FORMALIZED MATHEMATICS Volume 8, Number 1, 1999 University of Białystok

Bounded Domains and Unbounded Domains

Yatsuka Nakamura Shinshu University Nagano

Andrzej Trybulec University of Białystok

Czesław Byliński University of Białystok

Summary. First, notions of inside components and outside components are introduced for any subset of n-dimensional Euclid space. Next, notions of the bounded domain and the unbounded domain are defined using the above components. If the dimension is larger than 1, and if a subset is bounded, a unbounded domain of the subset coincides with an outside component (which is unique) of the subset. For a sphare in n-dimensional space, the similar fact is true for a bounded domain. In 2 dimensional space, any rectangle also has such property. We discussed relations between the Jordan property and the concept of boundary, which are necessary to find points in domains near a curve. In the last part, we gave the sufficient criterion for belonging to the left component of some clockwise oriented finite sequences.

MML Identifier: JORDAN2C.

The articles [44], [51], [12], [50], [53], [9], [10], [7], [22], [2], [1], [40], [54], [16], [27], [15], [24], [5], [38], [39], [20], [35], [32], [18], [42], [3], [8], [49], [46], [41], [21], [4], [26], [34], [37], [43], [6], [30], [52], [11], [25], [13], [17], [33], [14], [48], [47], [19], [23], [28], [29], [36], [45], and [31] provide the notation and terminology for this paper.

1. Definitions of Bounded Domain and Unbounded Domain

We follow the rules: m, n are natural numbers, r, s are real numbers, and x, y are sets.

The following propositions are true:

(1) If r 0, then |r| = − r.

(2) For all n, m such that n m and m n + 2 holds m = n or m = n + 1 or m = n + 2.

(3) For all n, m such that n m and m n + 3 holds m = n or m = n + 1 or m = n + 2 or m = n + 3.

(4) For all n, m such that n m and m n + 4 holds m = n or m = n + 1 or m = n + 2 or m = n + 3 or m = n + 4.

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

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

(7) For every finite sequence f such that rngf = {x, y} and lenf = 2 holds f(1) = x and f(2) = y or f(1) = y and f(2) = x.

(8) Let f be an increasing finite sequence of elements of ℝ. If rngf = {r, s} and lenf = 2 and r s, then f(1) = r and f(2) = s.

In the sequel p, p_{1}, p_{2}, p_{3}, q, q_{1}, q_{2} denote points of E_{T}^{n}. We now state several propositions:

(9) (p_{1} + p_{2}) − p_{3} = (p_{1} − p_{3}) + p_{2}.

(10) ||q|| = |q|.

(11) ||q_{1}| − |q_{2}|| |q_{1} − q_{2}|.

(12) ||[r]|| = |r|.

(13) q − 0_{E_{T}^{n}} = q and 0_{E_{T}^{n}} − q = − q.

Let us consider n and let P be a subset of E_{T}^{n}. We say that P is n-convex if and only if:

(Def. 1) For all points w_{1}, w_{2} of E_{T}^{n} such that w_{1} ∈ P and w_{2} ∈ P holds

L(w_{1}, w_{2}) ⊆ P. The following propositions are true:

(14) For every non empty subset P of E_{T}^{n} such that P is n-convex holds P is connected.

(15) Let G be a non empty topological space, P be a subset of G, A be a subset of the carrier of G, and Q be a subset of G↾A. If P ≠ ∅ and P = Q and P is connected, then Q is connected.

Let us consider n and let A be a subset of E_{T}^{n}. We say that A is Bounded if and only if:

(Def. 2) There exists a subset C of the carrier of E^{n} such that C = A and C is

bounded. One can prove the following proposition

(16) For all subsets A, B of E_{T}^{n} such that B is Bounded and A ⊆ B holds A is Bounded.

Let us consider n, let A be a subset of the carrier of E_{T}^{n}, and let B be a subset of E_{T}^{n}. We say that B is inside component of A if and only if:

(Def. 3) B is a component of A^{c} and Bounded.

Next we state the proposition

(17) Let A be a subset of the carrier of E_{T}^{n} and B be a subset of E_{T}^{n}. Then B is inside component of A if and only if there exists a subset C of (E_{T}^{n})↾A^{c} such that C = B and C is a component of (E_{T}^{n})↾A^{c} and for every subset D of the carrier of E^{n} such that D = C holds D is bounded.

Let us consider n, let A be a subset of the carrier of E_{T}^{n}, and let B be a subset of E_{T}^{n}. We say that B is outside component of A if and only if:

(Def. 4) B is a component of A^{c} and B is not Bounded.

Next we state three propositions:

(18) Let A be a subset of the carrier of E_{T}^{n} and B be a subset of E_{T}^{n}. Then B is outside component of A if and only if there exists a subset C of (E_{T}^{n})↾A^{c} such that C = B and C is a component of (E_{T}^{n})↾A^{c} and it is not true that for every subset D of the carrier of E^{n} such that D = C holds D is bounded.

(19) For all subsets A, B of E_{T}^{n} such that B is inside component of A holds B ⊆ A^{c}.

