider A. The functor 0_{^{A}} yields an element of ^{A} and is defined by:

0_{^{A}} = A ⟼ 0. The following proposition is true

(8) For every element f of ^{A} holds f = 0_{^{A}} if and only if f = A ⟼ 0. Let us consider A. The functor 1_{^{A}} yields an element of ^{A} and is defined

by:

1_{^{A}} = A ⟼ 1.

Real Functions Spaces 557

We now state several propositions:

(9) For every element f of ^{A} holds f = 1_{^{A}} if and only if f = A ⟼ 1.

(10) h = +_{^{A}} (f, g) if and only if for every element x of A holds h(x) = f(x) + g(x).

(11) h = ·_{^{A}} (f, g) if and only if for every element x of A holds h(x) = f(x) · g(x).

(12) For every element x of A holds 1_{^{A}} (x) = 1.

(13) For every element x of A holds 0_{^{A}} (x) = 0.

(14) 0_{^{A}} ≠ 1_{^{A}} .

In the sequel a, b are real numbers. The following proposition is true

(15) h = ·_{^{A}}^{}(⟨⟨a, f⟩⟩) if and only if for every element x of A holds h(x) =

a · f(x). One can prove the following propositions:

(16) +_{^{A}} (f, g) = +_{^{A}} (g, f).

(17) +_{^{A}} (f, +_{^{A}} (g, h)) = +_{^{A}} (+_{^{A}} (f, g), h).

(18) ·_{^{A}} (f, g) = ·_{^{A}} (g, f).

(19) ·_{^{A}} (f, ·_{^{A}} (g, h)) = ·_{^{A}} (·_{^{A}} (f, g), h).

(20) ·_{^{A}} (1_{^{A}} , f) = f.

(21) +_{^{A}} (0_{^{A}} , f) = f.

(22) +_{^{A}} (f, ·_{^{A}}^{}(⟨⟨ − 1, f⟩⟩)) = 0_{^{A}} .

(23) ·_{^{A}}^{}(⟨⟨1, f⟩⟩) = f.

(24) ·_{^{A}}^{}(⟨⟨a, ·_{^{A}}^{}(⟨⟨b, f⟩⟩)⟩⟩) = ·_{^{A}}^{}(⟨⟨a · b, f⟩⟩).

(25) +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, f⟩⟩)) = ·_{^{A}}^{}(⟨⟨a + b, f⟩⟩).

(26) ·_{^{A}} (f, +_{^{A}} (g, h)) = +_{^{A}} (·_{^{A}} (f, g), ·_{^{A}} (f, h)).

(27) ·_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), g) = ·_{^{A}}^{}(⟨⟨a, ·_{^{A}} (f, g)⟩⟩).

(28) Suppose x_{1} ∈ A and x_{2} ∈ A and x_{1} ≠ x_{2}. Then there exist f, g such that for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1}, then f(z) = 0 and for every z such that z ∈ A holds if z = x_{1}, then g(z) = 0 but if z ≠ x_{1}, then g(z) = 1.

(29) Suppose that (i) x_{1} ∈ A, (ii) x_{2} ∈ A, (iii) x_{1} ≠ x_{2}, (iv) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1},

then f(z) = 0,

(v) for every z such that z ∈ A holds if z = x_{1}, then g(z) = 0 but if z ≠ x_{1}, then g(z) = 1. Then for all a, b such that +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0.

(30) If x_{1} ∈ A and x_{2} ∈ A and x_{1} ≠ x_{2}, then there exist f, g such that for all a, b such that +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0.

(31) Suppose that (i) A = {x_{1}, x_{2}}, (ii) x_{1} ≠ x_{2}, (iii) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1},

then f(z) = 0, (iv) for every z such that z ∈ A holds if z = x_{1}, then g(z) = 0 but if z ≠ x_{1},

then g(z) = 1.

Then for every h there exist a, b such that

h = +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)) .

(32) If A = {x_{1}, x_{2}} and x_{1} ≠ x_{2}, then there exist f, g such that for every h there exist a, b such that h = +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)).

(33) Suppose A = {x_{1}, x_{2}} and x_{1} ≠ x_{2}. Then there exist f, g such that for all a, b such that +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0 and for every h there exist a, b such that h = +_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)).

(34) ⟨ ^{A}, 0_{^{A}} , +_{^{A}} , ·_{^{A}}^{}⟩ is a real linear space.

Let us consider A. The functor A_{}yields a real linear space and is defined by:

A_{}= ⟨ ^{A}, 0_{^{A}} , +_{^{A}} , ·_{^{A}}^{}⟩.

We now state two propositions:

(35) A_{}= ⟨ ^{A}, 0_{^{A}} , +_{^{A}} , ·_{^{A}}^{}⟩.

(36) A_{}is a real linear space.

In the sequel V will denote a real linear space and u, v, w will denote vectors of V. The following proposition is true

(37) There exists V and there exist u, v such that for all a, b such that a · u + b · v = 0_{V} holds a = 0 and b = 0 and for every w there exist a, b such that w = a · u + b · v.

We consider ring structures which are systems ⟨ a carrier, a multiplication, an addition, a unity, a zero ⟩ where the carrier is a non-empty set, the multiplication, the addition are

binary operations on the carrier, and the unity, the zero are elements of the carrier. In the sequel RS will be a ring structure. We now define four new functors. Let us consider RS. The functor 1_{RS} yields an element of the carrier of RS and is defined as follows:

1_{RS} = the unity of RS. The functor 0_{RS} yields an element of the carrier of RS and is defined as follows:

0_{RS} = the zero of RS. Let x, y be elements of the carrier of RS. The functor x · y yielding an element of the carrier of RS, is defined by:

x · y = (the multiplication of RS)(x, y). The functor x + y yielding an element of the carrier of RS, is defined by:

x + y = (the addition of RS)(x, y). In the sequel x, y denote elements of the carrier of RS. One can prove the

following four propositions:

(38) (the multiplication of RS)(x, y) = x · y.

(39) (the addition of RS)(x, y) = x + y.

(40) 1_{RS} = the unity of RS.

(41) 0_{RS} = the zero of RS.

Let us consider A. The functor RRing A yielding a ring structure, is defined by:

RRing A = ⟨ ^{A}, ·_{^{A}} , +_{^{A}} , 1_{^{A}} , 0_{^{A}} ⟩. Next we state a proposition

(42) Let x, y, z be elements of the carrier of RRing A. Then (i) x + y = y + x, (ii) (x + y) + z = x + (y + z), (iii) x + 0_{RRing A} = x, (iv) there exists an element t of the carrier of RRing A such that x + t =

0_{RRing A},

(v) x · y = y · x, (vi) (x · y) · z = x · (y · z), (vii) x · (1_{RRing A}) = x, (viii) x · (y + z) = x · y + x · z. A ring structure is said to be a ring if: Let x, y, z be elements of the carrier of it . Then

(i) x + y = y + x, (ii) (x + y) + z = x + (y + z), (iii) x + 0_{it} = x, (iv) there exists an element t of the carrier of it such that x + t = 0_{it},

(v) x · y = y · x, (vi) (x · y) · z = x · (y · z), (vii) x · (1_{it}) = x, (viii) x · (y + z) = x · y + x · z.

One can prove the following proposition

(43) RRing A is a ring.

We consider algebra structures which are systems ⟨ a carrier, a multiplication, an addition, a multiplication_{1}, a unity, a zero ⟩ where the carrier is a non-empty set, the multiplication, the addition are

binary operations on the carrier, the multiplication_{1} is a function from [: , the carrier :] into the carrier, and the unity, the zero are elements of the carrier. In the sequel AlS denotes an algebra structure. We now define four new functors. Let us consider AlS. The functor 1_{AlS} yielding an element of the carrier of AlS, is defined as follows:

1_{AlS} = the unity of AlS. The functor 0_{AlS} yielding an element of the carrier of AlS, is defined by:

0_{AlS} = the zero of AlS. Let x, y be elements of the carrier of AlS. The functor x · y yields an element of the carrier of AlS and is defined by:

x · y = (the multiplication of AlS)(x, y). The functor x+y yielding an element of the carrier of AlS, is defined as follows:

x + y = (the addition of AlS)(x, y). Let us consider AlS, and let x be an element of the carrier of AlS, and let a

be a real number. The functor a · x yields an element of the carrier of AlS and is defined as follows:

a · x = (the multiplication_{1} of AlS)(⟨⟨a, x⟩⟩). In the sequel x, y are elements of the carrier of AlS. Next we state several

propositions:

(44) (the multiplication of AlS)(x, y) = x · y.

(45) (the addition of AlS)(x, y) = x + y.

(46) (the multiplication_{1} of AlS)(⟨⟨a, x⟩⟩) = a · x.

(47) 0_{AlS} = the zero of AlS.

(48) 1_{AlS} = the unity of AlS.

Let us consider A. The functor RAlgebra A yielding an algebra structure, is defined as follows:

RAlgebra A = ⟨ ^{A}, ·_{^{A}} , +_{^{A}} , ·_{^{A}}^{}, 1_{^{A}} , 0_{^{A}} ⟩. The following proposition is true

(49) Let x, y, z be elements of the carrier of RAlgebra A. Given a, b. Then (i) x + y = y + x, (ii) (x + y) + z = x + (y + z), (iii) x + 0_{RAlgebra A} = x, (iv) there exists an element t of the carrier of RAlgebra A such that x+t =

0_{RAlgebra A},

(v) x · y = y · x, (vi) (x · y) · z = x · (y · z), (vii) x · (1_{RAlgebra A}) = x, (viii) x · (y + z) = x · y + x · z, (ix) a · (x · y) = (a · x) · y,

(x) a · (x + y) = a · x + a · y, (xi) (a + b) · x = a · x + b · x, (xii) (a · b) · x = a · (b · x). An algebra structure is said to be an algebra if: Let x, y, z be elements of the carrier of it . Given a, b. Then

(i) x + y = y + x, (ii) (x + y) + z = x + (y + z), (iii) x + 0_{it} = x, (iv) there exists an element t of the carrier of it such that x + t = 0_{it},

(v) x · y = y · x, (vi) (x · y) · z = x · (y · z), (vii) x · (1_{it}) = x, (viii) x · (y + z) = x · y + x · z, (ix) a · (x · y) = (a · x) · y, (x) a · (x + y) = a · x + a · y, (xi) (a + b) · x = a · x + b · x, (xii) (a · b) · x = a · (b · x).

The following proposition is true

(50) RAlgebra A is an algebra.

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

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

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

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

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

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

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

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

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

562

Tarski’s Classes and Ranks

Grzegorz Bancerek Warsaw University Białystok

Summary. In the article the Tarski’s classes (non-empty families of sets satisfying Tarski’s axiom A given in [9]) and the rank sets are introduced and some of their properties are shown. The transitive closure and the rank of a set is given here too.

MML Identifier: CLASSES1.

The terminology and notation used here have been introduced in the following articles: [9], [8], [7], [3], [4], [6], [5], [2], and [1]. For simplicity we adopt the following rules: W, X, Y, Z will denote sets, D will denote a non-empty set, f will denote a function, and x, y will be arbitrary. Let B be a set. We say that B is a Tarski-Class if and only if:

for all X, Y such that X ∈ B and Y ⊆ X holds Y ∈ B and for every X such that X ∈ B holds 2^{X} ∈ B and for every X such that X ⊆ B holds X ≈ B or X ∈ B.

Let A, B be sets. We say that B is Tarski-Class of A if and only if:

A ∈ B and B is a Tarski-Class.

Let A be a set. The functor T(A) yielding a non-empty family of sets, is defined as follows:

T(A) is Tarski-Class of A and for every D such that D is Tarski-Class of A holds T(A) ⊆ D.

We now state several propositions:

(1) W is a Tarski-Class if and only if for all X, Y such that X ∈ W and Y ⊆ X holds Y ∈ W and for every X such that X ∈ W holds 2^{X} ∈ W and for every X such that X ⊆ W holds X ≈ W or X ∈ W.

(2) W is a Tarski-Class if and only if for all X, Y such that X ∈ W and Y ⊆ X holds Y ∈ W and for every X such that X ∈ W holds 2^{X} ∈ W and for every X such that X ⊆ W and \overline{\overline{X}} < \overline{\overline{W}} holds X ∈ W.

(3) X is Tarski-Class of Y if and only if Y ∈ X and X is a Tarski-Class.

(4) For every non-empty family W of sets holds W = T(X) if and only if W is Tarski-Class of X and for every D such that D is Tarski-Class of X holds W ⊆ D.

(5) X ∈ T(X).

(6) If Y ∈ T(X) and Z ⊆ Y, then Z ∈ T(X).

(7) If Y ∈ T(X), then 2^{Y} ∈ T(X).

(8) If Y ⊆ T(X), then Y ≈ T(X) or Y ∈ T(X).

(9) If Y ⊆ T(X) and \overline{\overline{Y}} < \overline{\overline{T(X)}}, then Y ∈ T(X).

We follow a convention: u, v will denote elements of T(X), A, B, C will denote ordinal numbers, and L, L_{1} will denote transfinite sequences. Let us consider X, A. The functor T_{A}(X) is defined as follows:

there exists L such that T_{A}(X) = last L and dom L = succ A and L(0) = {X} and for all C, y such that succ C ∈ succ A and y = L(C) holds L(succ C) = ({u : ⋁_{v}[v ∈ [y] ∧ u ⊆ v]} ∪ {2^{v} : v ∈ [y]}) ∪ 2^{[y]} ∩ T(X) and for all C, L_{1} such that C ∈ succ A and C ≠ 0 and C is a limit ordinal number and L_{1} = L C holds L(C) = ⋃(rng L_{1}) ∩ T(X).

Let us consider X, A. Then T_{A}(X) is a subset of T(X).

Next we state a number of propositions:

(10) T_{0}(X) = {X}.

(11) T_{succ A}(X) = ({u : ⋁_{v}[v ∈ T_{A}(X) ∧ u ⊆ v]} ∪ {2^{v} : v ∈ T_{A}(X)}) ∪ 2^{T_{A}(X)} ∩ T(X).

(12) If A ≠ 0 and A is a limit ordinal number, then T_{A}(X) = {u : ⋁_{B}[B ∈ A ∧ u ∈ T_{B}(X)]}.

(13) Y ∈ T_{succ A}(X) if and only if Y ⊆ T_{A}(X) and Y ∈ T(X) or there exists Z such that Z ∈ T_{A}(X) but Y ⊆ Z or Y = 2^{Z}.

(14) If Y ⊆ Z and Z ∈ T_{A}(X), then Y ∈ T_{succ A}(X).

(15) If Y ∈ T_{A}(X), then 2^{Y} ∈ T_{succ A}(X).

(16) If A ≠ 0 and A is a limit ordinal number, then x ∈ T_{A}(X) if and only if there exists B such that B ∈ A and x ∈ T_{B}(X).

(17) If A ≠ 0 and A is a limit ordinal number and Y ∈ T_{A}(X) but Z ⊆ Y or Z = 2^{Y}, then Z ∈ T_{A}(X).

(18) T_{A}(X) ⊆ T_{succ A}(X).

(19) If A ⊆ B, then T_{A}(X) ⊆ T_{B}(X).

(20) There exists A such that T_{A}(X) = T_{succ A}(X).

(21) If T_{A}(X) = T_{succ A}(X), then T_{A}(X) = T(X).

(22) There exists A such that T_{A}(X) = T(X).

(23) There exists A such that T_{A}(X) = T(X) and for every B such that B ∈ A holds T_{B}(X) ≠ T(X).

(24) If Y ≠ X and Y ∈ T(X), then there exists A such that Y ∉ T_{A}(X) and Y ∈ T_{succ A}(X).

