FORMALIZED MATHEMATICS Volume 11, Number 3, 2003 University of Białystok

General Fashoda Meet Theorem for Unit Circle and Square

Yatsuka Nakamura Shinshu University Nagano

Summary. Here we will prove Fashoda meet theorem for the unit circle and for a square, when 4 points on the boundary are ordered cyclically. Also, the concepts of general rectangle and general circle are defined.

MML Identifier: JGRAPH 6.

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

1. Preliminaries

One can prove the following propositions: (2)^{1} For all real numbers a, b, r such that 0 r and r 1 and a b holds

a (1 − r) · a + r · b and (1 − r) · a + r · b b.

(3) For all real numbers a, b such that a 0 and b > 0 or a > 0 and b 0 holds a + b > 0.

(4) For all real numbers a, b such that − 1 a and a 1 and − 1 b and b 1 holds a^{2} · b^{2} 1.

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

(6) For all real numbers a, b such that − 1 a and a 1 and − 1 b and b 1 holds (− b) · \sqrt{1 + a^{2}} \sqrt{1 + b^{2}} and − \sqrt{1 + b^{2}} b · \sqrt{1 + a^{2}}.

1The proposition (1) has been removed.

(7) For all real numbers a, b such that − 1 a and a 1 and − 1 b and b 1 holds b · \sqrt{1 + a^{2}} \sqrt{1 + b^{2}}.

(8) For all real numbers a, b such that a b holds a · \sqrt{1 + b^{2}} b · \sqrt{1 + a^{2}}.

(9) Let a, c, d be real numbers and p be a point of E_{T}^{2}. If c d and p ∈ L([a, c], [a, d]), then p_{1} = a and c p_{2} and p_{2} d.

(10) For all real numbers a, c, d and for every point p of E_{T}^{2} such that c < d and p_{1} = a and c p_{2} and p_{2} d holds p ∈ L([a, c], [a, d]).

(11) Let a, b, d be real numbers and p be a point of E_{T}^{2}. If a b and p ∈ L([a, d], [b, d]), then p_{2} = d and a p_{1} and p_{1} b.

(12) For all real numbers a, b and for every subset B of 𝕀 such that B = [a, b] holds B is closed.

(13) Let X be a topological structure, Y, Z be non empty topological structures, f be a map from X into Y, and g be a map from X into Z. Then domf = domg and domf = the carrier of X and domf = Ω_{X}.

(14) Let X be a non empty topological space and B be a non empty subset of X. Then there exists a map f from X↾B into X such that for every point p of X↾B holds f(p) = p and f is continuous.

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

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

(17) Let X be a non empty topological space, n be a natural number, p be a point of E_{T}^{n}, and f be a map from X into ℝ^{1}. Suppose f is continuous. Then there exists a map g from X into E_{T}^{n} such that for every point r of X holds g(r) = f(r) · p and g is continuous.

(18) SqCirc([− 1, 0]) = [− 1, 0].

(19) For every compact non empty subset P of E_{T}^{2} such that P = {p; p ranges over points of E_{T}^{2}: |p| = 1} holds SqCirc([− 1, 0]) = W-minP.

(20) Let X be a non empty topological space, n be a natural number, and g_{1}, g_{2} be maps from X into E_{T}^{n}. Suppose g_{1} is continuous and g_{2} is continuous. Then there exists a map g from X into E_{T}^{n} such that for every point r of X holds g(r) = g_{1}(r) + g_{2}(r) and g is continuous.

(21) Let X be a non empty topological space, n be a natural number, p_{1}, p_{2} be points of E_{T}^{n}, and f_{1}, f_{2} be maps from X into ℝ^{1}. Suppose f_{1} is continuous and f_{2} is continuous. Then there exists a map g from X into E_{T}^{n} such that for every point r of X holds g(r) = f_{1}(r) · p_{1} + f_{2}(r) · p_{2} and

g is continuous.

(22) For every function f and for every set A such that f is one-to-one and A ⊆ domf holds (f^{−1} )^{∘}f^{∘}A = A.

2. General Fashoda Theorem for Unit Circle

In the sequel p, p_{1}, p_{2}, p_{3}, q, q_{1}, q_{2} are points of E_{T}^{2}. One can prove the following propositions:

(23) Let f, g be maps from 𝕀 into E_{T}^{2}, C_{0}, K_{1}, K_{2}, K_{3}, K_{4} be subsets of E_{T}^{2}, and O, I be points of 𝕀. Suppose that O = 0 and I = 1 and f is continuous and one-to-one and g is continuous and one-to-one and C_{0} = {p : |p| 1} and K_{1} = {q_{1}; q_{1} ranges over points of E_{T}^{2}: |q_{1}| = 1 ∧ (q_{1})_{2} (q_{1})_{1} ∧ (q_{1})_{2} − (q_{1})_{1}} and K_{2} = {q_{2}; q_{2} ranges over points of E_{T}^{2}: |q_{2}| = 1 ∧ (q_{2})_{2} (q_{2})_{1} ∧ (q_{2})_{2} − (q_{2})_{1}} and K_{3} = {q_{3}; q_{3} ranges over points of E_{T}^{2}: |q_{3}| = 1 ∧ (q_{3})_{2} (q_{3})_{1} ∧ (q_{3})_{2} − (q_{3})_{1}} and K_{4} = {q_{4}; q_{4} ranges over points of E_{T}^{2}: |q_{4}| = 1 ∧ (q_{4})_{2} (q_{4})_{1} ∧ (q_{4})_{2} − (q_{4})_{1}} and f(O) ∈ K_{1} and f(I) ∈ K_{2} and g(O) ∈ K_{3} and g(I) ∈ K_{4} and rngf ⊆ C_{0} and rngg ⊆ C_{0}. Then rngf meets rngg.

(24) Let f, g be maps from 𝕀 into E_{T}^{2}, C_{0}, K_{1}, K_{2}, K_{3}, K_{4} be subsets of E_{T}^{2}, and O, I be points of 𝕀. Suppose that O = 0 and I = 1 and f is continuous and one-to-one and g is continuous and one-to-one and C_{0} = {p : |p| 1} and K_{1} = {q_{1}; q_{1} ranges over points of E_{T}^{2}: |q_{1}| = 1 ∧ (q_{1})_{2} (q_{1})_{1} ∧ (q_{1})_{2} − (q_{1})_{1}} and K_{2} = {q_{2}; q_{2} ranges over points of E_{T}^{2}: |q_{2}| = 1 ∧ (q_{2})_{2} (q_{2})_{1} ∧ (q_{2})_{2} − (q_{2})_{1}} and K_{3} = {q_{3}; q_{3} ranges over points of E_{T}^{2}: |q_{3}| = 1 ∧ (q_{3})_{2} (q_{3})_{1} ∧ (q_{3})_{2} − (q_{3})_{1}} and K_{4} = {q_{4}; q_{4} ranges over points of E_{T}^{2}: |q_{4}| = 1 ∧ (q_{4})_{2} (q_{4})_{1} ∧ (q_{4})_{2} − (q_{4})_{1}} and f(O) ∈ K_{1} and f(I) ∈ K_{2} and g(O) ∈ K_{4} and g(I) ∈ K_{3} and rngf ⊆ C_{0} and rngg ⊆ C_{0}. Then rngf meets rngg.

(25) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, P be a compact non empty subset of E_{T}^{2}, and C_{0} be a subset of E_{T}^{2}. Suppose P = {p; p ranges over points of E_{T}^{2}: |p| = 1} and LE(p_{1}, p_{2}, P) and LE(p_{2}, p_{3}, P) and LE(p_{3}, p_{4}, P). Let f, g be maps from 𝕀 into E_{T}^{2}. Suppose that f is continuous and one-to-one and g is continuous and one-to-one and C_{0} = {p_{8}; p_{8} ranges over points of E_{T}^{2}: |p_{8}| 1} and f(0) = p_{3} and f(1) = p_{1} and g(0) = p_{2} and g(1) = p_{4} and rngf ⊆ C_{0} and rngg ⊆ C_{0}. Then rngf meets rngg.

(26) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, P be a compact non empty subset of E_{T}^{2}, and C_{0} be a subset of E_{T}^{2}. Suppose P = {p; p ranges over points of E_{T}^{2}: |p| = 1} and LE(p_{1}, p_{2}, P) and LE(p_{2}, p_{3}, P) and LE(p_{3}, p_{4}, P). Let f, g be maps from 𝕀 into E_{T}^{2}. Suppose that f is continuous and one-to-one and g is continuous and one-to-one and C_{0} = {p_{8}; p_{8} ranges over points of E_{T}^{2}:

|p_{8}| 1} and f(0) = p_{3} and f(1) = p_{1} and g(0) = p_{4} and g(1) = p_{2} and rngf ⊆ C_{0} and rngg ⊆ C_{0}. Then rngf meets rngg.

(27) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, P be a compact non empty subset of E_{T}^{2}, and C_{0} be a subset of E_{T}^{2}. Suppose P = {p; p ranges over points of E_{T}^{2}: |p| = 1} and p_{1}, p_{2}, p_{3}, p_{4} are in this order on P. Let f, g be maps from 𝕀 into E_{T}^{2}. Suppose that f is continuous and one-to-one and g is continuous and one-to-one and C_{0} = {p_{8}; p_{8} ranges over points of E_{T}^{2}: |p_{8}| 1} and f(0) = p_{1} and f(1) = p_{3} and g(0) = p_{2} and g(1) = p_{4} and rngf ⊆ C_{0} and rngg ⊆ C_{0}. Then rngf meets rngg.

3. General Rectangles and Circles

Let a, b, c, d be real numbers. The functor Rectangle(a, b, c, d) yielding a subset of E_{T}^{2} is defined by the condition (Def. 1).

(Def. 1) Rectangle(a, b, c, d) = {p : p_{1} = a ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = d ∧ a

p_{1} ∧ p_{1} b ∨ p_{1} = b ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = c ∧ a p_{1} ∧ p_{1} b}. The following proposition is true

(28) Let a, b, c, d be real numbers and p be a point of E_{T}^{2}. If a b and c d and p ∈ Rectangle(a, b, c, d), then a p_{1} and p_{1} b and c p_{2} and p_{2} d.

Let a, b, c, d be real numbers. The functor InsideOfRectangle(a, b, c, d) yields a subset of E_{T}^{2} and is defined as follows:

(Def. 2) InsideOfRectangle(a, b, c, d) = {p : a < p_{1} ∧ p_{1} < b ∧ c < p_{2} ∧ p_{2} < d}.

Let a, b, c, d be real numbers. The functor ClosedInsideOfRectangle(a, b, c, d) yielding a subset of E_{T}^{2} is defined as follows:

(Def. 3) ClosedInsideOfRectangle(a, b, c, d) = {p : a p_{1} ∧ p_{1} b ∧ c

p_{2} ∧ p_{2} d}. Let a, b, c, d be real numbers. The functor OutsideOfRectangle(a, b, c, d)

yields a subset of E_{T}^{2} and is defined by: (Def. 4) OutsideOfRectangle(a, b, c, d) = {p : a p̸_{1} ∨ p_{1} b̸ ∨ c p̸_{2} ∨ p_{2}

d}. Let a, b, c, d be real numbers. The functor ClosedOutsideOfRectangle(a, b, c, d)

yielding a subset of E_{T}^{2} is defined by: (Def. 5) ClosedOutsideOfRectangle(a, b, c, d) = {p : a ≮ p_{1} ∨ p_{1} ≮ b ∨ c ≮

p_{2} ∨ p_{2} ≮ d}. Next we state four propositions:

(29) Let a, b, r be real numbers and K_{5}, C_{1} be subsets of E_{T}^{2}. Suppose r 0 and K_{5} = {q : |q| = 1} and C_{1} = {p_{2}; p_{2} ranges over points of E_{T}^{2}: |p_{2} − [a, b]| = r}. Then (AffineMap(r, a, r, b))^{∘}K_{5} = C_{1}.

(30) Let P, Q be subsets of E_{T}^{2}. Suppose there exists a map from E_{T}^{2}↾P into E_{T}^{2}↾Q which is a homeomorphism and P is a simple closed curve. Then Q is a simple closed curve.

(31) For every subset P of E_{T}^{2} such that P satisfies conditions of simple closed curve holds P is compact.

(32) Let a, b, r be real numbers and C_{1} be a subset of E_{T}^{2}. Suppose r > 0 and C_{1} = {p; p ranges over points of E_{T}^{2}: |p − [a, b]| = r}. Then C_{1} is a simple closed curve.

Let a, b, r be real numbers. Let us assume that r > 0. The functor Circle(a, b, r) yielding a compact non empty subset of E_{T}^{2} is defined as follows:

(Def. 6) Circle(a, b, r) = {p; p ranges over points of E_{T}^{2}: |p − [a, b]| = r}.

Let a, b, r be real numbers. The functor InsideOfCircle(a, b, r) yielding a subset of E_{T}^{2} is defined by:

(Def. 7) InsideOfCircle(a, b, r) = {p; p ranges over points of E_{T}^{2}: |p − [a, b]| < r}.

Let a, b, r be real numbers. The functor ClosedInsideOfCircle(a, b, r) yields a subset of E_{T}^{2} and is defined as follows:

(Def. 8) ClosedInsideOfCircle(a, b, r) = {p; p ranges over points of E_{T}^{2}: |p − [a,

b]| r}. Let a, b, r be real numbers. The functor OutsideOfCircle(a, b, r) yielding a

subset of E_{T}^{2} is defined by: (Def. 9) OutsideOfCircle(a, b, r) = {p; p ranges over points of E_{T}^{2}: |p − [a, b]| > r}.

Let a, b, r be real numbers. The functor ClosedOutsideOfCircle(a, b, r) yielding a subset of E_{T}^{2} is defined as follows:

(Def. 10) ClosedOutsideOfCircle(a, b, r) = {p; p ranges over points of E_{T}^{2}: |p − [a,

b]| r}. One can prove the following propositions:

(33) Let r be a real number. Then InsideOfCircle(0, 0, r) = {p : |p| < r} and if r > 0, then Circle(0, 0, r) = {p_{2} : |p_{2}| = r} and OutsideOfCircle(0, 0, r) = {p_{3} : |p_{3}| > r} and ClosedInsideOfCircle(0, 0, r) = {q : |q| r} and ClosedOutsideOfCircle(0, 0, r) = {q_{2} : |q_{2}| r}.

(34) Let K_{5}, C_{1} be subsets of E_{T}^{2}. Suppose K_{5} = {p : − 1 < p_{1} ∧ p_{1} < 1 ∧ − 1 < p_{2} ∧ p_{2} < 1} and C_{1} = {p_{2}; p_{2} ranges over points of E_{T}^{2}: |p_{2}| < 1}. Then SqCirc^{∘} K_{5} = C_{1}.

(35) Let K_{5}, C_{1} be subsets of E_{T}^{2}. Suppose K_{5} = {p : − 1 p̸_{1} ∨ p_{1} 1 ∨ − 1 p̸_{2} ∨ p_{2} 1̸} and C_{1} = {p_{2}; p_{2} ranges over points of E_{T}^{2}: |p_{2}| > 1}. Then SqCirc^{∘} K_{5} = C_{1}.

(36) Let K_{5}, C_{1} be subsets of E_{T}^{2}. Suppose K_{5} = {p : − 1 p_{1} ∧ p_{1} 1 ∧ − 1 p_{2} ∧ p_{2} 1} and C_{1} = {p_{2}; p_{2} ranges over points of E_{T}^{2}: |p_{2}| 1}. Then SqCirc^{∘} K_{5} = C_{1}.

(37) Let K_{5}, C_{1} be subsets of E_{T}^{2}. Suppose K_{5} = {p : − 1 ≮ p_{1} ∨ p_{1} ≮ 1 ∨ − 1 ≮ p_{2} ∨ p_{2} ≮ 1} and C_{1} = {p_{2}; p_{2} ranges over points of E_{T}^{2}: |p_{2}| 1}. Then SqCirc^{∘} K_{5} = C_{1}.

(38) Let P_{0}, P_{1}, P_{2}, P_{11}, K_{0}, K_{6}, K_{7}, K_{11} be subsets of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose that P = Circle(0, 0, 1) and P_{0} = InsideOfCircle(0, 0, 1) and P_{1} = OutsideOfCircle(0, 0, 1) and P_{2} = ClosedInsideOfCircle(0, 0, 1) and P_{11} = ClosedOutsideOfCircle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and K_{0} = InsideOfRectangle(− 1, 1, − 1, 1) and K_{6} = OutsideOfRectangle(− 1, 1, − 1, 1) and K_{7} = ClosedInsideOfRectangle(− 1, 1, − 1, 1) and K_{11} = ClosedOutsideOfRectangle(− 1, 1, − 1, 1) and f = SqCirc. Then f^{∘}K = P and (f^{−1} )^{∘}P = K and f^{∘}K_{0} = P_{0} and (f^{−1} )^{∘}P_{0} = K_{0} and f^{∘}K_{6} = P_{1} and (f^{−1} )^{∘}P_{1} = K_{6} and f^{∘}K_{7} = P_{2} and f^{∘}K_{11} = P_{11} and (f^{−1} )^{∘}P_{2} = K_{7} and (f^{−1} )^{∘}P_{11} = K_{11}.

4. Order of Points on Rectangle

The following propositions are true:

(39) Let a, b, c, d be real numbers. Suppose a b and c d. Then (i) L([a, c], [a, d]) = {p_{1} : (p_{1})_{1} = a ∧ (p_{1})_{2} d ∧ (p_{1})_{2} c}, (ii) L([a, d], [b, d]) = {p_{2} : (p_{2})_{1} b ∧ (p_{2})_{1} a ∧ (p_{2})_{2} = d}, (iii) L([a, c], [b, c]) = {q_{1} : (q_{1})_{1} b ∧ (q_{1})_{1} a ∧ (q_{1})_{2} = c}, and (iv) L([b, c], [b, d]) = {q_{2} : (q_{2})_{1} = b ∧ (q_{2})_{2} d ∧ (q_{2})_{2} c}.

(40) Let a, b, c, d be real numbers. Suppose a b and c d. Then {p : p_{1} = a ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = d ∧ a p_{1} ∧ p_{1} b ∨ p_{1} = b ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = c ∧ a p_{1} ∧ p_{1} b} = L([a, c], [a, d]) ∪ L([a, d], [b, d]) ∪ (L([a, c], [b, c]) ∪ L([b, c], [b, d])).

(41) For all real numbers a, b, c, d such that a b and c d holds L([a, c], [a, d]) ∩ L([a, c], [b, c]) = {[a, c]}.

(42) For all real numbers a, b, c, d such that a b and c d holds L([a, c], [b, c]) ∩ L([b, c], [b, d]) = {[b, c]}.

(43) For all real numbers a, b, c, d such that a b and c d holds L([a, d], [b, d]) ∩ L([b, c], [b, d]) = {[b, d]}.

(44) For all real numbers a, b, c, d such that a b and c d holds L([a, c], [a, d]) ∩ L([a, d], [b, d]) = {[a, d]}.

(45) {q : − 1 = q_{1} ∧ − 1 q_{2} ∧ q_{2} 1 ∨ q_{1} = 1 ∧ − 1 q_{2} ∧ q_{2} 1 ∨ − 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1 ∨ 1 = q_{2} ∧ − 1 q_{1} ∧ q_{1} 1} = {p : p_{1} = − 1 ∧ − 1 p_{2} ∧ p_{2} 1 ∨ p_{2} = 1 ∧ − 1 p_{1} ∧ p_{1} 1 ∨ p_{1} = 1 ∧ − 1 p_{2} ∧ p_{2} 1 ∨ p_{2} = − 1 ∧ − 1 p_{1} ∧ p_{1} 1}.

(46) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then W-boundK = a.

(47) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then N-boundK = d.

(48) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then E-boundK = b.

(49) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then S-boundK = c.

(50) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then NW-cornerK = [a, d].

(51) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then NE-cornerK = [b, d].

(52) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then SW-cornerK = [a, c].

(53) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then SE-cornerK = [b, c].

(54) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then W-mostK = L([a, c], [a, d]).

(55) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then E-mostK = L([b, c], [b, d]).

(56) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a b and c d, then W-minK = [a, c] and E-maxK = [b, d].

(57) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. Suppose K = Rectangle(a, b, c, d) and a < b and c < d. Then L([a, c], [a, d]) ∪ L([a, d], [b, d]) is an arc from W-minK to E-maxK and L([a, c], [b, c]) ∪ L([b, c], [b, d]) is an arc from E-maxK to W-minK.

(58) Let P, P_{1}, P_{3} be subsets of E_{T}^{2}, a, b, c, d be real numbers, f_{1}, f_{2} be finite sequences of elements of E_{T}^{2}, and p_{0}, p_{1}, p_{5}, p_{10} be points of E_{T}^{2}. Suppose that a < b and c < d and P = {p : p_{1} = a ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = d ∧ a p_{1} ∧ p_{1} b ∨ p_{1} = b ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = c ∧ a p_{1} ∧ p_{1} b} and p_{0} = [a, c] and p_{1} = [b, d] and p_{5} = [a, d] and p_{10} = [b, c] and f_{1} = ⟨p_{0}, p_{5}, p_{1}⟩ and f_{2} = ⟨p_{0}, p_{10}, p_{1}⟩. Then f_{1} is a special sequence and L̃(f_{1}) = L(p_{0}, p_{5}) ∪ L(p_{5}, p_{1}) and f_{2} is a special sequence and L̃(f_{2}) = L(p_{0}, p_{10}) ∪ L(p_{10}, p_{1}) and P = L̃(f_{1}) ∪ L̃(f_{2}) and L̃(f_{1}) ∩ L̃(f_{2}) = {p_{0}, p_{1}} and (f_{1})_{1} = p_{0} and (f_{1})_{len f_{1}} = p_{1} and (f_{2})_{1} = p_{0} and (f_{2})_{len f_{2}} = p_{1}.

(59) Let P, P_{1}, P_{3} be subsets of E_{T}^{2}, a, b, c, d be real numbers, f_{1}, f_{2} be finite sequences of elements of E_{T}^{2}, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose that a < b and c < d and P = {p : p_{1} = a ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = d ∧ a p_{1} ∧ p_{1} b ∨ p_{1} = b ∧ c p_{2} ∧ p_{2} d ∨ p_{2} = c ∧ a p_{1} ∧ p_{1} b} and p_{1} = [a, c] and p_{2} = [b, d] and f_{1} = ⟨[a, c], [a, d], [b, d]⟩ and f_{2} = ⟨[a, c], [b, c], [b, d]⟩ and P_{1} = L̃(f_{1}) and P_{3} = L̃(f_{2}). Then P_{1} is an arc from p_{1} to p_{2} and P_{3} is an arc from p_{1} to p_{2} and P_{1} is non empty and P_{3} is non empty and P = P_{1} ∪ P_{3} and P_{1} ∩ P_{3} = {p_{1}, p_{2}}.

(60) For all real numbers a, b, c, d such that a < b and c < d holds Rectangle(a, b, c, d) is a simple closed curve.

(61) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a < b and c < d, then UpperArcK = L([a, c], [a, d]) ∪ L([a, d], [b, d]).

(62) Let K be a non empty compact subset of E_{T}^{2} and a, b, c, d be real numbers. If K = Rectangle(a, b, c, d) and a < b and c < d, then LowerArcK = L([a, c], [b, c]) ∪ L([b, c], [b, d]).

(63) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d. Then there exists a map f from 𝕀 into (E_{T}^{2})↾ UpperArcK such that f is a homeomorphism and f(0) = W-minK and f(1) = E-maxK and rngf = UpperArcK and for every real number r such that r ∈ [0, \frac{1}{2}] holds f(r) = (1 − 2 · r) · [a, c] + 2 · r · [a, d] and for every real number r such that r ∈ [\frac{1}{2}, 1] holds f(r) = (1 − (2 · r − 1)) · [a, d] + (2 · r − 1) · [b, d] and for every

point p of E_{T}^{2} such that p ∈ L([a, c], [a, d]) holds 0 ^{\frac{\frac{p_{2} −c}{d−c}}{2}} and ^{\frac{\frac{p_{2} −c}{d−c}}{2}} 1

and f(^{\frac{\frac{p_{2} −c}{d−c}}{2}} ) = p and for every point p of E_{T}^{2} such that p ∈ L([a, d], [b, d])

holds 0 ^{\frac{\frac{p_{1} −a}{b−a}}{2}} + \frac{1}{2} and ^{\frac{\frac{p_{1} −a}{b−a}}{2}} + \frac{1}{2} 1 and f(^{\frac{\frac{p_{1} −a}{b−a}}{2}} + \frac{1}{2}) = p.

(64) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d. Then there exists a map f from 𝕀 into (E_{T}^{2})↾ LowerArcK such that f is a homeomorphism and f(0) = E-maxK and f(1) = W-minK and rngf = LowerArcK and for every real number r such that r ∈ [0, \frac{1}{2}] holds f(r) = (1 − 2 · r) · [b, d] + 2 · r · [b, c] and for every real number r such that r ∈ [\frac{1}{2}, 1] holds f(r) = (1 − (2 · r − 1)) · [b, c] + (2 · r − 1) · [a, c] and for every point p of E_{T}^{2} such that p ∈ L([b, d], [b, c]) holds 0 ^{\frac{\frac{p_{2} −d}{c−d}}{2}} and ^{\frac{\frac{p_{2} −d}{c−d}}{2}} 1

and f(^{\frac{\frac{p_{2} −d}{c−d}}{2}} ) = p and for every point p of E_{T}^{2} such that p ∈ L([b, c], [a, c])

holds 0 ^{\frac{\frac{p_{1} −b}{a−b}}{2}} + \frac{1}{2} and ^{\frac{\frac{p_{1} −b}{a−b}}{2}} + \frac{1}{2} 1 and f(^{\frac{\frac{p_{1} −b}{a−b}}{2}} + \frac{1}{2}) = p.

(65) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([a, c], [a, d]) and p_{2} ∈ L([a, c], [a, d]). Then LE(p_{1}, p_{2}, K) if and only if (p_{1})_{2} (p_{2})_{2}.

(66) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([a, d], [b, d]) and p_{2} ∈ L([a, d], [b, d]). Then LE(p_{1}, p_{2}, K) if and only if (p_{1})_{1} (p_{2})_{1}.

(67) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([b, c], [b, d]) and p_{2} ∈ L([b, c], [b, d]). Then LE(p_{1}, p_{2}, K) if and only if (p_{1})_{2} (p_{2})_{2}.

(68) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([a, c], [b, c]) and p_{2} ∈ L([a, c], [b, c]). Then LE(p_{1}, p_{2}, K) and p_{1} ≠ W-minK if and only if (p_{1})_{1} (p_{2})_{1} and p_{2} ≠ W-minK.

(69) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([a, c], [a, d]). Then LE(p_{1}, p_{2}, K) if and only if one of the following conditions is satisfied:

(i) p_{2} ∈ L([a, c], [a, d]) and (p_{1})_{2} (p_{2})_{2}, or (ii) p_{2} ∈ L([a, d], [b, d]), or (iii) p_{2} ∈ L([b, d], [b, c]), or (iv) p_{2} ∈ L([b, c], [a, c]) and p_{2} ≠ W-minK.

(70) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([a, d], [b, d]). Then LE(p_{1}, p_{2}, K) if and only if one of the following conditions is satisfied:

(i) p_{2} ∈ L([a, d], [b, d]) and (p_{1})_{1} (p_{2})_{1}, or (ii) p_{2} ∈ L([b, d], [b, c]), or (iii) p_{2} ∈ L([b, c], [a, c]) and p_{2} ≠ W-minK.

(71) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([b, d], [b, c]). Then LE(p_{1}, p_{2}, K) if and only if one of the following conditions is satisfied:

(i) p_{2} ∈ L([b, d], [b, c]) and (p_{1})_{2} (p_{2})_{2}, or (ii) p_{2} ∈ L([b, c], [a, c]) and p_{2} ≠ W-minK.

(72) Let K be a non empty compact subset of E_{T}^{2}, a, b, c, d be real numbers, and p_{1}, p_{2} be points of E_{T}^{2}. Suppose K = Rectangle(a, b, c, d) and a < b and c < d and p_{1} ∈ L([b, c], [a, c]) and p_{1} ≠ W-minK. Then LE(p_{1}, p_{2}, K) if and only if the following conditions are satisfied:

(i) p_{2} ∈ L([b, c], [a, c]), (ii) (p_{1})_{1} (p_{2})_{1}, and (iii) p_{2} ≠ W-minK.

(73) Let x be a set and a, b, c, d be real numbers. Suppose x ∈ Rectangle(a, b, c, d) and a < b and c < d. Then x ∈ L([a, c], [a, d]) or x ∈ L([a, d], [b, d]) or x ∈ L([b, d], [b, c]) or x ∈ L([b, c], [a, c]).

5. General Fashoda Theorem for Square

The following propositions are true:

(74) Let p_{1}, p_{2} be points of E_{T}^{2} and K be a non empty compact subset of E_{T}^{2}. Suppose K = Rectangle(− 1, 1, − 1, 1) and LE(p_{1}, p_{2}, K) and p_{1} ∈ L([− 1, − 1], [− 1, 1]). Then p_{2} ∈ L([− 1, − 1], [− 1, 1]) and (p_{2})_{2} (p_{1})_{2} or p_{2} ∈ L([− 1, 1], [1, 1]) or p_{2} ∈ L([1, 1], [1, − 1]) or p_{2} ∈ L([1, − 1], [− 1, − 1]) and p_{2} ≠ [− 1, − 1].

(75) Let p_{1}, p_{2} be points of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc and p_{1} ∈ L([− 1, − 1], [− 1, 1]) and (p_{1})_{2} 0 and LE(p_{1}, p_{2}, K). Then LE(f(p_{1}), f(p_{2}), P).

(76) Let p_{1}, p_{2}, p_{3} be points of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc and p_{1} ∈ L([− 1, − 1], [− 1, 1]) and (p_{1})_{2} 0 and LE(p_{1}, p_{2}, K) and LE(p_{2}, p_{3}, K). Then LE(f(p_{2}), f(p_{3}), P).

(77) Let p be a point of E_{T}^{2} and f be a map from E_{T}^{2} into E_{T}^{2}. If f = SqCirc and p_{1} = − 1 and p_{2} < 0, then f(p)_{1} < 0 and f(p)_{2} < 0.