(20) For all subsets A, B of E_{T}^{n} such that B is outside component of A holds

B ⊆ A^{c}. Let us consider n and let A be a subset of the carrier of E_{T}^{n}. The functor

BDDA yields a subset of E_{T}^{n} and is defined by: (Def. 5) BDDA = ⋃{B; B ranges over subsets of E_{T}^{n}: B is inside component of

A}. Let us consider n and let A be a subset of the carrier of E_{T}^{n}. The functor

UBDA yielding a subset of E_{T}^{n} is defined by: (Def. 6) UBDA = ⋃{B; B ranges over subsets of E_{T}^{n}: B is outside component of

A}. One can prove the following propositions:

(21) Ω_{E_{T}^{n}} is n-convex.

(22) Ω_{E_{T}^{n}} is connected.

Let us consider n. One can check that Ω_{E_{T}^{n}} is connected. We now state several propositions:

(23) Ω_{E_{T}^{n}} is a component of E_{T}^{n}.

(24) For every subset A of the carrier of E_{T}^{n} holds BDDA is a union of components of (E_{T}^{n})↾A^{c}.

(25) For every subset A of the carrier of E_{T}^{n} holds UBDA is a union of components of (E_{T}^{n})↾A^{c}.

(26) Let A be a subset of the carrier of E_{T}^{n} and B be a subset of E_{T}^{n}. If B is inside component of A, then B ⊆ BDDA.

(27) Let A be a subset of the carrier of E_{T}^{n} and B be a subset of E_{T}^{n}. If B is outside component of A, then B ⊆ UBDA.

(28) For every subset A of the carrier of E_{T}^{n} holds BDDA ∩ UBDA = ∅.

(29) For every subset A of the carrier of E_{T}^{n} holds BDDA ⊆ A^{c}.

(30) For every subset A of the carrier of E_{T}^{n} holds UBDA ⊆ A^{c}.

(31) For every subset A of the carrier of E_{T}^{n} holds BDDA ∪ UBDA = A^{c}. In the sequel u is a point of E^{n}. One can prove the following propositions:

(32) Let G be a non empty topological space, w_{1}, w_{2}, w_{3} be points of G, h_{1} be a map from 𝕀 into G, and h_{2} be a map from 𝕀 into G. Suppose h_{1} is continuous and w_{1} = h_{1}(0) and w_{2} = h_{1}(1) and h_{2} is continuous and w_{2} = h_{2}(0) and w_{3} = h_{2}(1). Then there exists a map h_{3} from 𝕀 into G such that h_{3} is continuous and w_{1} = h_{3}(0) and w_{3} = h_{3}(1) and rngh_{3} ⊆ rngh_{1} ∪ rngh_{2}.

(33) For every subset P of E_{T}^{n} such that P = R^{n} holds P is connected. Let us consider n. The functor 1 ∗ n yielding a finite sequence of elements of

ℝ is defined by: (Def. 7) 1 ∗ n = n ↦ (1 qua real number).

Let us consider n. Then 1 ∗ n is an element of R^{n}.

Let us consider n. The functor 1.REALn yielding a point of E_{T}^{n} is defined by:

(Def. 8) 1.REALn = 1 ∗ n.

One can prove the following propositions:

(34) |1 ∗ n| = n ↦ (1 qua real number).

(35) |1 ∗ n| = \sqrt{n}.

(36) 1.REAL1 = ⟨(1 qua real number)⟩.

(37) | 1.REALn| = \sqrt{n}.

(38) If 1 n, then 1 | 1.REALn|.

(39) For every subset W of the carrier of E^{n} such that n 1 and W = R^{n} holds W is not bounded.

(40) Let A be a subset of E_{T}^{n}. Then A is Bounded if and only if there exists a real number r such that for every point q of E_{T}^{n} such that q ∈ A holds |q| < r.

(41) If n 1, then Ω_{E_{T}^{n}} is not Bounded.

(42) If n 1, then UBD∅_{E_{T}^{n}} = R^{n}.

(43) Let w_{1}, w_{2}, w_{3} be points of E_{T}^{n}, P be a non empty subset of the carrier of E_{T}^{n}, and h_{1}, h_{2} be maps from 𝕀 into (E_{T}^{n})↾P. Suppose h_{1} is continuous and w_{1} = h_{1}(0) and w_{2} = h_{1}(1) and h_{2} is continuous and w_{2} = h_{2}(0) and w_{3} = h_{2}(1). Then there exists a map h_{3} from 𝕀 into (E_{T}^{n})↾P such that h_{3} is continuous and w_{1} = h_{3}(0) and w_{3} = h_{3}(1).

(44) Let P be a subset of the carrier of E_{T}^{n} and w_{1}, w_{2}, w_{3} be points of E_{T}^{n}. Suppose w_{1} ∈ P and w_{2} ∈ P and w_{3} ∈ P and L(w_{1}, w_{2}) ⊆ P and L(w_{2}, w_{3}) ⊆ P. Then there exists a map h from 𝕀 into (E_{T}^{n})↾P such that h is continuous and w_{1} = h(0) and w_{3} = h(1).

(45) Let P be a subset of the carrier of E_{T}^{n} and w_{1}, w_{2}, w_{3}, w_{4} be points of E_{T}^{n}. Suppose w_{1} ∈ P and w_{2} ∈ P and w_{3} ∈ P and w_{4} ∈ P and L(w_{1}, w_{2}) ⊆ P and L(w_{2}, w_{3}) ⊆ P and L(w_{3}, w_{4}) ⊆ P. Then there exists a map h from 𝕀 into (E_{T}^{n})↾P such that h is continuous and w_{1} = h(0) and w_{4} = h(1).