Tarski’s Classes and Ranks 565

(25) If X is transitive, then for every A such that A ≠ 0 holds T_{A}(X) is transitive.

(26) T_{0}(X) ∈ T_{1}(X) and T_{0}(X) ≠ T_{1}(X).

(27) If X is transitive, then T(X) is transitive.

(28) If Y ∈ T(X), then \overline{\overline{Y}} < \overline{\overline{T(X)}}.

(29) If Y ∈ T(X), then Y ≉ T(X).

(30) If x ∈ T(X) and y ∈ T(X), then {x} ∈ T(X) and {x, y} ∈ T(X).

(31) If x ∈ T(X) and y ∈ T(X), then ⟨⟨x, y⟩⟩ ∈ T(X).

(32) If Y ⊆ T(X) and Z ⊆ T(X), then [: Y, Z:] ⊆ T(X).

Let us consider A. The functor R_{A} is defined as follows:

there exists L such that R_{A} = last L and dom L = succ A and L(0) = ∅ and for all C, y such that succ C ∈ succ A and y = L(C) holds L(succ C) = 2^{[y]} and for all C, L_{1} such that C ∈ succ A and C ≠ 0 and C is a limit ordinal number and L_{1} = L C holds L(C) = ⋃(rng L_{1}).

Let us consider A. Then R_{A} is a set.

One can prove the following propositions:

(33) R_{0} = ∅.

(34) R_{succ A} = 2^{R_{A}}.

(35) If A ≠ 0 and A is a limit ordinal number, then for every x holds x ∈ R_{A} if and only if there exists B such that B ∈ A and x ∈ R_{B}.

(36) X ⊆ R_{A} if and only if X ∈ R_{succ A}.

(37) R_{A} is transitive.

(38) If X ∈ R_{A}, then X ⊆ R_{A}.

(39) R_{A} ⊆ R_{succ A}.

(40) ⋃ R_{A} ⊆ R_{A}.

(41) If X ∈ R_{A}, then ⋃ X ∈ R_{A}.

(42) A ∈ B if and only if R_{A} ∈ R_{B}.

(43) A ⊆ B if and only if R_{A} ⊆ R_{B}.

(44) A ⊆ R_{A}.

(45) For all A, X such that X ∈ R_{A} holds X ≉ R_{A} and \overline{\overline{X}} < \overline{\overline{R_{A}}}.

(46) X ⊆ R_{A} if and only if 2^{X} ⊆ R_{succ A}.

(47) If X ⊆ Y and Y ∈ R_{A}, then X ∈ R_{A}.

(48) X ∈ R_{A} if and only if 2^{X} ∈ R_{succ A}.

(49) x ∈ R_{A} if and only if {x} ∈ R_{succ A}.

(50) x ∈ R_{A} and y ∈ R_{A} if and only if {x, y} ∈ R_{succ A}.

(51) x ∈ R_{A} and y ∈ R_{A} if and only if ⟨⟨x, y⟩⟩ ∈ R_{succ(succ A)}.

(52) If X is transitive and R_{A} ∩ T(X) = R_{succ A} ∩ T(X), then T(X) ⊆ R_{A}.

(53) If X is transitive, then there exists A such that T(X) ⊆ R_{A}.

(54) If X is transitive, then ⋃ X ⊆ X.

(55) If X is transitive and Y is transitive, then X ∪ Y is transitive.

(56) If X is transitive and Y is transitive, then X ∩ Y is transitive. In the sequel k, n denote natural numbers. Let us consider X. The functor

X^{∗_{∈}} yielding a set, is defined by:

x ∈ X^{∗_{∈}} if and only if there exist f, n, Y such that x ∈ Y and Y = f(n) and dom f = and f(0) = X and for all k, y such that y = f(k) holds f(k + 1) = ⋃[y].

Next we state a number of propositions:

(57) Z = X^{∗_{∈}} if and only if for every x holds x ∈ Z if and only if there exist f, n, Y such that x ∈ Y and Y = f(n) and dom f = and f(0) = X and for all k, y such that y = f(k) holds f(k + 1) = ⋃[y].

(58) X^{∗_{∈}} is transitive.

(59) X ⊆ X^{∗_{∈}}.

(60) If X ⊆ Y and Y is transitive, then X^{∗_{∈}} ⊆ Y.

(61) If for every Z such that X ⊆ Z and Z is transitive holds Y ⊆ Z and X ⊆ Y and Y is transitive, then X^{∗_{∈}} = Y.

(62) If X is transitive, then X^{∗_{∈}} = X.

(63) ∅^{∗_{∈}} = ∅.

(64) A^{∗_{∈}} = A.

(65) If X ⊆ Y, then X^{∗_{∈}} ⊆ Y^{∗_{∈}}.

(66) (X^{∗_{∈}})^{∗_{∈}} = X^{∗_{∈}}.

(67) (X ∪ Y)^{∗_{∈}} = X^{∗_{∈}} ∪ Y^{∗_{∈}}.

(68) (X ∩ Y)^{∗_{∈}} ⊆ X^{∗_{∈}} ∩ Y^{∗_{∈}}.

(69) There exists A such that X ⊆ R_{A}.

Let us consider X. The functor rk(X) yielding an ordinal number, is defined by:

X ⊆ R_{rk(X)} and for every B such that X ⊆ R_{B} holds rk(X) ⊆ B.

We now state a number of propositions:

(70) A = rk(X) if and only if X ⊆ R_{A} and for every B such that X ⊆ R_{B} holds A ⊆ B.

(71) rk(2^{X}) = succ rk(X).

(72) rk(R_{A}) = A.

(73) X ⊆ R_{A} if and only if rk(X) ⊆ A.

(74) X ∈ R_{A} if and only if rk(X) ∈ A.

(75) If X ⊆ Y, then rk(X) ⊆ rk(Y).

(76) If X ∈ Y, then rk(X) ∈ rk(Y).

(77) rk(X) ⊆ A if and only if for every Y such that Y ∈ X holds rk(Y) ∈ A.

(78) A ⊆ rk(X) if and only if for every B such that B ∈ A there exists Y such that Y ∈ X and B ⊆ rk(Y).

(79) rk(X) = 0 if and only if X = ∅.

(80) If rk(X) = succ A, then there exists Y such that Y ∈ X and rk(Y) = A.

(81) rk(A) = A.

(82) rk(T(X)) ≠ 0 and rk(T(X)) is a limit ordinal number.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

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

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

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

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

[6] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

568

Non-contiguous Substrings and One-to-one Finite Sequences

Wojciech A. Trybulec Warsaw University

Summary. This text is a continuation of [3]. We prove a number of theorems concerning both notions introduced there and one-to-one finite sequences. We introduce a function that removes from a string elements of the string that belongs to a given set.

MML Identifier: FINSEQ 3.

The notation and terminology used here have been introduced in the following articles: [9], [8], [5], [3], [4], [7], [6], [1], [2], and [10]. For simplicity we follow a convention: p, q, r are finite sequences, u, v, x, y, z are arbitrary, i, j, k, l, m, n are natural numbers, A, X, Y are sets, and D is a non-empty set. The following propositions are true:

(1) Seg 3 = {1, 2, 3}.

(2) Seg 4 = {1, 2, 3, 4}.

(3) Seg 5 = {1, 2, 3, 4, 5}.

(4) Seg 6 = {1, 2, 3, 4, 5, 6}.

(5) Seg 7 = {1, 2, 3, 4, 5, 6, 7}.

(6) Seg 8 = {1, 2, 3, 4, 5, 6, 7, 8}.

(7) Seg k = ∅ if and only if k ∉ Seg k.

(8) 0 ∉ Seg k.

(9) k + 1 ∉ Seg k.

(10) If k ≠ 0, then k ∈ Seg(k + n).

(11) If k + n ∈ Seg k, then n = 0.

(12) If k ∈ Seg n and k < n, then k + 1 ∈ Seg n.

(13) If k ∈ Seg n and m < k, then k − m ∈ Seg n.

(14) k − n ∈ Seg k if and only if n < k.

(15) Seg k misses {k + 1}.

570 Wojciech A. Trybulec

(16) Seg(k + 1) \ Seg k = {k + 1}.

(17) Seg k ≠ Seg(k + 1).

(18) If Seg k = Seg(k + n), then n = 0.

(19) Seg k ⊆ Seg(k + n).

(20) Seg k ⊆ Seg n or Seg n ⊆ Seg k.

(21) If Seg k = ∅, then k = 0.

(22) If Seg k = {y}, then k = 1 and y = 1.

(23) If Seg k = {x, y} and x ≠ y, then k = 2 and {x, y} = {1, 2}.

(24) If x ∈ dom p, then x ∈ dom(p^{}q).

(25) If x ∈ dom p, then x is a natural number.

(26) If x ∈ dom p, then x ≠ 0.

(27) n ∈ dom p if and only if 1 ≤ n and n ≤ len p.

(28) n ∈ dom p if and only if n − 1 is a natural number and len p − n is a natural number.

(29) dom⟨x, y⟩ = Seg 2.

(30) dom⟨x, y, z⟩ = Seg 3.

(31) len p = len q if and only if dom p = dom q.

(32) len p ≤ len q if and only if dom p ⊆ dom q.

(33) If x ∈ rng p, then 1 ∈ dom p.

(34) If rng p ≠ ∅, then 1 ∈ dom p.

(35) rng⟨x, y⟩ = {x, y}.

(36) rng⟨x, y, z⟩ = {x, y, z}.

(37) ε = .

(38) ε ≠ ⟨x, y⟩.

(39) ε ≠ ⟨x, y, z⟩.

(40) ⟨x⟩ ≠ ⟨y, z⟩.

(41) ⟨u⟩ ≠ ⟨x, y, z⟩.

(42) ⟨u, v⟩ ≠ ⟨x, y, z⟩.

(43) If len r = len p + len q and for every k such that k ∈ dom p holds r(k) = p(k) and for every k such that k ∈ dom q holds r(len p + k) = q(k), then r = p^{}q.

(44) If A ⊆ Seg k, then len(Sgm A) = card A.

(45) If A ⊆ Seg k, then dom(Sgm A) = Seg(card A).

(46) If X ⊆ Seg i and k < l and 1 ≤ n and m ≤ len(Sgm X) and Sgm X(m) = k and Sgm X(n) = l, then m < n.

(47) If X ⊆ Seg i and k ≤ l and 1 ≤ n and m ≤ len(Sgm X) and Sgm X(m) = k and Sgm X(n) = l, then m ≤ n.

(48) If X ⊆ Seg i and Y ⊆ Seg j, then for all m, n such that m ∈ X and n ∈ Y holds m < n if and only if Sgm(X ∪ Y) = Sgm X ^{}Sgm Y.

(49) Sgm ∅ = ε.

Non-contiguous Substrings and … 571

(50) If 0 ≠ n, then Sgm{n} = ⟨n⟩.

(51) If 0 < n and n < m, then Sgm{n, m} = ⟨n, m⟩.

(52) len(Sgm(Seg k)) = k.

(53) Sgm(Seg(k + n)) Seg k = Sgm(Seg k).

(54) Sgm(Seg k) = id_{k}.

(55) p Seg n = p if and only if len p ≤ n.

(56) id_{n+k} Seg n = id_{n}.

(57) id_{n} Seg m = id_{m} if and only if m ≤ n.

(58) id_{n} Seg m = id_{n} if and only if n ≤ m.

(59) If len p = k + l and q = p Seg k, then len q = k.

(60) If len p = k + l and q = p Seg k, then dom q = Seg k.

(61) If len p = k + 1 and q = p Seg k, then p = q^{}⟨p(k + 1)⟩.

(62) p X is a finite sequence if and only if there exists k such that X ∩ dom p = Seg k.

(63) card((p^{}q)^{−1} A) = card(p^{−1} A) + card(q^{−1} A).

(64) p^{−1} A ⊆ (p^{}q)^{−1} A.

Let us consider p, A. The functor p−A yields a finite sequence and is defined by:

p − A = p · Sgm(Seg(len p) \ p^{−1} A).

The following propositions are true:

(65) p − A = p · Sgm(Seg(len p) \ p^{−1} A).

(66) len(p − A) = len p − card(p^{−1} A).

(67) len(p − A) ≤ len p.

(68) If len(p − A) = len p, then A misses rng p.

(69) If n = len p − card(p^{−1} A), then dom(p − A) = Seg n.

(70) dom(p − A) ⊆ dom p.

(71) If dom(p − A) = dom p, then A misses rng p.

(72) rng(p − A) = rng p \ A.

(73) rng(p − A) ⊆ rng p.

(74) If rng(p − A) = rng p, then A misses rng p.

(75) p − A = ε if and only if rng p ⊆ A.

(76) p − A = p if and only if A misses rng p.

(77) p − {x} = p if and only if x ∉ rng p.

(78) p − ∅ = p.

(79) p − rng p = ε.

(80) p^{}q − A = (p − A)^{}(q − A).

(81) ε − A = ε.

(82) ⟨x⟩ − A = ⟨x⟩ if and only if x ∉ A.

(83) ⟨x⟩ − A = ε if and only if x ∈ A.

(84) ⟨x, y⟩ − A = ε if and only if x ∈ A and y ∈ A.

(85) If x ∈ A and y ∉ A, then ⟨x, y⟩ − A = ⟨y⟩.

(86) If ⟨x, y⟩ − A = ⟨y⟩ and x ≠ y, then x ∈ A and y ∉ A.

(87) If x ∉ A and y ∈ A, then ⟨x, y⟩ − A = ⟨x⟩.

(88) If ⟨x, y⟩ − A = ⟨x⟩ and x ≠ y, then x ∉ A and y ∈ A.

(89) ⟨x, y⟩ − A = ⟨x, y⟩ if and only if x ∉ A and y ∉ A.

(90) If len p = k + 1 and q = p Seg k, then p(k + 1) ∈ A if and only if p − A = q − A.

(91) If len p = k + 1 and q = p Seg k, then p(k + 1) ∉ A if and only if p − A = (q − A)^{}⟨p(k + 1)⟩.

(92) If n ∈ dom p, then p(n) ∈ A or (p − A)(n − card{k : k ∈ dom p ∧ k ≤ n ∧ p(k) ∈ A}) = p(n).

(93) If p is a finite sequence of elements of D, then p − A is a finite sequence of elements of D.

(94) If p is one-to-one, then p − A is one-to-one.

(95) If p is one-to-one, then len(p − A) = len p − card(A ∩ rng p).

(96) If p is one-to-one and A ⊆ rng p, then len(p − A) = len p − card A.

(97) If p is one-to-one and x ∈ rng p, then len(p − {x}) = len p − 1.

(98) rng p misses rng q and p is one-to-one and q is one-to-one if and only if p^{}q is one-to-one.

(99) If A ⊆ Seg k, then Sgm A is one-to-one.

(100) id_{n} is one-to-one.

(101) ε is one-to-one.

(102) ⟨x⟩ is one-to-one.

(103) x ≠ y if and only if ⟨x, y⟩ is one-to-one.

(104) x ≠ y and y ≠ z and z ≠ x if and only if ⟨x, y, z⟩ is one-to-one.

(105) If p is one-to-one and rng p = {x}, then len p = 1.

(106) If p is one-to-one and rng p = {x}, then p = ⟨x⟩.

(107) If p is one-to-one and rng p = {x, y} and x ≠ y, then len p = 2.

(108) If p is one-to-one and rng p = {x, y} and x ≠ y, then p = ⟨x, y⟩ or p = ⟨y, x⟩.

(109) If p is one-to-one and rng p = {x, y, z} and ⟨x, y, z⟩ is one-to-one, then len p = 3.