(78) Let p be a point of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. If P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc, then f(p)_{1} 0 iff p_{1} 0.

(79) Let p be a point of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. If P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc, then f(p)_{2} 0 iff p_{2} 0.

(80) Let p, q be points of E_{T}^{2} and f be a map from E_{T}^{2} into E_{T}^{2}. If f = SqCirc and p ∈ L([− 1, − 1], [− 1, 1]) and q ∈ L([1, − 1], [− 1, − 1]), then f(p)_{1} f(q)_{1}.

(81) Let p, q be points of E_{T}^{2} and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose f = SqCirc and p ∈ L([− 1, − 1], [− 1, 1]) and q ∈ L([− 1, − 1], [− 1, 1]) and p_{2} q_{2} and p_{2} < 0. Then f(p)_{2} f(q)_{2}.

(82) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc. Suppose LE(p_{1}, p_{2}, K) and LE(p_{2}, p_{3}, K) and LE(p_{3}, p_{4}, K). Then f(p_{1}), f(p_{2}), f(p_{3}), f(p_{4}) are in this order on P.

(83) Let p_{1}, p_{2} be points of E_{T}^{2} and P be a non empty compact subset of E_{T}^{2}. If P is a simple closed curve and p_{1} ∈ P and p_{2} ∈ P and not LE(p_{1}, p_{2}, P), then LE(p_{2}, p_{1}, P).

(84) Let p_{1}, p_{2}, p_{3} be points of E_{T}^{2} and P be a non empty compact subset of E_{T}^{2}. Suppose P is a simple closed curve and p_{1} ∈ P and p_{2} ∈ P and p_{3} ∈ P. Then LE(p_{1}, p_{2}, P) and LE(p_{2}, p_{3}, P) or LE(p_{1}, p_{3}, P) and LE(p_{3}, p_{2}, P) or LE(p_{2}, p_{1}, P) and LE(p_{1}, p_{3}, P) or LE(p_{2}, p_{3}, P) and LE(p_{3}, p_{1}, P) or LE(p_{3}, p_{1}, P) and LE(p_{1}, p_{2}, P) or LE(p_{3}, p_{2}, P) and LE(p_{2}, p_{1}, P).

(85) Let p_{1}, p_{2}, p_{3} be points of E_{T}^{2} and P be a non empty compact subset of E_{T}^{2}. Suppose P is a simple closed curve and p_{1} ∈ P and p_{2} ∈ P and p_{3} ∈ P and LE(p_{2}, p_{3}, P). Then LE(p_{1}, p_{2}, P) or LE(p_{2}, p_{1}, P) and LE(p_{1}, p_{3}, P) or LE(p_{3}, p_{1}, P).

(86) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2} and P be a non empty compact subset of E_{T}^{2}. Suppose P is a simple closed curve and p_{1} ∈ P and p_{2} ∈ P and p_{3} ∈ P and p_{4} ∈ P and LE(p_{2}, p_{3}, P) and LE(p_{3}, p_{4}, P). Then LE(p_{1}, p_{2}, P) or LE(p_{2}, p_{1}, P) and LE(p_{1}, p_{3}, P) or LE(p_{3}, p_{1}, P) and LE(p_{1}, p_{4}, P) or LE(p_{4}, p_{1}, P).

(87) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc and LE(f(p_{1}), f(p_{2}), P) and LE(f(p_{2}), f(p_{3}), P) and LE(f(p_{3}), f(p_{4}), P). Then p_{1}, p_{2}, p_{3}, p_{4} are in this order on K.

(88) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, P, K be non empty compact subsets of E_{T}^{2}, and f be a map from E_{T}^{2} into E_{T}^{2}. Suppose P = Circle(0, 0, 1) and K = Rectangle(− 1, 1, − 1, 1) and f = SqCirc. Then p_{1}, p_{2}, p_{3}, p_{4} are in this order on K if and only if f(p_{1}), f(p_{2}), f(p_{3}), f(p_{4}) are in this order on P.

(89) Let p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{2}, K be a compact non empty subset of E_{T}^{2}, and K_{0} be a subset of E_{T}^{2}. Suppose K = Rectangle(− 1, 1, − 1, 1) and p_{1}, p_{2}, p_{3}, p_{4} are in this order on K. Let f, g be maps from 𝕀 into E_{T}^{2}. Suppose that f is continuous and one-to-one and g is continuous and oneto-one and K_{0} = ClosedInsideOfRectangle(− 1, 1, − 1, 1) and f(0) = p_{1} and f(1) = p_{3} and g(0) = p_{2} and g(1) = p_{4} and rngf ⊆ K_{0} and rngg ⊆ K_{0}.

224 yatsuka nakamura

Then rngf meets rngg.

References

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

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

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

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

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

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

[9] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

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

[11] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Simple closed curves. Formalized Mathematics, 2(5):663–664, 1991.

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

[13] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[14] Artur Korniłowicz. The ordering of points on a curve. Part III. Formalized Mathematics, 10(3):169–171, 2002.

[15] Yatsuka Nakamura. On Outside Fashoda Meet Theorem. Formalized Mathematics, 9(4):697–704, 2001.

[16] Yatsuka Nakamura. On the simple closed curve property of the circle and the Fashoda Meet Theorem. Formalized Mathematics, 9(4):801–808, 2001.

[17] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of a simple closed curves and the order of their points. Formalized Mathematics, 6(4):563–572, 1997.

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

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

[20] Agnieszka Sakowicz, Jarosław Gryko, and Adam Grabowski. Sequences in E_{T}^{N}. Formalized Mathematics, 5(1):93–96, 1996.

[21] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

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

On Some Properties of Real Hilbert Space. Part I

Hiroshi Yamazaki Shinshu University Nagano Yasumasa Suzuki Take, Yokosuka-shi Japan

Takao Inoué The Iida Technical High School Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly summable ones are proved. In the last part of the paper, we give a necessary and sufficient condition for summability of an infinite set of vectors of real Hilbert space as our main theorem. The last theorem is due to [8].

MML Identifier: BHSP 6.

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

1. Preliminaries

In this paper X is a real unitary space, x is a point of X, and i is a natural number.

Let us consider X. Let us assume that the addition of X is commutative and associative and the addition of X has a unity. Let Y be a finite subset of the carrier of X. The functor Setsum(Y) yielding an element of the carrier of X is defined by the condition (Def. 1).

(Def. 1) There exists a finite sequence p of elements of the carrier of X such that

p is one-to-one and rngp = Y and Setsum(Y) = the addition of X ⊙ p. We now state two propositions:

(1) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let Y be a finite subset of the carrier of X and I be a function from the carrier of X into the carrier of X. Suppose Y ⊆ domI and for every set x such that x ∈ domI holds I(x) = x. Then Setsum(Y) = setopfunc(Y, the carrier of X, the carrier of X, I, the addition of X).

(2) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let Y_{1}, Y_{2} be finite subsets of the carrier of X. Suppose Y_{1} misses Y_{2}. Let Z be a finite subset of the carrier of X. If Z = Y_{1} ∪ Y_{2}, then Setsum(Z) = Setsum(Y_{1}) + Setsum(Y_{2}).

2. Summability

Let us consider X and let Y be a subset of the carrier of X. We say that Y is summable set if and only if the condition (Def. 2) is satisfied.

(Def. 2) There exists x such that for every real number e if e > 0, then there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and Y_{1} ⊆ Y holds ∥x − Setsum(Y_{1})∥ < e.

Let us consider X and let Y be a subset of the carrier of X. Let us assume that Y is summable set. The functor sumY yielding a point of X is defined by the condition (Def. 3).

(Def. 3) Let e be a real number. Suppose e > 0. Then there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and Y_{1} ⊆ Y holds ∥sumY − Setsum(Y_{1})∥ < e.

Let us consider X and let L be a linear functional in X. We say that L is Bounded if and only if:

(Def. 4) There exists a real number K such that K > 0 and for every x holds

|L(x)| K · ∥x∥. Let us consider X and let Y be a subset of the carrier of X. We say that Y

is weakly summable set if and only if the condition (Def. 5) is satisfied. (Def. 5) There exists x such that for every linear functional L in X if L is Bo-

unded, then for every real number e such that e > 0 there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and Y_{1} ⊆ Y holds |L(x − Setsum(Y_{1}))| < e.

Let us consider X, let Y be a subset of the carrier of X, and let L be a functional in X. We say that Y is summable set by L if and only if the condition (Def. 6) is satisfied.

(Def. 6) There exists a real number r such that for every real number e if e > 0, then there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and Y_{1} ⊆ Y holds |r − setopfunc(Y_{1}, the carrier of X, ℝ, L, +_{ℝ})| < e.

Let us consider X, let Y be a subset of the carrier of X, and let L be a functional in X. Let us assume that Y is summable set by L. The functor SumByfunc(Y, L) yielding a real number is defined by the condition (Def. 7).

(Def. 7) Let e be a real number. Suppose e > 0. Then there exists a finite subset

Y_{0} of the carrier of X such that

(i) Y_{0} is non empty, (ii) Y_{0} ⊆ Y, and (iii) for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and Y_{1} ⊆

Y holds |SumByfunc(Y, L)−setopfunc(Y_{1}, the carrier of X, ℝ, L, +_{ℝ})| < e. The following propositions are true:

(3) For every subset Y of the carrier of X such that Y is summable set holds Y is weakly summable set.

(4) Let L be a linear functional in X and p be a finite sequence of elements of the carrier of X. Suppose lenp 1. Let q be a finite sequence of elements of ℝ. Suppose domp = domq and for every i such that i ∈ domq holds q(i) = L(p(i)). Then L(the addition of X ⊙ p) = +_{ℝ} ⊙ q.

(5) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let S be a finite subset of the carrier of X. Suppose S is non empty. Let L be a linear functional in X. Then L(Setsum(S)) = setopfunc(S, the carrier of X, ℝ, L, +_{ℝ}).

(6) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let Y be a subset of the carrier of X. Suppose Y is weakly summable set. Then there exists x such that for every linear functional L in X if L is Bounded, then for every real number e such that e > 0 there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and Y_{1} ⊆ Y holds |L(x)−setopfunc(Y_{1}, the carrier of X, ℝ, L, +_{ℝ})| < e.

(7) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let Y be a subset of the carrier of X. Suppose Y is weakly summable set. Let L be a linear functional in X. If L is Bounded, then Y is summable set by L.

228 hiroshi yamazaki et al.

(8) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let Y be a subset of the carrier of X. Suppose Y is summable set. Let L be a linear functional in X. If L is Bounded, then Y is summable set by L.

(9) For every finite subset Y of the carrier of X such that Y is non empty holds Y is summable set.

3. Necessary and Sufficient Condition for Summability

One can prove the following proposition

(10) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity and X is a Hilbert space. Let Y be a subset of the carrier of X. Then Y is summable set if and only if for every real number e such that e > 0 there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{1} is non empty and Y_{1} ⊆ Y and Y_{0} misses Y_{1} holds ∥Setsum(Y_{1})∥ < e.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 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] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

[8] P. R. Halmos. Introduction to Hilbert Space. American Mathematical Society, 1987.

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

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

[11] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.

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

[13] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[14] Jan Popiołek. Introduction to Banach and Hilbert spaces - part III. Formalized Mathematics, 2(4):523–526, 1991.

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

[16] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

[17] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

on some properties of real hilbert space. … 229

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

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

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

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

[23] Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Bessel’s inequality. Formalized Mathematics, 11(2):169–173, 2003.

Full Subtracter Circuit. Part II

Shin’nosuke Yamaguchi Shinshu University Nagano

Grzegorz Bancerek Białystok Technical University

Katsumi Wasaki Shinshu University Nagano

Summary. In this article we continue investigations from [22] of verification of a design of subtracter circuit. We define it as a combination of multi cell circuit using schemes from [6]. As the main result we prove the stability of the circuit.

MML Identifier: FSCIRC 2.

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

Let n be a natural number and let x, y be finite sequences. The functor n-BitSubtracterStr(x, y) yields an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates and is defined by the condition (Def. 1).

(Def. 1) There exist many sorted sets f, g indexed by ℕ such that (i) n-BitSubtracterStr(x, y) = f(n), (ii) f(0) = 1GateCircStr(ε, Boolean^{0} ⟼→ true), (iii) g(0) = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩, and (iv) for every natural number n and for every non empty many sorted

signature S and for every set z such that S = f(n) and z = g(n) holds f(n + 1) = S+·BitSubtracterWithBorrowStr(x(n + 1), y(n + 1), z) and g(n + 1) = BorrowOutput(x(n + 1), y(n + 1), z).

Let n be a natural number and let x, y be finite sequences. The functor n-BitSubtracterCirc(x, y) yielding a Boolean strict circuit of n-BitSubtracterStr(x, y) with denotation held in gates is defined by the condition (Def. 2).

(Def. 2) There exist many sorted sets f, g, h indexed by ℕ such that (i) n-BitSubtracterStr(x, y) = f(n), (ii) n-BitSubtracterCirc(x, y) = g(n), (iii) f(0) = 1GateCircStr(ε, Boolean^{0} ⟼→ true), (iv) g(0) = 1GateCircuit(ε, Boolean^{0} ⟼→ true),

(v) h(0) = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩, and (vi) for every natural number n and for every non empty many sorted

signature S and for every non-empty algebra A over S and for every set z such that S = f(n) and A = g(n) and z = h(n) holds f(n + 1) = S+·BitSubtracterWithBorrowStr(x(n + 1), y(n + 1), z) and g(n + 1) = A+·BitSubtracterWithBorrowCirc(x(n + 1), y(n + 1), z) and h(n + 1) = BorrowOutput(x(n + 1), y(n + 1), z).

Let n be a natural number and let x, y be finite sequences. The functor n-BitBorrowOutput(x, y) yields an element of InnerVertices(n-BitSubtracterStr

(x, y)) and is defined by the condition (Def. 3). (Def. 3) There exists a many sorted set h indexed by ℕ such that

(i) n-BitBorrowOutput(x, y) = h(n), (ii) h(0) = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩, and (iii) for every natural number n and for every set z such that z = h(n) holds

h(n + 1) = BorrowOutput(x(n + 1), y(n + 1), z). One can prove the following propositions:

(1) Let x, y be finite sequences and f, g, h be many sorted sets indexed by ℕ. Suppose that

(i) f(0) = 1GateCircStr(ε, Boolean^{0} ⟼→ true), (ii) g(0) = 1GateCircuit(ε, Boolean^{0} ⟼→ true), (iii) h(0) = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩, and (iv) for every natural number n and for every non empty many sorted

signature S and for every non-empty algebra A over S and for every set z such that S = f(n) and A = g(n) and z = h(n) holds f(n + 1) = S+·BitSubtracterWithBorrowStr(x(n + 1), y(n + 1), z) and g(n + 1) = A+·BitSubtracterWithBorrowCirc(x(n + 1), y(n + 1), z) and h(n + 1) = BorrowOutput(x(n + 1), y(n + 1), z). Let n be a natural number. Then n-BitSubtracterStr(x, y) = f(n) and n-BitSubtracterCirc(x, y) = g(n) and n-BitBorrowOutput(x, y) = h(n).

(2) For all finite sequences a, b holds 0-BitSubtracterStr(a, b) = 1GateCircStr(ε, Boolean^{0} ⟼→ true) and 0-BitSubtracterCirc(a, b) = 1GateCircuit(ε, Boolean^{0} ⟼→ true) and 0-BitBorrowOutput(a, b) = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩.

(3) Let a, b be finite sequences and c be a set. Suppose c = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩. Then 1-BitSubtracterStr(a, b) = 1GateCircStr(ε, Boolean^{0} ⟼→ true)+·BitSubtracterWithBorrowStr(a(1), b(1), c) and 1-BitSubtracterCirc(a, b) = 1GateCircuit(ε, Boolean^{0} ⟼→ true)+· BitSubtracterWithBorrowCirc(a(1), b(1), c) and 1-BitBorrowOutput(a, b) = BorrowOutput(a(1), b(1), c).

(4) For all sets a, b, c such that c = ⟨⟨ε, Boolean^{0} ⟼→ true⟩⟩ holds 1-BitSubtracterStr(⟨a⟩, ⟨b⟩) = 1GateCircStr(ε, Boolean^{0} ⟼→ true)+· BitSubtracterWithBorrowStr(a, b, c) and 1-BitSubtracterCirc(⟨a⟩, ⟨b⟩) = 1GateCircuit(ε, Boolean^{0} ⟼→ true)+·BitSubtracterWithBorrowCirc(a, b, c) and 1-BitBorrowOutput(⟨a⟩, ⟨b⟩) = BorrowOutput(a, b, c).

(5) Let n be a natural number, p, q be finite sequences with length n, and p_{1}, p_{2}, q_{1}, q_{2} be finite sequences. Then n-BitSubtracterStr(p^{⌢} p_{1}, q^{⌢} q_{1}) = n-BitSubtracterStr(p^{⌢} p_{2}, q^{⌢} q_{2}) and n-BitSubtracterCirc(p^{⌢} p_{1}, q^{⌢} q_{1}) = n-BitSubtracterCirc(p^{⌢}p_{2}, q^{⌢}q_{2}) and n-BitBorrowOutput(p^{⌢}p_{1}, q^{⌢}q_{1}) = n-BitBorrowOutput(p^{⌢} p_{2}, q^{⌢} q_{2}).

(6) Let n be a natural number, x, y be finite sequences with length n, and a, b be sets. Then (n+1)-BitSubtracterStr(x^{⌢}⟨a⟩, y^{⌢}⟨b⟩) = (n-BitSubtracterStr(x, y))+· BitSubtracterWithBorrowStr(a, b, n-BitBorrowOutput(x, y)) and (n + 1)-BitSubtracterCirc(x^{⌢} ⟨a⟩, y^{⌢} ⟨b⟩) = (n-BitSubtracterCirc(x, y))+· BitSubtracterWithBorrowCirc(a, b, n-BitBorrowOutput(x, y)) and (n + 1)-BitBorrowOutput(x^{⌢} ⟨a⟩, y^{⌢} ⟨b⟩) = BorrowOutput(a, b, n-BitBorrowOutput(x, y)).

(7) Let n be a natural number and x, y be finite sequences. Then (n + 1)-BitSubtracterStr(x, y) = (n-BitSubtracterStr(x, y))+·BitSubtracterWithBorrowStr(x(n+1), y(n+ 1), n-BitBorrowOutput(x, y)) and (n + 1)-BitSubtracterCirc(x, y) = (n-BitSubtracterCirc(x, y))+·BitSubtracterWithBorrowCirc(x(n+1), y(n+ 1), n-BitBorrowOutput(x, y)) and (n + 1)-BitBorrowOutput(x, y) = BorrowOutput(x(n + 1), y(n + 1), n-BitBorrowOutput(x, y)).

(8) For all natural numbers n, m such that n m and for all finite sequences x, y holds InnerVertices(n-BitSubtracterStr(x, y)) ⊆ InnerVertices(m-BitSubtracterStr(x, y)).

(9) For every natural number n and for all finite sequences x, y holds InnerVertices((n + 1)-BitSubtracterStr(x, y)) = InnerVertices(n-BitSubtracterStr(x, y)) ∪ InnerVertices (BitSubtracterWithBorrowStr(x(n+1), y(n+1), n-BitBorrowOutput(x, y))).

Let k, n be natural numbers. Let us assume that k 1 and k n. Let x, y be finite sequences. The functor (k, n)-BitSubtracterOutput(x, y) yielding an element of InnerVertices(n-BitSubtracterStr(x, y)) is defined by:

(Def. 4) There exists a natural number i such that k = i + 1 and (k, n)-BitSubtracterOutput(x, y) = BitSubtracterOutput(x(k), y(k), i-BitBorrowOutput(x, y)).

One can prove the following propositions:

(10) For all natural numbers n, k such that k < n and for all finite sequences x, y holds (k + 1, n)-BitSubtracterOutput(x, y) = BitSubtracterOutput(x(k + 1), y(k + 1), k-BitBorrowOutput(x, y)).

(11) For every natural number n and for all finite sequences x, y holds InnerVertices(n-BitSubtracterStr(x, y)) is a binary relation.

(12) For all sets x, y, c holds InnerVertices(BorrowIStr(x, y, c)) = {⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩}.

(13) For all sets x, y, c such that x ≠ ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩and c ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩holds InputVertices(BorrowIStr(x, y, c)) = {x, y, c}.

(14) For all sets x, y, c holds InnerVertices(BorrowStr(x, y, c)) = {⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩} ∪ {BorrowOutput(x, y, c)}.

(15) For all sets x, y, c such that x ≠ ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ holds InputVertices(BorrowStr(x, y, c)) = {x, y, c}.

(16) For all sets x, y, c such that x ≠ ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, xor ⟩⟩ holds InputVertices(BitSubtracterWithBorrowStr(x, y, c)) = {x, y, c}.

(17) For all sets x, y, c holds InnerVertices(BitSubtracterWithBorrowStr(x, y, c)) = {⟨⟨⟨x, y⟩, xor ⟩⟩, 2GatesCircOutput(x, y, c, xor)}∪{⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩} ∪ {BorrowOutput(x, y, c)}.

Let n be a natural number and let x, y be finite sequences. Observe that n-BitBorrowOutput(x, y) is pair.

The following propositions are true:

(18) Let x, y be finite sequences and n be a natural number. Then (n-BitBorrowOutput(x, y))_{1} = ε and (n-BitBorrowOutput(x, y))_{2} = Boolean^{0} ⟼→ true and π_{1}((n-BitBorrowOutput(x, y))_{2}) = Boolean^{0} or \overline{\overline{(n-BitBorrowOutput(x, y))_{1}}} = 3 and (n-BitBorrowOutput(x, y))_{2} = or_{3} and π_{1}((n-BitBorrowOutput(x, y))_{2}) = Boolean^{3} .

(19) Let n be a natural number, x, y be finite sequences, and p be a set. Then n-BitBorrowOutput(x, y) ≠ ⟨⟨p, and_{2} ⟩⟩ and n-BitBorrowOutput(x, y) ≠ ⟨⟨p, and_{2a} ⟩⟩ and n-BitBorrowOutput(x, y) ≠ ⟨⟨p, xor ⟩⟩.

(20) Let f, g be nonpair yielding finite sequences and n be a natural number. Then InputVertices((n + 1)-BitSubtracterStr(f, g)) = InputVertices(n-BitSubtracterStr(f, g)) ∪ (InputVertices (BitSubtracterWithBorrowStr(f(n+1), g(n+1), n-BitBorrowOutput(f, g)))\ {n-BitBorrowOutput(f, g)}) and InnerVertices(n-BitSubtracterStr(f, g)) is a binary relation and InputVertices(n-BitSubtracterStr(f, g)) has no pairs.

(21) For every natural number n and for all nonpair yielding finite sequences x, y with length n holds InputVertices(n-BitSubtracterStr(x, y)) = rngx∪ rngy.

(22) Let x, y, c be sets, s be a state of BorrowCirc(x, y, c), and a_{1}, a_{2}, a_{3} be elements of Boolean. If a_{1} = s(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) and a_{2} = s(⟨⟨⟨y, c⟩, and_{2} ⟩⟩) and a_{3} = s(⟨⟨⟨x, c⟩, and_{2a} ⟩⟩), then (Following(s))(BorrowOutput(x, y, c)) = a_{1} ∨ a_{2} ∨ a_{3}.

(23) Let x, y, c be sets. Suppose x ≠ ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, xor ⟩⟩. Let s be a state of BorrowCirc(x, y, c). Then Following(s, 2) is stable.

(24) Let x, y, c be sets. Suppose x ≠ ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, xor ⟩⟩. Let s be a state of BitSubtracterWithBorrowCirc(x, y, c) and a_{1}, a_{2}, a_{3} be elements of Boolean. Suppose a_{1} = s(x) and a_{2} = s(y) and a_{3} = s(c). Then (Following(s, 2))(BitSubtracterOutput(x, y, c)) = a_{1} ⊕ a_{2} ⊕ a_{3} and (Following(s, 2))(BorrowOutput(x, y, c)) = ¬a_{1} ∧ a_{2} ∨ a_{2} ∧ a_{3} ∨ ¬a_{1} ∧ a_{3}.

(25) Let x, y, c be sets. Suppose x ≠ ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ and y ≠ ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ and c ≠ ⟨⟨⟨x, y⟩, xor ⟩⟩. Let s be a state of BitSubtracterWithBorrowCirc(x, y, c). Then Following(s, 2) is stable.

(26) Let n be a natural number, x, y be nonpair yielding finite sequences with length n, and s be a state of n-BitSubtracterCirc(x, y). Then Following(s, 1 + 2 · n) is stable.

References

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

[2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537– 541, 1990.

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

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

[5] Grzegorz Bancerek and Yatsuka Nakamura. Full adder circuit. Part I. Formalized Mathematics, 5(3):367–380, 1996.

[6] Grzegorz Bancerek, Shin’nosuke Yamaguchi, and Yasunari Shidama. Combining of multi cell circuits. Formalized Mathematics, 10(1):47–64, 2002.

[7] Grzegorz Bancerek, Shin’nosuke Yamaguchi, and Katsumi Wasaki. Full adder circuit. Part II. Formalized Mathematics, 10(1):65–71, 2002.

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

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

236 shin’nosuke yamaguchi et al.

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

[11] Yatsuka Nakamura and Grzegorz Bancerek. Combining of circuits. Formalized Mathematics, 5(2):283–295, 1996.

[12] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Introduction to circuits, II. Formalized Mathematics, 5(2):273–278, 1996.

[13] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, II. Formalized Mathematics, 5(2):215–220, 1996.

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

[15] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

[20] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37–42, 1996.

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

[22] Katsumi Wasaki and Noboru Endou. Full subtracter circuit. Part I. Formalized Mathematics, 8(1):77–81, 1999.

[23] Katsumi Wasaki and Pauline N. Kawamoto. 2’s complement circuit. Formalized Mathematics, 6(2):189–197, 1997.

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

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

Dijkstra’s Shortest Path Algorithm

Jing-Chao Chen Donghua University Shanghai

Summary. The article formalizes Dijkstra’s shortest path algorithm [11]. A path from a source vertex v to a target vertex u is said to be the shortest path if its total cost is minimum among all v-to-u paths. Dijkstra’s algorithm is based on the following assumptions:

• All edge costs are non-negative.

• The number of vertices is finite.

• The source is a single vertex, but the target may be all other vertices.

The underlying principle of the algorithm may be described as follows: the algorithm starts with the source; it visits the vertices in order of increasing cost, and maintains a set V of visited vertices (denoted by UsedVx in the article) whose cost from the source has been computed, and a tentative cost D(u) to each unvisited vertex u. In the article, the set of all unvisited vertices is denoted by UnusedVx. D(u) is the cost of the shortest path from the source to u in the subgraph induced by V ∪ {u}. We denote the set of all unvisited vertices whose D-values are not infinite (i.e. in the subgraph each of which has a path from the source to itself) by OuterVx. Dijkstra’s algorithm repeatedly searches OuterVx for the vertex with minimum tentative cost (this procedure is called findmin in the article), adds it to the set V and modifies D-values by a procedure, called Relax. Suppose the unvisited vertex with minimum tentative cost is x, the procedure Relax replaces D(u) with min{D(u), D(u) + cost(x, u)} where u is a vertex in UnusedVx, and cost(x, u) is the cost of edge (x, u). In the Mizar library, there are several computer models, e.g. SCMFSA and SCMPDS etc. However, it is extremely difficult to use these models to formalize the algorithm. Instead, we adopt functions in the Mizar library, which seem to be pseudo-codes, and are similar to those in the functional programming language, e.g. Lisp. To date, there is no rigorous justification with respect to the correctness of Dijkstra’s algorithm. The article presents first the rigorous justification.

MML Identifier: GRAPHSP.

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

1. Preliminaries

For simplicity, we adopt the following rules: X denotes a set, i, j, k, m, n denote natural numbers, p denotes a finite sequence of elements of X, and i_{1} denotes an integer.

We now state three propositions:

(1) For every finite sequence p and for every set x holds x ∉ rngp and p is one-to-one iff p^{⌢} ⟨x⟩ is one-to-one.

(2) If 1 i_{1} and i_{1} lenp, then p(i_{1}) ∈ X.

(3) If 1 i_{1} and i_{1} lenp, then p_{i_{1}} = p(i_{1}).

For simplicity, we adopt the following rules: G denotes a graph, p_{1}, q_{1} denote finite sequences of elements of the edges of G, p, q denote oriented chains of G, W denotes a function, U, V, e, e_{1} denote sets, and v_{1}, v_{2}, v_{3}, v_{4} denote vertices of G.

We now state three propositions:

(4) If W is weight of G and lenp_{1} = 1, then cost(p_{1}, W) = W (p_{1}(1)).

(5) If e ∈ the edges of G, then ⟨e⟩ is a Simple oriented chain of G.

(6) Let p be a Simple oriented chain of G. Suppose p = p_{1} ^{⌢} q_{1} and lenp_{1} 1 and lenq_{1} 1. Then (the target of G)(p(lenp)) ≠ (the target of G)(p_{1}(lenp_{1})) and (the source of G)(p(1)) ≠ (the source of G)(q_{1}(1)).

2. The Fundamental Properties of Directed Paths and Shortest Paths

We now state several propositions:

(7) p is oriented path from v_{1} to v_{2} in V iff p is oriented path from v_{1} to v_{2} in V ∪ {v_{2}}.

(8) p is shortest path from v_{1} to v_{2} in V w.r.t. W iff p is shortest path from v_{1} to v_{2} in V ∪ {v_{2}} w.r.t. W.

(9) Suppose p is shortest path from v_{1} to v_{2} in V w.r.t. W and q is shortest path from v_{1} to v_{2} in V w.r.t. W. Then cost(p, W) = cost(q, W).

(10) Let G be an oriented graph, v_{1}, v_{2} be vertices of G, and e_{2}, e_{3} be sets. Suppose e_{2} ∈ the edges of G and e_{3} ∈ the edges of G and e_{2} orientedly joins v_{1}, v_{2} and e_{3} orientedly joins v_{1}, v_{2}. Then e_{2} = e_{3}.

(11) Suppose that (i) the vertices of G = U ∪ V, (ii) v_{1} ∈ U, (iii) v_{2} ∈ V, and (iv) for all v_{3}, v_{4} such that v_{3} ∈ U and v_{4} ∈ V it is not true that there

exists e such that e ∈ the edges of G and e orientedly joins v_{3}, v_{4}. Then there exists no p which is oriented path from v_{1} to v_{2}.