(46) Let P be a subset of the carrier of E_{T}^{n} and w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}, w_{7} be points of E_{T}^{n}. Suppose w_{1} ∈ P and w_{2} ∈ P and w_{3} ∈ P and w_{4} ∈ P and w_{5} ∈ P and w_{6} ∈ P and w_{7} ∈ P and L(w_{1}, w_{2}) ⊆ P and L(w_{2}, w_{3}) ⊆ P and L(w_{3}, w_{4}) ⊆ P and L(w_{4}, w_{5}) ⊆ P and L(w_{5}, w_{6}) ⊆ P and L(w_{6}, w_{7}) ⊆ P. Then there exists a map h from 𝕀 into (E_{T}^{n})↾P such that h is continuous and w_{1} = h(0) and w_{7} = h(1).

(47) For all points w_{1}, w_{2} of E_{T}^{n} such that it is not true that there exists a real number r such that w_{1} = r · w_{2} or w_{2} = r · w_{1} holds 0_{E_{T}^{n}} ∉ L(w_{1}, w_{2}).

(48) Let w_{1}, w_{2} be points of E_{T}^{n} and P be a subset of (E^{n})_{top}. Suppose P = L(w_{1}, w_{2}) and 0_{E_{T}^{n}} ∉ L(w_{1}, w_{2}). Then there exists a point w_{0} of E_{T}^{n} such that w_{0} ∈ L(w_{1}, w_{2}) and |w_{0}| > 0 and |w_{0}| = (dist_{min}(P))(0_{E_{T}^{n}}).

(49) Let a be a real number, Q be a subset of the carrier of E_{T}^{n}, and w_{1}, w_{4} be points of E_{T}^{n}. Suppose Q = {q : |q| > a} and w_{1} ∈ Q and w_{4} ∈ Q and it is not true that there exists a real number r such that w_{1} = r · w_{4} or w_{4} = r · w_{1}. Then there exist points w_{2}, w_{3} of E_{T}^{n} such that w_{2} ∈ Q and w_{3} ∈ Q and L(w_{1}, w_{2}) ⊆ Q and L(w_{2}, w_{3}) ⊆ Q and L(w_{3}, w_{4}) ⊆ Q.

(50) Let a be a real number, Q be a subset of the carrier of E_{T}^{n}, and w_{1}, w_{4} be points of E_{T}^{n}. Suppose Q = R^{n} \ {q : |q| < a} and w_{1} ∈ Q and w_{4} ∈ Q and it is not true that there exists a real number r such that w_{1} = r · w_{4} or w_{4} = r · w_{1}. Then there exist points w_{2}, w_{3} of E_{T}^{n} such that w_{2} ∈ Q and w_{3} ∈ Q and L(w_{1}, w_{2}) ⊆ Q and L(w_{2}, w_{3}) ⊆ Q and L(w_{3}, w_{4}) ⊆ Q.

(51) Let x be an element of R^{n}. Then x is a finite sequence of elements of ℝ and for every finite sequence f such that f = x holds lenf = n.

(52) Every finite sequence f of elements of ℝ is an element of R^{len f} and a point of E_{T}^{len f}.

(53) Let x be an element of R^{n}, f, g be finite sequences of elements of ℝ, and r be a real number. Suppose f = x and g = r·x. Then lenf = leng and for

every natural number i such that 1 i and i lenf holds π_{i}g = r · π_{i}f.

(54) Let x be an element of R^{n} and f be a finite sequence. Suppose x ≠ ⟨\underbrace{0, …, 0}_{n}⟩ and x = f. Then there exists a natural number i such that 1 i

and i n and f(i) ≠ 0.

(55) Let x be an element of R^{n}. Suppose n 2 and x ≠ ⟨\underbrace{0, …, 0}_{n}⟩. Then it

is not true that there exists an element y of R^{n} and there exists a real number r such that y = r · x or x = r · y.

(56) Let a be a real number, Q be a subset of the carrier of E_{T}^{n}, and w_{1}, w_{7} be points of E_{T}^{n}. Suppose n 2 and Q = {q : |q| > a} and w_{1} ∈ Q and w_{7} ∈ Q and there exists a real number r such that w_{1} = r · w_{7} or w_{7} = r · w_{1}. Then there exist points w_{2}, w_{3}, w_{4}, w_{5}, w_{6} of E_{T}^{n} such that w_{2} ∈ Q and w_{3} ∈ Q and w_{4} ∈ Q and w_{5} ∈ Q and w_{6} ∈ Q and L(w_{1}, w_{2}) ⊆ Q and L(w_{2}, w_{3}) ⊆ Q and L(w_{3}, w_{4}) ⊆ Q and L(w_{4}, w_{5}) ⊆ Q and L(w_{5}, w_{6}) ⊆ Q and L(w_{6}, w_{7}) ⊆ Q.