(110) If p is one-to-one and rng p = {x, y, z} and x ≠ y and y ≠ z and x ≠ z, then len p = 3.

References

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

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

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

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

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

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

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

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

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

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

574

Pigeon Hole Principle

Wojciech A. Trybulec Warsaw University

Summary. We introduce the notion of a predicate that states that a function is one-to-one at a given element of it’s domain (i.e. counter image of image of the element is equal to its singleton). We also introduce some rather technical functors concerning finite sequences: the lowest index of the given element of the range of the finite sequence, the substring preceding (and succeeding) the first occurrence of given element of the range. At the end of the article we prove the pigeon hole principle.

MML Identifier: FINSEQ 4.

The notation and terminology used here are introduced in the following papers: [8], [4], [3], [6], [7], [1], [5], [9], [2], and [10]. For simplicity we adopt the following convention: f is a function, p, q are finite sequences, x, y, z are arbitrary, i, k, n are natural numbers, and A, B are sets. Let us consider f, x. We say that f is one-to-one at x if and only if:

f ^{−1} (f ^{∘} {x}) = {x}. We now state several propositions:

(1) f is one-to-one at x if and only if f ^{−1} (f ^{∘} {x}) = {x}.

(2) If f is one-to-one at x, then x ∈ dom f.

(3) f is one-to-one at x if and only if x ∈ dom f and f ^{−1} {f(x)} = {x}.

(4) f is one-to-one at x if and only if x ∈ dom f and for every z such that z ∈ dom f and x ≠ z holds f(x) ≠ f(z).

(5) For every x such that x ∈ dom f holds f is one-to-one at x if and only if f is one-to-one.

Let us consider f, y. We say that f yields y just once if and only if: f ^{−1} {y} is finite and card(f ^{−1} {y}) = 1. Next we state several propositions:

(6) f yields y just once if and only if f^{−1}{y} is finite and card(f^{−1}{y}) = 1.

(7) If f yields y just once, then y ∈ rng f.

(8) f yields y just once if and only if there exists x such that {x} = f^{−1}{y}.

(9) f yields y just once if and only if there exists x such that x ∈ dom f and y = f(x) and for every z such that z ∈ dom f and z ≠ x holds f(z) ≠ y.

(10) f is one-to-one if and only if for every y such that y ∈ rng f holds f yields y just once.

(11) f is one-to-one at x if and only if x ∈ dom f and f yields f(x) just

once. Let us consider f, y. Let us assume that f yields y just once. The functor

f^{−1}(y) is defined as follows:

f^{−1}(y) ∈ dom f and f(f^{−1}(y)) = y.

One can prove the following propositions:

(12) If f yields y just once and x ∈ dom f and f(x) = y, then x = f^{−1}(y).

(13) If f yields y just once, then f^{−1}(y) ∈ dom f.

(14) If f yields y just once, then f(f^{−1}(y)) = y.

(15) If f yields y just once, then for every x such that x ∈ dom f and x ≠ f^{−1}(y) holds f(x) ≠ y.

(16) If f yields y just once, then f ^{∘} {f^{−1}(y)} = {y}.

(17) If f yields y just once, then f ^{−1} {y} = {f^{−1}(y)}.

(18) If f is one-to-one and y ∈ rng f, then f^{−1}(y) = f^{−1}(y).

(19) If x ∈ dom f and f yields f(x) just once, then f^{−1}(f(x)) = x.

(20) If f is one-to-one at x, then f^{−1}(f(x)) = x.

(21) If f yields y just once, then f is one-to-one at f^{−1}(y).

We adopt the following convention: D will be a non-empty set, d, d_{1}, d_{2}, d_{3} will be elements of D, and P will be a finite sequence of elements of D. Let us consider D, d_{1}, d_{2}. Then ⟨d_{1}, d_{2}⟩ is a finite sequence of elements of D.

Let us consider D, d_{1}, d_{2}, d_{3}. Then ⟨d_{1}, d_{2}, d_{3}⟩ is a finite sequence of elements of D.

Let us consider D, P, i. Let us assume that i ∈ dom P. The functor π_{i}P yielding an element of D, is defined as follows:

π_{i}P = P (i).

Next we state several propositions:

(22) If i ∈ dom P, then π_{i}P = P (i).

(23) If i ∈ Seg(len P), then π_{i}P = P (i).

(24) If 1 ≤ i and i ≤ len P, then π_{i}P = P (i).

(25) π_{1}⟨d⟩ = d.

(26) π_{1}⟨d_{1}, d_{2}⟩ = d_{1} and π_{2}⟨d_{1}, d_{2}⟩ = d_{2}.

(27) π_{1}⟨d_{1}, d_{2}, d_{3}⟩ = d_{1} and π_{2}⟨d_{1}, d_{2}, d_{3}⟩ = d_{2} and π_{3}⟨d_{1}, d_{2}, d_{3}⟩ = d_{3}. Let us consider p, x. Let us assume that x ∈ rng p. The functor x p yields

a natural number and is defined by:

x p = Sgm(p^{−1} {x})(1).

Next we state a number of propositions:

(28) If x ∈ rng p, then x p = Sgm(p^{−1} {x})(1).

Pigeon Hole Principle 577

(29) If x ∈ rng p, then p(x p) = x.

(30) If x ∈ rng p, then x p ∈ dom p.

(31) If x ∈ rng p, then 1 ≤ x p and x p ≤ len p.

(32) If x ∈ rng p, then x p − 1 is a natural number and len p − x p is a natural number.

(33) If x ∈ rng p, then x p ∈ p^{−1} {x}.

(34) If x ∈ rng p, then for every k such that k ∈ dom p and k < x p holds p(k) ≠ x.

(35) If p yields x just once, then p^{−1}(x) = x p.

(36) If p yields x just once, then for every k such that k ∈ dom p and k ≠ x p holds p(k) ≠ x.

(37) If x ∈ rng p and for every k such that k ∈ dom p and k ≠ x p holds p(k) ≠ x, then p yields x just once.

(38) p yields x just once if and only if x ∈ rng p and {x p} = p^{−1} {x}.

(39) If p is one-to-one and x ∈ rng p, then {x p} = p^{−1} {x}.

(40) p yields x just once if and only if len(p − {x}) = len p − 1.

(41) If p yields x just once, then for every k such that k ∈ dom(p − {x}) holds if k < x p, then (p − {x})(k) = p(k) but if x p ≤ k, then (p − {x})(k) = p(k + 1).

(42) Suppose p is one-to-one and x ∈ rng p. Then for every k such that k ∈ dom(p − {x}) holds (p − {x})(k) = p(k) if and only if k < x p but (p − {x})(k) = p(k + 1) if and only if x p ≤ k.

Let us consider p, x. Let us assume that x ∈ rng p. The functor p ← x yields a finite sequence and is defined as follows:

there exists n such that n = x p − 1 and p ← x = p Seg n.

One can prove the following propositions:

(43) If x ∈ rng p, then there exists n such that n = x p−1 and p Seg n = p ← x.

(44) If x ∈ rng p and there exists n such that n = x p−1 and p Seg n = q, then q = p ← x.

(45) If x ∈ rng p and n = x p − 1, then p Seg n = p ← x.

(46) If x ∈ rng p, then len(p ← x) = x p − 1.

(47) If x ∈ rng p and n = x p − 1, then dom(p ← x) = Seg n.

(48) If x ∈ rng p and k ∈ dom(p ← x), then p(k) = (p ← x)(k).

(49) If x ∈ rng p, then x ∉ rng(p ← x).

(50) If x ∈ rng p, then rng(p ← x) misses {x}.

(51) If x ∈ rng p, then rng(p ← x) ⊆ rng p.

(52) If x ∈ rng p, then x p = 1 if and only if p ← x = ε.

(53) If x ∈ rng p and p is a finite sequence of elements of D, then p ← x is a finite sequence of elements of D.

Let us consider p, x. Let us assume that x ∈ rng p. The functor p → x yields a finite sequence and is defined as follows:

len(p → x) = len p − x p and for every k such that k ∈ dom(p → x) holds (p → x)(k) = p(k + x p).

One can prove the following propositions:

(54) If x ∈ rng p and len q = len p−x p and for every k such that k ∈ dom q holds q(k) = p(k + x p), then q = p → x.

(55) If x ∈ rng p, then len(p → x) = len p − x p.

(56) If x ∈ rng p, then for every k such that k ∈ dom(p → x) holds (p → x)(k) = p(k + x p).

(57) If x ∈ rng p and n = len p − x p, then dom(p → x) = Seg n.

(58) If x ∈ rng p and n ∈ dom(p → x), then n + x p ∈ dom p.

(59) If x ∈ rng p, then rng(p → x) ⊆ rng p.

(60) p yields x just once if and only if x ∈ rng p and x ∉ rng(p → x).

(61) If x ∈ rng p and p is one-to-one, then x ∉ rng(p → x).

(62) p yields x just once if and only if x ∈ rng p and rng(p → x) misses {x}.

(63) If x ∈ rng p and p is one-to-one, then rng(p → x) misses {x}.

(64) If x ∈ rng p, then x p = len p if and only if p → x = ε.

(65) If x ∈ rng p and p is a finite sequence of elements of D, then p → x is a finite sequence of elements of D.

(66) If x ∈ rng p, then p = ((p ← x)^{}⟨x⟩)^{}(p → x).

(67) If x ∈ rng p and p is one-to-one, then p ← x is one-to-one.

(68) If x ∈ rng p and p is one-to-one, then p → x is one-to-one.

(69) p yields x just once if and only if x ∈ rng p and p−{x} = (p ← x)^{}(p → x).

(70) If x ∈ rng p and p is one-to-one, then p − {x} = (p ← x)^{}(p → x).

(71) If x ∈ rng p and p − {x} is one-to-one and p − {x} = (p ← x)^{}(p → x), then p is one-to-one.

(72) If x ∈ rng p and p is one-to-one, then rng(p ← x) misses rng(p → x).

(73) If A is finite, then there exists p such that rng p = A and p is one-to-one.

(74) If rng p ⊆ dom p and p is one-to-one, then rng p = dom p.

(75) If rng p = dom p, then p is one-to-one.

(76) If rng p = rng q and len p = len q and q is one-to-one, then p is one-to-one.

(77) p is one-to-one if and only if card(rng p) = len p.

In the sequel f denotes a function from A into B. The following propositions are true:

(78) If card A = card B and A is finite and B is finite and f is one-to-one, then rng f = B.

(79) If card A = card B and A is finite and B is finite and rng f = B, then f is one-to-one.

(80) If \overline{\overline{B}} < \overline{\overline{A}} and B ≠ ∅, then there exist x, y such that x ∈ A and y ∈ A and x ≠ y and f(x) = f(y).

(81) If \overline{\overline{A}} < \overline{\overline{B}}, then there exists x such that x ∈ B and for every y such that y ∈ A holds f(y) ≠ x.

References

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

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

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

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

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

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

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

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

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

[10] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.

580

Linear Combinations in Real Linear Space

Wojciech A. Trybulec Warsaw University

Summary. The article is continuation of [14]. At the beginning we prove some theorems concerning sums of finite sequence of vectors. We introduce the following notions: sum of finite subset of vectors, linear combination, carrier of linear combination, linear combination of elements of a given set of vectors, sum of linear combination. We also show that the set of linear combinations is a real linear space. At the end of article we prove some auxiliary theorems that should be proved in [16], [5], [7],

[1] or [8].

MML Identifier: RLVECT 2.

The papers [16], [7], [5], [3], [6], [14], [8], [13], [15], [11], [9], [10], [4], [12], and [2] provide the notation and terminology for this paper. In the article we present several logical schemes. The scheme LambdaSep1 deals with a non-empty set A, a non-empty set B, an element C of A, an element D of B, and a unary functor F yielding an element of B and states that:

there exists a function f from A into B such that f(C) = D and for every element x of A such that x ≠ C holds f(x) = F(x) for all values of the parameters.

The scheme LambdaSep2 deals with a non-empty set A, a non-empty set B, an element C of A, an element D of A, an element E of B, an element F of B, and a unary functor F yielding an element of B and states that:

there exists a function f from A into B such that f(C) = E and f(D) = F and for every element x of A such that x ≠ C and x ≠ D holds f(x) = F(x) provided the following condition is satisfied:

• C ≠ D.

Let D be a non-empty set. Then ∅_{D} is a subset of D.

For simplicity we follow the rules: X, Y are sets, x is arbitrary, i, k, n are natural numbers, S is an RLS structure, V is a real linear space, u, v, v_{1}, v_{2}, v_{3} are vectors of V, a, b, r are real numbers, F, G, H are finite sequences of elements of the vectors of V, A, B are subsets of the vectors of V, and f is a function from the vectors of V into . Let us consider S, and let v be an element of the vectors of S. The functor @v yielding a vector of S, is defined as follows:

@v = v. One can prove the following proposition

(1) For every element v of the vectors of V holds v = @v. Let us consider S, x. Let us assume that x ∈ S. The functor x^{S} yielding a

vector of S, is defined as follows:

x^{S} = x. The following propositions are true:

(2) If x ∈ S, then x^{S} = x.

(3) For every vector v of S holds v^{S} = v.

(4) If len F = len G and len F = len H and for every k such that k ∈ Seg(len F) holds H(k) = @(π_{k}F) + @(π_{k}G), then ∑ H = ∑ F + ∑ G.

(5) If len F = len G and for every k such that k ∈ Seg(len F) holds G(k) = a · @(π_{k}F), then ∑ G = a · ∑ F.

(6) If len F = len G and for every k such that k ∈ Seg(len F) holds G(k) = −@(π_{k}F), then ∑ G = − ∑ F.

(7) If len F = len G and len F = len H and for every k such that k ∈ Seg(len F) holds H(k) = @(π_{k}F) − @(π_{k}G), then ∑ H = ∑ F − ∑ G.

(8) For all F, G and for every permutation f of dom F such that len F = len G and for every i such that i ∈ dom G holds G(i) = F (f(i)) holds ∑ F = ∑ G.

(9) For every permutation f of dom F such that G = F ·f holds ∑ F = ∑ G. Let us consider V. A subset of the vectors of V is called a finite subset of V

if:

it is finite. One can prove the following proposition

(10) A is a finite subset of V if and only if A is finite. In the sequel S, T will be finite subsets of V. Let us consider V, S, T. Then

S ∪ T is a finite subset of V. Then S ∩ T is a finite subset of V. Then S \ T is a finite subset of V. Then S− . T is a finite subset of V.

Let us consider V. The functor 0_{V} yielding a finite subset of V, is defined by:

0_{V} = ∅. One can prove the following proposition

(11) 0_{V} = ∅.

Let us consider V, T. The functor ∑ T yields a vector of V and is defined as follows:

there exists F such that rng F = T and F is one-to-one and ∑ T = ∑ F. One can prove the following propositions:

Linear Combinations in Real Linear Space 583

(12) There exists F such that rng F = T and F is one-to-one and ∑ T = ∑ F.

(13) If rng F = T and F is one-to-one and v = ∑ F, then v = ∑ T. Let us consider V, v. Then {v} is a finite subset of V. Let us consider V, v_{1}, v_{2}. Then {v_{1}, v_{2}} is a finite subset of V. Let us consider V, v_{1}, v_{2}, v_{3}. Then {v_{1}, v_{2}, v_{3}} is a finite subset of V. One can prove the following propositions:

(14) ∑(0_{V}) = 0_{V}.

(15) ∑{v} = v.

(16) If v_{1} ≠ v_{2}, then ∑{v_{1}, v_{2}} = v_{1} + v_{2}.