(12) Suppose that (i) the vertices of G = U ∪ V, (ii) v_{1} ∈ U, (iii) for all v_{3}, v_{4} such that v_{3} ∈ U and v_{4} ∈ V it is not true that there

exists e such that e ∈ the edges of G and e orientedly joins v_{3}, v_{4}, and (iv) p is oriented path from v_{1} to v_{2}.

Then p is oriented path from v_{1} to v_{2} in U.

3. The Basic Theorems for Dijkstra’s Shortest Path Algorithm (continue)

We adopt the following convention: G is a finite graph, P, Q are oriented chains of G, and v_{1}, v_{2}, v_{3} are vertices of G.

Next we state the proposition

(13) Suppose that W is nonnegative weight of G and P is shortest path from v_{1} to v_{2} in V w.r.t. W and v_{1} ≠ v_{2} and v_{1} ≠ v_{3} and Q is shortest path from v_{1} to v_{3} in V w.r.t. W and it is not true that there exists e such that e ∈ the edges of G and e orientedly joins v_{2}, v_{3} and P is longest in shortest path from v_{1} in V w.r.t. W. Then Q is shortest path from v_{1} to v_{3} in V ∪ {v_{2}} w.r.t. W.

For simplicity, we adopt the following rules: G is a finite oriented graph, P, Q are oriented chains of G, W is a function from the edges of G into ℝ_{0} , and v_{1}, v_{2}, v_{3}, v_{4} are vertices of G.

One can prove the following three propositions:

(14) Suppose e ∈ the edges of G and v_{1} ≠ v_{2} and P = ⟨e⟩ and e orientedly joins v_{1}, v_{2}. Then P is shortest path from v_{1} to v_{2} in {v_{1}} w.r.t. W.

(15) Suppose that e ∈ the edges of G and P is shortest path from v_{1} to v_{2} in V w.r.t. W and v_{1} ≠ v_{3} and Q = P ^{⌢} ⟨e⟩ and e orientedly joins v_{2}, v_{3} and v_{1} ∈ V and for every v_{4} such that v_{4} ∈ V it is not true that there exists e_{1} such that e_{1} ∈ the edges of G and e_{1} orientedly joins v_{4}, v_{3}. Then Q is shortest path from v_{1} to v_{3} in V ∪ {v_{2}} w.r.t. W.

(16) Suppose that (i) the vertices of G = U ∪ V, (ii) v_{1} ∈ U, and (iii) for all v_{3}, v_{4} such that v_{3} ∈ U and v_{4} ∈ V it is not true that there

exists e such that e ∈ the edges of G and e orientedly joins v_{3}, v_{4}.

Then P is shortest path from v_{1} to v_{2} in U w.r.t. W if and only if P is shortest path from v_{1} to v_{2} in W.

4. The Definition of Assignment Statement

Let f be a function and let i, x be sets. We introduce f_{i}:=x as a synonym of f +·(i, x).

We now state the proposition

(17) For all sets x, y and for every function f holds rng(f_{x}:=y) ⊆ rngf ∪{y}. Let f be a finite sequence of elements of ℝ, let x be a set, and let r be a real

number. Then f_{x}:=r is a finite sequence of elements of ℝ.

Let i, k be natural numbers, let f be a finite sequence of elements of ℝ, and let r be a real number. The functor (f, i) := (k, r) yielding a finite sequence of elements of ℝ is defined by:

(Def. 1) (f, i) := (k, r) = f_{i}:=k_{k}:=r.

In the sequel f, g, h denote elements of ℝ^{∗} and r denotes a real number. One can prove the following propositions:

(18) If i ≠ k and i ∈ domf, then ((f, i) := (k, r))(i) = k.

(19) If m ≠ i and m ≠ k and m ∈ domf, then ((f, i) := (k, r))(m) = f(m).

(20) If k ∈ domf, then ((f, i) := (k, r))(k) = r.

(21) dom((f, i) := (k, r)) = domf.

5. The Definition of Pascal–Like “while” - “do” Statement

Let X be a set. Then id_{X} is an element of X^{X}. Let X be a set and let f, g be functions from X into X. Then g · f is a

function from X into X.

Let X be a set and let f, g be elements of X^{X}. Then g · f is an element of X^{X}.

Let X be a set, let f be an element of X^{X}, and let g be an element of X. Then f(g) is an element of X.

Let X be a set and let f be an element of X^{X}. The functor repeatf yields a function from ℕ into X^{X} and is defined by:

(Def. 2) (repeatf)(0) = id_{X} and for every natural number i and for every element

x of X^{X} such that x = (repeatf)(i) holds (repeatf)(i + 1) = f · x. Next we state two propositions:

(22) For every element F of (ℝ^{∗})^{ℝ^{∗}} and for every element f of ℝ^{∗} and for all natural numbers n, i holds (repeatF)(0)(f) = f.

(23) Let F, G be elements of (ℝ^{∗})^{ℝ^{∗}}, f be an element of ℝ^{∗}, and i be a natural

number. Then (repeat(F · G))(i + 1)(f) = F (G((repeat(F · G))(i)(f))). Let g be an element of (ℝ^{∗})^{ℝ^{∗}} and let f be an element of ℝ^{∗}. Then g(f) is

an element of ℝ^{∗}.

Let f be an element of ℝ^{∗} and let n be a natural number. The functor OuterVx(f, n) yielding a subset of ℕ is defined by:

(Def. 3) OuterVx(f, n) = {i : i ∈ domf ∧ 1 i ∧ i n ∧ f(i) ≠ − 1 ∧ f(n+i) ≠

− 1}. Let f be an element of (ℝ^{∗})^{ℝ^{∗}}, let g be an element of ℝ^{∗}, and let

n be a natural number. Let us assume that there exists i such that OuterVx((repeatf)(i)(g), n) = ∅. The functor LifeSpan(f, g, n) yielding a natural number is defined by:

(Def. 4) OuterVx((repeatf)(LifeSpan(f, g, n))(g), n) = ∅ and for every natural number k such that OuterVx((repeatf)(k)(g), n) = ∅ holds LifeSpan(f, g, n) k.

Let f be an element of (ℝ^{∗})^{ℝ^{∗}} and let n be a natural number. The functor WhileDo(f, n) yielding an element of (ℝ^{∗})^{ℝ^{∗}} is defined as follows:

(Def. 5) domWhileDo(f, n) = ℝ^{∗} and for every element h of ℝ^{∗} holds (WhileDo(f, n))(h) = (repeatf)(LifeSpan(f, h, n))(h).

6. Defining a Weight Function for an Oriented Graph

Let G be an oriented graph and let v_{1}, v_{2} be vertices of G. Let us assume that there exists a set e such that e ∈ the edges of G and e orientedly joins v_{1}, v_{2}. The functor Edge(v_{1}, v_{2}) is defined as follows:

(Def. 6) There exists a set e such that Edge(v_{1}, v_{2}) = e and e ∈ the edges of G

and e orientedly joins v_{1}, v_{2}. Let G be an oriented graph, let v_{1}, v_{2} be vertices of G, and let W be a

function. The functor Weight(v_{1}, v_{2}, W) is defined as follows:

(Def. 7) Weight(v_{1}, v_{2}, W) = \{ \begin{matrix}W (Edge(v_{1}, v_{2})), if there exists a set e such \\ that e ∈ the edges of G and e orientedly joins \\ v_{1}, v_{2}, \\ − 1, otherwise. \end{matrix}

Let G be an oriented graph, let v_{1}, v_{2} be vertices of G, and let W be a function from the edges of G into ℝ_{0} . Then Weight(v_{1}, v_{2}, W) is a real number.

In the sequel G is an oriented graph, v_{1}, v_{2} are vertices of G, and W is a function from the edges of G into ℝ_{0} .

We now state three propositions:

(24) Weight(v_{1}, v_{2}, W) 0 iff there exists a set e such that e ∈ the edges of G and e orientedly joins v_{1}, v_{2}.

(25) Weight(v_{1}, v_{2}, W) = − 1 iff it is not true that there exists a set e such that e ∈ the edges of G and e orientedly joins v_{1}, v_{2}.

(26) If e ∈ the edges of G and e orientedly joins v_{1}, v_{2}, then Weight(v_{1}, v_{2}, W) = W (e).

7. Basic Operations for Dijkstra’s Shortest Path Algorithm

Let f be an element of ℝ^{∗} and let n be a natural number. The functor UnusedVx(f, n) yields a subset of ℕ and is defined as follows:

(Def. 8) UnusedVx(f, n) = {i : i ∈ domf ∧ 1 i ∧ i n ∧ f(i) ≠ − 1}.

Let f be an element of ℝ^{∗} and let n be a natural number. The functor UsedVx(f, n) yielding a subset of ℕ is defined as follows:

(Def. 9) UsedVx(f, n) = {i : i ∈ domf ∧ 1 i ∧ i n ∧ f(i) = − 1}.

The following proposition is true

(27) UnusedVx(f, n) ⊆ Segn.

Let f be an element of ℝ^{∗} and let n be a natural number. One can verify that UnusedVx(f, n) is finite.

Next we state two propositions:

(28) OuterVx(f, n) ⊆ UnusedVx(f, n).

(29) OuterVx(f, n) ⊆ Segn.

Let f be an element of ℝ^{∗} and let n be a natural number. Observe that OuterVx(f, n) is finite.

Let X be a finite subset of ℕ, let f be an element of ℝ^{∗}, and let us consider n. The functor Argmin(X, f, n) yielding a natural number is defined by the conditions (Def. 10).

(Def. 10)(i) If X ≠ ∅, then there exists i such that i = Argmin(X, f, n) and i ∈ X and for every k such that k ∈ X holds f_{2·n+i} f_{2·n+k} and for every k such that k ∈ X and f_{2·n+i}= f_{2·n+k} holds i k, and

(ii) if X = ∅, then Argmin(X, f, n) = 0. We now state two propositions:

(30) If OuterVx(f, n) ≠ ∅ and j = Argmin(OuterVx(f, n), f, n), then j ∈ domf and 1 j and j n and f(j) ≠ − 1 and f(n + j) ≠ − 1.

(31) Argmin(OuterVx(f, n), f, n) n.

Let n be a natural number. The functor findminn yields an element of (ℝ^{∗})^{ℝ^{∗}} and is defined as follows:

(Def. 11) domfindminn = ℝ^{∗} and for every element f of ℝ^{∗} holds (findminn)(f) = (f, n · n + 3 · n + 1) := (Argmin(OuterVx(f, n), f, n), − 1).

Next we state four propositions:

(32) If i ∈ domf and i > n and i ≠ n · n + 3 · n + 1, then (findminn)(f)(i) = f(i).

(33) If i ∈ domf and f(i) = − 1 and i ≠ n·n+3·n+1, then (findminn)(f)(i) = − 1.

(34) dom(findminn)(f) = domf.

(35) If OuterVx(f, n) ≠ ∅, then there exists j such that j ∈ OuterVx(f, n)

and 1 j and j n and (findminn)(f)(j) = − 1. Let f be an element of ℝ^{∗} and let n, k be natural numbers. The functor

newpathcost(f, n, k) yielding a real number is defined as follows: (Def. 12) newpathcost(f, n, k) = f_{2·n+f_{n·n+3·n+1}}+ f_{2·n+n·f_{n·n+3·n+1}+k}.

Let n, k be natural numbers and let f be an element of ℝ^{∗}. We say that f has better path at n, k if and only if:

(Def. 13) f(n + k) = − 1 or f_{2·n+k}> newpathcost(f, n, k) but f_{2·n+n·f_{n·n+3·n+1}+k}

0 but f(k) ≠ − 1. Let f be an element of ℝ^{∗} and let n be a natural number. The functor

Relax(f, n) yields an element of ℝ^{∗} and is defined by the conditions (Def. 14). (Def. 14)(i) domRelax(f, n) = domf, and

(ii) for every natural number k such that k ∈ domf holds if n < k and k 2 · n, then if f has better path at n, k − ′n, then (Relax(f, n))(k) = f_{n·n+3·n+1} and if f does not have better path at n, k − ′ n, then (Relax(f, n))(k) = f(k) and if 2 · n < k and k 3 · n, then if f has better path at n, k− ′2·n, then (Relax(f, n))(k) = newpathcost(f, n, k− ′2·n) and if f does not have better path at n, k − ′2 · n, then (Relax(f, n))(k) = f(k) and if k n or k > 3 · n, then (Relax(f, n))(k) = f(k).

Let n be a natural number. The functor Relaxn yields an element of (ℝ^{∗})^{ℝ^{∗}} and is defined by:

(Def. 15) domRelaxn = ℝ^{∗} and for every element f of ℝ^{∗} holds (Relaxn)(f) =

Relax(f, n). One can prove the following propositions:

(36) dom(Relaxn)(f) = domf.

(37) If i n or i > 3 · n and if i ∈ domf, then (Relaxn)(f)(i) = f(i).

(38) dom(repeat(Relaxn·findminn))(i)(f) = dom(repeat(Relaxn·findminn)) (i + 1)(f).

(39) If OuterVx((repeat(Relaxn · findminn))(i)(f), n) ≠ ∅, then UnusedVx((repeat(Relaxn · findminn))(i + 1)(f), n) ⊂ UnusedVx((repeat (Relaxn · findminn))(i)(f), n).

(40) If g = (repeat(Relaxn · findminn))(i)(f) and h = (repeat(Relaxn · findminn))(i + 1)(f) and k = Argmin(OuterVx(g, n), g, n) and OuterVx(g, n) ≠ ∅, then UsedVx(h, n) = UsedVx(g, n) ∪ {k} and k ∉ UsedVx(g, n).

(41) There exists i such that i n and OuterVx((repeat(Relaxn · findminn))(i)(f), n) = ∅.

(42) domf = dom(repeat(Relaxn · findminn))(i)(f).

Let f, g be elements of ℝ^{∗} and let us consider m, n. We say that f, g are equal at m, n if and only if:

(Def. 16) domf = domg and for every k such that k ∈ domf and m k and

k n holds f(k) = g(k). One can prove the following propositions:

(43) f, f are equal at m, n.

(44) If f, g are equal at m, n and g, h are equal at m, n, then f, h are equal at m, n.

(45) (repeat(Relaxn·findminn))(i)(f), (repeat(Relaxn·findminn))(i+1)(f) are equal at 3 · n + 1, n · n + 3 · n.

(46) Let F be an element of (ℝ^{∗})^{ℝ^{∗}}, f be an element of ℝ^{∗}, and n, i be natural numbers. If i < LifeSpan(F, f, n), then OuterVx((repeatF)(i)(f), n) ≠ ∅.

(47) f, (repeat(Relaxn · findminn))(i)(f) are equal at 3 · n + 1, n · n + 3 · n.

(48) Suppose that (i) 1 n, (ii) 1 ∈ domf, (iii) f(n + 1) ≠ − 1, (iv) for every i such that 1 i and i n holds f(i) = 1, and

(v) for every i such that 2 i and i n holds f(n + i) = − 1. Then 1 = Argmin(OuterVx(f, n), f, n) and UsedVx(f, n) = ∅ and {1} = UsedVx((repeat(Relaxn · findminn))(1)(f), n).

(49) If g = (repeat(Relaxn · findminn))(1)(f) and h = (repeat(Relaxn · findminn))(i)(f) and 1 i and i LifeSpan(Relaxn · findminn, f, n) and m ∈ UsedVx(g, n), then m ∈ UsedVx(h, n).

Let p be a finite sequence of elements of ℕ, let f be an element of ℝ^{∗}, and let i, n be natural numbers. We say that p is vertex sequence at f, i, n if and only if:

(Def. 17) p(lenp) = i and for every k such that 1 k and k < lenp holds p(lenp−

k) = f(n + p_{(len p−k)+1} ). Let p be a finite sequence of elements of ℕ, let f be an element of ℝ^{∗}, and

let i, n be natural numbers. We say that p is simple vertex sequence at f, i, n if and only if:

(Def. 18) p(1) = 1 and lenp > 1 and p is vertex sequence at f, i, n and one-to-one. Next we state the proposition (50) Let p, q be finite sequences of elements of ℕ, f be an element of ℝ^{∗}, and

i, n be natural numbers. Suppose p is simple vertex sequence at f, i, n and q is simple vertex sequence at f, i, n. Then p = q.

Let G be a graph, let p be a finite sequence of elements of the edges of G, and let v_{5} be a finite sequence. We say that p is oriented edge sequence at v_{5} if and only if:

(Def. 19) lenv_{5} = lenp + 1 and for every n such that 1 n and n lenp holds

(the source of G)(p(n)) = v_{5}(n) and (the target of G)(p(n)) = v_{5}(n + 1). One can prove the following two propositions:

(51) Let G be an oriented graph, v_{5} be a finite sequence, and p, q be oriented chains of G. Suppose p is oriented edge sequence at v_{5} and q is oriented edge sequence at v_{5}. Then p = q.

(52) Let G be a graph, v_{6}, v_{7} be finite sequences, and p be an oriented chain of G. Suppose p is oriented edge sequence at v_{6} and oriented edge sequence at v_{7} and lenp 1. Then v_{6} = v_{7}.

8. Data Structure for Dijkstra’s Shortest Path Algorithm

Let f be an element of ℝ^{∗}, let G be an oriented graph, let n be a natural number, and let W be a function from the edges of G into ℝ_{0} . We say that f is input of Dijkstra algorithm G to n in W if and only if the conditions (Def. 20) are satisfied.

(Def. 20)(i) lenf = n · n + 3 · n + 1, (ii) Segn = the vertices of G, (iii) for every i such that 1 i and i n holds f(i) = 1 and f(2·n+i) = 0, (iv) f(n + 1) = 0,

(v) for every i such that 2 i and i n holds f(n + i) = − 1, and (vi) for all vertices i, j of G and for all k, m such that k = i and m = j

holds f(2 · n + n · k + m) = Weight(i, j, W).

9. The Definition of Dijkstra’s Shortest Path Algorithm

Let n be a natural number. The functor DijkstraAlgorithmn yielding an element of (ℝ^{∗})^{ℝ^{∗}} is defined as follows:

(Def. 21) DijkstraAlgorithmn = WhileDo(Relaxn · findminn, n).

246 jing-chao chen

10. Justifying the Correctness of Dijkstra’s Shortest Path Algorithm

For simplicity, we adopt the following rules: p is a finite sequence of elements of ℕ, G is a finite oriented graph, P, Q are oriented chains of G, W is a function from the edges of G into ℝ_{0} , and v_{1}, v_{2} are vertices of G.

We now state the proposition

(53) Suppose f is input of Dijkstra algorithm G to n in W and v_{1} = 1 and 1 ≠ v_{2} and v_{2} = i and n 1 and g = (DijkstraAlgorithmn)(f). Then

(i) the vertices of G = UsedVx(g, n) ∪ UnusedVx(g, n), (ii) if v_{2} ∈ UsedVx(g, n), then there exist p, P such that p is simple vertex

sequence at g, i, n and P is oriented edge sequence at p and shortest path from v_{1} to v_{2} in W and cost(P, W) = g(2 · n + i), and

(iii) if v_{2} ∈ UnusedVx(g, n), then there exists no Q which is oriented path from v_{1} to v_{2}.

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

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

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

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

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

[9] Jing-Chao Chen and Yatsuka Nakamura. The underlying principle of Dijkstra’s shortest path algorithm. Formalized Mathematics, 11(2):143–152, 2003.

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

[11] E. W. Dijkstra. A note on two problems in connection with graphs. Numer. Math., 1:269–271, 1959.

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

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

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

[15] Yatsuka Nakamura and Piotr Rudnicki. Oriented chains. Formalized Mathematics, 7(2):189–192, 1998.

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

[17] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

dijkstra’s shortest path algorithm 247

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

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

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

Received March 17, 2003

Real Linear Space of Real Sequences

Noboru Endou Gifu National College of Technology Yasumasa Suzuki Take, Yokosuka-shi Japan

Yasunari Shidama Shinshu University Nagano

Summary. The article is a continuation of [14]. As the example of real linear spaces, we introduce the arithmetic addition in the set of real sequences and also introduce the multiplication. This set has the arithmetic structure which depends on these arithmetic operations.

MML Identifier: RSSPACE.

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

The non empty set the set of real sequences is defined by: (Def. 1) For every set x holds x ∈ the set of real sequences iff x is a sequence of

real numbers. Let a be a set. Let us assume that a ∈ the set of real sequences. The functor

id_{seq}(a) yields a sequence of real numbers and is defined by: (Def. 2) id_{seq}(a) = a.

Let a be a set. Let us assume that a ∈ ℝ. The functor id_{ℝ}(a) yielding a real number is defined by:

(Def. 3) id_{ℝ}(a) = a.

We now state two propositions:

(1) There exists a binary operation A_{1} on the set of real sequences such that for all elements a, b of the set of real sequences holds A_{1}(a, b) = id_{seq}(a) + id_{seq}(b) and A_{1} is commutative and associative.

(2) There exists a function f from [: ℝ, the set of real sequences :] into the set of real sequences such that for all sets r, x if r ∈ ℝ and x ∈ the set of real sequences, then f(⟨⟨r, x⟩⟩) = id_{ℝ}(r) id_{seq}(x).

The binary operation add_{seq} on the set of real sequences is defined as follows: (Def. 4) For all elements a, b of the set of real sequences holds add_{seq}(a, b) =

id_{seq}(a) + id_{seq}(b). The function mult_{seq} from [: ℝ, the set of real sequences :] into the set of real

sequences is defined by: (Def. 5) For all sets r, x such that r ∈ ℝ and x ∈ the set of real sequences holds

mult_{seq}(⟨⟨r, x⟩⟩) = id_{ℝ}(r) id_{seq}(x). The element Zeroseq of the set of real sequences is defined by:

(Def. 6) For every natural number n holds (id_{seq}(Zeroseq))(n) = 0.

One can prove the following propositions:

(3) For every sequence x of real numbers holds id_{seq}(x) = x.

(4) For all vectors v, w of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds v + w = id_{seq}(v) + id_{seq}(w).

(5) For every real number r and for every vector v of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds r · v = r id_{seq}(v).

One can verify that ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ is Abelian.

We now state several propositions:

(6) For all vectors u, v, w of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds (u + v) + w = u + (v + w).

(7) For every vector v of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds v + 0_{⟨the set of real sequences ,Zeroseq ,add_{seq} ,mult_{seq} ⟩}= v.

(8) Let v be a vector of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩. Then there exists a vector w of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ such that v + w = 0_{⟨the set of real sequences ,Zeroseq ,add_{seq} ,mult_{seq} ⟩}.

(9) For every real number a and for all vectors v, w of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds a · (v + w) = a · v + a · w.

(10) For all real numbers a, b and for every vector v of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds (a + b) · v = a · v + b · v.

(11) For all real numbers a, b and for every vector v of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩ holds (a · b) · v = a · (b · v).

(12) For every vector v of ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩

holds 1 · v = v. The real linear space the linear space of real sequences is defined by:

(Def. 7) The linear space of real sequences = ⟨the set of real sequences, Zeroseq, add_{seq}, mult_{seq}⟩.

Let X be a real linear space and let X_{1} be a subset of the carrier of X. Let us assume that X_{1} is linearly closed and non empty. The functor Add (X_{1}, X) yielding a binary operation on X_{1} is defined by:

(Def. 8) Add (X_{1}, X) = (the addition of X)↾[: X_{1}, X_{1} :].

Let X be a real linear space and let X_{1} be a subset of the carrier of X. Let us assume that X_{1} is linearly closed and non empty. The functor Mult (X_{1}, X) yielding a function from [: ℝ, X_{1} :] into X_{1} is defined as follows:

(Def. 9) Mult (X_{1}, X) = (the external multiplication of X)↾[: ℝ, X_{1} :].

Let X be a real linear space and let X_{1} be a subset of the carrier of X. Let us assume that X_{1} is linearly closed and non empty. The functor Zero (X_{1}, X) yields an element of X_{1} and is defined by:

(Def. 10) Zero (X_{1}, X) = 0_{X}.

We now state the proposition

(13) Let V be a real linear space and V_{1} be a subset of the carrier of V. Suppose V_{1} is linearly closed and non empty. Then ⟨V_{1}, Zero (V_{1}, V), Add (V_{1}, V), Mult (V_{1}, V)⟩ is a subspace of V.

The subset the set of l2-real sequences of the carrier of the linear space of real sequences is defined by the conditions (Def. 11).

(Def. 11)(i) The set of l2-real sequences is non empty, and

(ii) for every set x holds x ∈ the set of l2-real sequences iff x ∈ the set of

real sequences and id_{seq}(x) id_{seq}(x) is summable. Next we state several propositions:

(14) The set of l2-real sequences is linearly closed and the set of l2-real sequences is non empty.

(15) ⟨the set of l2-real sequences, Zero (the set of l2-real sequences, the linear space of real sequences), Add (the set of l2-real sequences, the linear space of real sequences), Mult (the set of l2-real sequences, the linear space of real sequences)⟩ is a subspace of the linear space of real sequences.

(16) ⟨the set of l2-real sequences, Zero (the set of l2-real sequences, the linear space of real sequences), Add (the set of l2-real sequences, the linear space of real sequences), Mult (the set of l2-real sequences, the linear space of real sequences)⟩ is a real linear space.

(17)(i) The carrier of the linear space of real sequences = the set of real

sequences, (ii) for every set x holds x is an element of the carrier of the linear space

of real sequences iff x is a sequence of real numbers, (iii) for every set x holds x is a vector of the linear space of real sequences

iff x is a sequence of real numbers, (iv) for every vector u of the linear space of real sequences holds u = id_{seq}(u), (v) for all vectors u, v of the linear space of real sequences holds u + v =

id_{seq}(u) + id_{seq}(v), and (vi) for every real number r and for every vector u of the linear space of

real sequences holds r · u = r id_{seq}(u).

(18) There exists a function f from [: the set of l2-real sequences, the set of l2-real sequences :] into ℝ such that for all sets x, y if x ∈ the set of l2-real sequences and y ∈ the set of l2-real sequences, then f(⟨⟨x, y⟩⟩) = ∑(id_{seq}(x) id_{seq}(y)).

The function scalar_{seq} from [: the set of l2-real sequences, the set of l2-real sequences :] into ℝ is defined by the condition (Def. 12).

(Def. 12) Let x, y be sets. Suppose x ∈ the set of l2-real sequences and y ∈ the

set of l2-real sequences. Then scalar_{seq}(⟨⟨x, y⟩⟩) = ∑(id_{seq}(x) id_{seq}(y)). One can check that ⟨the set of l2-real sequences, Zero (the set of

l2-real sequences, the linear space of real sequences), Add (the set of l2real sequences, the linear space of real sequences), Mult (the set of l2-real sequences, the linear space of real sequences), scalar_{seq}⟩ is non empty.

The non empty unitary space structure l2-Space is defined by the condition (Def. 13).

(Def. 13) l2-Space = ⟨the set of l2-real sequences, Zero (the set of l2-real sequences, the linear space of real sequences), Add (the set of l2-real sequences, the linear space of real sequences), Mult (the set of l2-real sequences, the linear space of real sequences), scalar_{seq}⟩.

One can prove the following propositions:

(19) Let l be a unitary space structure. Suppose ⟨the carrier of l, the zero of l, the addition of l, the external multiplication of l⟩ is a real linear space. Then l is a real linear space.

(20) Let r_{1} be a sequence of real numbers. If for every natural number n holds r_{1}(n) = 0, then r_{1} is summable and ∑ r_{1} = 0.

(21) Let r_{1} be a sequence of real numbers. Suppose for every natural number n holds 0 r_{1}(n) and r_{1} is summable and ∑ r_{1} = 0. Let n be a natural number. Then r_{1}(n) = 0.

Let us observe that l2-Space is Abelian, add-associative, right zeroed, right complementable, and real linear space-like.

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

real linear space of real sequences 253

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

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

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

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

[9] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

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

[11] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

[12] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 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] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Hilbert Space of Real Sequences

Noboru Endou

Gifu National College of Technology Yasumasa Suzuki Take, Yokosuka-shi Japan

Yasunari Shidama Shinshu University Nagano

Summary. A continuation of [16]. As the example of real unitary spaces, we introduce the arithmetic addition and multiplication in the set of square sum able real sequences and introduce the scaler products also. This set has the structure of the Hilbert space.

MML Identifier: RSSPACE2.

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

1. Hilbert Space of Real Sequences

One can prove the following two propositions:

(1) The carrier of l2-Space = the set of l2-real sequences and for every set x holds x is an element of the carrier of l2-Space iff x is a sequence of real numbers and id_{seq}(x) id_{seq}(x) is summable and for every set x holds x is a vector of l2-Space iff x is a sequence of real numbers and id_{seq}(x) id_{seq}(x) is summable and 0_{l2-Space} = Zeroseq and for every vector u of l2-Space holds u = id_{seq}(u) and for all vectors u, v of l2-Space holds u+v = id_{seq}(u)+id_{seq}(v) and for every real number r and for every vector u of l2-Space holds r · u = r id_{seq}(u) and for every vector u of l2-Space holds − u = − id_{seq}(u) and id_{seq}(− u) = − id_{seq}(u) and for all vectors u, v of l2-Space holds u − v = id_{seq}(u) − id_{seq}(v) and for all vectors v, w of l2-Space holds id_{seq}(v) id_{seq}(w) is summable and for all vectors v, w of l2-Space holds (v|w) = ∑(id_{seq}(v) id_{seq}(w)).

(2) Let x, y, z be points of l2-Space and a be a real number. Then (x|x) = 0 iff x = 0_{l2-Space} and 0 (x|x) and (x|y) = (y|x) and ((x + y)|z) = (x|z) + (y|z) and ((a · x)|y) = a · (x|y).

Let us note that l2-Space is real unitary space-like. One can prove the following proposition

(3) For every sequence v_{1} of l2-Space such that v_{1} is a Cauchy sequence holds v_{1} is convergent.

Let us mention that l2-Space is Hilbert and complete.

2. Miscellaneous

We now state several propositions:

(4) Let r_{1} be a sequence of real numbers. Suppose for every natural number n holds 0 r_{1}(n) and r_{1} is summable. Then

(i) for every natural number n holds r_{1}(n) (∑_{α=0}^{κ}(r_{1})(α))_{κ∈ℕ}(n), (ii) for every natural number n holds 0 (∑_{α=0}^{κ}(r_{1})(α))_{κ∈ℕ}(n), (iii) for every natural number n holds (∑_{α=0}^{κ}(r_{1})(α))_{κ∈ℕ}(n) ∑ r_{1}, and (iv) for every natural number n holds r_{1}(n) ∑ r_{1}.