(57) Let a be a real number, Q be a subset of the carrier of E_{T}^{n}, and w_{1}, w_{7} be points of E_{T}^{n}. Suppose n 2 and Q = R^{n} \ {q : |q| < a} and w_{1} ∈ Q and w_{7} ∈ Q and there exists a real number r such that w_{1} = r·w_{7} or w_{7} = r·w_{1}. Then there exist points w_{2}, w_{3}, w_{4}, w_{5}, w_{6} of E_{T}^{n} such that w_{2} ∈ Q and w_{3} ∈ Q and w_{4} ∈ Q and w_{5} ∈ Q and w_{6} ∈ Q and L(w_{1}, w_{2}) ⊆ Q and L(w_{2}, w_{3}) ⊆ Q and L(w_{3}, w_{4}) ⊆ Q and L(w_{4}, w_{5}) ⊆ Q and L(w_{5}, w_{6}) ⊆ Q and L(w_{6}, w_{7}) ⊆ Q.

(58) For every real number a such that n 1 holds {q : |q| > a} ≠ ∅.

(59) For every real number a and for every subset P of E_{T}^{n} such that n 2 and P = {q : |q| > a} holds P is connected.

(60) For every real number a such that n 1 holds R^{n} \ {q : |q| < a} ≠ ∅.

(61) For every real number a and for every subset P of E_{T}^{n} such that n 2 and P = R^{n} \ {q : |q| < a} holds P is connected.

(62) Let a be a real number, n be a natural number, and P be a subset of E_{T}^{n}. If n 1 and P = R^{n} \ {q; q ranges over points of E_{T}^{n}: |q| < a}, then P is not Bounded.

(63) Let a be a real number and P be a subset of E_{T}^{1}. Suppose P = {q; q ranges over points of E_{T}^{1}: ⋁_{r} (q = ⟨r⟩ ∧ r > a)}. Then P is n-convex.

(64) Let a be a real number and P be a subset of E_{T}^{1}. Suppose P = {q; q ranges over points of E_{T}^{1}: ⋁_{r} (q = ⟨r⟩ ∧ r < − a)}. Then P is n-convex.

(65) Let a be a real number and P be a subset of E_{T}^{1}. Suppose P = {q; q ranges over points of E_{T}^{1}: ⋁_{r} (q = ⟨r⟩ ∧ r > a)}. Then P is connected.

(66) Let a be a real number and P be a subset of E_{T}^{1}. Suppose P = {q; q ranges over points of E_{T}^{1}: ⋁_{r} (q = ⟨r⟩ ∧ r < − a)}. Then P is connected.

(67) Let W be a subset of the carrier of E^{1}, a be a real number, and P be a subset of E_{T}^{1}. Suppose W = {q; q ranges over points of E_{T}^{1}: ⋁_{r} (q = ⟨r⟩ ∧ r > a)} and P = W. Then P is connected and W is not bounded.

(68) Let W be a subset of the carrier of E^{1}, a be a real number, and P be a subset of E_{T}^{1}. Suppose W = {q; q ranges over points of E_{T}^{1}: ⋁_{r} (q = ⟨r⟩ ∧ r < − a)} and P = W. Then P is connected and W is not bounded.

(69) Let W be a subset of the carrier of E^{n}, a be a real number, and P be a subset of E_{T}^{n}. If n 2 and W = {q : |q| > a} and P = W, then P is connected and W is not bounded.

(70) Let W be a subset of the carrier of E^{n}, a be a real number, and P be a subset of E_{T}^{n}. If n 2 and W = R^{n} \ {q : |q| < a} and P = W, then P is connected and W is not bounded.

(71) Let P, P_{1} be subsets of E_{T}^{n}, Q be a subset of the carrier of E_{T}^{n}, and W be a subset of the carrier of E^{n}. Suppose P = W and P is connected and W is not bounded and P_{1} = Component(Down(P, Q^{c})) and W ∩ Q = ∅. Then P_{1} is outside component of Q.

Let S be a 1-sorted structure and let A be a subset of the carrier of S. The functor RACA yields a subset of S and is defined as follows:

(Def. 9) RACA = A.

The following propositions are true:

(72) Let A be a subset of the carrier of E^{n}, B be a non empty subset of the carrier of E^{n}, and C be a subset of the carrier of E^{n}↾B. If A ⊆ B and A = C and C is bounded, then A is bounded.

(73) For every subset A of E_{T}^{n} such that A is compact holds A is Bounded.

(74) For every subset A of E_{T}^{n} such that 1 n and A is Bounded holds A^{c} ≠ ∅.

(75) Let r be a real number. Then

(i) there exists a subset B of the carrier of E^{n} such that B = {q : |q| < r}, and

(ii) for every subset A of the carrier of E^{n} such that A = {q_{1} : |q_{1}| < r}

holds A is bounded.

(76) Let A be a subset of E_{T}^{n}. Suppose n 2 and A is Bounded. Then there exists a subset B of E_{T}^{n} such that B is outside component of A and B = UBDA.

(77) For every real number a and for every subset P of E_{T}^{n} such that P = {q : |q| < a} holds P is n-convex.

(78) For every real number a and for every subset P of E_{T}^{n} such that P = Ball(u, a) holds P is n-convex.

(79) For every real number a and for every subset P of E_{T}^{n} such that a > 0 and P = {q : |q| < a} holds P is connected.

In the sequel R denotes a subset of E_{T}^{n}, P denotes a subset of the carrier of E_{T}^{n}, and f denotes a finite sequence of elements of E_{T}^{n}.

Next we state a number of propositions:

(80) Suppose p ≠ q and p ∈ Ball(u, r) and q ∈ Ball(u, r). Then there exists a map h from 𝕀 into E_{T}^{n} such that h is continuous and h(0) = p and h(1) = q and rngh ⊆ Ball(u, r).