(17) If v_{1} ≠ v_{2} and v_{2} ≠ v_{3} and v_{1} ≠ v_{3}, then ∑{v_{1}, v_{2}, v_{3}} = (v_{1} + v_{2}) + v_{3}.

(18) If T misses S, then ∑(T ∪ S) = ∑ T + ∑ S.

(19) ∑(T ∪ S) = (∑ T + ∑ S) − ∑(T ∩ S).

(20) ∑(T ∩ S) = (∑ T + ∑ S) − ∑(T ∪ S).

(21) ∑(T \ S) = ∑(T ∪ S) − ∑ S.

(22) ∑(T \ S) = ∑ T − ∑(T ∩ S).

(23) ∑(T − . S) = ∑(T ∪ S) − ∑(T ∩ S).

(24) ∑(T − . S) = ∑(T \ S) + ∑(S \ T).

Let us consider V. An element of ^{the vectors of V} is called a linear combination of V if:

there exists T such that for every v such that v ∉ T holds it(v) = 0.

In the sequel K, L, L_{1}, L_{2}, L_{3} will be linear combinations of V. Next we state a proposition

(25) There exists T such that for every v such that v ∉ T holds L(v) = 0. In the sequel E denotes an element of ^{the vectors of V}. We now state a propo-

sition

(26) If there exists T such that for every v such that v ∉ T holds E(v) = 0,

then E is a linear combination of V. Let us consider V, L. The functor support L yields a finite subset of V and

is defined as follows: support L = {v : L(v) ≠ 0}. We now state two propositions:

(27) support L = {v : L(v) ≠ 0}.

(28) L(v) = 0 if and only if v ∉ support L.

Let us consider V. The functor 0_{LC_{V}} yields a linear combination of V and is defined as follows:

support 0_{LC_{V}} = ∅.

The following propositions are true:

(29) L = 0_{LC_{V}} if and only if support L = ∅.

(30) 0_{LC_{V}} (v) = 0.

Let us consider V, A. A linear combination of V is said to be a linear combination of A if:

support it ⊆ A.

One can prove the following proposition

(31) If support L ⊆ A, then L is a linear combination of A.

In the sequel l is a linear combination of A. The following propositions are true:

(32) support l ⊆ A.

(33) If A ⊆ B, then l is a linear combination of B.

(34) 0_{LC_{V}} is a linear combination of A.

(35) For every linear combination l of ∅_{the vectors of V} holds l = 0_{LC_{V}}.

(36) L is a linear combination of support L.

Let us consider V, F, f. The functor f ·F yields a finite sequence of elements of the vectors of V and is defined as follows:

len(f · F) = len F and for every i such that i ∈ dom(f · F) holds (f · F)(i) = f(@(π_{i}F)) · @(π_{i}F).

Next we state several propositions:

(37) len(f · F) = len F.

(38) For every i such that i ∈ dom(f·F) holds (f·F)(i) = f(@(π_{i}F))·@(π_{i}F).

(39) If len G = len F and for every i such that i ∈ dom G holds G(i) = f(@(π_{i}F)) · @(π_{i}F), then G = f · F.

(40) If i ∈ dom F and v = F (i), then (f · F)(i) = f(v) · v.

(41) f · ε_{the vectors of V} = ε_{the vectors of V}.

(42) f · ⟨v⟩ = ⟨f(v) · v⟩.

(43) f · ⟨v_{1}, v_{2}⟩ = ⟨f(v_{1}) · v_{1}, f(v_{2}) · v_{2}⟩.

(44) f · ⟨v_{1}, v_{2}, v_{3}⟩ = ⟨f(v_{1}) · v_{1}, f(v_{2}) · v_{2}, f(v_{3}) · v_{3}⟩.

Let us consider V, L. The functor ∑ L yields a vector of V and is defined by:

there exists F such that F is one-to-one and rng F = support L and ∑ L = ∑(L · F).

The following propositions are true:

(45) There exists F such that F is one-to-one and rng F = support L and ∑ L = ∑(L · F).

(46) If F is one-to-one and rng F = support L and u = ∑(L · F), then u = ∑ L.

(47) A ≠ ∅ and A is linearly closed if and only if for every l holds ∑ l ∈ A.

(48) ∑ 0_{LC_{V}} = 0_{V}.

(49) For every linear combination l of ∅_{the vectors of V} holds ∑ l = 0_{V}.

(50) For every linear combination l of {v} holds ∑ l = l(v) · v.

(51) If v_{1} ≠ v_{2}, then for every linear combination l of {v_{1}, v_{2}} holds ∑ l = l(v_{1}) · v_{1} + l(v_{2}) · v_{2}.

(52) If support L = ∅, then ∑ L = 0_{V}.

(53) If support L = {v}, then ∑ L = L(v) · v.

(54) If support L = {v_{1}, v_{2}} and v_{1} ≠ v_{2}, then ∑ L = L(v_{1}) · v_{1} + L(v_{2}) · v_{2}. Let us consider V, L_{1}, L_{2}. Let us note that one can characterize the predicate

L_{1} = L_{2} by the following (equivalent) condition: for every v holds L_{1}(v) = L_{2}(v).

One can prove the following proposition

(55) If for every v holds L_{1}(v) = L_{2}(v), then L_{1} = L_{2}.

Let us consider V, L_{1}, L_{2}. The functor L_{1} + L_{2} yields a linear combination of V and is defined as follows:

for every v holds (L_{1} + L_{2})(v) = L_{1}(v) + L_{2}(v).

The following propositions are true:

(56) If for every v holds L(v) = L_{1}(v) + L_{2}(v), then L = L_{1} + L_{2}.

(57) (L_{1} + L_{2})(v) = L_{1}(v) + L_{2}(v).

(58) support(L_{1} + L_{2}) ⊆ support L_{1} ∪ support L_{2}.

(59) If L_{1} is a linear combination of A and L_{2} is a linear combination of A, then L_{1} + L_{2} is a linear combination of A.

(60) L_{1} + L_{2} = L_{2} + L_{1}.

(61) L_{1} + (L_{2} + L_{3}) = (L_{1} + L_{2}) + L_{3}.

(62) L + 0_{LC_{V}} = L and 0_{LC_{V}} + L = L.

Let us consider V, a, L. The functor a · L yielding a linear combination of V, is defined by:

for every v holds (a · L)(v) = a · L(v).

The following propositions are true:

(63) If for every v holds K(v) = a · L(v), then K = a · L.

(64) (a · L)(v) = a · L(v).

(65) If a ≠ 0, then support(a · L) = support L.

(66) 0 · L = 0_{LC_{V}}.

(67) If L is a linear combination of A, then a · L is a linear combination of A.

(68) (a + b) · L = a · L + b · L.

(69) a · (L_{1} + L_{2}) = a · L_{1} + a · L_{2}.

(70) a · (b · L) = (a · b) · L.

(71) 1 · L = L.

Let us consider V, L. The functor −L yielding a linear combination of V, is defined as follows:

−L = (−1) · L.

Next we state several propositions:

(72) −L = (−1) · L.

(73) (−L)(v) = −L(v).

(74) If L_{1} + L_{2} = 0_{LC_{V}}, then L_{2} = −L_{1}.

(75) support(−L) = support L.

(76) If L is a linear combination of A, then −L is a linear combination of A.

(77) −(−L) = L.

Let us consider V, L_{1}, L_{2}. The functor L_{1} − L_{2} yields a linear combination of V and is defined by:

L_{1} − L_{2} = L_{1} + (−L_{2}). The following propositions are true:

(78) L_{1} − L_{2} = L_{1} + (−L_{2}).

(79) (L_{1} − L_{2})(v) = L_{1}(v) − L_{2}(v).

(80) support(L_{1} − L_{2}) ⊆ support L_{1} ∪ support L_{2}.

(81) If L_{1} is a linear combination of A and L_{2} is a linear combination of A, then L_{1} − L_{2} is a linear combination of A.

(82) L − L = 0_{LC_{V}}.

Let us consider V. The functor LC_{V} yields a non-empty set and is defined by:

x ∈ LC_{V} if and only if x is a linear combination of V. In the sequel D denotes a non-empty set and e, e_{1}, e_{2} denote elements of

LC_{V}. The following propositions are true:

(83) If for every x holds x ∈ D if and only if x is a linear combination of V, then D = LC_{V}.

(84) L ∈ LC_{V}.

Let us consider V, e. The functor @e yields a linear combination of V and is defined by:

@e = e. The following proposition is true

(85) @e = e.

Let us consider V, L. The functor @L yields an element of LC_{V} and is defined as follows:

@L = L. Next we state a proposition

(86) @L = L.

Let us consider V. The functor +_{LC_{V}} yields a binary operation on LC_{V} and is defined by:

for all e_{1}, e_{2} holds +_{LC_{V}} (e_{1}, e_{2}) = @e_{1} + @e_{2}. In the sequel o is a binary operation on LC_{V}. Next we state two propositions:

(87) If for all e_{1}, e_{2} holds o(e_{1}, e_{2}) = @e_{1} + @e_{2}, then o = +_{LC_{V}}.

(88) +_{LC_{V}} (e_{1}, e_{2}) = @e_{1} + @e_{2}.

Let us consider V. The functor ·_{LC_{V}} yields a function from [: , LC_{V} :] into LC_{V} and is defined as follows:

for all a, e holds ·_{LC_{V}} (⟨⟨a, e⟩⟩) = a · @e.

In the sequel g denotes a function from [: , LC_{V} :] into LC_{V}. We now state two propositions:

(89) If for all a, e holds g(⟨⟨a, e⟩⟩) = a · @e, then g = ·_{LC_{V}}.

(90) ·_{LC_{V}} (⟨⟨a, e⟩⟩) = a · @e.

Let us consider V. The functor _{V} yielding a real linear space, is defined as follows:

V = ⟨LC_{V} , @0_{LC_{V}} , +_{LC_{V}} , ·_{LC_{V}} ⟩.

Next we state several propositions:

(91) _{V} = ⟨LC_{V} , @0_{LC_{V}} , +_{LC_{V}} , ·_{LC_{V}} ⟩.

(92) The vectors of _{V} = LC_{V}.

(93) The zero of _{V} = 0_{LC_{V}}.

(94) The addition of _{V} = +_{LC_{V}}.

(95) The multiplication_{1} of _{V} = ·_{LC_{V}}.

(96) L_{1}^{^{V}} + L_{2}^{^{V}} = L_{1} + L_{2}.

(97) a · L^{^{V}} = a · L.

(98) −L^{^{V}} = −L.

(99) L_{1}^{^{V}} − L_{2}^{^{V}} = L_{1} − L_{2}.

Let us consider V, A. The functor _{A} yielding a subspace of _{V}, is defined by:

the vectors of _{A} = {l}.

In the sequel W denotes a subspace of _{V}. Next we state two propositions:

(100) If the vectors of W = {l}, then W = _{A}.

(101) The vectors of _{A} = {l}.

We now state several propositions:

(102) X \ Y misses Y \ X.

(103) If k < n, then n − 1 is a natural number.

(104) −1 ≠ 0.

(105) (−1) · r = −r.

(106) r − 1 < r.

(107) If X is finite and Y is finite, then X− . Y is finite.

(108) For every function f such that f ^{−1} X = f ^{−1} Y and X ⊆ rng f and Y ⊆ rng f holds X = Y.

References

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

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

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

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

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

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

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

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

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

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

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

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

[13] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

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

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

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

König’s Theorem

Grzegorz Bancerek Warsaw University Białystok

Summary. In the article the sum and product of any number of cardinals are introduced and their relationships to addition, multiplication and to other concepts are shown. Then the König’s theorem is proved. The theorem that the cardinal of union of increasing family of sets of power less than some cardinal m is not greater than m, is given too.

MML Identifier: CARD 3.

The papers [12], [6], [7], [3], [14], [13], [4], [2], [11], [9], [8], [10], [1], and [5] provide the terminology and notation for this paper. For simplicity we adopt the following rules: A, B are ordinal numbers, K, M, N are cardinal numbers, x, y, z are arbitrary, X, Y, Z, Z_{1}, Z_{2} are sets, n is a natural number, and f, g are functions. A function is said to be a function yielding cardinal numbers if:

for every x such that x ∈ dom it holds it(x) is a cardinal number.

Next we state a proposition

(1) f is a function yielding cardinal numbers if and only if for every x such that x ∈ dom f holds f(x) is a cardinal number.

In the sequel ff denotes a function yielding cardinal numbers. Let us consider ff, X. Then ff X is a function yielding cardinal numbers.

Let us consider ff, x. Then ff(x) is a set.

Let us consider X, K. Then X ⟼ K is a function yielding cardinal numbers. The following propositions are true:

(2) ff X is a function yielding cardinal numbers and X ⟼ K is a function yielding cardinal numbers.

(3) is a function yielding cardinal numbers.

The scheme CF Lambda concerns a set A and a unary functor F yielding a cardinal number and states that:

there exists ff such that dom ff = A and for every x such that x ∈ A holds ff(x) = F(x) for all values of the parameters.

We now define four new functors. Let us consider f. The functor \overline{\overline{f}} yields a function yielding cardinal numbers and is defined as follows:

dom \overline{\overline{f}} = dom f and for every x such that x ∈ dom f holds \overline{\overline{f}} (x) = \overline{\overline{[f(x)]}}. The functor disjoin f yielding a function, is defined as follows:

dom(disjoin f) = dom f and for every x such that x ∈ dom f holds (disjoin f)(x) = [: [f(x)], {x}:] .

The functor ⋃ f yields a set and is defined by:

⋃ f = ⋃(rng f). The functor ∏ f yielding a set, is defined by:

x ∈ ∏ f if and only if there exists g such that x = g and dom g = dom f and for every x such that x ∈ dom f holds g(x) ∈ [f(x)].

We now state a number of propositions:

(4) ff = \overline{\overline{f}} if and only if dom ff = dom f and for every x such that x ∈ dom f holds ff(x) = \overline{\overline{[f(x)]}}.

(5) g = disjoin f if and only if dom g = dom f and for every x such that x ∈ dom f holds g(x) = [: [f(x)], {x}:].

(6) ⋃ f = ⋃(rng f).

(7) X = ∏ f if and only if for every x holds x ∈ X if and only if there exists g such that x = g and dom g = dom f and for every x such that x ∈ dom f holds g(x) ∈ [f(x)].

(8) \overline{\overline{ff}} = ff.

(9) \overline{\overline{}}= .

(10) \overline{\overline{X ⟼ Y}} = X ⟼ \overline{\overline{Y}} .

(11) disjoin = .

(12) disjoin({x} ⟼ X) = {x} ⟼ [: X, {x}:].

(13) If x ∈ dom f and y ∈ dom f and x ≠ y, then [disjoin f(x)] ∩ [disjoin f(y)] = ∅ .

(14) ⋃ = ∅.

(15) ⋃(X ⟼ Y) ⊆ Y.

(16) If X ≠ ∅, then ⋃(X ⟼ Y) = Y.

(17) ⋃({x} ⟼ Y) = Y.

(18) g ∈ ∏ f if and only if dom g = dom f and for every x such that x ∈ dom f holds g(x) ∈ [f(x)].

(19) ∏ = { }.

(20) Y^{X} = ∏(X ⟼ Y).

Let us consider x, X. The functor π_{x}X yields a set and is defined by:

y ∈ π_{x}X if and only if there exists f such that f ∈ X and y = f(x).

König’s Theorem 591

Next we state a number of propositions:

(21) Y = π_{x}X if and only if for every y holds y ∈ Y if and only if there exists f such that f ∈ X and y = f(x).

(22) If x ∈ dom f and ∏ f ≠ ∅, then π_{x}(∏ f) = f(x).