(5) For all real numbers x, y holds (x + y) · (x + y) 2 · x · x + 2 · y · y and for all real numbers x, y holds x · x 2 · (x − y) · (x − y) + 2 · y · y.

(6) Let e be a real number and s_{1} be a sequence of real numbers. Suppose s_{1} is convergent and there exists a natural number k such that for every natural number i such that k i holds s_{1}(i) e. Then lims_{1} e.

(7) Let c be a real number and s_{1} be a sequence of real numbers. Suppose s_{1} is convergent. Let r_{1} be a sequence of real numbers. Suppose that for every natural number i holds r_{1}(i) = (s_{1}(i) − c) · (s_{1}(i) − c). Then r_{1} is convergent and limr_{1} = (lims_{1} − c) · (lims_{1} − c).

(8) Let c be a real number and s_{1}, s_{2} be sequences of real numbers. Suppose s_{1} is convergent and s_{2} is convergent. Let r_{1} be a sequence of real numbers. Suppose that for every natural number i holds r_{1}(i) = (s_{1}(i) − c) · (s_{1}(i) − c)+s_{2}(i). Then r_{1} is convergent and limr_{1} = (lims_{1}−c)·(lims_{1}−c)+lims_{2}.

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. Some basic properties of sets. Formalized Mathematics, 1(1):47–53,

hilbert space of real sequences 257

[4] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249–253, 2003.

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

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

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

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

[9] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[10] Jan Popiołek. Introduction to Banach and Hilbert spaces - part II. Formalized Mathematics, 2(4):517–521, 1991.

[11] Jan Popiołek. Introduction to Banach and Hilbert spaces - part III. Formalized Mathematics, 2(4):523–526, 1991.

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

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

[14] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator.

Part I

Takao Inoué The Iida Technical High School Nagano

Summary. In this paper, we develop intuitionistic propositional calculus IPC in the extended language with single modal operator. The formulation that we adopt in this paper is very useful not only to formalize the calculus but also to do a number of logics with essentially propositional character. In addition, it is much simpler than the past formalization for modal logic. In the first section, we give the mentioned formulation which the author heavily owes to the formalism of Adam Grabowski’s [4]. After the theoretical development of the logic, we prove a number of valid formulas of IPC in the sections 2–4. The last two sections are devoted to present classical propositional calculus and modal calculus S4 in our framework, as a preparation for future study. In the forthcoming Part II of this paper, we shall prove, among others, a number of intuitionistically valid formulas with negation.

MML Identifier: INTPRO 1.

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

1. Intuitionistic Propositional Calculus IPC in the Extended Language with Modal Operator

Let E be a set. We say that E has FALSUM if and only if: (Def. 1) ⟨0⟩ ∈ E.

Let E be a set. We say that E has intuitionistic implication if and only if:

(Def. 2) For all finite sequences p, q such that p ∈ E and q ∈ E holds ⟨1⟩^{⌢} p^{⌢} q ∈

E. Let E be a set. We say that E has intuitionistic conjunction if and only if:

(Def. 3) For all finite sequences p, q such that p ∈ E and q ∈ E holds ⟨2⟩^{⌢} p^{⌢} q ∈

E. Let E be a set. We say that E has intuitionistic disjunction if and only if:

(Def. 4) For all finite sequences p, q such that p ∈ E and q ∈ E holds ⟨3⟩^{⌢} p^{⌢} q ∈

E. Let E be a set. We say that E has intuitionistic propositional variables if

and only if: (Def. 5) For every natural number n holds ⟨5 + 2 · n⟩ ∈ E.

Let E be a set. We say that E has intuitionistic modal operator if and only if:

(Def. 6) For every finite sequence p such that p ∈ E holds ⟨6⟩^{⌢} p ∈ E.

Let E be a set. We say that E is MC-closed if and only if the conditions (Def. 7) are satisfied.

(Def. 7)(i) E ⊆ ℕ^{∗}, and

(ii) E has FALSUM, intuitionistic implication, intuitionistic conjunction, intuitionistic disjunction, intuitionistic propositional variables, and intuitionistic modal operator.

One can check that every set which is MC-closed is also non empty and has FALSUM, intuitionistic implication, intuitionistic conjunction, intuitionistic disjunction, intuitionistic propositional variables, and intuitionistic modal operator and every subset of ℕ^{∗} which has FALSUM, intuitionistic implication, intuitionistic conjunction, intuitionistic disjunction, intuitionistic propositional variables, and intuitionistic modal operator is also MC-closed.

The set MC-wff is defined by: (Def. 8) MC-wff is MC-closed and for every set E such that E is MC-closed holds

MC-wff ⊆ E. One can verify that MC-wff is MC-closed. Let us note that there exists a set which is MC-closed and non empty. One can verify that every element of MC-wff is relation-like and function-

like.

Let us note that every element of MC-wff is finite sequence-like. A MC-formula is an element of MC-wff. The MC-formula FALSUM is defined as follows:

(Def. 9) FALSUM = ⟨0⟩.

Let p, q be elements of MC-wff. The functor p ⇒ q yields a MC-formula and is defined as follows:

(Def. 10) p ⇒ q = ⟨1⟩^{⌢} p^{⌢} q.

The functor p ∧ q yields a MC-formula and is defined as follows: (Def. 11) p ∧ q = ⟨2⟩^{⌢} p^{⌢} q.

The functor p ∨ q yielding a MC-formula is defined by: (Def. 12) p ∨ q = ⟨3⟩^{⌢} p^{⌢} q.

Let p be an element of MC-wff. The functor Nes(p) yielding a MC-formula is defined by:

(Def. 13) Nes(p) = ⟨6⟩^{⌢} p.

We use the following convention: T, X, Y denote subsets of MC-wff and p, q, r, s denote elements of MC-wff.

Let T be a subset of MC-wff. We say that T is IPC theory if and only if the condition (Def. 14) is satisfied.

(Def. 14) Let p, q, r be elements of MC-wff. Then p ⇒ (q ⇒ p) ∈ T and (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ T and p∧q ⇒ p ∈ T and p∧q ⇒ q ∈ T and p ⇒ (q ⇒ p ∧ q) ∈ T and p ⇒ p ∨ q ∈ T and q ⇒ p ∨ q ∈ T and (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) ∈ T and FALSUM ⇒ p ∈ T and if p ∈ T and p ⇒ q ∈ T, then q ∈ T.

Let us consider X. The functor CnIPC(X) yielding a subset of MC-wff is defined as follows:

(Def. 15) r ∈ CnIPC(X) iff for every T such that T is IPC theory and X ⊆ T

holds r ∈ T. The subset IPC-Taut of MC-wff is defined as follows:

(Def. 16) IPC-Taut = CnIPC(∅_{MC-wff} ).

Let p be an element of MC-wff. The functor neg(p) yields a MC-formula and is defined as follows:

(Def. 17) neg(p) = p ⇒ FALSUM.

The MC-formula IVERUM is defined by: (Def. 18) IVERUM = FALSUM ⇒ FALSUM.

The following propositions are true:

(1) p ⇒ (q ⇒ p) ∈ CnIPC(X).

(2) (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ CnIPC(X).

(3) p ∧ q ⇒ p ∈ CnIPC(X).

(4) p ∧ q ⇒ q ∈ CnIPC(X).

(5) p ⇒ (q ⇒ p ∧ q) ∈ CnIPC(X).

(6) p ⇒ p ∨ q ∈ CnIPC(X).

(7) q ⇒ p ∨ q ∈ CnIPC(X).

(8) (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) ∈ CnIPC(X).

(9) FALSUM ⇒ p ∈ CnIPC(X).

(10) If p ∈ CnIPC(X) and p ⇒ q ∈ CnIPC(X), then q ∈ CnIPC(X).

(11) If T is IPC theory and X ⊆ T, then CnIPC(X) ⊆ T.

(12) X ⊆ CnIPC(X).

(13) If X ⊆ Y, then CnIPC(X) ⊆ CnIPC(Y).

(14) CnIPC(CnIPC(X)) = CnIPC(X).

Let X be a subset of MC-wff. Observe that CnIPC(X) is IPC theory. The following propositions are true:

(15) T is IPC theory iff CnIPC(T) = T.

(16) If T is IPC theory, then IPC-Taut ⊆ T.

One can verify that IPC-Taut is IPC theory.

2. Formulas Provable in IPC: Implication

We now state a number of propositions:

(17) p ⇒ p ∈ IPC-Taut.

(18) If q ∈ IPC-Taut, then p ⇒ q ∈ IPC-Taut.

(19) IVERUM ∈ IPC-Taut.

(20) (p ⇒ q) ⇒ (p ⇒ p) ∈ IPC-Taut.

(21) (q ⇒ p) ⇒ (p ⇒ p) ∈ IPC-Taut.

(22) (q ⇒ r) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ IPC-Taut.

(23) If p ⇒ (q ⇒ r) ∈ IPC-Taut, then q ⇒ (p ⇒ r) ∈ IPC-Taut.

(24) (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒ r)) ∈ IPC-Taut.

(25) If p ⇒ q ∈ IPC-Taut, then (q ⇒ r) ⇒ (p ⇒ r) ∈ IPC-Taut.

(26) If p ⇒ q ∈ IPC-Taut and q ⇒ r ∈ IPC-Taut, then p ⇒ r ∈ IPC-Taut.

(27) (p ⇒ (q ⇒ r)) ⇒ ((s ⇒ q) ⇒ (p ⇒ (s ⇒ r))) ∈ IPC-Taut.

(28) ((p ⇒ q) ⇒ r) ⇒ (q ⇒ r) ∈ IPC-Taut.

(29) (p ⇒ (q ⇒ r)) ⇒ (q ⇒ (p ⇒ r)) ∈ IPC-Taut.

(30) (p ⇒ (p ⇒ q)) ⇒ (p ⇒ q) ∈ IPC-Taut.

(31) q ⇒ ((q ⇒ p) ⇒ p) ∈ IPC-Taut.

(32) If s ⇒ (q ⇒ p) ∈ IPC-Taut and q ∈ IPC-Taut, then s ⇒ p ∈ IPC-Taut.

3. Formulas Provable in IPC: Conjunction

The following propositions are true:

(33) p ⇒ p ∧ p ∈ IPC-Taut.

(34) p ∧ q ∈ IPC-Taut iff p ∈ IPC-Taut and q ∈ IPC-Taut.

(35) p ∧ q ∈ IPC-Taut iff q ∧ p ∈ IPC-Taut.

(36) (p ∧ q ⇒ r) ⇒ (p ⇒ (q ⇒ r)) ∈ IPC-Taut.

(37) (p ⇒ (q ⇒ r)) ⇒ (p ∧ q ⇒ r) ∈ IPC-Taut.

(38) (r ⇒ p) ⇒ ((r ⇒ q) ⇒ (r ⇒ p ∧ q)) ∈ IPC-Taut.

(39) (p ⇒ q) ∧ p ⇒ q ∈ IPC-Taut.

(40) (p ⇒ q) ∧ p ∧ s ⇒ q ∈ IPC-Taut.

(41) (q ⇒ s) ⇒ (p ∧ q ⇒ s) ∈ IPC-Taut.

(42) (q ⇒ s) ⇒ (q ∧ p ⇒ s) ∈ IPC-Taut.

(43) (p ∧ s ⇒ q) ⇒ (p ∧ s ⇒ q ∧ s) ∈ IPC-Taut.

(44) (p ⇒ q) ⇒ (p ∧ s ⇒ q ∧ s) ∈ IPC-Taut.

(45) (p ⇒ q) ∧ (p ∧ s) ⇒ q ∧ s ∈ IPC-Taut.

(46) p ∧ q ⇒ q ∧ p ∈ IPC-Taut.

(47) (p ⇒ q) ∧ (p ∧ s) ⇒ s ∧ q ∈ IPC-Taut.

(48) (p ⇒ q) ⇒ (p ∧ s ⇒ s ∧ q) ∈ IPC-Taut.

(49) (p ⇒ q) ⇒ (s ∧ p ⇒ s ∧ q) ∈ IPC-Taut.

(50) p ∧ (s ∧ q) ⇒ p ∧ (q ∧ s) ∈ IPC-Taut.

(51) (p ⇒ q) ∧ (p ⇒ s) ⇒ (p ⇒ q ∧ s) ∈ IPC-Taut.

(52) p ∧ q ∧ s ⇒ p ∧ (q ∧ s) ∈ IPC-Taut.

(53) p ∧ (q ∧ s) ⇒ p ∧ q ∧ s ∈ IPC-Taut.

4. Formulas Provable in IPC: Disjunction

We now state a number of propositions:

(54) p ∨ p ⇒ p ∈ IPC-Taut.

(55) If p ∈ IPC-Taut or q ∈ IPC-Taut, then p ∨ q ∈ IPC-Taut.

(56) p ∨ q ⇒ q ∨ p ∈ IPC-Taut.

(57) p ∨ q ∈ IPC-Taut iff q ∨ p ∈ IPC-Taut.

(58) (p ⇒ q) ⇒ (p ⇒ q ∨ s) ∈ IPC-Taut.

(59) (p ⇒ q) ⇒ (p ⇒ s ∨ q) ∈ IPC-Taut.

(60) (p ⇒ q) ⇒ (p ∨ s ⇒ q ∨ s) ∈ IPC-Taut.

(61) If p ⇒ q ∈ IPC-Taut, then p ∨ s ⇒ q ∨ s ∈ IPC-Taut.

(62) (p ⇒ q) ⇒ (s ∨ p ⇒ s ∨ q) ∈ IPC-Taut.

(63) If p ⇒ q ∈ IPC-Taut, then s ∨ p ⇒ s ∨ q ∈ IPC-Taut.

(64) p ∨ (q ∨ s) ⇒ q ∨ (p ∨ s) ∈ IPC-Taut.

(65) p ∨ (q ∨ s) ⇒ p ∨ q ∨ s ∈ IPC-Taut.

(66) p ∨ q ∨ s ⇒ p ∨ (q ∨ s) ∈ IPC-Taut.

5. Classical Propositional Calculus CPC

We use the following convention: T, X, Y are subsets of MC-wff and p, q, r are elements of MC-wff.

Let T be a subset of MC-wff. We say that T is CPC theory if and only if the condition (Def. 19) is satisfied.

(Def. 19) Let p, q, r be elements of MC-wff. Then p ⇒ (q ⇒ p) ∈ T and (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ T and p∧q ⇒ p ∈ T and p∧q ⇒ q ∈ T and p ⇒ (q ⇒ p ∧ q) ∈ T and p ⇒ p ∨ q ∈ T and q ⇒ p ∨ q ∈ T and (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) ∈ T and FALSUM ⇒ p ∈ T and p ∨ (p ⇒ FALSUM) ∈ T and if p ∈ T and p ⇒ q ∈ T, then q ∈ T.

One can prove the following proposition

(67) If T is CPC theory, then T is IPC theory.

Let us consider X. The functor CnCPC(X) yielding a subset of MC-wff is defined by:

(Def. 20) r ∈ CnCPC(X) iff for every T such that T is CPC theory and X ⊆ T

holds r ∈ T. The subset CPC-Taut of MC-wff is defined by:

(Def. 21) CPC-Taut = CnCPC(∅_{MC-wff} ).

Next we state several propositions:

(68) CnIPC(X) ⊆ CnCPC(X).

(69) p ⇒ (q ⇒ p) ∈ CnCPC(X) and (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ CnCPC(X) and p∧q ⇒ p ∈ CnCPC(X) and p∧q ⇒ q ∈ CnCPC(X) and p ⇒ (q ⇒ p ∧ q) ∈ CnCPC(X) and p ⇒ p ∨ q ∈ CnCPC(X) and q ⇒ p ∨ q ∈ CnCPC(X) and (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) ∈ CnCPC(X) and FALSUM ⇒ p ∈ CnCPC(X) and p ∨ (p ⇒ FALSUM) ∈ CnCPC(X).

(70) If p ∈ CnCPC(X) and p ⇒ q ∈ CnCPC(X), then q ∈ CnCPC(X).

(71) If T is CPC theory and X ⊆ T, then CnCPC(X) ⊆ T.

(72) X ⊆ CnCPC(X).

(73) If X ⊆ Y, then CnCPC(X) ⊆ CnCPC(Y).

(74) CnCPC(CnCPC(X)) = CnCPC(X).

Let X be a subset of MC-wff. Note that CnCPC(X) is CPC theory.

Next we state two propositions:

(75) T is CPC theory iff CnCPC(T) = T.

(76) If T is CPC theory, then CPC-Taut ⊆ T.

Let us note that CPC-Taut is CPC theory.

The following proposition is true

(77) IPC-Taut ⊆ CPC-Taut.

6. Modal Calculus S4

We use the following convention: T, X, Y are subsets of MC-wff and p, q, r are elements of MC-wff.

Let T be a subset of MC-wff. We say that T is S4 theory if and only if the condition (Def. 22) is satisfied.

(Def. 22) Let p, q, r be elements of MC-wff. Then p ⇒ (q ⇒ p) ∈ T and (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ T and p∧q ⇒ p ∈ T and p∧q ⇒ q ∈ T and p ⇒ (q ⇒ p ∧ q) ∈ T and p ⇒ p ∨ q ∈ T and q ⇒ p ∨ q ∈ T and (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) ∈ T and FALSUM ⇒ p ∈ T and p ∨ (p ⇒ FALSUM) ∈ T and Nes(p ⇒ q) ⇒ (Nes(p) ⇒ Nes(q)) ∈ T and Nes(p) ⇒ p ∈ T and Nes(p) ⇒ Nes(Nes(p)) ∈ T and if p ∈ T and p ⇒ q ∈ T, then q ∈ T and if p ∈ T, then Nes(p) ∈ T.

Next we state two propositions:

(78) If T is S4 theory, then T is CPC theory.

(79) If T is S4 theory, then T is IPC theory.

Let us consider X. The functor CnS4(X) yielding a subset of MC-wff is defined by:

(Def. 23) r ∈ CnS4(X) iff for every T such that T is S4 theory and X ⊆ T holds

r ∈ T. The subset S4-Taut of MC-wff is defined by:

(Def. 24) S4-Taut = CnS4(∅_{MC-wff} ).

Next we state a number of propositions:

(80) CnCPC(X) ⊆ CnS4(X).

(81) CnIPC(X) ⊆ CnS4(X).

(82) p ⇒ (q ⇒ p) ∈ CnS4(X) and (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ CnS4(X) and p ∧ q ⇒ p ∈ CnS4(X) and p ∧ q ⇒ q ∈ CnS4(X) and p ⇒ (q ⇒ p ∧ q) ∈ CnS4(X) and p ⇒ p ∨ q ∈ CnS4(X) and q ⇒ p ∨ q ∈ CnS4(X) and (p ⇒ r) ⇒ ((q ⇒ r) ⇒ (p ∨ q ⇒ r)) ∈ CnS4(X) and FALSUM ⇒ p ∈ CnS4(X) and p ∨ (p ⇒ FALSUM) ∈ CnS4(X).

(83) If p ∈ CnS4(X) and p ⇒ q ∈ CnS4(X), then q ∈ CnS4(X).

(84) Nes(p ⇒ q) ⇒ (Nes(p) ⇒ Nes(q)) ∈ CnS4(X).

(85) Nes(p) ⇒ p ∈ CnS4(X).

(86) Nes(p) ⇒ Nes(Nes(p)) ∈ CnS4(X).

(87) If p ∈ CnS4(X), then Nes(p) ∈ CnS4(X).

(88) If T is S4 theory and X ⊆ T, then CnS4(X) ⊆ T.

(89) X ⊆ CnS4(X).

(90) If X ⊆ Y, then CnS4(X) ⊆ CnS4(Y).

(91) CnS4(CnS4(X)) = CnS4(X).

266 takao inoué

Let X be a subset of MC-wff. One can verify that CnS4(X) is S4 theory. Next we state two propositions:

(92) T is S4 theory iff CnS4(T) = T.

(93) If T is S4 theory, then S4-Taut ⊆ T. Let us note that S4-Taut is S4 theory. The following propositions are true:

(94) CPC-Taut ⊆ S4-Taut.

(95) IPC-Taut ⊆ S4-Taut.

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] Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1):69–72, 1999.

[5] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

Some Properties for Convex Combinations

Noboru Endou

Gifu National College of Technology Yasumasa Suzuki Miyagi University

Yasunari Shidama Shinshu University Nagano

Summary. This is a continuation of [6]. In this article, we proved that convex combination on convex family is convex.

MML Identifier: CONVEX2.

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

1. Convex Combinations on Convex Family

The following propositions are true:

(1) For every non empty RLS structure V and for all convex subsets M, N of V holds M ∩ N is convex.

(2) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, F be a finite sequence of elements of the carrier of V, and B be a finite sequence of elements of ℝ. Suppose M = {u; u ranges over vectors of V: ⋀_{i: natural number} (i ∈ domF ∩ domB ⇒ ⋁_{v: vector of V} (v = F (i) ∧ (u|v) B(i)))}. Then M is convex.

(3) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, F be a finite sequence of elements of the carrier of V, and B be a finite sequence of elements of ℝ. Suppose M = {u; u ranges over vectors of V: ⋀_{i: natural number} (i ∈ domF ∩ domB ⇒ ⋁_{v: vector of V} (v = F (i) ∧ (u|v) < B(i)))}. Then M is convex.

(4) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, F be a finite sequence of elements of the carrier of V, and B be a finite sequence of elements of ℝ. Suppose M = {u; u ranges over vectors of V: ⋀_{i: natural number} (i ∈ domF ∩ domB ⇒ ⋁_{v: vector of V} (v = F (i) ∧ (u|v) B(i)))}. Then M is convex.

(5) Let V be a real unitary space-like non empty unitary space structure, M be a subset of V, F be a finite sequence of elements of the carrier of V, and B be a finite sequence of elements of ℝ. Suppose M = {u; u ranges over vectors of V: ⋀_{i: natural number} (i ∈ domF ∩ domB ⇒ ⋁_{v: vector of V} (v = F (i) ∧ (u|v) > B(i)))}. Then M is convex.

(6) Let V be a real linear space and M be a subset of V. Then for every subset N of V and for every linear combination L of N such that L is convex and N ⊆ M holds ∑ L ∈ M if and only if M is convex.

Let V be a real linear space and let M be a subset of V. The functor LC_{M} yielding a set is defined as follows:

(Def. 1) For every set L holds L ∈ LC_{M} iff L is a linear combination of M.

Let V be a real linear space. Observe that there exists a linear combination of V which is convex.

Let V be a real linear space. A convex combination of V is a convex linear combination of V.

Let V be a real linear space and let M be a non empty subset of V. One can verify that there exists a linear combination of M which is convex.

Let V be a real linear space and let M be a non empty subset of V. A convex combination of M is a convex linear combination of M.

The following propositions are true:

(7) For every real linear space V and for every subset M of V holds Convex-FamilyM ≠ ∅.

(8) For every real linear space V and for every subset M of V holds M ⊆ convM.

(9) Let V be a real linear space, L_{1}, L_{2} be convex combinations of V, and r be a real number. If 0 < r and r < 1, then r · L_{1} + (1 − r) · L_{2} is a convex combination of V.

(10) Let V be a real linear space, M be a non empty subset of V, L_{1}, L_{2} be convex combinations of M, and r be a real number. If 0 < r and r < 1, then r · L_{1} + (1 − r) · L_{2} is a convex combination of M.

(11) For every real linear space V holds there exists a linear combination of V which is convex.

(12) For every real linear space V and for every non empty subset M of V holds there exists a linear combination of M which is convex. some properties for convex combinations 269

2. Miscellaneous

We now state several propositions:

(13) For every real linear space V and for all subspaces W_{1}, W_{2} of V holds Up(W_{1} + W_{2}) = Up(W_{1}) + Up(W_{2}).

(14) For every real linear space V and for all subspaces W_{1}, W_{2} of V holds Up(W_{1} ∩ W_{2}) = Up(W_{1}) ∩ Up(W_{2}).

(15) Let V be a real linear space, L_{1}, L_{2} be convex combinations of V, and a, b be real numbers. Suppose a·b > 0. Then the support of a·L_{1}+b·L_{2} = (the support of a · L_{1}) ∪ (the support of b · L_{2}).

(16) Let F, G be functions. Suppose F and G are fiberwise equipotent. Let x_{1}, x_{2} be sets. Suppose x_{1} ∈ domF and x_{2} ∈ domF and x_{1} ≠ x_{2}. Then there exist sets z_{1}, z_{2} such that z_{1} ∈ domG and z_{2} ∈ domG and z_{1} ≠ z_{2} and F (x_{1}) = G(z_{1}) and F (x_{2}) = G(z_{2}).

(17) Let V be a real linear space, L be a linear combination of V, and A be a subset of V. Suppose A ⊆ the support of L. Then there exists a linear combination L_{1} of V such that the support of L_{1} = A.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 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] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[6] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53–58, 2003.

[7] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23–28, 2003.

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

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

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

[11] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[12] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

[14] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.

[15] Wojciech A. Trybulec. Operations on subspaces in real linear space. Formalized Mathematics, 1(2):395–399, 1990.

[16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

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

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

270 noboru endou et al.

[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received April 3, 2003

On Some Properties of Real Hilbert Space. Part II

Hiroshi Yamazaki Shinshu University Nagano Yasumasa Suzuki Take, Yokosuka-shi Japan

Takao Inoué

The Iida Technical High School Nagano Yasunari Shidama Shinshu University Nagano

Summary. This paper is a continuation of our paper [21]. We give an analogue of the necessary and sufficient condition for summable set (i.e. the main theorem of [21]) with respect to summable set by a functional L in real Hilbert space. After presenting certain useful lemmas, we prove our main theorem that the summability for an orthonormal infinite set in real Hilbert space is equivalent to its summability with respect to the square of norm, say H(x) = (x, x). Then we show that the square of norm H commutes with infinite sum operation if the orthonormal set under our consideration is summable. Our main theorem is due to [7].

MML Identifier: BHSP 7.

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

1. Necessary and Sufficient Condition for Summable Set

In this paper X is a real unitary space and x, y are points of X. The following propositions are true:

(1) Let Y be a subset of the carrier of X and L be a functional in X. Then Y is summable set by L if and only if for every real number e such that 0 < e there exists a finite subset Y_{0} of the carrier of X such that Y_{0} is non empty and Y_{0} ⊆ Y and for every finite subset Y_{1} of the carrier of X such that Y_{1} is non empty and Y_{1} ⊆ Y and Y_{0} misses Y_{1} holds |setopfunc(Y_{1}, the carrier of X, ℝ, L, +_{ℝ})| < e.

(2) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let S be a finite orthogonal family of X. Suppose S is non empty. Let I be a function from the carrier of X into the carrier of X. Suppose S ⊆ domI and for every y such that y ∈ S holds I(y) = y. Let H be a function from the carrier of X into ℝ. Suppose S ⊆ domH and for every y such that y ∈ S holds H(y) = (y|y). Then (setopfunc(S, the carrier of X, the carrier of X, I, the addition of X)| setopfunc(S, the carrier of X, the carrier of X, I, the addition of X)) = setopfunc(S, the carrier of X, ℝ, H, +_{ℝ}).

(3) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity. Let S be a finite orthogonal family of X. Suppose S is non empty. Let H be a functional in X. Suppose S ⊆ domH and for every x such that x ∈ S holds H(x) = (x|x). Then (Setsum(S)| Setsum(S)) = setopfunc(S, the carrier of X, ℝ, H, +_{ℝ}).

(4) Let Y be an orthogonal family of X and Z be a subset of the carrier of X. If Z is a subset of Y, then Z is an orthogonal family of X.

(5) Let Y be an orthonormal family of X and Z be a subset of the carrier of X. If Z is a subset of Y, then Z is an orthonormal family of X.

2. Equivalence of Summability

Next we state three propositions:

(6) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity and X is a Hilbert space. Let S be an orthonormal family of X and H be a functional in X. Suppose S ⊆ domH and for every x such that x ∈ S holds H(x) = (x|x). Then S is summable set if and only if S is summable set by H.

(7) Let S be a subset of the carrier of X. Suppose S is non empty and summable set. Let e be a real number. Suppose 0 < e. Then there exists a finite subset Y_{0} of the carrier of X such that

(i) Y_{0} is non empty, (ii) Y_{0} ⊆ S, and (iii) for every finite subset Y_{1} of the carrier of X such that Y_{0} ⊆ Y_{1} and

Y_{1} ⊆ S holds |(sumS| sumS) − (Setsum(Y_{1})| Setsum(Y_{1}))| < e.

(8) Let given X. Suppose the addition of X is commutative and associative and the addition of X has a unity and X is a Hilbert space. Let S be an orthonormal family of X. Suppose S is non empty. Let H be a functional

on some properties of real hilbert space. … 273

in X. Suppose S ⊆ domH and for every x such that x ∈ S holds H(x) = (x|x). If S is summable set, then (sumS| sumS) = SumByfunc(S, H).

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

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

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

[7] P. R. Halmos. Introduction to Hilbert Space. American Mathematical Society, 1987.

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

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

[10] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.

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

[12] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[13] Jan Popiołek. Introduction to Banach and Hilbert spaces - part III. Formalized Mathematics, 2(4):523–526, 1991.

[14] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

[15] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

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

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

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

[20] Hiroshi Yamazaki, Yasunari Shidama, and Yatsuka Nakamura. Bessel’s inequality. Formalized Mathematics, 11(2):169–173, 2003.

[21] Hiroshi Yamazaki, Yasumasa Suzuki, Takao Inoué, and Yasunari Shidama. On some properties of real Hilbert space. Part I. Formalized Mathematics, 11(3):225–229, 2003.

Inner Products and Angles of Complex Numbers

Wenpai Chang Shinshu University

Nagano Yatsuka Nakamura Shinshu University Nagano

Piotr Rudnicki University of Alberta Edmonton

Summary. An inner product of complex numbers is defined and used to characterize the (counter-clockwise) angle between (a,0) and (0,b) in the complex plane. For complex a, b and c we then define the (counter-clockwise) angle between (a,c) and (c, b) and prove theorems about the sum of internal and external angles of a triangle.

MML Identifier: COMPLEX2.