(81) Let f be a map from 𝕀 into E_{T}^{n}. Suppose f is continuous and f(0) = p_{1} and f(1) = p_{2} and p ∈ Ball(u, r) and p_{2} ∈ Ball(u, r). Then there exists a map h from 𝕀 into E_{T}^{n} such that h is continuous and h(0) = p_{1} and h(1) = p and rngh ⊆ rngf ∪ Ball(u, r).

(82) Let f be a map from 𝕀 into E_{T}^{n}. Suppose p ≠ p_{1} and f is continuous and rngf ⊆ P and f(0) = p_{1} and f(1) = p_{2} and p ∈ Ball(u, r) and p_{2} ∈ Ball(u, r) and Ball(u, r) ⊆ P. Then there exists a map f_{1} from 𝕀 into E_{T}^{n} such that f_{1} is continuous and rngf_{1} ⊆ P and f_{1}(0) = p_{1} and f_{1}(1) = p.

(83) Let given p and P be a subset of E_{T}^{n}. Suppose that (i) R is connected and open, and (ii) P = {q : q ≠ p ∧ q ∈ R ∧ ¬ ⋁_{f : map from 𝕀 into E_{T}^{n}} (f is

continuous ∧ rngf ⊆ R ∧ f(0) = p ∧ f(1) = q)}. Then P is open.

(84) Let P be a subset of E_{T}^{n}. Suppose that (i) R is connected and open, (ii) p ∈ R, and (iii) P = {q : q = p ∨ ⋁_{f : map from 𝕀 into E_{T}^{n}} (f is continuous ∧ rngf ⊆

R ∧ f(0) = p ∧ f(1) = q)}. Then P is open.

(85) Let R be a subset of the carrier of E_{T}^{n}. Suppose p ∈ R and P = {q : q = p ∨ ⋁_{f : map from 𝕀 into E_{T}^{n}} (f is continuous ∧ rngf ⊆ R ∧ f(0) = p ∧ f(1) = q)}. Then P ⊆ R.

(86) Let R be a subset of E_{T}^{n} and p be a point of E_{T}^{n}. Suppose that (i) R is connected and open, (ii) p ∈ R, and (iii) P = {q : q = p ∨ ⋁_{f : map from 𝕀 into E_{T}^{n}} (f is continuous ∧ rngf ⊆

R ∧ f(0) = p ∧ f(1) = q)}. Then R ⊆ P.

(87) Let R be a subset of E_{T}^{n} and p, q be points of E_{T}^{n}. Suppose R is connected and open and p ∈ R and q ∈ R and p ≠ q. Then there exists a map f from 𝕀 into E_{T}^{n} such that f is continuous and rngf ⊆ R and f(0) = p and f(1) = q.

(88) For every subset A of E_{T}^{n} and for every real number a such that A = {q : |q| = a} holds − A is open and A is closed.

(89) For every non empty subset B of E_{T}^{n} such that B is open holds (E_{T}^{n})↾B is locally connected.

(90) Let B be a non empty subset of the carrier of E_{T}^{n}, A be a subset of the carrier of E_{T}^{n}, and a be a real number. If A = {q : |q| = a} and A^{c} = B, then (E_{T}^{n})↾B is locally connected.

(91) For every map f from E_{T}^{n} into ℝ^{1} such that for every q holds f(q) = |q| holds f is continuous.

(92) There exists a map f from E_{T}^{n} into ℝ^{1} such that for every q holds f(q) =

|q| and f is continuous. Let X, Y be non empty 1-sorted structures, let f be a map from X into

Y, and let x be a set. Let us assume that x is a point of X. The functor π_{x}f yielding a point of Y is defined as follows:

(Def. 10) π_{x}f = f(x).

We now state four propositions:

(93) Let g be a map from 𝕀 into E_{T}^{n}. Suppose g is continuous. Then there exists a map f from 𝕀 into ℝ^{1} such that for every point t of 𝕀 holds f(t) = |g(t)| and f is continuous.

(94) Let g be a map from 𝕀 into E_{T}^{n} and a be a real number. Suppose g is continuous and |π_{0}g| a and a |π_{1}g|. Then there exists a point s of 𝕀 such that |π_{s}g| = a.

(95) If q = ⟨r⟩, then |q| = |r|.

(96) Let A be a subset of the carrier of E_{T}^{n} and a be a real number. Suppose n 1 and a > 0 and A = {q : |q| = a}. Then there exists a subset B of E_{T}^{n} such that B is inside component of A and B = BDDA.

2. Bounded and Unbounded Domains of Rectangles

In the sequel D is a non vertical non horizontal non empty compact subset of E_{T}^{2}.

Next we state several propositions:

(97) lenthe Go-board of SpStSeqD = 2 and widththe Go-board of SpStSeqD = 2 and π_{1} SpStSeqD = (the Go-board of SpStSeqD)_{1,2} and π_{2} SpStSeqD = (the Go-board of SpStSeqD)_{2,2} and π_{3} SpStSeqD = (the Go-board of SpStSeqD)_{2,1} and π_{4} SpStSeqD = (the Go-board of SpStSeqD)_{1,1} and π_{5} SpStSeqD = (the Go-board of SpStSeqD)_{1,2}.