(23) If f ∈ X, then f(x) ∈ π_{x}X.

(24) π_{x}∅ = ∅.

(25) π_{x}{g} = {g(x)}.

(26) π_{x}{f, g} = {f(x), g(x)}.

(27) π_{x}(X ∪ Y) = π_{x}X ∪ π_{x}Y.

(28) π_{x}(X ∩ Y) ⊆ π_{x}X ∩ π_{x}Y.

(29) π_{x}X \ π_{x}Y ⊆ π_{x}(X \ Y).

(30) π_{x}X− . π_{x}Y ⊆ π_{x}(X− . Y).

(31) \overline{\overline{π_{x}X}} ≤ \overline{\overline{X}}.

(32) If x ∈ ⋃(disjoin f), then there exist y, z such that x = ⟨⟨y, z⟩⟩.

(33) x ∈ ⋃(disjoin f) if and only if x_{2} ∈ dom f and x_{1} ∈ [f(x_{2})] and x = ⟨⟨x_{1}, x_{2}⟩⟩.

(34) If f ≤ g, then disjoin f ≤ disjoin g.

(35) If f ≤ g, then ⋃ f ⊆ ⋃ g.

(36) ⋃(disjoin(Y ⟼ X)) = [: X, Y :].

(37) ∏ f = ∅ if and only if ∅ ∈ rng f.

(38) If dom f = dom g and for every x such that x ∈ dom f holds [f(x)] ⊆

[g(x)], then ∏ f ⊆ ∏ g. In the sequel F, G will denote functions yielding cardinal numbers. The

following two propositions are true:

(39) For every x such that x ∈ dom F holds \overline{\overline{F (x)}} = F (x).

(40) For every x such that x ∈ dom F holds \overline{\overline{[disjoin F (x)]}} = F (x). We now define two new functors. Let us consider F. The functor ∑ F yields

a cardinal number and is defined as follows:

∑ F = \overline{\overline{⋃(disjoin F)}}. The functor ∏ F yielding a cardinal number, is defined as follows:

∏ F = \overline{\overline{∏ F}}.

The following propositions are true:

(41) ∑ F = \overline{\overline{⋃(disjoin F)}}.

(42) ∏ F = \overline{\overline{∏ F}}.

(43) If dom F = dom G and for every x such that x ∈ dom F holds F (x) ⊆ G(x), then ∑ F ≤ ∑ G.

(44) ∅ ∈ rng F if and only if ∏ F = \overline{0}.

(45) If dom F = dom G and for every x such that x ∈ dom F holds F (x) ⊆ G(x), then ∏ F ≤ ∏ G.

(46) If F ≤ G, then ∑ F ≤ ∑ G.

(47) If F ≤ G and \overline{0} ∉ rng G, then ∏ F ≤ ∏ G.

(48) ∑(∅ ⟼ K) = \overline{0}.

(49) ∏(∅ ⟼ K) = \overline{1}.

(50) ∑({x} ⟼ K) = K.

(51) ∏({x} ⟼ K) = K.

(52) ∑(M ⟼ N) = M · N.

(53) ∏(N ⟼ M) = M^{N}.

(54) \overline{\overline{⋃ f}} ≤ ∑ \overline{\overline{f}}.

(55) \overline{\overline{⋃ F}} ≤ ∑ F.

(56) If dom F = dom G and for every x such that x ∈ dom F holds F (x) ∈

G(x), then ∑ F < ∏ G. Now we present three schemes. The scheme FinRegularity deals with a set

A, and a binary predicate P, and states that:

there exists x such that x ∈ A and for every y such that y ∈ A and y ≠ x holds not P[y, x] provided the following conditions are fulfilled:

• A is finite and A ≠ ∅,

• for all x, y such that P[x, y] and P[y, x] holds x = y,

• for all x, y, z such that P[x, y] and P[y, z] holds P[x, z].

The scheme MaxFinSetElem concerns a set A, and a binary predicate P, and states that:

there exists x such that x ∈ A and for every y such that y ∈ A holds P[x, y] provided the following requirements are fulfilled:

• A is finite and A ≠ ∅,

• for all x, y holds P[x, y] or P[y, x],

• for all x, y, z such that P[x, y] and P[y, z] holds P[x, z].

The scheme FuncSeparation deals with a set A, a unary functor F yielding a set, and a binary predicate P, and states that:

there exists f such that dom f = A and for every x such that x ∈ A for every y holds y ∈ [f(x)] if and only if y ∈ F(x) and P[x, y] for all values of the parameters.

We now state several propositions:

(57) R_{ord(n)} is finite.

(58) If X is finite, then \overline{\overline{X}} < \overline{\overline{ω}}.

(59) If \overline{\overline{A}} < \overline{\overline{B}}, then A ∈ B.

(60) If \overline{\overline{A}} < M, then A ∈ M.

(61) Suppose for all Z_{1}, Z_{2} such that Z_{1} ∈ X and Z_{2} ∈ X holds Z_{1} ⊆ Z_{2} or Z_{2} ⊆ Z_{1}. Then there exists Y such that Y ⊆ X and ⋃ Y = ⋃ X and for every Z such that Z ⊆ Y and Z ≠ ∅ there exists Z_{1} such that Z_{1} ∈ Z and for every Z_{2} such that Z_{2} ∈ Z holds Z_{1} ⊆ Z_{2}.

(62) If for every Z such that Z ∈ X holds \overline{\overline{Z}} < M and for all Z_{1}, Z_{2} such that Z_{1} ∈ X and Z_{2} ∈ X holds Z_{1} ⊆ Z_{2} or Z_{2} ⊆ Z_{1}, then \overline{\overline{⋃ X}} ≤ M.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 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] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

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

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

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

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

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

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

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

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

594

Universal Classes

Bogdan Nowak Łódź University Grzegorz Bancerek Warsaw University Białystok

Summary. In the article we have shown that there exist universal classes, i.e. there are sets which are closed w.r.t. basic set theory operations.

MML Identifier: CLASSES2.

The articles [11], [8], [4], [7], [10], [9], [5], [2], [1], [6], and [3] provide the terminology and notation for this paper. For simplicity we adopt the following convention: m is a cardinal number, A, B, C are ordinal numbers, x, y are arbitrary, and X, Y, W are sets. One can prove the following propositions:

(1) If W is a Tarski-Class and X ∈ W, then X ≉ W and \overline{\overline{X}} < \overline{\overline{W}}.

(2) If W is a Tarski-Class and X ⊆ W and \overline{\overline{X}} < \overline{\overline{W}}, then X ∈ W.

(3) If W is a Tarski-Class and x ∈ W and y ∈ W, then {x} ∈ W and {x, y} ∈ W.

(4) If W is a Tarski-Class and x ∈ W and y ∈ W, then ⟨⟨x, y⟩⟩ ∈ W.

(5) If W is a Tarski-Class and X ∈ W, then T(X) ⊆ W. The scheme TC deals with a unary predicate P, and states that: for every X holds P[T(X)]

provided the parameter fulfills the following condition:

• for every X such that X is a Tarski-Class holds P[X]. Next we state a number of propositions:

(6) If W is a Tarski-Class and A ∈ W, then succ A ∈ W and A ⊆ W.

(7) If A ∈ T(W), then succ A ∈ T(W) and A ⊆ T(W).

(8) If W is a Tarski-Class and X is transitive and X ∈ W, then X ⊆ W.

(9) If X is transitive and X ∈ T(W), then X ⊆ T(W).

(10) If W is a Tarski-Class, then On W = \overline{\overline{W}}.

(11) On T(W) = \overline{\overline{T(W)}}.

596 Bogdan Nowak and Grzegorz Bancerek

(12) If W is a Tarski-Class and X ∈ W, then \overline{\overline{X}} ∈ W.

(13) If X ∈ T(W), then \overline{\overline{X}} ∈ T(W).

(14) If W is a Tarski-Class and x ∈ ord(\overline{\overline{W}}), then x ∈ W.

(15) If x ∈ ord(\overline{\overline{T(W)}}), then x ∈ T(W).

(16) If W is a Tarski-Class and m < \overline{\overline{W}}, then m ∈ W.

(17) If m < \overline{\overline{T(W)}}, then m ∈ T(W).

(18) If W is a Tarski-Class and m ∈ W, then m ⊆ W.

(19) If m ∈ T(W), then m ⊆ T(W).

(20) If W is a Tarski-Class, then ord(\overline{\overline{W}}) is a limit ordinal number.

(21) If W is a Tarski-Class and W ≠ ∅, then \overline{\overline{W}} ≠ \overline{0} and ord(\overline{\overline{W}}) ≠ 0 and ord(\overline{\overline{W}}) is a limit ordinal number.

(22) \overline{\overline{T(W)}} ≠ \overline{0} and ord(\overline{\overline{T(W)}}) ≠ 0 and ord(\overline{\overline{T(W)}}) is a limit ordinal

number. In the sequel L, L_{1} are transfinite sequences. We now state a number of

propositions:

(23) If W is a Tarski-Class but X ∈ W and W is transitive or X ∈ W and X ⊆ W or \overline{\overline{X}} < \overline{\overline{W}} and X ⊆ W, then W^{X} ⊆ W.

(24) If X ∈ T(W) and W is transitive or X ∈ T(W) and X ⊆ T(W) or \overline{\overline{X}} < \overline{\overline{T(W)}} and X ⊆ T(W), then T(W)^{X} ⊆ T(W).

(25) If dom L is a limit ordinal number and for every A such that A ∈ dom L holds L(A) = R_{A}, then R_{dom L} = ⋃ L.

(26) If W is a Tarski-Class and A ∈ On W, then \overline{\overline{R_{A}}} < \overline{\overline{W}} and R_{A} ∈ W.

(27) If A ∈ On T(W), then \overline{\overline{R_{A}}} < \overline{\overline{T(W)}} and R_{A} ∈ T(W).

(28) If W is a Tarski-Class, then R_{ord(\overline{\overline{W}} )} ⊆ W.

(29) R_{ord(\overline{\overline{T(W)}})} ⊆ T(W).

(30) If W is a Tarski-Class and W is transitive and X ∈ W, then rk(X) ∈ W.

(31) If W is a Tarski-Class and W is transitive, then W ⊆ R_{ord(\overline{\overline{W}} )}.

(32) If W is a Tarski-Class and W is transitive, then R_{ord(\overline{\overline{W}} )} = W.

(33) If W is a Tarski-Class and A ∈ On W, then \overline{\overline{R_{A}}} ≤ \overline{\overline{W}}.

(34) If A ∈ On T(W), then \overline{\overline{R_{A}}} ≤ \overline{\overline{T(W)}}.

(35) If W is a Tarski-Class, then \overline{\overline{W}} = \overline{\overline{R_{ord(\overline{\overline{W}} )}}} .

(36) \overline{\overline{T(W)}} = \overline{\overline{R_{ord(\overline{\overline{T(W)}})}}} .

(37) If W is a Tarski-Class and X ⊆ R_{ord(\overline{\overline{W}} )}, then X ≈ R_{ord(\overline{\overline{W}} )} or X ∈ R_{ord(\overline{\overline{W}} )}.

Universal Classes 597

(38) If X ⊆ R_{ord(\overline{\overline{T(W)}})}, then X ≈ R_{ord(\overline{\overline{T(W)}})} or X ∈ R_{ord(\overline{\overline{T(W)}})}.

(39) If W is a Tarski-Class, then R_{ord(\overline{\overline{W}} )} is a Tarski-Class.

(40) R_{ord(\overline{\overline{T(W)}})} is a Tarski-Class.

(41) If X is transitive and A ∈ rk(X), then there exists Y such that Y ∈ X and rk(Y) = A.

(42) If X is transitive, then \overline{\overline{rk(X)}} ≤ \overline{\overline{X}}.

(43) If W is a Tarski-Class and X is transitive and X ∈ W, then X ∈ R_{ord(\overline{\overline{W}} )}.

(44) If X is transitive and X ∈ T(W), then X ∈ R_{ord(\overline{\overline{T(W)}})}.

(45) If W is transitive, then R_{ord(\overline{\overline{T(W)}})} is Tarski-Class of W.

(46) If W is transitive, then R_{ord(\overline{\overline{T(W)}})} = T(W).

A non-empty family of sets is called a universal class if:

it is transitive and it is a Tarski-Class.

In the sequel M denotes a non-empty family of sets. The following proposition is true

(47) For every M holds M is a universal class if and only if M is transitive

and M is a Tarski-Class. In the sequel U_{1}, U_{2}, U_{3}, Universum will be universal classes. We now state

several propositions:

(48) If X ∈ Universum, then X ⊆ Universum.

(49) If X ∈ Universum and Y ⊆ X, then Y ∈ Universum.

(50) On Universum is an ordinal number.

(51) If X is transitive, then T(X) is a universal class.

(52) T(Universum) is a universal class.

Let us consider Universum. Then On Universum is an ordinal number. Then T(Universum) is a universal class.

Next we state a proposition

(53) T(A) is a universal class.

Let us consider A. Then T(A) is a universal class.

Next we state a number of propositions:

(54) Universum = R_{On Universum}.

(55) On Universum ≠ 0 and On Universum is a limit ordinal number.

(56) U_{1} ∈ U_{2} or U_{1} = U_{2} or U_{2} ∈ U_{1}.

(57) U_{1} ⊆ U_{2} or U_{2} ∈ U_{1}.

(58) U_{1} ⊆ U_{2} or U_{2} ⊆ U_{1}.

(59) If U_{1} ∈ U_{2} and U_{2} ∈ U_{3}, then U_{1} ∈ U_{3}.

(60) If U_{1} ⊆ U_{2} and U_{2} ∈ U_{3}, then U_{1} ∈ U_{3}.

(61) U_{1} ∪ U_{2} is a universal class and U_{1} ∩ U_{2} is a universal class.

(62) ∅ ∈ Universum.

(63) If x ∈ Universum, then {x} ∈ Universum.

(64) If x ∈ Universum and y ∈ Universum, then {x, y} ∈ Universum and ⟨⟨x, y⟩⟩ ∈ Universum.

(65) If X ∈ Universum, then 2^{X} ∈ Universum and ⋃ X ∈ Universum and ⋂ X ∈ Universum.

(66) If X ∈ Universum and Y ∈ Universum, then X ∪Y ∈ Universum and X ∩ Y ∈ Universum and X \ Y ∈ Universum and X− . Y ∈ Universum.

(67) If X ∈ Universum and Y ∈ Universum, then [: X, Y :] ∈ Universum

and Y^{X} ∈ Universum. In the sequel u, v are elements of Universum. Let us consider Universum,

u. Then {u} is an element of Universum. Then 2^{u} is an element of Universum. Then ⋃ u is an element of Universum. Then ⋂ u is an element of Universum. Let us consider v. Then {u, v} is an element of Universum. Then ⟨⟨u, v⟩⟩ is an element of Universum. Then u ∪ v is an element of Universum. Then u ∩ v is an element of Universum. Then u \ v is an element of Universum. Then u− . v is an element of Universum. Then [: u, v:] is an element of Universum. Then v^{u} is an element of Universum.

The universal class U_{0} is defined as follows:

U_{0} = T(0).

We now state four propositions:

(68) U_{0} = T(0).

(69) \overline{\overline{R_{ω}}} = \overline{\overline{ω}}.

(70) R_{ω} is a Tarski-Class.

(71) U_{0} = R_{ω}.

The universal class U_{1} is defined by:

U_{1} = T(U_{0}).

The following proposition is true

(72) U_{1} = T(U_{0}).

We now define three new constructions. A set of a finite rank is an element of U_{0}.

A Set is an element of U_{1}.