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

1. Preliminaries

One can prove the following propositions:

(1) For all real numbers a, b holds − (a + bi) = − a + (− b)i.

(2) For all real numbers a, b such that b > 0 there exists a real number r such that r = b · − ⌊\frac{a}{b}⌋+ a and 0 r and r < b.

(3) Let a, b, c be real numbers. Suppose a > 0 and b 0 and c 0 and b < a and c < a. Let i be an integer. If b = c + a · i, then b = c.

(4) For all real numbers a, b holds sin(a − b) = sina · cosb − cosa · sinb and cos(a − b) = cosa · cosb + sina · sinb.

(5) For every real number a holds sin(a − π) = − sin(a) and cos(a − π) = − cos(a).

(6) For every real number a holds sin(a − π) = − sina and cos(a − π) = − cosa.

(7) For all real numbers a, b such that a ∈ ]0, \frac{π}{2} [ and b ∈ ]0, \frac{π}{2} [ holds a < b iff sina < sinb.

(8) For all real numbers a, b such that a ∈ ]\frac{π}{2} , π[ and b ∈ ]\frac{π}{2} , π[ holds a < b iff sina > sinb.

(9) For every real number a and for every integer i holds sina = sin(2·π·i+a).

(10) For every real number a and for every integer i holds cosa = cos(2 · π · i + a).

(11) For every real number a such that sina = 0 holds cosa ≠ 0.

(12) For all real numbers a, b such that 0 a and a < 2 · π and 0 b and b < 2 · π and sina = sinb and cosa = cosb holds a = b.

2. More on the Argument of a Complex Number

Let us observe that ℂ_{F} is non empty.

Let z be an element of ℂ. The functor Ftize(z) yields an element of the carrier of ℂ_{F} and is defined as follows:

(Def. 1) Ftize(z) = z.

We now state four propositions:

(13) For every element z of ℂ holds ℜ(z) = ℜ(Ftize(z)) and ℑ(z) = ℑ(Ftize(z)).

(14) For all elements x, y of ℂ holds Ftize(x + y) = Ftize(x) + Ftize(y).

(15) For every element z of ℂ holds z = 0_{ℂ} iff Ftize(z) = 0_{ℂ_{F}}.

(16) For every element z of ℂ holds |z| = | Ftize(z)|.

Let z be an element of ℂ. The functor Argz yields a real number and is defined as follows:

(Def. 2) Argz = ArgFtize(z).

One can prove the following propositions:

(17) For every element z of ℂ and for every element u of the carrier of ℂ_{F} such that z = u holds Argz = Argu.

(18) For every element z of ℂ holds 0 Argz and Argz < 2 · π.

(19) For every element z of ℂ holds z = |z| · cosArgz + (|z| · sinArgz)i.

(20) Arg(0_{ℂ}) = 0.

(21) Let z be an element of ℂ and r be a real number. If z ≠ 0 and z = |z| · cosr + (|z| · sinr)i and 0 r and r < 2 · π, then r = Argz.

(22) For every element z of ℂ such that z ≠ 0_{ℂ} holds if Argz < π, then Arg(− z) = Argz + π and if Argz π, then Arg(− z) = Argz − π.

(23) For every real number r such that r 0 holds Arg(r + 0i) = 0.

(24) For every element z of ℂ holds Argz = 0 iff z = |z| + 0i.

(25) For every element z of ℂ such that z ≠ 0_{ℂ} holds Argz < π iff Arg(− z) π.

(26) For all elements x, y of ℂ such that x ≠ y or x−y ≠ 0_{ℂ} holds Arg(x−y) < π iff Arg(y − x) π.

(27) For every element z of ℂ holds Argz ∈ ]0, π[ iff ℑ(z) > 0.

(28) For every element z of ℂ such that Argz ≠ 0 holds Argz < π iff sinArgz > 0.

(29) For all elements x, y of ℂ such that Argx < π and Argy < π holds Arg(x + y) < π.

(30) For every real number x such that x > 0 holds Arg(0 + xi) = \frac{π}{2} .

(31) For every element z of ℂ holds Argz ∈ ]0, \frac{π}{2} [ iff ℜ(z) > 0 and ℑ(z) > 0.

(32) For every element z of ℂ holds Argz ∈ ]\frac{π}{2} , π[ iff ℜ(z) < 0 and ℑ(z) > 0.

(33) For every element z of ℂ such that ℑ(z) > 0 holds sinArgz > 0.

(34) For every element z of ℂ holds Argz = 0 iff ℜ(z) 0 and ℑ(z) = 0.

(35) For every element z of ℂ holds Argz = π iff ℜ(z) < 0 and ℑ(z) = 0.

(36) For every element z of ℂ holds ℑ(z) = 0 iff Argz = 0 or Argz = π.

(37) For every element z of ℂ such that Argz π holds ℑ(z) 0.

(38) For every element z of ℂ such that z ≠ 0 holds cosArg(− z) = − cosArgz and sinArg(− z) = − sinArgz.

(39) For every element a of ℂ such that a ≠ 0 holds cosArga = \frac{ℜ(a)}{|a|} and sinArga = \frac{ℑ(a)}{|a|} .

(40) For every element a of ℂ and for every real number r such that r > 0 holds Arg(a · (r + 0i)) = Arga.

(41) For every element a of ℂ and for every real number r such that r < 0 holds Arg(a · (r + 0i)) = Arg(− a).

3. Inner Product

Let x, y be elements of ℂ. The functor (x|y) yielding an element of ℂ is defined by:

(Def. 3) (x|y) = x · \overline{y}. In the sequel a, b, c, d, x, y, z are elements of ℂ. The following propositions are true:

(42) (x|y) = (ℜ(x) · ℜ(y) + ℑ(x) · ℑ(y)) + (− ℜ(x) · ℑ(y) + ℑ(x) · ℜ(y))i.

(43) (z|z) = (ℜ(z) · ℜ(z) + ℑ(z) · ℑ(z)) + 0i and (z|z) = (ℜ(z)^{2} + ℑ(z)^{2}) + 0i.

(44) (z|z) = |z|^{2} + 0i and |z|^{2} = ℜ((z|z)).

(45) |(x|y)| = |x| · |y|.

(46) If (x|x) = 0, then x = 0.

(47) (y|x) = \overline{(x|y)}.

(48) ((x + y)|z) = (x|z) + (y|z).

(49) (x|(y + z)) = (x|y) + (x|z).

(50) ((a · x)|y) = a · (x|y).

(51) (x|(a · y)) = \overline{a} · (x|y).

(52) ((a · x)|y) = (x|(\overline{a} · y)).

(53) ((a · x + b · y)|z) = a · (x|z) + b · (y|z).

(54) (x|(a · y + b · z)) = \overline{a} · (x|y) + \overline{b} · (x|z).

(55) ((− x)|y) = (x|− y).

(56) ((− x)|y) = − (x|y).

(57) − (x|y) = (x|− y).

(58) ((− x)|− y) = (x|y).

(59) ((x − y)|z) = (x|z) − (y|z).

(60) (x|(y − z)) = (x|y) − (x|z).

(61) (0_{ℂ}|x) = 0_{ℂ} and (x|0_{ℂ}) = 0_{ℂ}.

(62) ((x + y)|(x + y)) = (x|x) + (x|y) + (y|x) + (y|y).

(63) ((x − y)|(x − y)) = ((x|x) − (x|y) − (y|x)) + (y|y).

(64) ℜ((x|y)) = 0 iff ℑ((x|y)) = |x| · |y| or ℑ((x|y)) = − |x| · |y|.

4. Rotation

Let a be an element of ℂ and let r be a real number. The functor a ⟲ r yielding an element of ℂ is defined as follows:

(Def. 4) a ⟲ r = |a| · cos(r + Arga) + (|a| · sin(r + Arga))i. In the sequel r denotes a real number. We now state a number of propositions:

(65) a ⟲ 0 = a.

(66) a ⟲ r = 0_{ℂ} iff a = 0_{ℂ}.

(67) |a ⟲ r| = |a|.

(68) If a ≠ 0_{ℂ}, then there exists an integer i such that Arg(a ⟲ r) = 2 · π · i + (r + Arga).

(69) a ⟲ − Arga = |a| + 0i.

(70) ℜ(a ⟲ r) = ℜ(a)·cosr−ℑ(a)·sinr and ℑ(a ⟲ r) = ℜ(a)·sinr+ℑ(a)·cosr.

(71) a + b ⟲ r = (a ⟲ r) + (b ⟲ r).

(72) − a ⟲ r = − (a ⟲ r).

(73) a − b ⟲ r = (a ⟲ r) − (b ⟲ r).

(74) a ⟲ π = − a.

5. Angles

Let a, b be elements of ℂ. The functor ∡(a, b) yielding a real number is defined by:

(Def. 5) ∡(a, b) = \{ \begin{matrix}Arg(b ⟲ − Arga), if Arga = 0 or b ≠ 0, \\ 2 · π − Arga, otherwise. \end{matrix}

Next we state several propositions:

(75) If r 0, then ∡(r + 0i, a) = Arga.

(76) If Arga = Argb and a ≠ 0 and b ≠ 0, then Arg(a ⟲ r) = Arg(b ⟲ r).

(77) If r > 0, then ∡(a, b) = ∡(a · (r + 0i), b · (r + 0i)).

(78) If a ≠ 0 and b ≠ 0 and Arga = Argb, then Arg(− a) = Arg(− b).

(79) If a ≠ 0 and b ≠ 0, then ∡(a, b) = ∡(a ⟲ r, b ⟲ r).

(80) If r < 0 and a ≠ 0 and b ≠ 0, then ∡(a, b) = ∡(a · (r + 0i), b · (r + 0i)).

(81) If a ≠ 0 and b ≠ 0, then ∡(a, b) = ∡(− a, − b).

(82) If b ≠ 0 and ∡(a, b) = 0, then ∡(a, − b) = π.

(83) If a ≠ 0 and b ≠ 0, then cos∡(a, b) = \frac{ℜ((a|b))}{|a|·|b|} and sin∡(a, b) = − \frac{ℑ((a|b))}{|a|·|b|} . Let x, y, z be elements of ℂ. The functor ∡(x, y, z) yielding a real number

is defined as follows:

(Def. 6) ∡(x, y, z) = \{ \begin{matrix}Arg(z − y) − Arg(x − y), if Arg(z − y) − Arg(x − y) 0, \\ 2 · π + (Arg(z − y) − Arg(x − y)), otherwise. \end{matrix}

One can prove the following propositions:

(84) 0 ∡(x, y, z) and ∡(x, y, z) < 2 · π.

(85) ∡(x, y, z) = ∡(x − y, 0_{ℂ}, z − y).

(86) ∡(a, b, c) = ∡(a + d, b + d, c + d).

(87) ∡(a, b) = ∡(a, 0_{ℂ}, b).

(88) If ∡(x, y, z) = 0, then Arg(x − y) = Arg(z − y) and ∡(z, y, x) = 0.

(89) If a ≠ 0_{ℂ} and b ≠ 0_{ℂ}, then ℜ((a|b)) = 0 iff ∡(a, 0_{ℂ}, b) = \frac{π}{2} or ∡(a, 0_{ℂ}, b) = \frac{3}{2} · π.

(90) If a ≠ 0_{ℂ} and b ≠ 0_{ℂ}, then ℑ((a|b)) = |a| · |b| or ℑ((a|b)) = − |a| · |b|iff ∡(a, 0_{ℂ}, b) = \frac{π}{2} or ∡(a, 0_{ℂ}, b) = \frac{3}{2} · π.

(91) If x ≠ y and if z ≠ y and if ∡(x, y, z) = \frac{π}{2} or ∡(x, y, z) = \frac{3}{2} · π, then |x − y|^{2} + |z − y|^{2} = |x − z|^{2}.

(92) If a ≠ b and b ≠ c, then ∡(a, b, c) = ∡(a ⟲ r, b ⟲ r, c ⟲ r).

(93) ∡(a, b, a) = 0.

(94) ∡(a, b, c) ≠ 0 iff ∡(a, b, c) + ∡(c, b, a) = 2 · π.

(95) If ∡(a, b, c) ≠ 0, then ∡(c, b, a) ≠ 0.

(96) If ∡(a, b, c) = π, then ∡(c, b, a) = π.

280 wenpai chang et al.

(97) If a ≠ b and a ≠ c and b ≠ c, then ∡(a, b, c) ≠ 0 or ∡(b, c, a) ≠ 0 or ∡(c, a, b) ≠ 0.

(98) If a ≠ b and b ≠ c and 0 < ∡(a, b, c) and ∡(a, b, c) < π, then ∡(a, b, c) + ∡(b, c, a) + ∡(c, a, b) = π and 0 < ∡(b, c, a) and 0 < ∡(c, a, b).

(99) If a ≠ b and b ≠ c and ∡(a, b, c) > π, then ∡(a, b, c) + ∡(b, c, a) + ∡(c, a, b) = 5 · π and ∡(b, c, a) > π and ∡(c, a, b) > π.

(100) If a ≠ b and b ≠ c and ∡(a, b, c) = π, then ∡(b, c, a) = 0 and ∡(c, a, b) = 0.

(101) If a ≠ b and a ≠ c and b ≠ c and ∡(a, b, c) = 0, then ∡(b, c, a) = 0 and ∡(c, a, b) = π or ∡(b, c, a) = π and ∡(c, a, b) = 0.

(102) ∡(a, b, c)+∡(b, c, a)+∡(c, a, b) = π or ∡(a, b, c)+∡(b, c, a)+∡(c, a, b) = 5 · π iff a ≠ b and a ≠ c and b ≠ c.

References

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

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

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

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

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

[6] Robert Milewski. Trigonometric form of complex numbers. Formalized Mathematics, 9(3):455–460, 2001.

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

[8] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

[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.

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

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

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

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

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

Angle and Triangle in Euclidean Topological Space

Akihiro Kubo Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. Two transformations between the complex space and 2dimensional Euclidean topological space are defined. By them, the concept of argument is induced to 2-dimensional vectors using argument of complex number. Similarly, the concept of an angle is introduced using the angle of two complex numbers. The concept of a triangle and related concepts are also defined in n-dimensional Euclidean topological spaces.

MML Identifier: EUCLID 3.

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

We follow the rules: z, z_{1}, z_{2} are elements of ℂ, r, r_{1}, r_{2}, x_{1}, x_{2} are real numbers, and p, p_{1}, p_{2}, p_{3}, q are points of E_{T}^{2}.

Let z be an element of ℂ. The functor cpx2euc(z) yielding a point of E_{T}^{2} is defined by:

(Def. 1) cpx2euc(z) = [ℜ(z), ℑ(z)].

Let p be a point of E_{T}^{2}. The functor euc2cpx(p) yields an element of ℂ and is defined as follows:

(Def. 2) euc2cpx(p) = p_{1} + p_{2}i. One can prove the following propositions:

(1) euc2cpx(cpx2euc(z)) = z.

(2) cpx2euc(euc2cpx(p)) = p.

(3) For every p there exists z such that p = cpx2euc(z).

(4) For every z there exists p such that z = euc2cpx(p).

(5) For all z_{1}, z_{2} such that cpx2euc(z_{1}) = cpx2euc(z_{2}) holds z_{1} = z_{2}.

(6) For all p_{1}, p_{2} such that euc2cpx(p_{1}) = euc2cpx(p_{2}) holds p_{1} = p_{2}.

(7) (cpx2euc(z))_{1} = ℜ(z) and (cpx2euc(z))_{2} = ℑ(z).

(8) ℜ(euc2cpx(p)) = p_{1} and ℑ(euc2cpx(p)) = p_{2}.

(9) cpx2euc(x_{1} + x_{2}i) = [x_{1}, x_{2}].

(10) [ℜ(z_{1} + z_{2}), ℑ(z_{1} + z_{2})] = [ℜ(z_{1}) + ℜ(z_{2}), ℑ(z_{1}) + ℑ(z_{2})].

(11) cpx2euc(z_{1} + z_{2}) = cpx2euc(z_{1}) + cpx2euc(z_{2}).

(12) (p_{1} + p_{2})_{1} + (p_{1} + p_{2})_{2}i = ((p_{1})_{1} + (p_{2})_{1}) + ((p_{1})_{2} + (p_{2})_{2})i.

(13) euc2cpx(p_{1} + p_{2}) = euc2cpx(p_{1}) + euc2cpx(p_{2}).

(14) [ℜ(− z), ℑ(− z)] = [− ℜ(z) , − ℑ(z) ].

(15) cpx2euc(− z) = − cpx2euc(z).

(16) (− p)_{1} + (− p)_{2}i = − p_{1} + (− p_{2})i.

(17) euc2cpx(− p) = − euc2cpx(p).

(18) cpx2euc(z_{1} − z_{2}) = cpx2euc(z_{1}) − cpx2euc(z_{2}).

(19) euc2cpx(p_{1} − p_{2}) = euc2cpx(p_{1}) − euc2cpx(p_{2}).

(20) cpx2euc(0_{ℂ}) = 0_{E_{T}^{2}}.

(21) euc2cpx(0_{E_{T}^{2}}) = 0_{ℂ}.

(22) If euc2cpx(p) = 0_{ℂ}, then p = 0_{E_{T}^{2}}.

(23) cpx2euc((r + 0i) · z) = r · cpx2euc(z).

(24) (r + 0i) · (r_{1} + r_{2}i) = r · r_{1} + (r · r_{2})i.

(25) euc2cpx(r · p) = (r + 0i) · euc2cpx(p).

(26) | euc2cpx(p)| = \sqrt{(p_{1})^{2} + (p_{2})^{2}}.

(27) For every finite sequence f of elements of ℝ such that lenf = 2 holds |f| = \sqrt{f(1)^{2} + f(2)^{2}}.

(28) For every finite sequence f of elements of ℝ and for every point p of E_{T}^{2} such that lenf = 2 and p = f holds |p| = |f|.

(29) | cpx2euc(z)| = \sqrt{ℜ(z)^{2} + ℑ(z)^{2}} .

(30) | cpx2euc(z)| = |z|.

(31) | euc2cpx(p)| = |p|.

Let us consider p. The functor Argp yields a real number and is defined as follows:

(Def. 3) Argp = Argeuc2cpx(p).

We now state a number of propositions:

(32) For every element z of ℂ and for every p such that z = euc2cpx(p) or p = cpx2euc(z) holds Argz = Argp.

(33) For every p holds 0 Argp and Argp < 2 · π.

(34) For all real numbers x_{1}, x_{2} and for every p such that x_{1} = |p| · cosArgp and x_{2} = |p| · sinArgp holds p = [x_{1}, x_{2}].

(35) Arg(0_{E_{T}^{2}}) = 0.

(36) For every p such that p ≠ 0_{E_{T}^{2}} holds if Argp < π, then Arg(− p) = Argp + π and if Argp π, then Arg(− p) = Argp − π.

(37) For every p such that Argp = 0 holds p = [|p|, 0] and p_{2} = 0.

(38) For every p such that p ≠ 0_{E_{T}^{2}} holds Argp < π iff Arg(− p) π.

(39) For all p_{1}, p_{2} such that p_{1} ≠ p_{2} or p_{1} − p_{2} ≠ 0_{E_{T}^{2}} holds Arg(p_{1} − p_{2}) < π iff Arg(p_{2} − p_{1}) π.

(40) For every p holds Argp ∈ ]0, π[ iff p_{2} > 0.

(41) For every p such that Argp ≠ 0 holds Argp < π iff sinArgp > 0.

(42) For all p_{1}, p_{2} such that Argp_{1} < π and Argp_{2} < π holds Arg(p_{1}+p_{2}) < π. Let us consider p_{1}, p_{2}, p_{3}. The functor ∡(p_{1}, p_{2}, p_{3}) yielding a real number

is defined as follows: (Def. 4) ∡(p_{1}, p_{2}, p_{3}) = ∡(euc2cpx(p_{1}), euc2cpx(p_{2}), euc2cpx(p_{3})).

The following propositions are true:

(43) For all p_{1}, p_{2}, p_{3} holds 0 ∡(p_{1}, p_{2}, p_{3}) and ∡(p_{1}, p_{2}, p_{3}) < 2 · π.

(44) For all p_{1}, p_{2}, p_{3} holds ∡(p_{1}, p_{2}, p_{3}) = ∡(p_{1} − p_{2}, 0_{E_{T}^{2}}, p_{3} − p_{2}).

(45) For all p_{1}, p_{2}, p_{3} such that ∡(p_{1}, p_{2}, p_{3}) = 0 holds Arg(p_{1}−p_{2}) = Arg(p_{3}− p_{2}) and ∡(p_{3}, p_{2}, p_{1}) = 0.

(46) For all p_{1}, p_{2}, p_{3} such that ∡(p_{1}, p_{2}, p_{3}) ≠ 0 holds ∡(p_{3}, p_{2}, p_{1}) = 2 · π − ∡(p_{1}, p_{2}, p_{3}).

(47) For all p_{1}, p_{2}, p_{3} such that ∡(p_{3}, p_{2}, p_{1}) ≠ 0 holds ∡(p_{3}, p_{2}, p_{1}) = 2 · π − ∡(p_{1}, p_{2}, p_{3}).

(48) For all elements x, y of ℂ holds ℜ((x|y)) = ℜ(x) · ℜ(y) + ℑ(x) · ℑ(y).

(49) For all elements x, y of ℂ holds ℑ((x|y)) = − ℜ(x) · ℑ(y) + ℑ(x) · ℜ(y).

(50) For all p, q holds |(p, q)| = p_{1} · q_{1} + p_{2} · q_{2}.

(51) For all p_{1}, p_{2} holds |(p_{1}, p_{2})| = ℜ((euc2cpx(p_{1})| euc2cpx(p_{2}))).

(52) For all p_{1}, p_{2}, p_{3} such that p_{1} ≠ 0_{E_{T}^{2}} and p_{2} ≠ 0_{E_{T}^{2}} holds |(p_{1}, p_{2})| = 0 iff ∡(p_{1}, 0_{E_{T}^{2}}, p_{2}) = \frac{π}{2} or ∡(p_{1}, 0_{E_{T}^{2}}, p_{2}) = \frac{3}{2} · π.

(53) Let given p_{1}, p_{2}. Suppose p_{1} ≠ 0_{E_{T}^{2}} and p_{2} ≠ 0_{E_{T}^{2}}. Then − (p_{1})_{1} · (p_{2})_{2}+ (p_{1})_{2} · (p_{2})_{1} = |p_{1}| · |p_{2}| or − (p_{1})_{1} · (p_{2})_{2}+ (p_{1})_{2} · (p_{2})_{1} = − |p_{1}| · |p_{2}|if and only if ∡(p_{1}, 0_{E_{T}^{2}}, p_{2}) = \frac{π}{2} or ∡(p_{1}, 0_{E_{T}^{2}}, p_{2}) = \frac{3}{2} · π.

(54) For all p_{1}, p_{2}, p_{3} such that p_{1} ≠ p_{2} and p_{3} ≠ p_{2} holds |(p_{1}−p_{2}, p_{3}−p_{2})| = 0 iff ∡(p_{1}, p_{2}, p_{3}) = \frac{π}{2} or ∡(p_{1}, p_{2}, p_{3}) = \frac{3}{2} · π.

(55) For all p_{1}, p_{2}, p_{3} such that p_{1} ≠ p_{2} but p_{3} ≠ p_{2} but ∡(p_{1}, p_{2}, p_{3}) = \frac{π}{2} or ∡(p_{1}, p_{2}, p_{3}) = \frac{3}{2} · π holds |p_{1} − p_{2}|^{2} + |p_{3} − p_{2}|^{2} = |p_{1} − p_{3}|^{2}.

(56) For all p_{1}, p_{2}, p_{3} such that p_{2} ≠ p_{1} and p_{1} ≠ p_{3} and p_{3} ≠ p_{2} and ∡(p_{2}, p_{1}, p_{3}) < π and ∡(p_{1}, p_{3}, p_{2}) < π and ∡(p_{3}, p_{2}, p_{1}) < π holds ∡(p_{2}, p_{1}, p_{3}) + ∡(p_{1}, p_{3}, p_{2}) + ∡(p_{3}, p_{2}, p_{1}) = π.

Let n be a natural number and let p_{1}, p_{2}, p_{3} be points of E_{T}^{n}. The functor Triangle(p_{1}, p_{2}, p_{3}) yields a subset of E_{T}^{n} and is defined as follows:

(Def. 5) Triangle(p_{1}, p_{2}, p_{3}) = L(p_{1}, p_{2}) ∪ L(p_{2}, p_{3}) ∪ L(p_{3}, p_{1}).

Let n be a natural number and let p_{1}, p_{2}, p_{3} be points of E_{T}^{n}. The functor ClInsideOfTriangle(p_{1}, p_{2}, p_{3}) yields a subset of E_{T}^{n} and is defined as follows:

(Def. 6) ClInsideOfTriangle(p_{1}, p_{2}, p_{3}) = {p; p ranges over points of E_{T}^{n}: ⋁_{a_{1},a_{2},a_{3} : real number} (0 a_{1} ∧ 0 a_{2} ∧ 0 a_{3} ∧ a_{1} + a_{2} + a_{3} = 1 ∧ p = a_{1} · p_{1} + a_{2} · p_{2} + a_{3} · p_{3})}.

Let n be a natural number and let p_{1}, p_{2}, p_{3} be points of E_{T}^{n}. The functor InsideOfTriangle(p_{1}, p_{2}, p_{3}) yielding a subset of E_{T}^{n} is defined by:

(Def. 7) InsideOfTriangle(p_{1}, p_{2}, p_{3}) = ClInsideOfTriangle(p_{1}, p_{2}, p_{3})\Triangle(p_{1},

p_{2}, p_{3}). Let n be a natural number and let p_{1}, p_{2}, p_{3} be points of E_{T}^{n}. The functor

OutsideOfTriangle(p_{1}, p_{2}, p_{3}) yielding a subset of E_{T}^{n} is defined by the condition (Def. 8).

(Def. 8) OutsideOfTriangle(p_{1}, p_{2}, p_{3}) = {p; p ranges over points of E_{T}^{n}: ⋁_{a_{1},a_{2},a_{3} : real number} ((0 > a_{1} ∨ 0 > a_{2} ∨ 0 > a_{3}) ∧ a_{1} + a_{2} + a_{3} = 1 ∧ p = a_{1} · p_{1} + a_{2} · p_{2} + a_{3} · p_{3})}.

Let n be a natural number and let p_{1}, p_{2}, p_{3} be points of E_{T}^{n}. The functor plane(p_{1}, p_{2}, p_{3}) yielding a subset of E_{T}^{n} is defined as follows:

(Def. 9) plane(p_{1}, p_{2}, p_{3}) = OutsideOfTriangle(p_{1}, p_{2}, p_{3})∪ClInsideOfTriangle(p_{1},

p_{2}, p_{3}). One can prove the following propositions:

(57) Let n be a natural number and p_{1}, p_{2}, p_{3}, p be points of E_{T}^{n}. Suppose p ∈ plane(p_{1}, p_{2}, p_{3}). Then there exist real numbers a_{1}, a_{2}, a_{3} such that a_{1} + a_{2} + a_{3} = 1 and p = a_{1} · p_{1} + a_{2} · p_{2} + a_{3} · p_{3}.

(58) For every natural number n and for all points p_{1}, p_{2}, p_{3} of E_{T}^{n} holds

Triangle(p_{1}, p_{2}, p_{3}) ⊆ ClInsideOfTriangle(p_{1}, p_{2}, p_{3}). Let n be a natural number and let q_{1}, q_{2} be points of E_{T}^{n}. We say that q_{1}, q_{2}

are lindependent2 if and only if: (Def. 10) For all real numbers a_{1}, a_{2} such that a_{1} · q_{1} + a_{2} · q_{2} = 0_{E_{T}^{n}} holds a_{1} = 0

and a_{2} = 0. We introduce q_{1}, q_{2} are ldependent2 as an antonym of q_{1}, q_{2} are lindependent2.

One can prove the following propositions:

(59) Let n be a natural number and q_{1}, q_{2} be points of E_{T}^{n}. If q_{1}, q_{2} are lindependent2, then q_{1} ≠ q_{2} and q_{1} ≠ 0_{E_{T}^{n}} and q_{2} ≠ 0_{E_{T}^{n}}.

(60) Let n be a natural number and p_{1}, p_{2}, p_{3}, p_{0} be points of E_{T}^{n}. Suppose p_{2} − p_{1}, p_{3} − p_{1} are lindependent2 and p_{0} ∈ plane(p_{1}, p_{2}, p_{3}). Then there exist real numbers a_{1}, a_{2}, a_{3} such that

(i) p_{0} = a_{1} · p_{1} + a_{2} · p_{2} + a_{3} · p_{3}, (ii) a_{1} + a_{2} + a_{3} = 1, and (iii) for all real numbers b_{1}, b_{2}, b_{3} such that p_{0} = b_{1} · p_{1} + b_{2} · p_{2} + b_{3} · p_{3} and

b_{1} + b_{2} + b_{3} = 1 holds b_{1} = a_{1} and b_{2} = a_{2} and b_{3} = a_{3}.

(61) Let n be a natural number and p_{1}, p_{2}, p_{3}, p_{0} be points of E_{T}^{n}. Given real numbers a_{1}, a_{2}, a_{3} such that p_{0} = a_{1}·p_{1}+a_{2}·p_{2}+a_{3}·p_{3} and a_{1}+a_{2}+a_{3} = 1. Then p_{0} ∈ plane(p_{1}, p_{2}, p_{3}).

(62) Let n be a natural number and p_{1}, p_{2}, p_{3} be points of E_{T}^{n}. Then plane(p_{1}, p_{2}, p_{3}) = {p; p ranges over points of E_{T}^{n}: ⋁_{a_{1},a_{2},a_{3} : real number} (a_{1} + a_{2} + a_{3} = 1 ∧ p = a_{1} · p_{1} + a_{2} · p_{2} + a_{3} · p_{3})}.