(98) LeftComp(SpStSeqD) is not Bounded.

(99) LeftComp(SpStSeqD) ⊆ UBD L̃(SpStSeqD).

(100) Let G be a topological space and A, B, C be subsets of G. Suppose A is a component of G and B is a component of G and C is connected and A ∩ C ≠ ∅ and B ∩ C ≠ ∅. Then A = B.

(101) For every subset B of E_{T}^{2} such that B is a component of (L̃(SpStSeqD))^{c} and B is not Bounded holds B = LeftComp(SpStSeqD).

(102) RightComp(SpStSeqD) ⊆ BDD L̃(SpStSeqD) and RightComp(SpStSeqD) is Bounded.

(103) LeftComp(SpStSeqD) = UBD L̃(SpStSeqD) and RightComp(SpStSeqD) = BDD L̃(SpStSeqD).

(104) UBD L̃(SpStSeqD) ≠ ∅ and UBD L̃(SpStSeqD) is outside component of L̃(SpStSeqD) and BDD L̃(SpStSeqD) ≠ ∅ and BDD L̃(SpStSeqD) is inside component of L̃(SpStSeqD).

3. Jordan Property and Boundary Property

One can prove the following propositions:

(105) Let G be a non empty topological space and A be a subset of G. Suppose A^{c} ≠ ∅. Then A is boundary if and only if for every set x and for every subset V of G such that x ∈ A and x ∈ V and V is open there exists a subset B of the carrier of G such that B is a component of A^{c} and V ∩ B ≠ ∅.

(106) Let A be a subset of E_{T}^{2}. Suppose A^{c} ≠ ∅. Then A is boundary and Jordan if and only if there exist subsets A_{1}, A_{2} of E_{T}^{2} such that A^{c} = A_{1} ∪ A_{2} and A_{1} ∩A_{2} = ∅ and \overline{A_{1}} \A_{1} = \overline{A_{2}} \A_{2} and A = \overline{A_{1}} \A_{1} and for all subsets C_{1}, C_{2} of (E_{T}^{2})↾A^{c} such that C_{1} = A_{1} and C_{2} = A_{2} holds C_{1} is a component of (E_{T}^{2})↾A^{c} and C_{2} is a component of (E_{T}^{2})↾A^{c}.

(107) For every point p of E_{T}^{n} and for every subset P of E_{T}^{n} such that n 1 and P = {p} holds P is boundary.

(108) For all points p, q of E_{T}^{2} and for every r such that p_{1} = q_{2} and − p_{2} = q_{1} and p = r · q holds p_{1} = 0 and p_{2} = 0 and p = 0_{E_{T}^{2}}.

(109) For all points q_{1}, q_{2} of E_{T}^{2} holds L(q_{1}, q_{2}) is boundary.

Let q_{1}, q_{2} be points of E_{T}^{2}. Observe that L(q_{1}, q_{2}) is boundary. One can prove the following proposition

(110) For every finite sequence f of elements of E_{T}^{2} holds L̃(f) is boundary. Let f be a finite sequence of elements of E_{T}^{2}. Note that L̃(f) is boundary. We now state several propositions:

(111) For every point e_{1} of E^{n} and for all points p, q of E_{T}^{n} such that p = e_{1} and q ∈ Ball(e_{1}, r) holds |p − q| < r and |q − p| < r.

(112) Let a be a real number and p be a point of E_{T}^{2}. Suppose a > 0 and p ∈ L̃(SpStSeqD). Then there exists a point q of E_{T}^{2} such that q ∈ UBD L̃(SpStSeqD) and |p − q| < a.

(113) R^{0} = {0_{E_{T}^{0}}}.

(114) For every subset A of E_{T}^{n} such that A is Bounded holds BDDA is Bounded.

(115) Let G be a non empty topological space and A, B, C, D be subsets of G. Suppose A is a component of G and B is a component of G and C is a component of G and A ∪ B = the carrier of G and C ∩ A = ∅. Then C = B.

(116) For every subset A of E_{T}^{2} such that A is Bounded and Jordan holds BDDA is inside component of A.

(117) Let a be a real number and p be a point of E_{T}^{2}. Suppose a > 0 and p ∈ L̃(SpStSeqD). Then there exists a point q of E_{T}^{2} such that q ∈ BDD L̃(SpStSeqD) and |p − q| < a.

4. Points in LeftComp

In the sequel f denotes a clockwise oriented non constant standard special circular sequence.

Next we state four propositions:

(118) For every point p of E_{T}^{2} such that π_{1}f = N-min L̃(f) and p_{1} < W-bound L̃(f) holds p ∈ LeftComp(f).

(119) For every point p of E_{T}^{2} such that π_{1}f = N-min L̃(f) and p_{1} > E-bound L̃(f) holds p ∈ LeftComp(f).

(120) For every point p of E_{T}^{2} such that π_{1}f = N-min L̃(f) and p_{2} < S-bound L̃(f) holds p ∈ LeftComp(f).

(121) For every point p of E_{T}^{2} such that π_{1}f = N-min L̃(f) and p_{2} > N-bound L̃(f) holds p ∈ LeftComp(f).

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] Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353–359, 1996.

12 yatsuka nakamura et al.

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

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

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

[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.

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

[11] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

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

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

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

[19] 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.

[20] Alicia de la Cruz. Totally bounded metric spaces. Formalized Mathematics, 2(4):559–562, 1991.