Let us consider A. The functor U_{A} is defined as follows:

there exists L such that U_{A} = last L and dom L = succ A and L(0) = U_{0} and for all C, y such that succ C ∈ succ A and y = L(C) holds L(succ C) = T([y]) and for all C, L_{1} such that C ∈ succ A and C ≠ 0 and C is a limit ordinal number and L_{1} = L C holds L(C) = T(⋃ L_{1}).

The following two propositions are true:

(73) For every element u of U_{0} holds u is a set of a finite rank.

(74) For every element u of U_{1} holds u is a Set.

Let u be a set of a finite rank. Then {u} is a set of a finite rank. Then 2^{u} is a set of a finite rank. Then ⋃ u is a set of a finite rank. Then ⋂ u is a set of a finite rank. Let v be a set of a finite rank. Then {u, v} is a set of a finite rank. Then ⟨⟨u, v⟩⟩ is a set of a finite rank. Then u ∪ v is a set of a finite rank. Then u ∩ v is a set of a finite rank. Then u \ v is a set of a finite rank. Then u− . v is a set of a finite rank. Then [: u, v:] is a set of a finite rank. Then v^{u} is a set of a finite rank.

Let u be a Set. Then {u} is a Set. Then 2^{u} is a Set. Then ⋃ u is a Set. Then ⋂ u is a Set. Let v be a Set. Then {u, v} is a Set. Then ⟨⟨u, v⟩⟩ is a Set. Then u ∪ v is a Set. Then u ∩ v is a Set. Then u \ v is a Set. Then u− . v is a Set. Then [: u, v:] is a Set. Then v^{u} is a Set.

Let us consider A. Then U_{A} is a universal class.

We now state several propositions:

(75) U_{0} = U_{0}.

(76) U_{succ A} = T(U_{A}).

(77) U_{1} = U_{1}.

(78) If A ≠ 0 and A is a limit ordinal number and dom L = A and for every B such that B ∈ A holds L(B) = U_{B}, then U_{A} = T(⋃ L).

(79) U_{0} ⊆ Universum and T(0) ⊆ Universum and U_{0} ⊆ Universum.

(80) A ∈ B if and only if U_{A} ∈ U_{B}.

(81) If U_{A} = U_{B}, then A = B.

(82) A ⊆ B if and only if U_{A} ⊆ U_{B}.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

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

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

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

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

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

[7] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

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

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

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

Analytical Ordered Affine Spaces ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. In the article with a given arbitrary real linear space we correlate the (ordered) affine space defined in terms of a directed parallelity of segments. The abstract contains a construction of the ordered affine structure associated with a vector space; this is a structure of the type which frequently occurs in geometry and consists of the set of points and a binary relation on segments. For suitable underlying vector spaces we prove that the corresponding affine structures are ordered affine spaces or ordered affine planes, i.e. that they satisfy appropriate axioms. A formal definition of an arbitrary ordered affine space and an arbitrary ordered affine plane is given.

MML Identifier: ANALOAF.

The notation and terminology used here have been introduced in the following articles: [4], [3], [2], [1], and [5]. We adopt the following rules: V will denote a real linear space, p, q, u, v, w, y will denote vectors of V, and a, b will denote real numbers. Let us consider V, u, v, w, y. The predicate u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y is defined by:

u = v or w = y or there exist a, b such that 0 < a and 0 < b and a · (v − u) = b · (y − w).

Next we state a number of propositions:

(1) u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y if and only if u = v or w = y or there exist a, b such that 0 < a and 0 < b and a · (v − u) = b · (y − w).

(2) If 0 < a and 0 < b, then 0 < a + b.

(3) If a ≠ b, then 0 < a − b or 0 < b − a.

(4) (w − v) + (v − u) = w − u.

(5) −(u − v) = v − u.

(6) w − (u − v) = w + (v − u).

(7) (w − u) + u = w.

1Supported by RPBP.III.24-C6

(8) (w + u) − u = w.

(9) If y + u = v + w, then y − w = v − u.

(10) a · (u − v) = −a · (v − u).

(11) (a − b) · (u − v) = (b − a) · (v − u).

(12) If a ≠ 0 and a · u = v, then u = a^{−1} · v.

(13) If a ≠ 0 and a · u = v, then u = a^{−1} · v but if a ≠ 0 and u = a^{−1} · v, then a · u = v.

(14) If u = v or w = y, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

(15) If a · (v − u) = b · (y − w) and 0 < a and 0 < b, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

(16) If u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y and u ≠ v and w ≠ y, then there exist a, b such that a · (v − u) = b · (y − w) and 0 < a and 0 < b.

(17) u, v \begin{matrix} \\ \\ ∥ \end{matrix} u, v.

(18) u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, w and u, u \begin{matrix} \\ \\ ∥ \end{matrix} v, w.

(19) If u, v \begin{matrix} \\ \\ ∥ \end{matrix} v, u, then u = v.

(20) If p ≠ q and p, q \begin{matrix} \\ \\ ∥ \end{matrix} u, v and p, q \begin{matrix} \\ \\ ∥ \end{matrix} w, y, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

(21) If u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y, then v, u \begin{matrix} \\ \\ ∥ \end{matrix} y, w and w, y \begin{matrix} \\ \\ ∥ \end{matrix} u, v.

(22) If u, v \begin{matrix} \\ \\ ∥ \end{matrix} v, w, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} u, w.

(23) If u, v \begin{matrix} \\ \\ ∥ \end{matrix} u, w, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} v, w or u, w \begin{matrix} \\ \\ ∥ \end{matrix} w, v.

(24) If v − u = y − w, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

(25) If y = (v + w) − u, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y and u, w \begin{matrix} \\ \\ ∥ \end{matrix} v, y.

(26) If there exist p, q such that p ≠ q, then for every u, v, w there exists y

such that u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y and u, w \begin{matrix} \\ \\ ∥ \end{matrix} v, y and v ≠ y.

(27) If p ≠ v and v, p \begin{matrix} \\ \\ ∥ \end{matrix} p, w, then there exists y such that u, p \begin{matrix} \\ \\ ∥ \end{matrix} p, y and

u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

(28) If for all a, b such that a · u + b · v = 0_{V} holds a = 0 and b = 0, then u ≠ v and u ≠ 0_{V} and v ≠ 0_{V}.

(29) If there exist u, v such that for all a, b such that a · u + b · v = 0_{V} holds a = 0 and b = 0, then there exist u, v, w, y such that u, v \begin{matrix} \\ \\ \end{matrix}w, y and

u, v \begin{matrix} \\ \\ \end{matrix}y, w.

(30) If a − b = 0, then a = b.

Next we state a proposition

(31) Suppose there exist p, q such that for every w there exist a, b such that a · p + b · q = w. Then for all u, v, w, y such that u, v \begin{matrix} \\ \\ \end{matrix}w, y and u, v \begin{matrix} \\ \\ \end{matrix}y, w there exists a vector z of V such that u, v \begin{matrix} \\ \\ ∥ \end{matrix} u, z or u, v \begin{matrix} \\ \\ ∥ \end{matrix} z, u

but w, y \begin{matrix} \\ \\ ∥ \end{matrix} w, z or w, y \begin{matrix} \\ \\ ∥ \end{matrix} z, w.

We consider affine structures which are systems

⟨ points, a congruence ⟩

where the points is a non-empty set and the congruence is a relation on [: the points, the points :]. We adopt the following convention: AS will denote an affine structure and a, b, c, d will denote elements of the points of AS. Let us consider AS, a, b, c, d. The predicate a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d is defined by:

Analytical Ordered Affine Spaces 603

⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of AS. We now state a proposition

(32) a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of AS. In the sequel x, z are arbitrary. Let us consider V. The functor _{V} yields a

relation on [: the vectors of V, the vectors of V :] and is defined as follows:

⟨⟨x, z⟩⟩ ∈ _{V} if and only if there exist u, v, w, y such that x = ⟨⟨u, v⟩⟩ and

z = ⟨⟨w, y⟩⟩ and u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

One can prove the following proposition

(33) ⟨⟨⟨⟨u, v⟩⟩, ⟨⟨w, y⟩⟩⟩⟩ ∈ _{V} if and only if u, v \begin{matrix} \\ \\ ∥ \end{matrix} w, y.

Let us consider V. The functor OASpace V yields an affine structure and is defined as follows:

OASpace V = ⟨ the vectors of V, _{V} ⟩. Next we state three propositions:

(34) OASpace V = ⟨ the vectors of V, _{V} ⟩.

(35) Suppose there exist u, v such that for all real numbers a, b such that

a · u + b · v = 0_{V} holds a = 0 and b = 0. Then

(i) there exist elements a, b of the points of OASpace V such that a ≠ b, (ii) for all elements a, b, c, d, p, q, r, s of the points of OASpace V holds

a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, a, then a = b but if a ≠ b and a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, q and

a, b \begin{matrix} \\ \\ ∥ \end{matrix} r, s, then p, q \begin{matrix} \\ \\ ∥ \end{matrix} r, s but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d, then b, a \begin{matrix} \\ \\ ∥ \end{matrix} d, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c,

then a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c or a, c \begin{matrix} \\ \\ ∥ \end{matrix} c, b,

(iii) there exist elements a, b, c, d of the points of OASpace V such that

a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c,

(iv) for every elements a, b, c of the points of OASpace V there exists an element d of the points of OASpace V such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and b ≠ d,

(v) for all elements p, a, b, c of the points of OASpace V such that p ≠ b and b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, c there exists an element d of the points of OASpace V such

that a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, d and a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d.

(36) Suppose there exist vectors p, q of V such that for every vector w of V there exist real numbers a, b such that a · p + b · q = w. Let a, b, c, d be elements of the points of OASpace V. Then if a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c, then there exists an element t of the points of OASpace V such

that a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, t or a, b \begin{matrix} \\ \\ ∥ \end{matrix} t, a but c, d \begin{matrix} \\ \\ ∥ \end{matrix} c, t or c, d \begin{matrix} \\ \\ ∥ \end{matrix} t, c.

An affine structure is called an ordered affine space if:

(i) there exist elements a, b of the points of it such that a ≠ b, (ii) for all elements a, b, c, d, p, q, r, s of the points of it holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, c but

if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, a, then a = b but if a ≠ b and a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, q and a, b \begin{matrix} \\ \\ ∥ \end{matrix} r, s, then p, q \begin{matrix} \\ \\ ∥ \end{matrix} r, s

but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d, then b, a \begin{matrix} \\ \\ ∥ \end{matrix} d, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c,

then a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c or a, c \begin{matrix} \\ \\ ∥ \end{matrix} c, b,

(iii) there exist elements a, b, c, d of the points of it such that a, b \begin{matrix} \\ \\ \end{matrix}c, d and

a, b \begin{matrix} \\ \\ \end{matrix}d, c,

(iv) for every elements a, b, c of the points of it there exists an element d of the points of it such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and b ≠ d,

(v) for all elements p, a, b, c of the points of it such that p ≠ b and b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, c there exists an element d of the points of it such that a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, d and a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d.

One can prove the following propositions:

(37) The following conditions are equivalent:

(i) there exist elements a, b of the points of AS such that a ≠ b and for all elements a, b, c, d, p, q, r, s of the points of AS holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, c but

if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, a, then a = b but if a ≠ b and a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, q and a, b \begin{matrix} \\ \\ ∥ \end{matrix} r, s, then

p, q \begin{matrix} \\ \\ ∥ \end{matrix} r, s but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d, then b, a \begin{matrix} \\ \\ ∥ \end{matrix} d, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c

but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c or a, c \begin{matrix} \\ \\ ∥ \end{matrix} c, b and there exist elements a, b, c, d of the points of AS such that a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c and for every elements a, b, c of the points of AS there exists an element d of the points of AS such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and b ≠ d and for all elements p, a, b, c of the points of AS such that p ≠ b and b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, c there exists an element d of the points of AS such that a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, d and a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d,

(ii) AS is an ordered affine space.

(38) If there exist u, v such that for all real numbers a, b such that a·u+b·v =

0_{V} holds a = 0 and b = 0, then OASpace V is an ordered affine space. We adopt the following rules: A will denote an ordered affine space and a, b,

c, d, p, q, r, s will denote elements of the points of A. We now state a number of propositions:

(39) There exist a, b such that a ≠ b.

(40) a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, c.

(41) If a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, a, then a = b.

(42) If a ≠ b and a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, q and a, b \begin{matrix} \\ \\ ∥ \end{matrix} r, s, then p, q \begin{matrix} \\ \\ ∥ \end{matrix} r, s.

(43) If a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d, then b, a \begin{matrix} \\ \\ ∥ \end{matrix} d, c.

(44) If a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c.

(45) If a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c or a, c \begin{matrix} \\ \\ ∥ \end{matrix} c, b.

(46) There exist a, b, c, d such that a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c.

(47) There exists d such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and b ≠ d.

(48) If p ≠ b and b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, c, then there exists d such that a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, d and

a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d.

An ordered affine space is said to be an ordered affine plane if: Let a, b, c, d be elements of the points of it . Then if a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c,

then there exists an element p of the points of it such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, p or a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, a

but c, d \begin{matrix} \\ \\ ∥ \end{matrix} c, p or c, d \begin{matrix} \\ \\ ∥ \end{matrix} p, c.

We now state three propositions:

(49) The following conditions are equivalent:

(i) for all elements a, b, c, d of the points of A such that a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c there exists an element p of the points of A such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, p

or a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, a but c, d \begin{matrix} \\ \\ ∥ \end{matrix} c, p or c, d \begin{matrix} \\ \\ ∥ \end{matrix} p, c,

(ii) A is an ordered affine plane.

(50) The following conditions are equivalent:

(i) there exist elements a, b of the points of AS such that a ≠ b and for all elements a, b, c, d, p, q, r, s of the points of AS holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, c but

if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, a, then a = b but if a ≠ b and a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, q and a, b \begin{matrix} \\ \\ ∥ \end{matrix} r, s, then

p, q \begin{matrix} \\ \\ ∥ \end{matrix} r, s but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d, then b, a \begin{matrix} \\ \\ ∥ \end{matrix} d, c but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c

but if a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c, then a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c or a, c \begin{matrix} \\ \\ ∥ \end{matrix} c, b and there exist elements a, b, c, d of the points of AS such that a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c and for every elements a, b, c of the points of AS there exists an element d of the points of AS such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and b ≠ d and for all elements p, a, b, c of the points of AS such that p ≠ b and b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, c there exists an element d of the points of AS such that a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, d and a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and for all elements a, b, c, d of the points of AS such that a, b \begin{matrix} \\ \\ \end{matrix}c, d and a, b \begin{matrix} \\ \\ \end{matrix}d, c there exists an element p of the points of AS such that

a, b \begin{matrix} \\ \\ ∥ \end{matrix} a, p or a, b \begin{matrix} \\ \\ ∥ \end{matrix} p, a but c, d \begin{matrix} \\ \\ ∥ \end{matrix} c, p or c, d \begin{matrix} \\ \\ ∥ \end{matrix} p, c,

(ii) AS is an ordered affine plane.

(51) If there exist u, v such that for all real numbers a, b such that a·u+b·v = 0_{V} holds a = 0 and b = 0 and for every w there exist real numbers a, b such that w = a · u + b · v, then OASpace V is an ordered affine plane.

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 from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

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

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

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

606

Metric Spaces ^{1}

Stanisława Kanas Technical University in Rzeszów

Department of Mathematics

Adam Lecko Technical University in Rzeszów Department of Mathematics

Mariusz Startek Technical University in Rzeszów Department of Mathematics

Summary. In this paper we define the metric spaces. Two examples of metric spaces are given. We define the discrete metric and the metric on the real axis. Moreover the open ball, the close ball and the sphere in metric spaces are introduced. We also prove some theorems concerning these concepts.