(63) For all p_{1}, p_{2}, p_{3} such that p_{2} − p_{1}, p_{3} − p_{1} are lindependent2 holds

plane(p_{1}, p_{2}, p_{3}) = R^{2}. Let n be a natural number and let p_{1}, p_{2}, p_{3}, p be points of E_{T}^{n}. Let us assume

that p_{2} − p_{1}, p_{3} − p_{1} are lindependent2 and p ∈ plane(p_{1}, p_{2}, p_{3}). The functor tricord1(p_{1}, p_{2}, p_{3}, p) yields a real number and is defined as follows:

(Def. 11) There exist real numbers a_{2}, a_{3} such that tricord1(p_{1}, p_{2}, p_{3}, p)+a_{2}+a_{3} =

1 and p = tricord1(p_{1}, p_{2}, p_{3}, p) · p_{1} + a_{2} · p_{2} + a_{3} · p_{3}. Let n be a natural number and let p_{1}, p_{2}, p_{3}, p be points of E_{T}^{n}. Let us assume

that p_{2} − p_{1}, p_{3} − p_{1} are lindependent2 and p ∈ plane(p_{1}, p_{2}, p_{3}). The functor tricord2(p_{1}, p_{2}, p_{3}, p) yielding a real number is defined as follows:

(Def. 12) There exist real numbers a_{1}, a_{3} such that a_{1}+tricord2(p_{1}, p_{2}, p_{3}, p)+a_{3} =

1 and p = a_{1} · p_{1} + tricord2(p_{1}, p_{2}, p_{3}, p) · p_{2} + a_{3} · p_{3}. Let n be a natural number and let p_{1}, p_{2}, p_{3}, p be points of E_{T}^{n}. Let us assume

that p_{2} − p_{1}, p_{3} − p_{1} are lindependent2 and p ∈ plane(p_{1}, p_{2}, p_{3}). The functor tricord2(p_{1}, p_{2}, p_{3}, p) yielding a real number is defined as follows:

(Def. 13) There exist real numbers a_{1}, a_{2} such that a_{1}+a_{2}+tricord2(p_{1}, p_{2}, p_{3}, p) =

1 and p = a_{1} · p_{1} + a_{2} · p_{2} + tricord2(p_{1}, p_{2}, p_{3}, p) · p_{3}. Let us consider p_{1}, p_{2}, p_{3}. The functor trcmap1(p_{1}, p_{2}, p_{3}) yielding a map

from E_{T}^{2} into ℝ^{1} is defined as follows: (Def. 14) For every p holds (trcmap1(p_{1}, p_{2}, p_{3}))(p) = tricord1(p_{1}, p_{2}, p_{3}, p).

Let us consider p_{1}, p_{2}, p_{3}. The functor trcmap2(p_{1}, p_{2}, p_{3}) yields a map from E_{T}^{2} into ℝ^{1} and is defined as follows:

(Def. 15) For every p holds (trcmap2(p_{1}, p_{2}, p_{3}))(p) = tricord2(p_{1}, p_{2}, p_{3}, p).

Let us consider p_{1}, p_{2}, p_{3}. The functor trcmap3(p_{1}, p_{2}, p_{3}) yielding a map from E_{T}^{2} into ℝ^{1} is defined by:

(Def. 16) For every p holds (trcmap3(p_{1}, p_{2}, p_{3}))(p) = tricord2(p_{1}, p_{2}, p_{3}, p).

Next we state several propositions:

(64) Let given p_{1}, p_{2}, p_{3}, p. Suppose p_{2} − p_{1}, p_{3} − p_{1} are lindependent2. Then p ∈ OutsideOfTriangle(p_{1}, p_{2}, p_{3}) if and only if one of the following conditions is satisfied:

(i) tricord1(p_{1}, p_{2}, p_{3}, p) < 0, or (ii) tricord2(p_{1}, p_{2}, p_{3}, p) < 0, or (iii) tricord2(p_{1}, p_{2}, p_{3}, p) < 0.

(65) Let given p_{1}, p_{2}, p_{3}, p. Suppose p_{2} − p_{1}, p_{3} − p_{1} are lindependent2. Then

p ∈ Triangle(p_{1}, p_{2}, p_{3}) if and only if the following conditions are satisfied:

(i) tricord1(p_{1}, p_{2}, p_{3}, p) 0, (ii) tricord2(p_{1}, p_{2}, p_{3}, p) 0, (iii) tricord2(p_{1}, p_{2}, p_{3}, p) 0, and (iv) tricord1(p_{1}, p_{2}, p_{3}, p) = 0 or tricord2(p_{1}, p_{2}, p_{3}, p) = 0 or

tricord2(p_{1}, p_{2}, p_{3}, p) = 0.

(66) Let given p_{1}, p_{2}, p_{3}, p. Suppose p_{2} − p_{1}, p_{3} − p_{1} are lindependent2. Then p ∈ Triangle(p_{1}, p_{2}, p_{3}) if and only if one of the following conditions is satisfied:

(i) tricord1(p_{1}, p_{2}, p_{3}, p) = 0 and tricord2(p_{1}, p_{2}, p_{3}, p) 0 and tricord2(p_{1}, p_{2}, p_{3}, p) 0, or

(ii) tricord1(p_{1}, p_{2}, p_{3}, p) 0 and tricord2(p_{1}, p_{2}, p_{3}, p) = 0 and

tricord2(p_{1}, p_{2}, p_{3}, p) 0, or (iii) tricord1(p_{1}, p_{2}, p_{3}, p) 0 and tricord2(p_{1}, p_{2}, p_{3}, p) 0 and

tricord2(p_{1}, p_{2}, p_{3}, p) = 0.

(67) Let given p_{1}, p_{2}, p_{3}, p. Suppose p_{2} − p_{1}, p_{3} − p_{1} are lindependent2. Then p ∈ InsideOfTriangle(p_{1}, p_{2}, p_{3}) if and only if the following conditions are satisfied:

(i) tricord1(p_{1}, p_{2}, p_{3}, p) > 0, (ii) tricord2(p_{1}, p_{2}, p_{3}, p) > 0, and (iii) tricord2(p_{1}, p_{2}, p_{3}, p) > 0.

(68) For all p_{1}, p_{2}, p_{3} such that p_{2} − p_{1}, p_{3} − p_{1} are lindependent2 holds InsideOfTriangle(p_{1}, p_{2}, p_{3}) is non empty.

References

[1] Kanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of n-dimensional topological space. Formalized Mathematics, 11(2):179–183, 2003.

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

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

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

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

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

angle and triangle in euclidean … 287

[7] Wenpai Chang, Yatsuka Nakamura, and Piotr Rudnicki. Inner products and angles of complex numbers. Formalized Mathematics, 11(3):275–280, 2003.

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

[9] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

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

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

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

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

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

[15] Agnieszka Sakowicz, Jarosław Gryko, and Adam Grabowski. Sequences in E_{T}^{N}. Formalized Mathematics, 5(1):93–96, 1996.

[16] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

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

The Class of Series-Parallel Graphs. Part II^{1}

Krzysztof Retel University of Białystok

Summary. In this paper we introduce two new operations on graphs: sum and union corresponding to parallel and series operation respectively. We determine N-free graph as the graph that does not embed Necklace 4. We define “fin RelStr” as the set of all graphs with finite carriers. We also define the smallest class of graphs which contains the one-element graph and which is closed under parallel and series operations. The goal of the article is to prove the theorem that the class of finite series-parallel graphs is the class of finite N-free graphs. This paper formalizes the next part of [12].

MML Identifier: NECKLA 2.

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

In this paper U denotes a universal class. Next we state two propositions:

(1) For all sets X, Y such that X ∈ U and Y ∈ U and for every relation R between X and Y holds R ∈ U.

(2) The internal relation of Necklace4 = {⟨⟨0, 1⟩⟩, ⟨⟨1, 0⟩⟩, ⟨⟨1, 2⟩⟩, ⟨⟨2, 1⟩⟩, ⟨⟨2, 3⟩⟩, ⟨⟨3, 2⟩⟩}.

Let n be a natural number. One can check that every element of R_{n} is finite. Next we state the proposition

(3) For every set x such that x ∈ U_{0} holds x is finite. Let us mention that every element of U_{0} is finite. Let us note that every number which is finite and ordinal is also natural.

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

Let G be a non empty relational structure. We say that G is N-free if and only if:

(Def. 1) G does not embed Necklace4.

Let us mention that there exists a non empty relational structure which is N-free.

Let R, S be relational structures. The functor UnionOf(R, S) yielding a strict relational structure is defined by the conditions (Def. 2).

(Def. 2)(i) The carrier of UnionOf(R, S) = (the carrier of R)∪(the carrier of S),

and (ii) the internal relation of UnionOf(R, S) = (the internal relation of R) ∪

(the internal relation of S). Let R, S be relational structures. The functor SumOf(R, S) yielding a strict

relational structure is defined by the conditions (Def. 3). (Def. 3)(i) The carrier of SumOf(R, S) = (the carrier of R) ∪ (the carrier of S),

and (ii) the internal relation of SumOf(R, S) = (the internal relation of R)∪(the

internal relation of S) ∪ [: the carrier of R, the carrier of S:] ∪ [: the carrier of S, the carrier of R:].

The functor FinRelStr is defined by the condition (Def. 4). (Def. 4) Let X be a set. Then X ∈ FinRelStr if and only if there exists a strict

relational structure R such that X = R and the carrier of R ∈ U_{0}. Let us mention that FinRelStr is non empty. The subset FinRelStrSp of FinRelStr is defined by the conditions (Def. 5).

(Def. 5)(i) For every strict relational structure R such that the carrier of R is

non empty and trivial and the carrier of R ∈ U_{0} holds R ∈ FinRelStrSp, (ii) for all strict relational structures H_{1}, H_{2} such that the carrier of H_{1} mis-

ses the carrier of H_{2} and H_{1} ∈ FinRelStrSp and H_{2} ∈ FinRelStrSp holds UnionOf(H_{1}, H_{2}) ∈ FinRelStrSp and SumOf(H_{1}, H_{2}) ∈ FinRelStrSp, and

(iii) for every subset M of FinRelStr such that for every strict relational structure R such that the carrier of R is non empty and trivial and the carrier of R ∈ U_{0} holds R ∈ M and for all strict relational structures H_{1}, H_{2} such that the carrier of H_{1} misses the carrier of H_{2} and H_{1} ∈ M and H_{2} ∈ M holds UnionOf(H_{1}, H_{2}) ∈ M and SumOf(H_{1}, H_{2}) ∈ M holds FinRelStrSp ⊆ M.

One can verify that FinRelStrSp is non empty. We now state four propositions:

(4) For every set X such that X ∈ FinRelStrSp holds X is a finite strict non empty relational structure.

(5) For every relational structure R such that R ∈ FinRelStrSp holds the carrier of R ∈ U_{0}.

the class of series-parallel graphs. … 291

(6) Let X be a set. Suppose X ∈ FinRelStrSp. Then

(i) X is a strict non empty trivial relational structure, or (ii) there exist strict relational structures H_{1}, H_{2} such that the carrier of

H_{1} misses the carrier of H_{2} and H_{1} ∈ FinRelStrSp and H_{2} ∈ FinRelStrSp and X = UnionOf(H_{1}, H_{2}) or X = SumOf(H_{1}, H_{2}).

(7) For every strict non empty relational structure R such that R ∈ FinRelStrSp holds R is N-free.

References

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

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

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

[4] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 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] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

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

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

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

[11] Krzysztof Retel. The class of series – parallel graphs. Part I. Formalized Mathematics, 11(1):99–103, 2003.

[12] Stephan Thomasse. On better-quasi-ordering countable series-parallel orders. Transactions of the American Mathematical Society, 352(6):2491–2505, 2000.

[13] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

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

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

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

Characterization and Existence of Gröbner Bases

Christoph Schwarzweller University of Tübingen

Summary. We continue the Mizar formalization of Gröbner bases following [8]. In this article we prove a number of characterizations of Gröbner bases among them that Gröbner bases are convergent rewriting systems. We also show the existence and uniqueness of reduced Gröbner bases.

MML Identifier: GROEB 1.

The papers [24], [31], [33], [32], [10], [5], [17], [29], [28], [11], [13], [4], [2], [30], [9], [7], [15], [16], [12], [20], [19], [25], [27], [18], [1], [6], [14], [22], [26], [23], [3], and [21] provide the terminology and notation for this paper.

1. Preliminaries

Let n be an ordinal number, let L be a right zeroed add-associative right complementable unital distributive non trivial double loop structure, and let p be a polynomial of n, L. Then {p} is a non empty finite subset of Polynom-Ring(n, L).

We now state several propositions:

(1) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and f, p, g be polynomials of n, L. Suppose f reduces to g, p, T. Then there exists a monomial m of n, L such that g = f − m ∗ p.

(2) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and f, p, g be polynomials of n, L. Suppose f reduces to g, p, T. Then there exists a monomial m of n, L such that g = f −m∗p and HT(m∗p, T) ∉ Supportg and HT(m∗p, T) _{T} HT(f, T).

(3) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, f, g be polynomials of n, L, and P, Q be subsets of Polynom-Ring(n, L). If P ⊆ Q, then if f reduces to g, P, T, then f reduces to g, Q, T.

(4) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and P, Q be subsets of Polynom-Ring(n, L). If P ⊆ Q, then PolyRedRel(P, T) ⊆ PolyRedRel(Q, T).

(5) Let n be an ordinal number, L be a right zeroed add-associative right complementable non empty double loop structure, and p be a polynomial of n, L. Then Support(− p) = Supportp.

(6) Let n be an ordinal number, T be a connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, and p be a polynomial of n, L. Then HT(− p, T) = HT(p, T).

(7) Let n be an ordinal number, T be an admissible connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, and p, q be polynomials of n, L. Then HT(p − q, T) _{T} max_{T} (HT(p, T), HT(q, T)).

(8) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and p, q be polynomials of n, L. If Supportq ⊆ Supportp, then q _{T} p.

(9) Let n be an ordinal number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and f, p be non-zero polynomials of n, L. If f is reducible wrt p, T, then HT(p, T) _{T} HT(f, T).

2. Characterization of Gröbner Bases

Next we state two propositions:

(10) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure, and p be a polynomial of n, L. Then PolyRedRel({p}, T) is locally-confluent.

(11) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and P be a subset of Polynom-Ring(n, L). Given a polynomial p of n, L such that p ∈ P and P –ideal = {p}–ideal. Then PolyRedRel(P, T) is locally-confluent.

Let n be an ordinal number, let T be a connected term order of n, let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, and let P be a subset of Polynom-Ring(n, L). The functor HT(P, T) yields a subset of Bagsn and is defined as follows:

(Def. 1) HT(P, T) = {HT(p, T); p ranges over polynomials of n, L: p ∈ P ∧ p ≠

0_{n}L}. Let n be an ordinal number and let S be a subset of Bagsn. The functor

multiples(S) yields a subset of Bagsn and is defined by: (Def. 2) multiples(S) = {b; b ranges over elements of Bagsn : ⋁_{b′: bag of n} (b′ ∈

S ∧ b′| b)}. We now state several propositions:

(12) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and P be a subset of Polynom-Ring(n, L). If PolyRedRel(P, T) is locally-confluent, then PolyRedRel(P, T) is confluent.

(13) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and P be a subset of Polynom-Ring(n, L). If PolyRedRel(P, T) is confluent, then PolyRedRel(P, T) has unique normal form property.

(14) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and P be a subset of Polynom-Ring(n, L). Suppose PolyRedRel(P, T) has unique normal form property. Then PolyRedRel(P, T) has Church-Rosser property.

(15) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and P be a non empty subset of Polynom-Ring(n, L). Suppose PolyRedRel(P, T) has Church-Rosser property. Let f be a polynomial of n, L. If f ∈ P –ideal, then PolyRedRel(P, T) reduces f to 0_{n}L.

(16) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and P be a subset of Polynom-Ring(n, L). Suppose that for every polynomial f of n, L such that f ∈ P –ideal holds PolyRedRel(P, T) reduces f to 0_{n}L. Let f be a non-zero polynomial of n, L. If f ∈ P –ideal, then f is reducible wrt P, T.

(17) Let n be a natural number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and P be a subset of Polynom-Ring(n, L). Suppose that for every non-zero polynomial f of n, L such that f ∈ P –ideal holds f is reducible wrt P, T. Let f be a non-zero polynomial of n, L. If f ∈ P –ideal, then f is top reducible wrt P, T.

(18) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and P be a subset of Polynom-Ring(n, L). Suppose that for every non-zero polynomial f of n, L such that f ∈ P –ideal holds f is top reducible wrt P, T. Let b be a bag of n. If b ∈ HT(P –ideal, T), then there exists a bag b′ of n such that b′∈ HT(P, T) and b′| b.

(19) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and P be a subset of Polynom-Ring(n, L). Suppose that for every bag b of n such that b ∈ HT(P –ideal, T) there exists a bag b′ of n such that b′∈ HT(P, T) and b′| b. Then HT(P –ideal, T) ⊆ multiples(HT(P, T)).

(20) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and P be a subset of Polynom-Ring(n, L). If HT(P –ideal, T) ⊆ multiples(HT(P, T)), then PolyRedRel(P, T) is locallyconfluent.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let G be a subset of Polynom-Ring(n, L). We say that G is a Groebner basis wrt T if and only if:

(Def. 3) PolyRedRel(G, T) is locally-confluent.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let G, I be subsets of Polynom-Ring(n, L). We say that G is a Groebner basis of I, T if and only if:

(Def. 4) G–ideal = I and PolyRedRel(G, T) is locally-confluent.

One can prove the following propositions:

(21) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and G, P be non empty subsets of Polynom-Ring(n, L). If G is a Groebner basis of P, T, then PolyRedRel(G, T) is a completion of PolyRedRel(P, T).

(22) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, p, q be elements of Polynom-Ring(n, L), and G be a non empty subset of Polynom-Ring(n, L). Suppose G is a Groebner basis wrt T. Then p ≡ q(modG–ideal) if and only if nf_{PolyRedRel (G,T)}(p) = nf_{PolyRedRel (G,T)}(q).

(23) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, f be a polynomial of n, L, and P be a non empty subset of Polynom-Ring(n, L). Suppose P is a Groebner basis wrt T. Then f ∈ P –ideal if and only if PolyRedRel(P, T) reduces f to 0_{n}L.

(24) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, I be a subset of Polynom-Ring(n, L), and G be a non empty subset of Polynom-Ring(n, L). Suppose G is a Groebner basis of I, T. Let f be a polynomial of n, L. If f ∈ I, then PolyRedRel(G, T) reduces f to 0_{n}L.

(25) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and G, I be subsets of Polynom-Ring(n, L). Suppose that for every polynomial f of n, L such that f ∈ I holds PolyRedRel(G, T) reduces f to 0_{n}L. Let f be a non-zero polynomial of n, L. If f ∈ I, then f is reducible wrt G, T.

(26) Let n be a natural number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, I be an add closed left ideal subset of Polynom-Ring(n, L), and G be a subset of Polynom-Ring(n, L). Suppose G ⊆ I. Suppose that for every non-zero polynomial f of n, L such that f ∈ I holds f is reducible wrt G, T. Let f be a non-zero polynomial of n, L. If f ∈ I, then f is top reducible wrt G, T.

(27) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and G, I be subsets of Polynom-Ring(n, L). Suppose that for every non-zero polynomial f of n, L such that f ∈ I holds f is top reducible wrt G, T. Let b be a bag of n. If b ∈ HT(I, T), then there exists a bag b′ of n such that b′∈ HT(G, T) and b′| b.

(28) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and G, I be subsets of Polynom-Ring(n, L). Suppose that for every bag b of n such that b ∈ HT(I, T) there exists a bag b′ of n such that b′∈ HT(G, T) and b′| b. Then HT(I, T) ⊆ multiples(HT(G, T)).

(29) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, I be an add closed left ideal non empty subset of Polynom-Ring(n, L), and G be a non empty subset of Polynom-Ring(n, L). If G ⊆ I, then if HT(I, T) ⊆ multiples(HT(G, T)), then G is a Groebner basis of I, T.

3. Existence of Gröbner Bases

Next we state four propositions:

(30) Let n be a natural number, T be a connected admissible term order of n, and L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure. Then {0_{n}L} is a Groebner basis of {0_{n}L}, T.

(31) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non trivial double loop structure, and p be a polynomial of n, L. Then {p} is a Groebner basis of {p}–ideal, T.

(32) Let T be an admissible connected term order of ∅, L be an addassociative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, I be an add closed left ideal non empty subset of Polynom-Ring(∅, L), and P be a non empty subset of Polynom-Ring(∅, L). If P ⊆ I and P ≠ {0_{∅}L}, then P is a Groebner basis of I, T.

(33) Let n be a non empty ordinal number, T be an admissible connected term order of n, and L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure. Then there exists a subset P of Polynom-Ring(n, L) such that P is not a Groebner basis wrt T.

Let n be an ordinal number. The functor DivOrder(n) yields an order in Bagsn and is defined by:

(Def. 5) For all bags b_{1}, b_{2} of n holds ⟨⟨b_{1}, b_{2}⟩⟩∈ DivOrder(n) iff b_{1} | b_{2}.

Let n be a natural number. One can check that ⟨Bagsn, DivOrder(n)⟩ is Dickson.

The following propositions are true:

(34) For every natural number n and for every subset N of the carrier of ⟨Bagsn, DivOrder(n)⟩ holds there exists a finite subset of Bagsn which is Dickson basis of N, ⟨Bagsn, DivOrder(n)⟩.

(35) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and I be an add closed left ideal non empty subset of Polynom-Ring(n, L). Then there exists a finite subset of Polynom-Ring(n, L) which is a Groebner basis of I, T.

(36) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and I be an add closed left ideal non empty subset of Polynom-Ring(n, L). Suppose I ≠ {0_{n}L}. Then there exists a finite subset G of Polynom-Ring(n, L) such that G is a Groebner basis of I, T and 0_{n}L ∉ G.

Let n be an ordinal number, let T be a connected term order of n, let L be a non empty multiplicative loop with zero structure, and let p be a polynomial of n, L. We say that p is monic wrt T if and only if:

(Def. 6) HC(p, T) = 1_{L}.

Let n be an ordinal number, let T be a connected term order of n, let L be a right zeroed add-associative right complementable commutative associative well unital distributive field-like non trivial non empty double loop structure, and let P be a subset of Polynom-Ring(n, L). We say that P is reduced wrt T if and only if:

(Def. 7) For every polynomial p of n, L such that p ∈ P holds p is monic wrt T

and irreducible wrt P \ {p}, T. We introduce P is autoreduced wrt T as a synonym of P is reduced wrt T.

Next we state four propositions:

(37) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, I be an add closed left ideal subset of Polynom-Ring(n, L), m be a monomial of n, L, and f, g be polynomials of n, L. Suppose f ∈ I and g ∈ I and HM(f, T) = m and HM(g, T) = m. Suppose that

(i) it is not true that there exists a polynomial p of n, L such that p ∈ I and p <_{T} f and HM(p, T) = m, and

(ii) it is not true that there exists a polynomial p of n, L such that p ∈ I and p <_{T} g and HM(p, T) = m. Then f = g.

(38) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, I be an add closed left ideal non empty subset of Polynom-Ring(n, L), G be a subset of Polynom-Ring(n, L), p be a polynomial of n, L, and q be a non-zero polynomial of n, L. Suppose p ∈ G and q ∈ G and p ≠ q and HT(q, T) | HT(p, T). If G is a Groebner basis of I, T, then G \ {p} is a Groebner basis of I, T.

(39) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and I be an add closed left ideal non empty subset of Polynom-Ring(n, L). If I ≠ {0_{n}L}, then there exists a finite subset G of Polynom-Ring(n, L) which is a Groebner basis of I, T and reduced wrt T.

(40) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, I be an add closed left ideal non empty subset of Polynom-Ring(n, L), and G_{1}, G_{2} be non empty finite subsets of Polynom-Ring(n, L). Suppose G_{1} is a Groebner basis of I, T and reduced wrt T and G_{2} is a Groebner basis of I, T and reduced wrt T. Then G_{1} = G_{2}. characterization and existence of … 301

References

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.

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

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

[6] Grzegorz Bancerek. Reduction relations. Formalized Mathematics, 5(4):469–478, 1996.

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

[8] Thomas Becker and Volker Weispfenning. Gröbner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.

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

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

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

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

[13] Gilbert Lee and Piotr Rudnicki. Dickson’s lemma. Formalized Mathematics, 10(1):29–37, 2002.

[14] Gilbert Lee and Piotr Rudnicki. On ordering of bags. Formalized Mathematics, 10(1):39– 46, 2002.

[15] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

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

[17] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

[18] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[19] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.

[20] Christoph Schwarzweller. More on multivariate polynomials: Monomials and constant polynomials. Formalized Mathematics, 9(4):849–855, 2001.

[21] Christoph Schwarzweller. Polynomial reduction. Formalized Mathematics, 11(1):113–123, 2003.

[22] Christoph Schwarzweller. Term orders. Formalized Mathematics, 11(1):105–111, 2003.

[23] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

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

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

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

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

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

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

302 christoph schwarzweller Received June 11, 2003

Construction of Gröbner bases. S-Polynomials and Standard Representations

Christoph Schwarzweller University of Tübingen

Summary. We continue the Mizar formalization of Gröbner bases following [6]. In this article we introduce S-polynomials and standard representations and show how these notions can be used to characterize Gröbner bases.

MML Identifier: GROEB 2.

The notation and terminology used here are introduced in the following papers: [24], [31], [32], [34], [33], [8], [3], [15], [30], [29], [9], [7], [5], [14], [12], [19], [18], [25], [28], [17], [1], [4], [13], [22], [21], [27], [26], [16], [10], [23], [2], [20], [11], and [35].

1. Preliminaries

One can prove the following propositions:

(1) For every set X and for every finite sequence p of elements of X such that p ≠ ∅ holds p↾1 = ⟨p_{1}⟩.

(2) Let L be a non empty loop structure, p be a finite sequence of elements of L, and n, m be natural numbers. If m n, then p↾n↾m = p↾m.

(3) Let L be an add-associative right zeroed right complementable non empty loop structure, p be a finite sequence of elements of L, and n be a natural number. Suppose that for every natural number k such that k ∈ domp and k > n holds p(k) = 0_{L}. Then ∑ p = ∑(p↾n).

(4) Let L be an add-associative right zeroed Abelian non empty loop structure, f be a finite sequence of elements of L, and i, j be natural numbers. Then ∑ Swap(f, i, j) = ∑ f.

(5) Let n be an ordinal number, T be a term order of n, and b_{1}, b_{2}, b_{3} be bags of n. If b_{1} _{T} b_{3} and b_{2} _{T} b_{3}, then max_{T} (b_{1}, b_{2}) _{T} b_{3}.

(6) Let n be an ordinal number, T be a term order of n, and b_{1}, b_{2}, b_{3} be bags of n. If b_{3} _{T} b_{1} and b_{3} _{T} b_{2}, then b_{3} _{T} min_{T} (b_{1}, b_{2}).

Let X be a set and let b_{1}, b_{2} be bags of X. Let us assume that b_{2} | b_{1}. The functor \frac{b_{1}}{b_{2}} yields a bag of X and is defined by:

(Def. 1) b_{2} + \frac{b_{1}}{b_{2}} = b_{1}.

Let X be a set and let b_{1}, b_{2} be bags of X. The functor lcm(b_{1}, b_{2}) yields a bag of X and is defined as follows:

(Def. 2) For every set k holds lcm(b_{1}, b_{2})(k) = max(b_{1}(k), b_{2}(k)). Let us observe that the functor lcm(b_{1}, b_{2}) is commutative and idempotent. We introduce lcm(b_{1}, b_{2}) as a synonym of lcm(b_{1}, b_{2}).

Let X be a set and let b_{1}, b_{2} be bags of X. We say that b_{1}, b_{2} are disjoint if and only if:

(Def. 3) For every set i holds b_{1}(i) = 0 or b_{2}(i) = 0.

We introduce b_{1}, b_{2} are non disjoint as an antonym of b_{1}, b_{2} are disjoint. We now state several propositions:

(7) For every set X and for all bags b_{1}, b_{2} of X holds b_{1} | lcm(b_{1}, b_{2}) and b_{2} | lcm(b_{1}, b_{2}).

(8) For every set X and for all bags b_{1}, b_{2}, b_{3} of X such that b_{1} | b_{3} and b_{2} | b_{3} holds lcm(b_{1}, b_{2}) | b_{3}.

(9) Let X be a set, T be a term order of X, and b_{1}, b_{2} be bags of X. Then b_{1}, b_{2} are disjoint if and only if lcm(b_{1}, b_{2}) = b_{1} + b_{2}.

(10) For every set X and for every bag b of X holds \frac{b}{b} = EmptyBagX.

(11) For every set X and for all bags b_{1}, b_{2} of X holds b_{2} | b_{1} iff lcm(b_{1}, b_{2}) = b_{1}.

(12) For every set X and for all bags b_{1}, b_{2}, b_{3} of X such that b_{1} | lcm(b_{2}, b_{3}) holds lcm(b_{2}, b_{1}) | lcm(b_{2}, b_{3}).

(13) For every set X and for all bags b_{1}, b_{2}, b_{3} of X such that lcm(b_{2}, b_{1}) | lcm(b_{2}, b_{3}) holds lcm(b_{1}, b_{3}) | lcm(b_{2}, b_{3}).

(14) For every set n and for all bags b_{1}, b_{2}, b_{3} of n such that lcm(b_{1}, b_{3}) | lcm(b_{2}, b_{3}) holds b_{1} | lcm(b_{2}, b_{3}).

(15) Let n be a natural number, T be a connected admissible term order of n, and P be a non empty subset of Bagsn. Then there exists a bag b of n such that b ∈ P and for every bag b′ of n such that b′∈ P holds b _{T} b′.

Let L be an add-associative right zeroed right complementable non trivial loop structure and let a be a non-zero element of L. Note that − a is non-zero.

Let X be a set, let L be a left zeroed right zeroed add-cancelable distributive non empty double loop structure, let m be a monomial of X, L, and let a be an element of L. One can verify that a · m is monomial-like.

Let n be an ordinal number, let L be a left zeroed right zeroed add-cancelable distributive integral domain-like non trivial double loop structure, let p be a nonzero polynomial of n, L, and let a be a non-zero element of L. One can verify that a · p is non-zero.

Next we state several propositions:

(16) Let n be an ordinal number, T be a term order of n, L be a right zeroed right distributive non empty double loop structure, p, q be series of n, L, and b be a bag of n. Then b ∗ (p + q) = b ∗ p + b ∗ q.

(17) Let n be an ordinal number, T be a term order of n, L be an addassociative right zeroed right complementable non empty loop structure, p be a series of n, L, and b be a bag of n. Then b ∗ − p = − b ∗ p.