[21] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

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

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

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

[25] Zbigniew Karno. Continuity of mappings over the union of subspaces. Formalized Mathematics, 3(1):1–16, 1992.

[26] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

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

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

[29] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

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

[31] Roman Matuszewski and Yatsuka Nakamura. Projections in n-dimensional Euclidean space to each coordinates. Formalized Mathematics, 6(4):505–509, 1997.

[32] Yatsuka Nakamura. Graph theoretical properties of arcs in the plane and Fashoda Meet Theorem. Formalized Mathematics, 7(2):193–201, 1998.

[33] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[34] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Formalized Mathematics, 6(2):255–263, 1997.

[35] Yatsuka Nakamura and Andrzej Trybulec. Components and unions of components. Formalized Mathematics, 5(4):513–517, 1996.

bounded domains and unbounded domains 13

[36] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

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

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

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

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

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

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

[43] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

[45] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

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

[47] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

[48] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Formalized Mathematics, 6(4):531–539, 1997.

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

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

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

[52] Toshihiko Watanabe. The Brouwer fixed point theorem for intervals. Formalized Mathematics, 3(1):85–88, 1992.

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

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

14 yatsuka nakamura et al.

Rotating and Reversing

Andrzej Trybulec University of Białystok

Summary. Quite a number of lemmas for the Jordan curve theorem, as yet in the case of the special polygonal curves, have been proved. By ”special” we mean, that it is a polygonal curve with edges parallel to axes and actually the lemmas have been proved, mostly, for the triangulations i.e. for finite sequences that define the curve. Moreover some of the results deal only with a special case:

- finite sequences are clockwise oriented,

- the first member of the sequence is the member with the lowest ordinate among those with the highest abscissa (N-min f, where f is a finite sequence, in the Mizar jargon).

In the change of the orientation one has to reverse the sequence (the operation introduced in [7]) and to change the second restriction one has to rotate the sequence (the operation introduced in [26]). The goal of the paper is to prove, mostly simple, facts about the relationship between properties and attributes of the finite sequence and its rotation (similar results about reversing had been proved in [7]). Some of them deal with recounting parameters, others with properties that are invariant under the rotation. We prove also that the finite sequence is either clockwise oriented or it is such after reversing. Everything is proved for the so called standard finite sequences, which means that if a point belongs to it then every point with the same abscissa or with the same ordinate, that belongs to the polygon, belongs also to the finite sequence. It does not seem that this requirement causes serious technical obstacles.

MML Identifier: REVROT 1.

The terminology and notation used here are introduced in the following articles: [24], [29], [12], [2], [23], [20], [1], [4], [6], [3], [5], [13], [28], [14], [7], [26], [22], [30], [21], [9], [10], [11], [15], [16], [18], [25], [8], [17], [27], and [19].

1. Preliminaries

For simplicity, we use the following convention: i, k, m, n are natural numbers, D is a non empty set, p is an element of D, and f is a finite sequence of elements of D.

Let S be a non trivial 1-sorted structure. Observe that the carrier of S is non trivial.

Let D be a non empty set and let f be a finite sequence of elements of D. Let us observe that f is constant if and only if:

(Def. 1) For all n, m such that n ∈ domf and m ∈ domf holds π_{n}f = π_{m}f. One can prove the following three propositions:

(1) Let D be a non empty set and f be a finite sequence of elements of D. If f yields π_{len f}f just once, then (π_{len f}f) ↫ f = lenf.

(2) For every non empty set D and for every finite sequence f of elements of D holds f_{⇂len f} = ∅.

(3) For every non empty set D and for every non empty finite sequence f of elements of D holds π_{len f}f ∈ rngf.

Let D be a non empty set, let f be a finite sequence of elements of D, and let y be a set. Let us observe that f yields y just once if and only if:

(Def. 2) There exists a set x such that x ∈ domf and y = π_{x}f and for every set

z such that z ∈ domf and z ≠ x holds π_{z}f ≠ y. The following propositions are true:

(4) Let D be a non empty set and f be a finite sequence of elements of D. If f yields π_{len f}f just once, then f − : π_{len f}f = f.

(5) Let D be a non empty set and f be a finite sequence of elements of D. If f yields π_{len f}f just once, then f :− π_{len f}f = ⟨π_{len f}f⟩.

(6) 1 len(f :− p).

(7) Let D be a non empty set, p be an element of D, and f be a finite sequence of elements of D. If p ∈ rngf, then len(f :− p) lenf.

(8) For every non empty set D and for every circular non empty finite sequence f of elements of D holds Rev(f) is circular.

2. About the Rotation

In the sequel D denotes a non empty set, p denotes an element of D, and f denotes a finite sequence of elements of D.

We now state several propositions:

(9) If p ∈ rngf and 1 i and i len(f :− p), then π_{i}f_{⟲}^{p} = π_{(i− ′1)+p↫f}f.

(10) If p ∈ rngf and p ↫ f i and i lenf, then π_{i}f = π_{(i+1)− ′p↫f}f_{⟲}^{p}.

(11) If p ∈ rngf, then π_{len(f:− p)}f_{⟲}^{p} = π_{len f}f.

(12) If p ∈ rngf and len(f :− p) < i and i lenf, then π_{i}f_{⟲}^{p} = π_{(i+p↫f)− ′len f}f.