MML Identifier: METRIC 1.

The papers [3], [7], [2], [1], [5], [6], and [4] provide the notation and terminology for this paper. We consider metric structures which are systems

⟨ a carrier, a distance ⟩ where the carrier is a non-empty set and the distance is a function from [: the

carrier, the carrier :] into . In the sequel M will be a metric structure. Let us consider M. A point of M is an element of the carrier of M.

Next we state a proposition

(1) For every element x of the carrier of M holds x is a point of M. Let us consider M, and let a, b be elements of the carrier of M. The functor

ρ(a, b) yielding a real number, is defined by:

ρ(a, b) = (the distance of M)(a, b). We now state a proposition

(2) For all elements x, y of the carrier of M holds ρ(x, y) = (the distance of M)(x, y).

1Supported by RPBP.III.24-B.5

608 Stanisława Kanas et al.

In the sequel x will be arbitrary. Let us consider x. Then {x} is a non-empty set.

The function {[∅, ∅]} ↦ 0 from [: {∅}, {∅}:] into is defined by: {[∅, ∅]} ↦ 0 = [: {∅}, {∅}:] ⟼ 0. Next we state a proposition

(3) {[∅, ∅]} ↦ 0 = [: {∅}, {∅}:] ⟼ 0. A metric structure is said to be a metric space if: for all elements a, b, c of the carrier of it holds ρ(a, b) = 0 if and only if a = b

but ρ(a, b) = ρ(b, a) and ρ(a, c) ≤ ρ(a, b) + ρ(b, c). We now state three propositions:

(4) For every M being a metric structure holds M is a metric space if and only if for all elements a, b, c of the carrier of M holds ρ(a, b) = 0 if and only if a = b but ρ(a, b) = ρ(b, a) and ρ(a, c) ≤ ρ(a, b) + ρ(b, c).

(5) For every metric space M and for all elements a, b of the carrier of M holds ρ(a, b) = ρ(b, a).

(6) For every metric space M and for all elements a, b, c of the carrier of M holds ρ(a, c) ≤ ρ(a, b) + ρ(b, c).

In the sequel P M denotes a metric space and p_{1}, p_{2} denote elements of the carrier of P M. Next we state a proposition

(7) 0 ≤ ρ(p_{1}, p_{2}). Let A be a non-empty set. The discrete metric of A yielding a function from

[: A, A:] into , is defined by:

for all elements x, y of A holds (the discrete metric of A)(x, x) = 0 but if x ≠ y, then (the discrete metric of A)(x, y) = 1.

In the sequel A denotes a non-empty set and x, y denote elements of A. Next we state two propositions:

(8) (The discrete metric of A)(x, x) = 0.

(9) If x ≠ y, then (the discrete metric of A)(x, y) = 1. Let A be a non-empty set. The discrete space on A yielding a metric space,

is defined as follows:

the discrete space on A = ⟨A, the discrete metric of A⟩. In the sequel x will be an element of . Let us consider x. The functor @x

yielding a real number, is defined by:

@x = x. Next we state a proposition

(10) x = @x.

The function ρ_{}from [: , :] into is defined as follows: for all elements x, y of holds ρ_{}(x, y) = |@x − @y|. Next we state several propositions:

(11) For every function F from [: , :] into holds F = ρ_{}if and only if for all elements x, y of holds F (x, y) = |@x − @y|.

Metric Spaces 609

(12) For all real numbers x, y holds ρ_{}(x, y) = |x − y|.

(13) For all elements x, y of holds ρ_{}(x, y) = 0 if and only if x = y.

(14) For all elements x, y of holds ρ_{}(x, y) = ρ_{}(y, x).

(15) For all elements x, y, z of holds ρ_{}(x, y) ≤ ρ_{}(x, z) + ρ_{}(z, y). The metric space of real numbers a metric space is defined as follows: the metric space of real numbers = ⟨ , ρ_{}⟩. Let M be a metric structure, and let p be an element of the carrier of M, and

let r be a real number. The functor Ball(p, r) yielding a subset of the carrier of M, is defined as follows:

Ball(p, r) = {q : ρ(p, q) < r}. We now state a proposition

(16) For every M being a metric structure and for every element p of the carrier of M and for every real number r holds Ball(p, r) = {q : ρ(p, q) < r}.

Let M be a metric structure, and let p be an element of the carrier of M, and let r be a real number. The functor \overline{Ball}(p, r) yields a subset of the carrier of M and is defined as follows:

\overline{Ball}(p, r) = \{q : ρ(p, q) ≤ r\}. We now state a proposition

(17) For every M being a metric structure and for every element p of the carrier of M and for every real number r holds \overline{Ball}(p, r) = \{q : ρ(p, q) ≤ r}.

Let M be a metric structure, and let p be an element of the carrier of M, and let r be a real number. The functor Sphere(p, r) yielding a subset of the carrier of M, is defined by:

Sphere(p, r) = {q : ρ(p, q) = r}. Next we state several propositions:

(18) For every M being a metric structure and for every element p of the carrier of M and for every real number r holds Sphere(p, r) = {q : ρ(p, q) = r}.

(19) For every M being a metric structure and for all elements p, x of the carrier of M and for every real number r holds x ∈ Ball(p, r) if and only if ρ(p, x) < r.

(20) For every M being a metric structure and for all elements p, x of the carrier of M and for every real number r holds x ∈ \overline{Ball}(p, r) if and only if ρ(p, x) ≤ r.

(21) For every M being a metric structure and for all elements p, x of the carrier of M and for every real number r holds x ∈ Sphere(p, r) if and only if ρ(p, x) = r.

(22) For every M being a metric structure and for every element p of the carrier of M and for every real number r holds Ball(p, r) ⊆ \overline{Ball}(p, r).

610 Stanisława Kanas et al.

(23) For every M being a metric structure and for every element p of the carrier of M and for every real number r holds Sphere(p, r) ⊆ \overline{Ball}(p, r).

(24) For every M being a metric structure and for every element p of the carrier of M and for every real number r holds Sphere(p, r) ∪ Ball(p, r) = \overline{Ball}(p, r).

References

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

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

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

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

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

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

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

Ordered Affine Spaces Defined in Terms of Directed Parallelity - part I ^{1}

Henryk Oryszczyszyn^{2} Warsaw University Białystok Krzysztof Prażmowski^{3} Warsaw University Białystok

Summary. In the article we consider several geometrical relations in given arbitrary ordered affine space defined in terms of directed parallelity. In particular we introduce the notions of the nondirected parallelity of segments, of collinearity, and the betweenness relation determined by the given relation of directed parallelity. The obtained structures satisfy commonly accepted axioms for affine spaces. At the end of the article we introduce a formal definition of affine space and affine plane (defined in terms of parallelity of segments).

MML Identifier: DIRAF.

The notation and terminology used in this paper are introduced in the articles [2] and [1]. In the sequel X is a non-empty set. Let us consider X, and let R be a relation on [: X, X:]. The functor λ(R) yielding a relation on [: X, X:], is defined as follows:

for all elements a, b, c, d of X holds ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ λ(R) if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ R or ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨d, c⟩⟩⟩⟩ ∈ R.

One can prove the following two propositions:

(1) For all relations R, R′ on [: X, X:] holds R′ = λ(R) if and only if for all elements a, b, c, d of X holds ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ R′ if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ R or ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨d, c⟩⟩⟩⟩ ∈ R.

(2) For every relation R on [: X, X:] and for all elements a, b, c, d of X holds ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ λ(R) if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ R or ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨d, c⟩⟩⟩⟩ ∈ R.

1Supported by RPBP.III.24-C.2

2Supported by RPBP.III-24.C2.

3Supported by RPBP.III-24.C2.

Let S be an affine structure. The functor Λ(S) yielding an affine structure, is defined as follows:

Λ(S) = ⟨ the points of S, λ( the congruence of S)⟩.

One can prove the following proposition

(3) For all S, S′ being affine structures holds Λ(S) = S′ if and only if S′ = ⟨ the points of S, λ( the congruence of S)⟩.

We adopt the following convention: S will be an ordered affine space and a, b, c, d, x, y, z, t, u, w will be elements of the points of S. The following propositions are true:

(4) x, y \begin{matrix} \\ \\ ∥ \end{matrix} x, y.

(5) If x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t, then y, x \begin{matrix} \\ \\ ∥ \end{matrix} t, z and z, t \begin{matrix} \\ \\ ∥ \end{matrix} x, y and t, z \begin{matrix} \\ \\ ∥ \end{matrix} y, x.

(6) If z ≠ t and x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t and z, t \begin{matrix} \\ \\ ∥ \end{matrix} u, w, then x, y \begin{matrix} \\ \\ ∥ \end{matrix} u, w.

(7) x, x \begin{matrix} \\ \\ ∥ \end{matrix} y, z and y, z \begin{matrix} \\ \\ ∥ \end{matrix} x, x.

(8) If x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t and x, y \begin{matrix} \\ \\ ∥ \end{matrix} t, z, then x = y or z = t.

(9) x, y \begin{matrix} \\ \\ ∥ \end{matrix} x, z if and only if x, y \begin{matrix} \\ \\ ∥ \end{matrix} y, z or x, z \begin{matrix} \\ \\ ∥ \end{matrix} z, y.

Let us consider S, a, b, c. The predicate B(a, b, c) is defined as follows:

a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c.

The following propositions are true:

(10) B(a, b, c) if and only if a, b \begin{matrix} \\ \\ ∥ \end{matrix} b, c.

(11) x, y \begin{matrix} \\ \\ ∥ \end{matrix} x, z if and only if B(x, y, z) or B(x, z, y).

(12) If B(a, b, a), then a = b.

(13) If B(a, b, c), then B(c, b, a).

(14) B(x, x, y) and B(x, y, y).

(15) If B(a, b, c) and B(a, c, d), then B(b, c, d).

(16) If b ≠ c and B(a, b, c) and B(b, c, d), then B(a, c, d).

(17) There exists z such that B(x, y, z) and y ≠ z.

(18) If B(x, y, z) and B(y, x, z), then x = y.

(19) If x ≠ y and B(x, y, z) and B(x, y, t), then B(y, z, t) or B(y, t, z).

(20) If x ≠ y and B(x, y, z) and B(x, y, t), then B(x, z, t) or B(x, t, z).

(21) If B(x, y, t) and B(x, z, t), then B(x, y, z) or B(x, z, y).

Let us consider S, a, b, c, d. The predicate a, b |||| c, d is defined as follows:

a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d or a, b \begin{matrix} \\ \\ ∥ \end{matrix} d, c.

One can prove the following propositions:

(22) a, b |||| c, d if and only if a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d or a, b \begin{matrix} \\ \\ ∥ \end{matrix} d, c.

(23) a, b |||| c, d if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ λ( the congruence of S).

(24) x, y |||| y, x and x, y |||| x, y.

(25) x, y |||| z, z and z, z |||| x, y.

(26) If x, y |||| x, z, then y, x |||| y, z.

(27) If x, y |||| z, t, then x, y |||| t, z and y, x |||| z, t and y, x |||| t, z and z, t |||| x, y and z, t |||| y, x and t, z |||| x, y and t, z |||| y, x.

Ordered Affine Spaces Defined in Terms of … 613

(28) Suppose that (i) a ≠ b, (ii) a, b |||| x, y and a, b |||| z, t or a, b |||| x, y and z, t |||| a, b or x, y |||| a, b and

z, t |||| a, b or x, y |||| a, b and a, b |||| z, t.

Then x, y |||| z, t.

(29) There exist x, y, z such that x, y |||| x, z.

(30) There exists t such that x, z |||| y, t and y ≠ t.

(31) There exists t such that x, y |||| z, t and x, z |||| y, t.

(32) If z, x |||| x, t and x ≠ z, then there exists u such that y, x |||| x, u and

y, z |||| t, u. Let us consider S, a, b, c. The predicate L(a, b, c) is defined as follows: a, b |||| a, c. One can prove the following propositions:

(33) L(a, b, c) if and only if a, b |||| a, c.

(34) If B(a, b, c), then L(a, b, c).

(35) If L(a, b, c), then B(a, b, c) or B(b, a, c) or B(a, c, b).

(36) If L(x, y, z), then L(x, z, y) and L(y, x, z) and L(y, z, x) and L(z, x, y) and L(z, y, x).

(37) L(x, x, y) and L(x, y, y) and L(x, y, x).

(38) If x ≠ y and L(x, y, z) and L(x, y, t) and L(x, y, u), then L(z, t, u).

(39) If x ≠ y and L(x, y, z) and x, y |||| z, t, then L(x, y, t).

(40) If L(x, y, z) and L(x, y, t), then x, y |||| z, t.

(41) If u ≠ z and L(x, y, u) and L(x, y, z) and L(u, z, w), then L(x, y, w).

(42) There exist x, y, z such that not L(x, y, z).

(43) If x ≠ y, then there exists z such that not L(x, y, z).

In the sequel AS will denote an affine structure. Let us consider AS, and let a, b, c, d be elements of the points of AS. The predicate a, b ∥ c, d is defined as follows:

⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of AS.

The following propositions are true:

(44) For all elements a, b, c, d of the points of AS holds a, b ∥ c, d if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of AS.

(45) If AS = Λ(S), then for all elements a, b, c, d of the points of S and for all elements a′, b′, c′, d′ of the points of AS such that a = a′ and b = b′ and c = c′ and d = d′ holds a′, b′ ∥ c′, d′ if and only if a, b |||| c, d.

(46) Suppose AS = Λ(S). Then (i) there exist elements x, y of the points of AS such that x ≠ y, (ii) for all elements x, y, z, t, u, w of the points of AS holds x, y ∥ y, x and

x, y ∥ z, z but if x ≠ y and x, y ∥ z, t and x, y ∥ u, w, then z, t ∥ u, w but if x, y ∥ x, z, then y, x ∥ y, z,

(iii) there exist elements x, y, z of the points of AS such that x, y x, z, (iv) for every elements x, y, z of the points of AS there exists an element

t of the points of AS such that x, z ∥ y, t and y ≠ t,

(v) for every elements x, y, z of the points of AS there exists an element t of the points of AS such that x, y ∥ z, t and x, z ∥ y, t,

(vi) for all elements x, y, z, t of the points of AS such that z, x ∥ x, t and x ≠ z there exists an element u of the points of AS such that y, x ∥ x, u and y, z ∥ t, u.

An affine structure is said to be an affine space if:

(i) there exist elements x, y of the points of it such that x ≠ y, (ii) for all elements x, y, z, t, u, w of the points of it holds x, y ∥ y, x and x, y ∥ z, z but if x ≠ y and x, y ∥ z, t and x, y ∥ u, w, then z, t ∥ u, w but if x, y ∥ x, z, then y, x ∥ y, z, (iii) there exist elements x, y, z of the points of it such that x, y x, z, (iv) for every elements x, y, z of the points of it there exists an element t of the points of it such that x, z ∥ y, t and y ≠ t,

(v) for every elements x, y, z of the points of it there exists an element t of the points of it such that x, y ∥ z, t and x, z ∥ y, t, (vi) for all elements x, y, z, t of the points of it such that z, x ∥ x, t and x ≠ z there exists an element u of the points of it such that y, x ∥ x, u and y, z ∥ t, u.