(18) Let n be an ordinal number, T be a term order of n, L be a left zeroed add-right-cancelable right distributive non empty double loop structure, p be a series of n, L, b be a bag of n, and a be an element of L. Then b ∗ (a · p) = a · (b ∗ p).

(19) Let n be an ordinal number, T be a term order of n, L be a right distributive non empty double loop structure, p, q be series of n, L, and a be an element of L. Then a · (p + q) = a · p + a · q.

(20) Let X be a set, L be an add-associative right zeroed right complementable non empty double loop structure, and a be an element of L. Then − (a (X, L)) = − a (X, L).

2. S-Polynomials

The following proposition is true

(21) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and P be a subset of Polynom-Ring(n, L). Suppose 0_{n}L ∉ P. Suppose that for all polynomials p_{1}, p_{2} of n, L such that p_{1} ≠ p_{2} and p_{1} ∈ P and p_{2} ∈ P and for all monomials m_{1}, m_{2} of n, L such that HM(m_{1} ∗ p_{1}, T) = HM(m_{2} ∗ p_{2}, T) holds PolyRedRel(P, T) reduces m_{1} ∗ p_{1} − m_{2} ∗ p_{2} to 0_{n}L. Then P is a Groebner basis wrt T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let p_{1}, p_{2} be polynomials of n, L. The functor S-Poly(p_{1}, p_{2}, T) yielding a polynomial of n, L is defined by:

(Def. 4) S-Poly(p_{1}, p_{2}, T) = HC(p_{2}, T) · (\frac{lcm(HT(p_{1},T),HT(p_{2},T))}{HT(p_{1},T)} ∗ p_{1}) − HC(p_{1}, T) ·

(\frac{lcm(HT(p_{1},T),HT(p_{2},T))}{HT(p_{2},T)} ∗ p_{2}). One can prove the following propositions:

(22) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like Abelian non trivial double loop structure, P be a subset of Polynom-Ring(n, L), and p_{1}, p_{2} be polynomials of n, L. If p_{1} ∈ P and p_{2} ∈ P, then S-Poly(p_{1}, p_{2}, T) ∈ P –ideal.

(23) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and p_{1}, p_{2} be polynomials of n, L. If p_{1} = p_{2}, then S-Poly(p_{1}, p_{2}, T) = 0_{n}L.

(24) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and m_{1}, m_{2} be monomials of n, L. Then S-Poly(m_{1}, m_{2}, T) = 0_{n}L.

(25) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and p be a polynomial of n, L. Then S-Poly(p, 0_{n}L, T) = 0_{n}L and S-Poly(0_{n}L, p, T) = 0_{n}L.

(26) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and p_{1}, p_{2} be polynomials of n, L. Then S-Poly(p_{1}, p_{2}, T) = 0_{n}L or HT(S-Poly(p_{1}, p_{2}, T), T) <_{T} lcm(HT(p_{1}, T), HT(p_{2}, T)).

(27) Let n be an ordinal number, T be a connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and p_{1}, p_{2} be non-zero polynomials of n, L. If HT(p_{2}, T) | HT(p_{1}, T), then HC(p_{2}, T) · p_{1} top reduces to S-Poly(p_{1}, p_{2}, T), p_{2}, T.

(28) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose G is a Groebner basis wrt T. Let g_{1}, g_{2}, h be polynomials of n, L. If g_{1} ∈ G and g_{2} ∈ G and h is a normal form of S-Poly(g_{1}, g_{2}, T) w.r.t. PolyRedRel(G, T), then h = 0_{n}L.

(29) Let n be a natural number, T be a connected admissible term order of n, L be an Abelian add-associative right complementable right zeroed commutative associative well unital distributive field-like non degenerated non empty double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose that for all polynomials g_{1}, g_{2}, h of n, L such that g_{1} ∈ G and g_{2} ∈ G and h is a normal form of S-Poly(g_{1}, g_{2}, T) w.r.t. PolyRedRel(G, T) holds h = 0_{n}L. Let g_{1}, g_{2} be polynomials of n, L. If g_{1} ∈ G and g_{2} ∈ G, then PolyRedRel(G, T) reduces S-Poly(g_{1}, g_{2}, T) to 0_{n}L.

(30) Let n be a natural number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose 0_{n}L ∉ G. Suppose that for all polynomials g_{1}, g_{2} of n, L such that g_{1} ∈ G and g_{2} ∈ G holds PolyRedRel(G, T) reduces S-Poly(g_{1}, g_{2}, T) to 0_{n}L. Then G is a Groebner basis wrt T.

Let n be an ordinal number, let T be a connected term order of n, let L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let P be a subset of Polynom-Ring(n, L). The functor S-Poly(P, T) yielding a subset of Polynom-Ring(n, L) is defined by:

(Def. 5) S-Poly(P, T) = {S-Poly(p_{1}, p_{2}, T); p_{1} ranges over polynomials of n, L, p_{2}

ranges over polynomials of n, L: p_{1} ∈ P ∧ p_{2} ∈ P }. Let n be an ordinal number, let T be a connected term order of n, let L be

an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, and let P be a finite subset of Polynom-Ring(n, L). One can check that S-Poly(P, T) is finite.

One can prove the following proposition

(31) Let n be a natural number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and G be a subset of Polynom-Ring(n, L). Suppose 0_{n}L ∉ G and for every polynomial g of n, L such that g ∈ G holds g is a monomial of n, L. Then G is a Groebner basis wrt T.

3. Standard Representations

The following three propositions are true:

(32) Let L be a non empty multiplicative loop structure, P be a non empty subset of L, A be a left linear combination of P, and i be a natural number. Then A↾i is a left linear combination of P.

(33) Let L be a non empty multiplicative loop structure, P be a non empty subset of L, A be a left linear combination of P, and i be a natural number. Then A_{⇂i} is a left linear combination of P.

(34) Let L be a non empty multiplicative loop structure, P, Q be non empty subsets of the carrier of L, and A be a left linear combination of P. If P ⊆ Q, then A is a left linear combination of Q.

Let n be an ordinal number, let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, let P be a non empty subset of Polynom-Ring(n, L), and let A, B be left linear combinations of P. Then A^{⌢} B is a left linear combination of P.

Let n be an ordinal number, let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, let f be a polynomial of n, L, let P be a non empty subset of Polynom-Ring(n, L), and let A be a left linear combination of P. We say that A is a monomial representation of f if and only if the conditions (Def. 6) are satisfied.

(Def. 6)(i) ∑ A = f, and

(ii) for every natural number i such that i ∈ domA there exists a monomial m of n, L and there exists a polynomial p of n, L such that p ∈ P and A_{i} = m ∗ p.

Next we state two propositions:

(35) Let n be an ordinal number, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f be a polynomial of n, L, P be a non empty subset of Polynom-Ring(n, L), and A be a left linear combination of P. Suppose A is a monomial representation of f. Then Supportf ⊆ ⋃{Support(m ∗ p); m ranges over monomials of n, L, p ranges over polynomials of n, L: ⋁_{i: natural number} (i ∈ domA ∧ A_{i} = m ∗ p)}.

(36) Let n be an ordinal number, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f, g be polynomials of n, L, P be a non empty subset of Polynom-Ring(n, L), and A, B be left linear combinations of P. Suppose A is a monomial representation of f and B is a monomial representation of g. Then A^{⌢} B is a monomial representation of f + g.

Let n be an ordinal number, let T be a connected term order of n, let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, let f be a polynomial of n, L, let P be a non empty subset of Polynom-Ring(n, L), let A be a left linear combination of P, and let b be a bag of n. We say that A is a standard representation of f, P, b, T if and only if the conditions (Def. 7) are satisfied.

(Def. 7)(i) ∑ A = f, and

(ii) for every natural number i such that i ∈ domA there exists a non-zero monomial m of n, L and there exists a non-zero polynomial p of n, L such that p ∈ P and A_{i} = m ∗ p and HT(m ∗ p, T) _{T} b.

Let n be an ordinal number, let T be a connected term order of n, let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, let f be a polynomial of n, L, let P be a non empty subset of Polynom-Ring(n, L), and let A be a left linear combination of P. We say that A is a standard representation of f, P, T if and only if:

(Def. 8) A is a standard representation of f, P, HT(f, T), T.

Let n be an ordinal number, let T be a connected term order of n, let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, let f be a polynomial of n, L, let P be a non empty subset of Polynom-Ring(n, L), and let b be a bag of n. We say that f has a standard representation of P, b, T if and only if:

(Def. 9) There exists a left linear combination of P which is a standard represen-

tation of f, P, b, T. Let n be an ordinal number, let T be a connected term order of n, let L

be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, let f be a polynomial of n, L, and let P be a non empty subset of Polynom-Ring(n, L). We say that f has a standard representation of P, T if and only if:

(Def. 10) There exists a left linear combination of P which is a standard represen-

tation of f, P, T. One can prove the following propositions:

(37) Let n be an ordinal number, T be a connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f be a polynomial of n, L, P be a non empty subset of Polynom-Ring(n, L), A be a left linear combination of P, and b be a bag of n. Suppose A is a standard representation of f, P, b, T. Then A is a monomial representation of f.

(38) Let n be an ordinal number, T be a connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f, g be polynomials of n, L, P be a non empty subset of Polynom-Ring(n, L), A, B be left linear combinations of P, and b be a bag of n. Suppose A is a standard representation of f, P, b, T and B is a standard representation of g, P, b, T. Then A^{⌢} B is a standard representation of f + g, P, b, T.

(39) Let n be an ordinal number, T be a connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f, g be polynomials of n, L, P be a non empty subset of Polynom-Ring(n, L), A, B be left linear combinations of P, b be a bag of n, and i be a natural number. Suppose A is a standard representation of f, P, b, T and B = A↾i and g = ∑(A_{⇂i}).

Then B is a standard representation of f − g, P, b, T.

(40) Let n be an ordinal number, T be a connected term order of n, L be an Abelian right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f, g be polynomials of n, L, P be a non empty subset of Polynom-Ring(n, L), A, B be left linear combinations of P, b be a bag of n, and i be a natural number. Suppose A is a standard representation of f, P, b, T and B = A_{⇂i} and g = ∑(A↾i) and i lenA. Then B is a standard representation of f − g, P, b, T.

(41) Let n be an ordinal number, T be a connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f be a non-zero polynomial of n, L, P be a non empty subset of Polynom-Ring(n, L), and A be a left linear combination of P. Suppose A is a monomial representation of f. Then there exists a natural number i and there exists a non-zero monomial m of n, L and there exists a non-zero polynomial p of n, L such that i ∈ domA and p ∈ P and A(i) = m ∗ p and HT(f, T) _{T} HT(m ∗ p, T).

(42) Let n be an ordinal number, T be a connected term order of n, L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure, f be a non-zero polynomial of n, L, P be a non empty subset of Polynom-Ring(n, L), and A be a left linear combination of P. Suppose A is a standard representation of f, P, T. Then there exists a natural number i and there exists a non-zero monomial m of n, L and there exists a non-zero polynomial p of n, L such that p ∈ P and i ∈ domA and A_{i} = m ∗ p and HT(f, T) = HT(m ∗ p, T).

(43) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, f be a polynomial of n, L, and P be a non empty subset of Polynom-Ring(n, L) such that PolyRedRel(P, T) reduces f to 0_{n}L. Then f has a standard representation of P, T.

(44) Let n be an ordinal number, T be an admissible connected term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive field-like non trivial double loop structure, f be a non-zero polynomial of n, L, and P be a non empty subset of Polynom-Ring(n, L). If f has a standard representation of P, T, then f is top reducible wrt P, T.

(45) Let n be a natural number, T be a connected admissible term order of n, L be an add-associative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and G be a non empty subset of Polynom-Ring(n, L). Then G is a Groebner basis wrt T if and only if for

construction of gröbner bases. … 311

every non-zero polynomial f of n, L such that f ∈ G–ideal holds f has a standard representation of G, T.

References

[1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.

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

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

[4] Grzegorz Bancerek. Reduction relations. Formalized Mathematics, 5(4):469–478, 1996.

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

[6] Thomas Becker and Volker Weispfenning. Gröbner Bases: A Computational Approach to Commutative Algebra. Springer-Verlag, New York, Berlin, 1993.

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

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

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

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

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

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

[13] Gilbert Lee and Piotr Rudnicki. On ordering of bags. Formalized Mathematics, 10(1):39– 46, 2002.

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

[15] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

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

[17] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95–110, 2001.

[18] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.

[19] Christoph Schwarzweller. More on multivariate polynomials: Monomials and constant polynomials. Formalized Mathematics, 9(4):849–855, 2001.

[20] Christoph Schwarzweller. Characterization and existence of Gröbner bases. Formalized Mathematics, 11(3):293–301, 2003.

[21] Christoph Schwarzweller. Polynomial reduction. Formalized Mathematics, 11(1):113–123, 2003.

[22] Christoph Schwarzweller. Term orders. Formalized Mathematics, 11(1):105–111, 2003.

[23] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

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

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

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

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

312 christoph schwarzweller

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

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

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

[35] Hiroshi Yamazaki, Yoshinori Fujisawa, and Yatsuka Nakamura. On replace function and swap function for finite sequences. Formalized Mathematics, 9(3):471–474, 2001.

On the Subcontinua of a Real Line^{1}

Adam Grabowski University of Białystok

Summary. In [10] we showed that the only proper subcontinua of the simple closed curve are arcs and single points. In this article we prove that the only proper subcontinua of the real line are closed intervals. We introduce some auxiliary notions such as ]a, b[_{ℚ}, ]a, b[_{𝕀ℚ} – intervals consisting of rational and irrational numbers respectively. We show also some basic topological properties of intervals.

MML Identifier: BORSUK 5.

The notation and terminology used in this paper are introduced in the following papers: [24], [27], [22], [23], [18], [25], [28], [3], [4], [26], [19], [6], [21], [13], [16], [17], [1], [8], [5], [9], [14], [7], [20], [15], [12], [11], and [2].

1. Preliminaries

The following three propositions are true:

(1) For all sets A, B, C, D holds (A ∪ B ∪ C) ∪ D = A ∪ (B ∪ C ∪ D).

(2) For all sets A, B, a such that A ⊆ B and B ⊆ A∪{a} holds A∪{a} = B or A = B.

(3) For all sets x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} holds {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} = {x_{1}, x_{3}, x_{6}} ∪ {x_{2}, x_{4}, x_{5}}.

In the sequel x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} are sets.

Let x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} be sets. We say that x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} are mutually different if and only if the conditions (Def. 1) are satisfied.

(Def. 1) x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{1} ≠ x_{5} and x_{1} ≠ x_{6} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{2} ≠ x_{5} and x_{2} ≠ x_{6} and x_{3} ≠ x_{4} and x_{3} ≠ x_{5} and x_{3} ≠ x_{6} and x_{4} ≠ x_{5} and x_{4} ≠ x_{6} and x_{5} ≠ x_{6}.

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

Let x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} be sets. We say that x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} are mutually different if and only if the conditions (Def. 2) are satisfied.

(Def. 2) x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{1} ≠ x_{5} and x_{1} ≠ x_{6} and x_{1} ≠ x_{7} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{2} ≠ x_{5} and x_{2} ≠ x_{6} and x_{2} ≠ x_{7} and x_{3} ≠ x_{4} and x_{3} ≠ x_{5} and x_{3} ≠ x_{6} and x_{3} ≠ x_{7} and x_{4} ≠ x_{5} and x_{4} ≠ x_{6} and x_{4} ≠ x_{7} and x_{5} ≠ x_{6} and x_{5} ≠ x_{7} and x_{6} ≠ x_{7}.

One can prove the following propositions:

(4) For all sets x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} such that x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} are mutually different holds card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} = 6.

(5) For all sets x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} such that x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} are mutually different holds card{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}} = 7.

(6) If {x_{1}, x_{2}, x_{3}} misses {x_{4}, x_{5}, x_{6}}, then x_{1} ≠ x_{4} and x_{1} ≠ x_{5} and x_{1} ≠ x_{6} and x_{2} ≠ x_{4} and x_{2} ≠ x_{5} and x_{2} ≠ x_{6} and x_{3} ≠ x_{4} and x_{3} ≠ x_{5} and x_{3} ≠ x_{6}.

(7) Suppose x_{1}, x_{2}, x_{3} are mutually different and x_{4}, x_{5}, x_{6} are mutually different and {x_{1}, x_{2}, x_{3}} misses {x_{4}, x_{5}, x_{6}}. Then x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} are mutually different.

(8) Suppose x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} are mutually different and {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} misses {x_{7}}. Then x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} are mutually different.

(9) If x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} are mutually different, then x_{7}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} are mutually different.

(10) If x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} are mutually different, then x_{1}, x_{2}, x_{5}, x_{3}, x_{6}, x_{7}, x_{4} are mutually different.

(11) Let T be a non empty topological space and a, b be points of T. Given a map f from 𝕀 into T such that f is continuous and f(0) = a and f(1) = b. Then there exists a map g from 𝕀 into T such that g is continuous and g(0) = b and g(1) = a.

Let us observe that ℝ^{1} is arcwise connected. Let us note that there exists a topological space which is connected and non

empty.

2. Intervals

The following two propositions are true:

(12) Every subset of ℝ is a subset of ℝ^{1}.

(13) Ω_{ℝ^{1}} = ℝ.

Let a be a real number. We introduce ] − ∞, a] as a synonym of ]−∞, a]. We introduce ]−∞, a[ as a synonym of ]−∞, a[. We introduce [a, +∞[ as a synonym of [a, +∞[. We introduce ]a, +∞[ as a synonym of ]a, +∞[.

Next we state a number of propositions:

(14) For all real numbers a, b holds a ∈]b, +∞[ iff a > b.

(15) For all real numbers a, b holds a ∈ [b, +∞[ iff a b.

(16) For all real numbers a, b holds a ∈] − ∞, b] iff a b.

(17) For all real numbers a, b holds a ∈] − ∞, b[ iff a < b.

(18) For every real number a holds ℝ \ {a} =] − ∞, a[∪]a, +∞[.

(19) For all real numbers a, b, c, d such that a < b and b c holds [a, b] misses ]c, d].

(20) For all real numbers a, b, c, d such that a < b and b c holds [a, b[ misses [c, d].

(21) Let A, B be subsets of the carrier of ℝ^{1} and a, b, c, d be real numbers. Suppose a < b and b c and c < d and A = [a, b[ and B = ]c, d]. Then A and B are separated.

(22) For every real number a holds ℝ\] − ∞, a[= [a, +∞[.

(23) For every real number a holds ℝ\] − ∞, a] =]a, +∞[.

(24) For every real number a holds ℝ\]a, +∞[=] − ∞, a].

(25) For every real number a holds ℝ \ [a, +∞[=] − ∞, a[.

(26) For every real number a holds ] − ∞, a] misses ]a, +∞[.

(27) For every real number a holds ] − ∞, a[ misses [a, +∞[.

(28) For all real numbers a, b, c such that a c and c b holds [a, b] ∪ [c, +∞[= [a, +∞[.

(29) For all real numbers a, b, c such that a c and c b holds ] − ∞, c] ∪ [a, b] =] − ∞, b].

(30) For every 1-sorted structure T and for every subset A of T holds {A} is a family of subsets of T.

(31) For every 1-sorted structure T and for all subsets A, B of T holds {A, B} is a family of subsets of T.

(32) For every 1-sorted structure T and for all subsets A, B, C of T holds

{A, B, C} is a family of subsets of T. Let us observe that every element of ℚ is real. Let us observe that every element of the carrier of the metric space of real

numbers is real.

Next we state four propositions:

(33) Let A be a subset of the carrier of ℝ^{1} and p be a point of the metric space of real numbers. Then p ∈ \overline{A} if and only if for every real number r such that r > 0 holds Ball(p, r) meets A.

(34) For all elements p, q of the carrier of the metric space of real numbers such that q p holds ρ(p, q) = q − p.

(35) For every subset A of the carrier of ℝ^{1} such that A = ℚ holds \overline{A} = the carrier of ℝ^{1}.

(36) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that A = ]a, b[ and a ≠ b holds \overline{A} = [a, b].

3. Rational and Irrational Numbers

Let us mention that e is irrational.

The subset 𝕀ℚ of ℝ is defined by: (Def. 3) 𝕀ℚ = ℝ \ ℚ.

Let a, b be real numbers. The functor ]a, b[_{ℚ} yielding a subset of ℝ is defined by:

(Def. 4) ]a, b[_{ℚ} = ℚ ∩]a, b[.

The functor ]a, b[_{𝕀ℚ} yielding a subset of ℝ is defined as follows: (Def. 5) ]a, b[_{𝕀ℚ} = 𝕀ℚ ∩]a, b[.

One can prove the following proposition

(37) For every real number x holds x is irrational iff x ∈ 𝕀ℚ.

Let us observe that there exists a real number which is irrational.

Let us note that 𝕀ℚ is non empty.

Next we state several propositions:

(38) For every rational number a and for every irrational real number b holds a + b is irrational.

(39) For every irrational real number a holds − a is irrational.

(40) For every rational number a and for every irrational real number b holds a − b is irrational.

(41) For every rational number a and for every irrational real number b holds b − a is irrational.

(42) For every rational number a and for every irrational real number b such that a ≠ 0 holds a · b is irrational.

(43) For every rational number a and for every irrational real number b such

that a ≠ 0 holds \frac{b}{a} is irrational. One can check that every real number which is irrational is also non zero. The following propositions are true:

(44) For every rational number a and for every irrational real number b such that a ≠ 0 holds \frac{a}{b} is irrational.

(45) For every irrational real number r holds fracr is irrational.

Let r be an irrational real number. Note that fracr is irrational.

Let a be an irrational real number. Note that − a is irrational.

Let a be a rational number and let b be an irrational real number. One can verify the following observations:

∗ a + b is irrational,

∗ b + a is irrational,

∗ a − b is irrational, and

∗ b − a is irrational.

Let us observe that there exists a rational number which is non zero.

Let a be a non zero rational number and let b be an irrational real number. One can check the following observations:

∗ a · b is irrational,

∗ b · a is irrational,

∗ \frac{a}{b} is irrational, and

∗ \frac{b}{a} is irrational.

The following propositions are true:

(46) For every irrational real number r holds 0 < fracr.

(47) For all real numbers a, b such that a < b there exist rational numbers p_{1}, p_{2} such that a < p_{1} and p_{1} < p_{2} and p_{2} < b.

(48) For all real numbers s_{1}, s_{3}, s_{4}, l such that s_{1} s_{3} and s_{1} < s_{4} and 0 < l and l < 1 holds s_{1} < (1 − l) · s_{3} + l · s_{4}.

(49) For all real numbers s_{1}, s_{3}, s_{4}, l such that s_{3} < s_{1} and s_{4} s_{1} and 0 < l and l < 1 holds (1 − l) · s_{3} + l · s_{4} < s_{1}.

(50) For all real numbers a, b such that a < b there exists an irrational real number p such that a < p and p < b.

(51) For every subset A of the carrier of ℝ^{1} such that A = 𝕀ℚ holds \overline{A} = the carrier of ℝ^{1}.

(52) For all real numbers a, b, c such that a < b holds c ∈]a, b[_{ℚ} iff c is rational and a < c and c < b.

(53) For all real numbers a, b, c such that a < b holds c ∈]a, b[_{𝕀ℚ} iff c is irrational and a < c and c < b.

(54) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that a < b and A =]a, b[_{ℚ} holds \overline{A} = [a, b].

(55) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that a < b and A =]a, b[_{𝕀ℚ} holds \overline{A} = [a, b].

(56) For every connected topological space T and for every closed open subset A of T holds A = ∅ or A = Ω_{T} .

(57) For every subset A of ℝ^{1} such that A is closed and open holds A = ∅ or A = ℝ.

4. Topological Properties of Intervals

We now state a number of propositions:

(58) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that A = [a, b[ and a ≠ b holds \overline{A} = [a, b].

(59) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that A = ]a, b] and a ≠ b holds \overline{A} = [a, b].

(60) Let A be a subset of the carrier of ℝ^{1} and a, b, c be real numbers. If A = [a, b[∪]b, c] and a < b and b < c, then \overline{A} = [a, c].

(61) For every subset A of the carrier of ℝ^{1} and for every real number a such that A = \{a\} holds \overline{A} = \{a\}.

(62) For every subset A of ℝ and for every subset B of ℝ^{1} such that A = B holds A is open iff B is open.

(63) For every subset A of ℝ^{1} and for every real number a such that A = ]a, +∞[ holds A is open.

(64) For every subset A of ℝ^{1} and for every real number a such that A = ] − ∞, a[ holds A is open.

(65) For every subset A of ℝ^{1} and for every real number a such that A = ] − ∞, a] holds A is closed.

(66) For every subset A of ℝ^{1} and for every real number a such that A = [a, +∞[ holds A is closed.

(67) For every real number a holds [a, +∞[= {a}∪]a, +∞[.

(68) For every real number a holds ] − ∞, a] = {a}∪] − ∞, a[.

(69) For every real number a holds ]a, +∞[⊆ [a, +∞[.

(70) For every real number a holds ] − ∞, a[⊆] − ∞, a].

Let a be a real number. One can check the following observations:

∗ ]a, +∞[ is non empty,

∗ ] − ∞, a] is non empty,

∗ ] − ∞, a[ is non empty, and

∗ [a, +∞[ is non empty.

The following propositions are true:

(71) For every real number a holds ]a, +∞[≠ ℝ.

(72) For every real number a holds [a, +∞[≠ ℝ.

(73) For every real number a holds ] − ∞, a] ≠ ℝ.

(74) For every real number a holds ] − ∞, a[≠ ℝ.

(75) For every subset A of the carrier of ℝ^{1} and for every real number a such that A =]a, +∞[ holds \overline{A} = [a, +∞[.

(76) For every real number a holds \overline{]a, +∞[} = [a, +∞[.

(77) For every subset A of the carrier of ℝ^{1} and for every real number a such that A =] − ∞, a[ holds \overline{A} =] − ∞, a].

(78) For every real number a holds \overline{] − ∞, a[} =] − ∞, a].

(79) Let A, B be subsets of the carrier of ℝ^{1} and b be a real number. If A =] − ∞, b[ and B =]b, +∞[, then A and B are separated.

(80) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that a < b and A = [a, b[∪]b, +∞[ holds \overline{A} = [a, +∞[.

(81) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that a < b and A = ]a, b[∪]b, +∞[ holds \overline{A} = [a, +∞[.

(82) Let A be a subset of the carrier of ℝ^{1} and a, b, c be real numbers. If a < b and b < c and A =]a, b[_{ℚ} ∪]b, c[∪]c, +∞[, then \overline{A} = [a, +∞[.

(83) For every subset A of the carrier of ℝ^{1} holds − A = ℝ \ A.

(84) For all real numbers a, b such that a < b holds ]a, b[_{𝕀ℚ} misses ]a, b[_{ℚ}.

(85) For all real numbers a, b such that a < b holds ℝ\]a, b[_{ℚ} =] − ∞, a]∪]a, b[_{𝕀ℚ} ∪ [b, +∞[.

(86) For all real numbers a, b, c such that a b and b < c holds a ∉ ]b, c[∪]c, +∞[.

(87) For all real numbers a, b such that a < b holds b ∉ ]a, b[∪]b, +∞[.

(88) For all real numbers a, b such that a < b holds [a, +∞[\(]a, b[∪]b, +∞[) = {a} ∪ {b}.

(89) For every subset A of the carrier of ℝ^{1} such that A =]2, 3[_{ℚ} ∪ ]3, 4[∪]4, +∞[ holds − A =] − ∞, 2]∪]2, 3[_{𝕀ℚ} ∪ {3} ∪ {4}.

(90) For every subset A of the carrier of ℝ^{1} and for every real number a such that A = {a} holds − A =] − ∞, a[∪]a, +∞[.

(91) For all real numbers a, b such that a < b holds ]a, +∞[∩] − ∞, b] = ]a, b].

(92) (] − ∞, 1[∪]1, +∞[) ∩ (] − ∞, 2]∪]2, 3[_{𝕀ℚ} ∪ {3} ∪ {4}) =] − ∞, 1[∪]1, 2]∪]2, 3[_{𝕀ℚ} ∪ {3} ∪ {4}.

(93) For all real numbers a, b such that a b holds ] − ∞, b[\{a} =] − ∞, a[∪]a, b[.

(94) For all real numbers a, b such that a b holds ]a, +∞[\{b} = ]a, b[∪]b, +∞[.

(95) Let A be a subset of the carrier of ℝ^{1} and a, b be real numbers. If a b and A = {a} ∪ [b, +∞[, then − A =] − ∞, a[∪]a, b[.

(96) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that a < b and A =] − ∞, a[∪]a, b[ holds \overline{A} =] − ∞, b].

(97) For every subset A of the carrier of ℝ^{1} and for all real numbers a, b such that a < b and A =] − ∞, a[∪]a, b] holds \overline{A} =] − ∞, b].

(98) For every subset A of the carrier of ℝ^{1} and for every real number a such that A =] − ∞, a] holds − A =]a, +∞[.

(99) For every subset A of the carrier of ℝ^{1} and for every real number a such that A = [a, +∞[ holds − A =] − ∞, a[.

(100) Let A be a subset of the carrier of ℝ^{1} and a, b, c be real numbers. If a < b and b < c and A =] − ∞, a[∪]a, b]∪]b, c[_{𝕀ℚ} ∪ \{c\}, then \overline{A} =] − ∞, c].

(101) Let A be a subset of the carrier of ℝ^{1} and a, b, c, d be real numbers. If a < b and b < c and A =] − ∞, a[∪]a, b]∪]b, c[_{𝕀ℚ} ∪ {c} ∪ {d}, then \overline{A} =] − ∞, c] ∪ \{d\}.

(102) Let A be a subset of the carrier of ℝ^{1} and a, b be real numbers. If a b and A =] − ∞, a] ∪ {b}, then − A = ]a, b[∪]b, +∞[.

(103) For all real numbers a, b holds [a, +∞[∪{b} ≠ ℝ.

(104) For all real numbers a, b holds ] − ∞, a] ∪ {b} ≠ ℝ.

(105) For every topological structure T_{1} and for all subsets A, B of the carrier of T_{1} such that A ≠ B holds − A ≠ − B.

(106) For every subset A of the carrier of ℝ^{1} such that ℝ = − A holds A = ∅.

5. Subcontinua of a Real Line

Let us mention that 𝕀 is arcwise connected.