The following three propositions are true:

(47) Let AS be an affine space. Then (i) there exist elements x, y of the points of AS such that x ≠ y, (ii) for all elements x, y, z, t, u, w of the points of AS holds x, y ∥ y, x and

x, y ∥ z, z but if x ≠ y and x, y ∥ z, t and x, y ∥ u, w, then z, t ∥ u, w but if x, y ∥ x, z, then y, x ∥ y, z,

(iii) there exist elements x, y, z of the points of AS such that x, y x, z, (iv) for every elements x, y, z of the points of AS there exists an element

t of the points of AS such that x, z ∥ y, t and y ≠ t,

(v) for every elements x, y, z of the points of AS there exists an element t of the points of AS such that x, y ∥ z, t and x, z ∥ y, t,

(vi) for all elements x, y, z, t of the points of AS such that z, x ∥ x, t and x ≠ z there exists an element u of the points of AS such that y, x ∥ x, u and y, z ∥ t, u.

(48) Λ(S) is an affine space.

(49) The following conditions are equivalent:

(i) there exist elements x, y of the points of AS such that x ≠ y and for all elements x, y, z, t, u, w of the points of AS holds x, y ∥ y, x and x, y ∥ z, z but if x ≠ y and x, y ∥ z, t and x, y ∥ u, w, then z, t ∥ u, w but if x, y ∥ x, z, then y, x ∥ y, z and there exist elements x, y, z of the points of AS such that x, y x, z and for every elements x, y, z of the points of AS there exists an element t of the points of AS such that x, z ∥ y, t and y ≠ t and for every elements x, y, z of the points of AS there exists an element t of the points of AS such that x, y ∥ z, t and x, z ∥ y, t and for all elements x, y, z, t of the points of AS such that z, x ∥ x, t and x ≠ z there exists an element u of the points of AS such that y, x ∥ x, u and y, z ∥ t, u,

(ii) AS is an affine space. We follow the rules: S will be an ordered affine plane and x, y, z, t, u will

be elements of the points of S. We now state two propositions:

(50) If x, y |||| z, t, then there exists u such that x, y |||| x, u and z, t |||| z, u.

(51) If AS = Λ(S), then for all elements x, y, z, t of the points of AS such that x, y z, t there exists an element u of the points of AS such that x, y ∥ x, u and z, t ∥ z, u.

An affine space is said to be an affine plane if: for all elements x, y, z, t of the points of it such that x, y z, t there exists

an element u of the points of it such that x, y ∥ x, u and z, t ∥ z, u.

In the sequel ASP will denote an affine space. Next we state three propositions:

(52) ASP is an affine plane if and only if for all elements x, y, z, t of the points of ASP such that x, y z, t there exists an element u of the points of ASP such that x, y ∥ x, u and z, t ∥ z, u.

(53) Λ(S) is an affine plane.

(54) AS is an affine plane if and only if the following conditions are satisfied: (i) there exist elements x, y of the points of AS such that x ≠ y, (ii) for all elements x, y, z, t, u, w of the points of AS holds x, y ∥ y, x and

x, y ∥ z, z but if x ≠ y and x, y ∥ z, t and x, y ∥ u, w, then z, t ∥ u, w but if x, y ∥ x, z, then y, x ∥ y, z,

(iii) there exist elements x, y, z of the points of AS such that x, y x, z, (iv) for every elements x, y, z of the points of AS there exists an element

t of the points of AS such that x, z ∥ y, t and y ≠ t,

(v) for every elements x, y, z of the points of AS there exists an element t of the points of AS such that x, y ∥ z, t and x, z ∥ y, t,

(vi) for all elements x, y, z, t of the points of AS such that z, x ∥ x, t and x ≠ z there exists an element u of the points of AS such that y, x ∥ x, u and y, z ∥ t, u,

(vii) for all elements x, y, z, t of the points of AS such that x, y z, t there exists an element u of the points of AS such that x, y ∥ x, u and z, t ∥ z, u.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

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

616 Henryk Oryszczyszyn et al.

Parallelity and Lines in Affine Spaces ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. In the article we introduce basic notions concerning affine spaces and investigate their fundamental properties. We define the function which to every nondegenerate pair of points assigns the line joining them and we extend the relation of parallelity to a relation between segments and lines, and between lines.

MML Identifier: AFF 1.

The papers [3], [1], and [2] provide the notation and terminology for this paper. We adopt the following convention: AS will be an affine space and a, a′, b, b′, c, d, o, p, q, x, y, z, t, u, w will be elements of the points of AS. One can prove the following propositions:

(1) There exist elements x, y of the points of AS such that x ≠ y.

(2) x, y ∥ y, x and x, y ∥ z, z.

(3) If x ≠ y and x, y ∥ z, t and x, y ∥ u, w, then z, t ∥ u, w.

(4) If x, y ∥ x, z, then y, x ∥ y, z.

(5) There exist x, y, z such that x, y x, z.

(6) There exists t such that x, z ∥ y, t and y ≠ t.

(7) There exists t such that x, y ∥ z, t and x, z ∥ y, t.

(8) If z, x ∥ x, t and x ≠ z, then there exists u such that y, x ∥ x, u and y, z ∥ t, u.

Let us consider AS, a, b, c. The predicate L(a, b, c) is defined as follows: a, b ∥ a, c. The following propositions are true:

(9) L(a, b, c) if and only if a, b ∥ a, c.

(10) For every a there exists b such that a ≠ b.

(11) x, y ∥ y, x and x, y ∥ x, y.

1Supported by RPBP.III.24-C.2

(12) x, y ∥ z, z and z, z ∥ x, y.

(13) If x, y ∥ z, t, then x, y ∥ t, z and y, x ∥ z, t and y, x ∥ t, z and z, t ∥ x, y and z, t ∥ y, x and t, z ∥ x, y and t, z ∥ y, x.

(14) Suppose that (i) a ≠ b, (ii) a, b ∥ x, y and a, b ∥ z, t or a, b ∥ x, y and z, t ∥ a, b or x, y ∥ a, b and

z, t ∥ a, b or x, y ∥ a, b and a, b ∥ z, t.

Then x, y ∥ z, t.

(15) If L(x, y, z), then L(x, z, y) and L(y, x, z) and L(y, z, x) and L(z, x, y) and L(z, y, x).

(16) L(x, x, y) and L(x, y, y) and L(x, y, x).

(17) If x ≠ y and L(x, y, z) and L(x, y, t) and L(x, y, u), then L(z, t, u).

(18) If x ≠ y and L(x, y, z) and x, y ∥ z, t, then L(x, y, t).

(19) If L(x, y, z) and L(x, y, t), then x, y ∥ z, t.

(20) If u ≠ z and L(x, y, u) and L(x, y, z) and L(u, z, w), then L(x, y, w).

(21) There exist x, y, z such that not L(x, y, z).

(22) If x ≠ y, then there exists z such that not L(x, y, z).

(23) If not L(o, a, b) and L(o, b, b′) and a, b ∥ a, b′, then b = b′.

Let us consider AS, a, b. The functor Line(a, b) yielding a subset of the points of AS, is defined as follows:

for every x holds x ∈ Line(a, b) if and only if L(a, b, x).

In the sequel A, C, D, K are subsets of the points of AS. We now state several propositions:

(24) A = Line(a, b) if and only if for every x holds x ∈ A if and only if L(a, b, x).

(25) Line(a, b) = Line(b, a).

(26) a ∈ Line(a, b) and b ∈ Line(a, b).

(27) If c ∈ Line(a, b) and d ∈ Line(a, b) and c ≠ d, then Line(c, d) ⊆ Line(a, b).

(28) If c ∈ Line(a, b) and d ∈ Line(a, b) and a ≠ b, then Line(a, b) ⊆

Line(c, d). Let us consider AS, A. We say that A is a line if and only if: there exist a, b such that a ≠ b and A = Line(a, b). One can prove the following propositions:

(29) A is a line if and only if there exist a, b such that a ≠ b and A = Line(a, b).

(30) For all a, b, A, C such that A is a line and C is a line and a ∈ A and b ∈ A and a ∈ C and b ∈ C holds a = b or A = C.

(31) If A is a line, then there exist a, b such that a ∈ A and b ∈ A and a ≠ b.

(32) If A is a line and a ∈ A, then there exists b such that a ≠ b and b ∈ A.

Parallelity and Lines in Affine Spaces 619

(33) L(a, b, c) if and only if there exists A such that A is a line and a ∈ A

and b ∈ A and c ∈ A. Let us consider AS, a, b, A. The predicate a, b ∥ A is defined by: there exist c, d such that c ≠ d and A = Line(c, d) and a, b ∥ c, d. The following proposition is true

(34) a, b ∥ A if and only if there exist c, d such that c ≠ d and A = Line(c, d)

and a, b ∥ c, d. Let us consider AS, A, C. The predicate A ∥ C is defined as follows: there exist a, b such that A = Line(a, b) and a ≠ b and a, b ∥ C. We now state a number of propositions:

(35) A ∥ C if and only if there exist a, b such that A = Line(a, b) and a ≠ b and a, b ∥ C.

(36) If c ∈ Line(a, b) and a ≠ b, then d ∈ Line(a, b) if and only if a, b ∥ c, d.

(37) If A is a line and a ∈ A, then b ∈ A if and only if a, b ∥ A.

(38) a ≠ b and A = Line(a, b) if and only if A is a line and a ∈ A and b ∈ A and a ≠ b.

(39) If A is a line and a ∈ A and b ∈ A and a ≠ b and L(a, b, x), then x ∈ A.

(40) If there exist a, b such that a, b ∥ A, then A is a line.

(41) If c ∈ A and d ∈ A and A is a line and c ≠ d, then a, b ∥ A if and only if a, b ∥ c, d.

(42) If c ≠ d and a, b ∥ c, d, then a, b ∥ Line(c, d).

(43) If a ≠ b, then a, b ∥ Line(a, b).

(44) If A is a line, then a, b ∥ A if and only if there exist c, d such that c ≠ d and c ∈ A and d ∈ A and a, b ∥ c, d.

(45) If A is a line and a, b ∥ A and c, d ∥ A, then a, b ∥ c, d.

(46) If a, b ∥ A and a, b ∥ p, q and a ≠ b, then p, q ∥ A.

(47) If A is a line, then a, a ∥ A.

(48) If a, b ∥ A, then b, a ∥ A.

(49) If a, b ∥ A and a ∉ A, then b ∉ A.

(50) If A ∥ C, then A is a line and C is a line.

(51) A ∥ C if and only if there exist a, b, c, d such that a ≠ b and c ≠ d and a, b ∥ c, d and A = Line(a, b) and C = Line(c, d).

(52) If A is a line and C is a line and a ∈ A and b ∈ A and c ∈ C and d ∈ C and a ≠ b and c ≠ d, then A ∥ C if and only if a, b ∥ c, d.

(53) If a ∈ A and b ∈ A and c ∈ C and d ∈ C and A ∥ C, then a, b ∥ c, d.

(54) If a ∈ A and b ∈ A and A ∥ C, then a, b ∥ C.

(55) If A is a line, then A ∥ A.

(56) If A ∥ C, then C ∥ A.

(57) If a, b ∥ A and A ∥ C, then a, b ∥ C.

(58) If A ∥ C and C ∥ D or A ∥ C and D ∥ C or C ∥ A and C ∥ D or C ∥ A and D ∥ C, then A ∥ D.

(59) If A ∥ C and p ∈ A and p ∈ C, then A = C.

(60) If x ∈ K and a ∉ K and a, b ∥ K, then a = b or not L(x, a, b).

(61) If a, b ∥ K and a′, b′ ∥ K and L(p, a, a′) and L(p, b, b′) and p ∈ K and a ∉ K and a = b, then a′ = b′.

(62) If A is a line and a ∈ A and b ∈ A and c ∈ A and a ≠ b and a, b ∥ c, d, then d ∈ A.

(63) For all a, A such that A is a line there exists C such that a ∈ C and A ∥ C.

(64) If A ∥ C and A ∥ D and p ∈ C and p ∈ D, then C = D.

(65) If A is a line and a ∈ A and b ∈ A and c ∈ A and d ∈ A, then a, b ∥ c, d.

(66) If A is a line and a ∈ A and b ∈ A, then a, b ∥ A.

(67) If a, b ∥ A and a, b ∥ C and a ≠ b, then A ∥ C.

(68) If not L(o, a, b) and L(o, a, a′) and L(o, b, b′) and a, b ∥ a′, b′ and a′ = b′, then a′ = o and b′ = o.

(69) If not L(o, a, b) and L(o, a, a′) and L(o, b, b′) and a, b ∥ a′, b′ and a′ = o, then b′ = o.

(70) If not L(o, a, b) and L(o, a, a′) and L(o, b, b′) and L(o, b, x) and a, b ∥ a′, b′ and a, b ∥ a′, x, then b′ = x.

(71) For all a, b, A such that A is a line and a ∈ A and b ∈ A and a ≠ b

holds A = Line(a, b). We adopt the following convention: AP will be an affine plane, a, b, c, d, x,

p will be elements of the points of AP, and A, C will be subsets of the points of AP. One can prove the following three propositions:

(72) If A is a line and C is a line and A C, then there exists x such that x ∈ A and x ∈ C.

(73) If A is a line and a, b A, then there exists x such that x ∈ A and L(a, b, x).

(74) If a, b c, d, then there exists p such that L(a, b, p) and L(c, d, p).

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part i. Formalized Mathematics, 1(3):611–615, 1990.

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

Parallelity and Lines in Affine Spaces 621 Received May 4, 1990

Index of MML Identifiers

AFF 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .617 ANALOAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 CARD 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543 CARD 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589 CLASSES1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 CLASSES2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 COMPLEX1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 DIRAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 EQREL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 FINSEQ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 FINSEQ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 FINSEQ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 FRAENKEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 FUNCSDOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 FUNCT 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 FUNCT 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 INT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .501 METRIC 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 MIDSP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 ORDINAL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 PARSP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 QC LANG2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 QC LANG3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 QMAX 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489 REALSET1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 RLVECT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 SEQ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .477 SEQM 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .471 SQUARE 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 WELLSET1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Contents Form. Math. 1 (3)

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Zermelo’s Theorem

By Bogdan Nowak and Sławomir Białecki . . . . . . . . . . . . . . . 431

Group and Field Definitions

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Equivalence Relations and Classes of Abstraction

By Konrad Raczkowski and Paweł Sadowski . . . . . . . . . . . . 441

Some Properties of Real Numbers

By Andrzej Trybulec and Czesław Byliński . . . . . . . . . . . . 445

Connectives and Subformulae of the First Order Language

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Variables in Formulae of the First Order Language

By Czesław Byliński and Grzegorz Bancerek . . . . . . . . . . 459

Monotone Real Sequences. Subsequences

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Midpoint algebras

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

The Fundamental Logic Structure in Quantum Mechanics

By Paweł Sadowski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

Function Domains and Frænkel Operator

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Continued on inside back cover

Integers

By Michał J. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

The Complex Numbers

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Ordinal Arithmetics

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

The Modification of a Function by a Function and the Iteration of the Composition of a Function

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Finite Sequences and Tuples of Elements of a Non-empty Sets

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

Curried and Uncurried Functions

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Cardinal Arithmetics

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

Fano-Desargues Parallelity Spaces

By Eugeniusz Kusak and Wojciech Leończuk . . . . . . . . . . . 549

Real Functions Spaces

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

Tarski’s Classes and Ranks

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

Non-contiguous Substrings and One-to-one Finite Sequences

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Pigeon Hole Principle

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

Linear Combinations in Real Linear Space

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

König’s Theorem

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

Universal Classes

By Bogdan Nowak and Grzegorz Bancerek . . . . . . . . . . . . . 595

Analytical Ordered Affine Spaces

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Metric Spaces

By Stanisława Kanas et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

Ordered Affine Spaces Defined in Terms of Directed Parallelity - part I

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

Parallelity and Lines in Affine Spaces

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

Continued on inside back cover