We now state several propositions:

(107) Let X be a compact subset of ℝ^{1} and X′ be a subset of ℝ. If X′ = X, then X′ is upper bounded and lower bounded.

(108) Let X be a compact subset of ℝ^{1}, X′ be a subset of ℝ, and x be a real number. If x ∈ X′ and X′= X, then inf X′ x and x supX′.

(109) Let C be a non empty compact connected subset of ℝ^{1} and C′ be a subset of ℝ. If C = C′ and [inf C′, supC′] ⊆ C′, then [inf C′, supC′] = C′.

(110) Let A be a connected subset of ℝ^{1} and a, b, c be real numbers. If a b and b c and a ∈ A and c ∈ A, then b ∈ A.

(111) For every connected subset A of ℝ^{1} and for all real numbers a, b such that a ∈ A and b ∈ A holds [a, b] ⊆ A.

(112) Every non empty compact connected subset of ℝ^{1} is a non empty closedinterval subset of ℝ.

(113) For every non empty compact connected subset A of ℝ^{1} there exist real numbers a, b such that a b and A = [a, b].

6. Sets with Proper Subsets Only

Let T_{1} be a topological structure and let F be a family of subsets of T_{1}. We say that F has proper subsets if and only if:

(Def. 6) The carrier of T_{1} ∉ F.

on the subcontinua of a real line 321

One can prove the following proposition

(114) Let T_{1} be a topological structure and F, G be families of subsets of T_{1} such that F has proper subsets and G ⊆ F. Then G has proper subsets.

Let T_{1} be a non empty topological structure. Observe that there exists a family of subsets of T_{1} which has proper subsets.

We now state the proposition

(115) Let T_{1} be a non empty topological structure and A, B be families of subsets of T_{1} with proper subsets. Then A ∪ B has proper subsets.

Let T be a topological structure and let F be a family of subsets of T. We say that F is open if and only if:

(Def. 7) For every subset P of T such that P ∈ F holds P is open. We say that F is closed if and only if:

(Def. 8) For every subset P of T such that P ∈ F holds P is closed. Let T be a topological space. Note that there exists a family of subsets of T

which is open, closed, and non empty.

References

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

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

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

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

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

[8] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[9] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93–102, 1999.

[10] Adam Grabowski. On the decompositions of intervals and simple closed curves. Formalized Mathematics, 10(3):145–151, 2002.

[11] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

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

[13] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[14] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[15] Wojciech Leończuk and Krzysztof Prażmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.

[16] Yatsuka Nakamura. Half open intervals in real numbers. Formalized Mathematics, 10(1):21–22, 2002.

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

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

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

322 adam grabowski

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

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

[22] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

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

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

On the Kuratowski Closure-Complement Problem

Lilla Krystyna Bagińska University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. In this article we formalize the Kuratowski closure-complement result: there is at most 14 distinct sets that one can produce from a given subset A of a topological space T by applying closure and complement operators and that all 14 can be obtained from a suitable subset of ℝ, namely KuratExSet = {1} ∪ ℚ(2, 3) ∪ (3, 4) ∪ (4, ∞).

The second part of the article deals with the maximal number of distinct sets which may be obtained from a given subset A of T by applying closure and interior operators. The subset KuratExSet of ℝ is also enough to show that 7 can be achieved.

MML Identifier: KURATO 1.

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

1. Fourteen Kuratowski Sets

In this paper T is a non empty topological space and A is a subset of T. The following proposition is true

(1) \overline{− \overline{− \overline{− \overline{A}}}} = \overline{− \overline{A}}. Let us consider T, A. The functor Kurat14Part(A) is defined as follows:

(Def. 1) Kurat14Part(A) = \{A, \overline{A}, − \overline{A}, \overline{− \overline{A}}, − \overline{− \overline{A}}, \overline{− \overline{− \overline{A}}}, − \overline{− \overline{− \overline{A}}}\}. Let us consider T, A. One can check that Kurat14Part(A) is finite. Let us consider T, A. The functor Kurat14Set(A) yields a family of subsets

of T and is defined by:

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

(Def. 2) Kurat14Set(A) = \{A, \overline{A}, − \overline{A}, \overline{− \overline{A}}, − \overline{− \overline{A}}, \overline{− \overline{− \overline{A}}}, − \overline{− \overline{− \overline{A}}}\}∪

\{− A, \overline{− A}, − \overline{− A}, \overline{− \overline{− A}}, − \overline{− \overline{− A}}, \overline{− \overline{− \overline{− A}}}, − \overline{− \overline{− \overline{− A}}}\}. We now state three propositions:

(2) Kurat14Set(A) = Kurat14Part(A) ∪ Kurat14Part(− A).

(3) A ∈ Kurat14Set(A) and \overline{A} ∈ Kurat14Set(A) and − \overline{A} ∈ Kurat14Set(A) and \overline{− \overline{A}} ∈ Kurat14Set(A) and − \overline{− \overline{A}} ∈ Kurat14Set(A) and \overline{− \overline{− \overline{A}}} ∈ Kurat14Set(A) and − \overline{− \overline{− \overline{A}}} ∈ Kurat14Set(A).

(4) − A ∈ Kurat14Set(A) and \overline{− A} ∈ Kurat14Set(A) and − \overline{− A} ∈ Kurat14Set(A) and \overline{− \overline{− A}} ∈ Kurat14Set(A) and − \overline{− \overline{− A}} ∈ Kurat14Set(A) and \overline{− \overline{− \overline{− A}}} ∈ Kurat14Set(A) and − \overline{− \overline{− \overline{− A}}} ∈ Kurat14Set(A).

Let us consider T, A. The functor Kurat14ClosedPart(A) yielding a family of subsets of T is defined by:

(Def. 3) Kurat14ClosedPart(A) = \{\overline{A}, \overline{− \overline{A}}, \overline{− \overline{− \overline{A}}}, \overline{− A}, \overline{− \overline{− A}}, \overline{− \overline{− \overline{− A}}}\}. The functor Kurat14OpenPart(A) yields a family of subsets of T and is defined as follows:

(Def. 4) Kurat14OpenPart(A) = \{− \overline{A}, − \overline{− \overline{A}}, − \overline{− \overline{− \overline{A}}}, − \overline{− A}, − \overline{− \overline{− A}}, − \overline{− \overline{− \overline{− A}}}\}. We now state the proposition

(5) Kurat14Set(A) = {A, − A}∪Kurat14ClosedPart(A)∪Kurat14OpenPart(A). Let us consider T, A. One can verify that Kurat14Set(A) is finite. Next we state two propositions:

(6) For every subset Q of the carrier of T such that Q ∈ Kurat14Set(A) holds − Q ∈ Kurat14Set(A) and \overline{Q} ∈ Kurat14Set(A).

(7) cardKurat14Set(A) 14.

2. Seven Kuratowski Sets

Let us consider T, A. The functor Kurat7Set(A) yielding a family of subsets of T is defined as follows:

(Def. 5) Kurat7Set(A) = \{A, IntA, \overline{A}, Int \overline{A}, \overline{IntA}, \overline{Int \overline{A}}, Int \overline{IntA}\}. We now state two propositions:

(8) A ∈ Kurat7Set(A) and IntA ∈ Kurat7Set(A) and \overline{A} ∈ Kurat7Set(A) and Int \overline{A} ∈ Kurat7Set(A) and \overline{IntA} ∈ Kurat7Set(A) and \overline{Int \overline{A}} ∈ Kurat7Set(A) and Int \overline{IntA} ∈ Kurat7Set(A).

(9) Kurat7Set(A) = \{A\} ∪ \{IntA, Int \overline{A}, Int \overline{IntA}\} ∪ \{\overline{A}, \overline{IntA}, \overline{Int \overline{A}}\}. Let us consider T, A. Note that Kurat7Set(A) is finite. We now state two propositions:

(10) For every subset Q of the carrier of T such that Q ∈ Kurat7Set(A) holds IntQ ∈ Kurat7Set(A) and \overline{Q} ∈ Kurat7Set(A).

(11) cardKurat7Set(A) 7.

3. The Set Generating Exactly Fourteen Kuratowski Sets

The subset KuratExSet of ℝ^{1} is defined as follows: (Def. 6) KuratExSet = {1}∪]2, 3[_{ℚ} ∪]3, 4[∪]4, +∞[.

Next we state a number of propositions:

(12) \overline{KuratExSet} = \{1\} ∪ [2, +∞[.

(13) − \overline{KuratExSet} =] − ∞, 1[∪]1, 2[.

(14) \overline{− \overline{KuratExSet}} =] − ∞, 2].

(15) − \overline{− \overline{KuratExSet}} =]2, +∞[.

(16) \overline{− \overline{− \overline{KuratExSet}}} = [2, +∞[.

(17) − \overline{− \overline{− \overline{KuratExSet}}} =] − ∞, 2[.

(18) − KuratExSet =] − ∞, 1[∪]1, 2]∪]2, 3[_{𝕀ℚ} ∪ {3} ∪ {4}.

(19) \overline{− KuratExSet} =] − ∞, 3] ∪ \{4\}.

(20) − \overline{− KuratExSet} = ]3, 4[∪]4, +∞[.

(21) \overline{− \overline{− KuratExSet}} = [3, +∞[.

(22) − \overline{− \overline{− KuratExSet}} =] − ∞, 3[.

(23) \overline{− \overline{− \overline{− KuratExSet}}} =] − ∞, 3].

(24) − \overline{− \overline{− \overline{− KuratExSet}}} =]3, +∞[.

4. The Set Generating Exactly Seven Kuratowski Sets

Next we state several propositions:

(25) IntKuratExSet = ]3, 4[∪]4, +∞[.

(26) \overline{IntKuratExSet} = [3, +∞[.

(27) Int \overline{IntKuratExSet} =]3, +∞[.

(28) Int \overline{KuratExSet} =]2, +∞[.

(29) \overline{Int \overline{KuratExSet}} = [2, +∞[.

5. The Difference Between Chosen Kuratowski Sets

One can prove the following propositions:

(30) \overline{Int \overline{KuratExSet}} ≠ Int \overline{KuratExSet}.

(31) \overline{Int \overline{KuratExSet}} ≠ \overline{KuratExSet}.

(32) \overline{Int \overline{KuratExSet}} ≠ Int \overline{IntKuratExSet}.

(33) \overline{Int \overline{KuratExSet}} ≠ \overline{IntKuratExSet}.

(34) \overline{Int \overline{KuratExSet}} ≠ IntKuratExSet.

(35) Int \overline{KuratExSet} ≠ \overline{KuratExSet}.

(36) Int \overline{KuratExSet} ≠ Int \overline{IntKuratExSet}.

(37) Int \overline{KuratExSet} ≠ \overline{IntKuratExSet}.

(38) Int \overline{KuratExSet} ≠ IntKuratExSet.

(39) Int \overline{IntKuratExSet} ≠ \overline{KuratExSet}.

(40) \overline{IntKuratExSet} ≠ \overline{KuratExSet}.

(41) IntKuratExSet ≠ \overline{KuratExSet}.

(42) \overline{KuratExSet} ≠ KuratExSet.

(43) KuratExSet ≠ IntKuratExSet.

(44) \overline{IntKuratExSet} ≠ Int \overline{IntKuratExSet}.

(45) Int \overline{IntKuratExSet} ≠ IntKuratExSet.

(46) \overline{IntKuratExSet} ≠ IntKuratExSet.

6. Final Proofs For Seven Sets

The following propositions are true:

(47) Int \overline{IntKuratExSet} ≠ Int \overline{KuratExSet}.

(48) IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet} are mutually different.

(49) \overline{KuratExSet}, \overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}} are mutually different.

(50) For every set X such that X ∈ \{IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet}\} holds X is an open non empty subset of ℝ^{1}.

(51) For every set X such that X ∈ \{\overline{KuratExSet}, \overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}}\} holds X is a closed subset of ℝ^{1}.

(52) For every set X such that X ∈ \{IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet}\} holds X ≠ ℝ.

(53) For every set X such that X ∈ \{\overline{KuratExSet}, \overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}}\} holds X ≠ ℝ.

(54) \{IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet}\} misses \{\overline{KuratExSet}, \overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}}\}.

(55) IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet}, \overline{KuratExSet},

\overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}} are mutually different. Let us note that KuratExSet is non closed and non open. Next we state three propositions:

(56) \{IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet}, \overline{KuratExSet}, \overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}}\} misses \{KuratExSet\}.

(57) KuratExSet, IntKuratExSet, Int \overline{KuratExSet}, Int \overline{IntKuratExSet}, \overline{KuratExSet}, \overline{IntKuratExSet}, \overline{Int \overline{KuratExSet}} are mutually different.

(58) cardKurat7Set(KuratExSet) = 7.

7. Final Proofs For Fourteen Sets

One can check that Kurat14ClosedPart(KuratExSet) has proper subsets and Kurat14OpenPart(KuratExSet) has proper subsets.

One can verify that Kurat14Set(KuratExSet) has proper subsets. Let us note that Kurat14Set(KuratExSet) has non empty elements. We now state the proposition

(59) For every set A with non empty elements and for every set B such that

B ⊆ A holds B has non empty elements. Let us note that Kurat14ClosedPart(KuratExSet) has non empty elements

and Kurat14OpenPart(KuratExSet) has non empty elements.

Let us note that there exists a family of subsets of ℝ^{1} which has proper subsets and non empty elements.

We now state the proposition

(60) Let F, G be families of subsets of ℝ^{1} with proper subsets and non empty

elements. If F is open and G is closed, then F misses G. Let us mention that Kurat14ClosedPart(KuratExSet) is closed and

Kurat14OpenPart(KuratExSet) is open.

One can prove the following proposition

(61) Kurat14ClosedPart(KuratExSet) misses Kurat14OpenPart(KuratExSet). Let us consider T, A. Observe that Kurat14ClosedPart(A) is finite and

Kurat14OpenPart(A) is finite.

We now state three propositions:

(62) cardKurat14ClosedPart(KuratExSet) = 6.

(63) cardKurat14OpenPart(KuratExSet) = 6.

(64) {KuratExSet, − KuratExSet} misses Kurat14ClosedPart(KuratExSet).

Let us observe that KuratExSet is non empty.

The following three propositions are true:

(65) KuratExSet ≠ − KuratExSet.

(66) {KuratExSet, − KuratExSet} misses Kurat14OpenPart(KuratExSet).

(67) cardKurat14Set(KuratExSet) = 14.

8. Properties of Kuratowski Sets

Let T be a topological structure and let A be a family of subsets of T. We say that A is closed for closure operator if and only if:

(Def. 7) For every subset P of the carrier of T such that P ∈ A holds \overline{P} ∈ A.

We say that A is closed for interior operator if and only if: (Def. 8) For every subset P of the carrier of T such that P ∈ A holds IntP ∈ A.

Let T be a 1-sorted structure and let A be a family of subsets of T. We say that A is closed for complement operator if and only if:

(Def. 9) For every subset P of the carrier of T such that P ∈ A holds − P ∈ A. Let us consider T, A. One can verify the following observations:

∗ Kurat14Set(A) is non empty,

∗ Kurat14Set(A) is closed for closure operator, and

∗ Kurat14Set(A) is closed for complement operator.

Let us consider T, A. One can check the following observations:

∗ Kurat7Set(A) is non empty,

∗ Kurat7Set(A) is closed for interior operator, and

∗ Kurat7Set(A) is closed for closure operator.

Let us consider T. One can check that there exists a family of subsets of T which is closed for interior operator, closed for closure operator, and non empty and there exists a family of subsets of T which is closed for complement operator, closed for closure operator, and non empty.

References

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

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

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

[4] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[5] Adam Grabowski. On the subcontinua of a real line. Formalized Mathematics, 11(3):313– 322, 2003.

[6] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[7] Wojciech Leończuk and Krzysztof Prażmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.

on the kuratowski closure-complement … 329

[8] Yatsuka Nakamura. Half open intervals in real numbers. Formalized Mathematics, 10(1):21–22, 2002.

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

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

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

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

[13] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

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

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

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

Convex Hull, Set of Convex Combinations and Convex Cone

Noboru Endou Gifu National College of Technology Yasunari Shidama Shinshu University Nagano

Summary. In this article, there are two themes. One of them is the proof that convex hull of a given subset M consists of all convex combinations of M. Another is definitions of cone and convex cone and some properties of them.

MML Identifier: CONVEX3.

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

1. Equality of Convex Hull and Set of Convex Combinations

Let V be a real linear space. The functor ConvexComb(V) yielding a set is defined by:

(Def. 1) For every set L holds L ∈ ConvexComb(V) iff L is a convex combination

of V. Let V be a real linear space and let M be a non empty subset of V. The

functor ConvexComb(M) yielding a set is defined as follows: (Def. 2) For every set L holds L ∈ ConvexComb(M) iff L is a convex combination

of M. We now state several propositions:

(1) Let V be a real linear space and v be a vector of V. Then there exists a convex combination L of V such that ∑ L = v and for every non empty subset A of V such that v ∈ A holds L is a convex combination of A.

(2) Let V be a real linear space and v_{1}, v_{2} be vectors of V. Suppose v_{1} ≠ v_{2}. Then there exists a convex combination L of V such that for every non empty subset A of V if {v_{1}, v_{2}} ⊆ A, then L is a convex combination of A.

(3) Let V be a real linear space and v_{1}, v_{2}, v_{3} be vectors of V. Suppose v_{1} ≠ v_{2} and v_{1} ≠ v_{3} and v_{2} ≠ v_{3}. Then there exists a convex combination L of V such that for every non empty subset A of V if {v_{1}, v_{2}, v_{3}} ⊆ A, then L is a convex combination of A.

(4) Let V be a real linear space and M be a non empty subset of V. Then M is convex if and only if {∑ L; L ranges over convex combinations of M: L ∈ ConvexComb(V)} ⊆ M.

(5) Let V be a real linear space and M be a non empty subset of V. Then convM = {∑ L; L ranges over convex combinations of M: L ∈ ConvexComb(V)}.

2. Cone and Convex Cone

Let V be a non empty RLS structure and let M be a subset of V. We say that M is cone if and only if:

(Def. 3) For every real number r and for every vector v of V such that r > 0 and

v ∈ M holds r · v ∈ M. One can prove the following proposition

(6) For every non empty RLS structure V and for every subset M of V such that M = ∅ holds M is cone.

Let V be a non empty RLS structure. Observe that there exists a subset of V which is cone.

Let V be a non empty RLS structure. Observe that there exists a subset of V which is empty and cone.

Let V be a real linear space. Observe that there exists a subset of V which is non empty and cone.

The following propositions are true:

(7) Let V be a non empty RLS structure and M be a cone subset of V. Suppose V is real linear space-like. Then M is convex if and only if for all vectors u, v of V such that u ∈ M and v ∈ M holds u + v ∈ M.

(8) Let V be a real linear space and M be a subset of V. Then M is convex and cone if and only if for every linear combination L of M such that the support of L ≠ ∅ and for every vector v of V such that v ∈ the support of L holds L(v) > 0 holds ∑ L ∈ M.

(9) For every non empty RLS structure V and for all subsets M, N of V such that M is cone and N is cone holds M ∩ N is cone. convex hull, set of convex combinations and … 333

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 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] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[6] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53–58, 2003.

[7] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

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

[9] Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581–588, 1990.

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

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

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

On the Two Short Axiomatizations of Ortholattices

Wioletta Truszkowska University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. In the paper, two short axiom systems for Boolean algebras are introduced. In the first section we show that the single axiom (DN_{1}) proposed in

[2] in terms of disjunction and negation characterizes Boolean algebras. To prove that (DN_{1}) is a single axiom for Robbins algebras (that is, Boolean algebras as well), we use the Otter theorem prover. The second section contains proof that the two classical axioms (Meredith_{1}), (Meredith_{2}) proposed by Meredith [3] may also serve as a basis for Boolean algebras. The results will be used to characterize ortholattices.

MML Identifier: ROBBINS2.

The terminology and notation used in this paper have been introduced in the following articles: [4], [5], and [1].

1. Single Axiom for Boolean Algebras

Let L be a non empty complemented lattice structure. We say that L satisfies (DN_{1}) if and only if:

(Def. 1) For all elements x, y, z, u of the carrier of L holds (((x + y)^{c} + z)^{c} + (x +

(z^{c} + (z + u)^{c})^{c})^{c})^{c} = z. Let us observe that TrivComplLat satisfies (DN_{1}) and TrivOrtLat satisfies

(DN_{1}).

Let us observe that there exists a non empty complemented lattice structure which is join-commutative and join-associative and satisfies (DN_{1}).

Next we state a number of propositions:

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

(1) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z, u, v be elements of the carrier of L. Then ((x + y)^{c} + (((z + u)^{c} + x)^{c} + (y^{c} + (y + v)^{c})^{c})^{c})^{c} = y.

(2) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z, u be elements of the carrier of L. Then ((x + y)^{c} + ((z + x)^{c} + (y^{c} + (y + u)^{c})^{c})^{c})^{c} = y.

(3) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x be an element of the carrier of L. Then ((x + x^{c})^{c} + x)^{c} = x^{c}.

(4) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z, u be elements of the carrier of L. Then ((x + y)^{c} + ((z + x)^{c} + (((y + y^{c})^{c} + y)^{c} + (y + u)^{c})^{c})^{c})^{c} = y.

(5) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then ((x+y)^{c}+((z+x)^{c}+y)^{c})^{c} = y.

(6) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then ((x + y)^{c} + (x^{c} + y)^{c})^{c} = y.

(7) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (((x+y)^{c}+x)^{c}+(x+y)^{c})^{c} = x.

(8) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x + ((x + y)^{c} + x)^{c})^{c} = (x + y)^{c}.

(9) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (((x+y)^{c}+z)^{c}+(x+z)^{c})^{c} = z.

(10) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x+((y+z)^{c}+(y+x)^{c})^{c})^{c} = (y + x)^{c}.

(11) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then ((((x + y)^{c} + z)^{c} + (x^{c} + y)^{c})^{c} + y)^{c} = (x^{c} + y)^{c}.

(12) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x+((y+z)^{c}+(z+x)^{c})^{c})^{c} = (z + x)^{c}.

(13) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z, u be elements of the carrier of L. Then ((x + y)^{c} + ((z + x)^{c} + (y^{c} + (u + y)^{c})^{c})^{c})^{c} = y.

(14) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x + y)^{c} = (y + x)^{c}.

(15) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (((x+y)^{c}+(y+z)^{c})^{c}+z)^{c} = (y + z)^{c}.

(16) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then ((x+((x+y)^{c}+z)^{c})^{c}+z)^{c} = ((x + y)^{c} + z)^{c}.

(17) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (((x+y)^{c}+x)^{c}+y)^{c} = (y+y)^{c}.

(18) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x^{c} + (y + x)^{c})^{c} = x.

(19) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then ((x + y)^{c} + y^{c})^{c} = y.

(20) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x + (y + x^{c})^{c})^{c} = x^{c}.

(21) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x be an element of the carrier of L. Then (x + x)^{c} = x^{c}.

(22) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (((x + y)^{c} + x)^{c} + y)^{c} = y^{c}.

(23) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x be an element of the carrier of L. Then (x^{c})^{c} = x.

(24) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then ((x + y)^{c} + x)^{c} + y = (y^{c})^{c}.

(25) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then ((x + y)^{c})^{c} = y + x.

(26) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x+((y +z)^{c}+(y +x)^{c})^{c} = ((y + x)^{c})^{c}.

(27) Let L be a non empty complemented lattice structure satisfying (DN_{1})

and x, y be elements of the carrier of L. Then x + y = y + x. One can verify that every non empty complemented lattice structure which

satisfies (DN_{1}) is also join-commutative.

Next we state a number of propositions:

(28) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then ((x + y)^{c} + x)^{c} + y = y.

(29) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then ((x + y)^{c} + y)^{c} + x = x.

(30) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then x + ((y + x)^{c} + y)^{c} = x.

(31) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x+y^{c})^{c}+(y^{c}+y)^{c} = (x+y^{c})^{c}.

(32) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x+y)^{c}+(y+y^{c})^{c} = (x+y)^{c}.

(33) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x+y)^{c}+(y^{c}+y)^{c} = (x+y)^{c}.

(34) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (((x+y^{c})^{c})^{c}+y)^{c} = (y^{c}+y)^{c}.

(35) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x + y^{c} + y)^{c} = (y^{c} + y)^{c}.

(36) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (((x + y^{c} + z)^{c} + y)^{c} + (y^{c} + y)^{c})^{c} = y.

(37) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x+((y +z)^{c}+(y +x)^{c})^{c} = y + x.

(38) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x+(y +((z +y)^{c}+x)^{c})^{c} = (z + y)^{c} + x.

(39) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x+((y +x)^{c}+(y +z)^{c})^{c} = y + x.

(40) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then ((x + y)^{c} + ((x + y)^{c} + (x + z)^{c})^{c})^{c} + y = y.

(41) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (((x+y^{c}+z)^{c}+y)^{c})^{c} = y.

(42) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x + (y + x^{c} + z)^{c} = x.

(43) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x^{c} + (y + x + z)^{c} = x^{c}.

(44) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x + y)^{c} + x = x + y^{c}.

(45) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y be elements of the carrier of L. Then (x + (x + y^{c})^{c})^{c} = (x + y)^{c}.

(46) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then ((x+y)^{c}+(x+z))^{c}+y = y.

(47) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (((x + y)^{c} + z)^{c} + (x^{c} + y)^{c})^{c} + y = ((x^{c} + y)^{c})^{c}.

(48) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (((x + y)^{c} + z)^{c} + (x^{c} + y)^{c})^{c} + y = x^{c} + y.

(49) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x^{c} + (((y + x)^{c})^{c} + (y + z))^{c})^{c} + (y + z) = ((y + x)^{c})^{c} + (y + z).

(50) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x^{c}+(y +x+(y +z))^{c})^{c}+ (y + z) = ((y + x)^{c})^{c} + (y + z).

(51) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x^{c}+(y +x+(y +z))^{c})^{c}+ (y + z) = (y + x) + (y + z).

(52) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x^{c})^{c} + (y + z) = (y + x) + (y + z).

(53) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x+y)+(x+z) = y+(x+z).

(54) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then (x+y)+(x+z) = z+(x+y).

(55) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x + (y + z) = z + (y + x).

(56) Let L be a non empty complemented lattice structure satisfying (DN_{1}) and x, y, z be elements of the carrier of L. Then x + (y + z) = y + (z + x).

(57) Let L be a non empty complemented lattice structure satisfying (DN_{1})

and x, y, z be elements of the carrier of L. Then (x + y) + z = x + (y + z). Let us observe that every non empty complemented lattice structure which

satisfies (DN_{1}) is also join-associative and every non empty complemented lattice structure which satisfies (DN_{1}) is also Robbins.

One can prove the following propositions:

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

(59) Let L be a non empty complemented lattice structure such that L is

join-commutative, join-associative, and Robbins. Then L satisfies (DN_{1}). Let us mention that every non empty complemented lattice structure which

is join-commutative, join-associative, and Robbins satisfies also (DN_{1}).

Let us observe that there exists a pre-ortholattice which is de Morgan and satisfies (DN_{1}).

One can verify that every pre-ortholattice which is de Morgan satisfies (DN_{1}) is also Boolean and every well-complemented pre-ortholattice which is Boolean satisfies also (DN_{1}).

2. Meredith Two Axioms for Boolean Algebras

Let L be a non empty complemented lattice structure. We say that L satisfies (Meredith_{1}) if and only if:

(Def. 2) For all elements x, y of the carrier of L holds (x^{c} + y)^{c} + x = x.

We say that L satisfies (Meredith_{2}) if and only if: (Def. 3) For all elements x, y, z of the carrier of L holds (x^{c} + y)^{c} + (z + y) =

y + (z + x). Let us note that every non empty complemented lattice structure which

satisfies (Meredith_{1}) and (Meredith_{2}) is also join-commutative, join-associative, and Huntington and every non empty complemented lattice structure which is join-commutative, join-associative, and Huntington satisfies also (Meredith_{1}) and (Meredith_{2}).

Let us note that there exists a pre-ortholattice which is de Morgan and satisfies (Meredith_{1}), (Meredith_{2}), and (DN_{1}).

Let us observe that every pre-ortholattice which is de Morgan satisfies (Meredith_{1}) and (Meredith_{2}) is also Boolean and every well-complemented preortholattice which is Boolean satisfies also (Meredith_{1}) and (Meredith_{2}).

References

[1] Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4):681–690, 2001.

[2] W. McCune, R. Veroff, B. Fitelson, K. Harris, A. Feist, and L. Wos. Short single axioms for Boolean algebra. Journal of Automated Reasoning, 29(1):1–16, 2002.

[3] C. A. Meredith and A. N. Prior. Equational logic. Notre Dame Journal of Formal Logic, 9:212–226, 1968.

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

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

Contents Formaliz. Math. 11 (3)

General Fashoda Meet Theorem for Unit Circle and Square

By Yatsuka Nakamura  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

On Some Properties of Real Hilbert Space. Part I

By Hiroshi Yamazaki et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Full Subtracter Circuit. Part II

By Shin’nosuke Yamaguchi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Dijkstra’s Shortest Path Algorithm

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Real Linear Space of Real Sequences

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Hilbert Space of Real Sequences

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Intuitionistic Propositional Calculus in the Extended Framework

with Modal Operator. Part I

By Takao Inoué . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Some Properties for Convex Combinations

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

On Some Properties of Real Hilbert Space. Part II

By Hiroshi Yamazaki et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Inner Products and Angles of Complex Numbers

By Wenpai Chang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Angle and Triangle in Euclidean Topological Space

By Akihiro Kubo and Yatsuka Nakamura  . . . . . . . . . . . . . . . . 281

Continued on inside back cover

The Class of Series-Parallel Graphs. Part II

By Krzysztof Retel  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Characterization and Existence of Gröbner Bases

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Construction of Gröbner bases. S-Polynomials and Standard Re-

presentations

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

On the Subcontinua of a Real Line

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

On the Kuratowski Closure-Complement Problem

By Lilla Krystyna Bagińska and Adam Grabowski  . . . . . 323

Convex Hull, Set of Convex Combinations and Convex Cone

By Noboru Endou and Yasunari Shidama  . . . . . . . . . . . . . . . . . 331

On the Two Short Axiomatizations of Ortholattices

By Wioletta Truszkowska and Adam Grabowski  . . . . . . . . 335