(13) If p ∈ rngf and 1 < i and i p ↫ f, then π_{i}f = π_{(i+len f)− ′p↫f}f_{⟲}^{p}.

(14) len(f_{⟲}^{p}) = lenf.

(15) dom(f_{⟲}^{p}) = domf.

(16) Let D be a non empty set, f be a circular finite sequence of elements of D, and p be an element of D. If for every i such that 1 < i and i < lenf holds π_{i}f ≠ π_{1}f, then (f_{⟲}^{p})_{⟲}^{π_{1}f} = f.

3. Rotating Circular Ones

In the sequel f is a circular finite sequence of elements of D.

The following propositions are true:

(17) If p ∈ rngf and len(f :− p) i and i lenf, then π_{i}f_{⟲}^{p} = π_{(i+p↫f)− ′len f}f.

(18) If p ∈ rngf and 1 i and i p ↫ f, then π_{i}f = π_{(i+len f)− ′p↫f}f_{⟲}^{p}. Let D be a non trivial set. Note that there exists a finite sequence of elements

of D which is non constant and circular.

Let D be a non trivial set, let p be an element of D, and let f be a non constant circular finite sequence of elements of D. Note that f_{⟲}^{p} is non constant.

4. Finite Sequence on the Plane

The following proposition is true

(19) For every non empty natural number n holds 0_{E_{T}^{n}} ≠ 1.REALn. Let n be a non empty natural number. Note that E_{T}^{n} is non trivial. In the sequel f, g are finite sequences of elements of E_{T}^{2}. Next we state four propositions:

(20) If rngf ⊆ rngg, then rngX-coordinate(f) ⊆ rngX-coordinate(g).

(21) If rngf = rngg, then rngX-coordinate(f) = rngX-coordinate(g).

(22) If rngf ⊆ rngg, then rngY-coordinate(f) ⊆ rngY-coordinate(g).

(23) If rngf = rngg, then rngY-coordinate(f) = rngY-coordinate(g).

5. Rotating Finite Sequence on the Plane

In the sequel p denotes a point of E_{T}^{2} and f denotes a finite sequence of elements of E_{T}^{2}.

Let p be a point of E_{T}^{2} and let f be a special circular finite sequence of elements of E_{T}^{2}. Observe that f_{⟲}^{p} is special.

The following propositions are true:

(24) If p ∈ rngf and 1 i and i < len(f :− p), then L(f_{⟲}^{p}, i) = L(f, (i − ′1) + p ↫ f).

(25) If p ∈ rngf and p ↫ f i and i < lenf, then L(f, i) = L(f_{⟲}^{p}, (i − ′p ↫ f) + 1).

(26) For every circular finite sequence f of elements of E_{T}^{2} holds Inc(X-coordinate(f)) = Inc(X-coordinate(f_{⟲}^{p})).

(27) For every circular finite sequence f of elements of E_{T}^{2} holds Inc(Y-coordinate(f)) = Inc(Y-coordinate(f_{⟲}^{p})).

(28) For every non empty circular finite sequence f of elements of E_{T}^{2} holds the Go-board of f_{⟲}^{p} = the Go-board of f.

(29) For every non constant standard special circular sequence f holds Rev(f_{⟲}^{p}) = (Rev(f))_{⟲}^{p}.

6. Rotating Circular Ones (on the Plane)

In the sequel f is a circular finite sequence of elements of E_{T}^{2}.

We now state two propositions:

(30) For every circular s.c.c. finite sequence f of elements of E_{T}^{2} such that lenf > 4 holds L(f, lenf − ′1) ∩ L(f, 1) = {π_{1}f}.

(31) If p ∈ rngf and len(f:− p) i and i < lenf, then L(f_{⟲}^{p}, i) = L(f, (i+p ↫

f) − ′lenf). Let p be a point of E_{T}^{2} and let f be a circular s.c.c. finite sequence of elements

of E_{T}^{2}. One can check that f_{⟲}^{p} is s.c.c..

Let p be a point of E_{T}^{2} and let f be a non constant standard special circular sequence. Observe that f_{⟲}^{p} is unfolded.

Next we state three propositions:

(32) If p ∈ rngf and 1 i and i < p ↫ f, then L(f, i) = L(f_{⟲}^{p}, (i + lenf) − ′ p ↫ f).

(33) L̃(f_{⟲}^{p}) = L̃(f).

(34) Let G be a Go-board. Then f is a sequence which elements belong to G if and only if f_{⟲}^{p} is a sequence which elements belong to G.

rotating and reversing 19

Let p be a point of E_{T}^{2} and let f be a standard non empty circular finite sequence of elements of E_{T}^{2}. One can verify that f_{⟲}^{p} is standard.

One can prove the following three propositions:

(35) Let f be a non constant standard special circular sequence and given p, k. If p ∈ rngf and 1 k and k < p ↫ f, then leftcell(f, k) = leftcell(f_{⟲}^{p}, (k + lenf) − ′p ↫ f).

(36) For every non constant standard special circular sequence f holds LeftComp(f_{⟲}^{p}) = LeftComp(f).

(37) For every non constant standard special circular sequence f holds RightComp(f_{⟲}^{p}) = RightComp(f).

7. The Orientation

Let p be a point of E_{T}^{2} and let f be a clockwise oriented non constant standard special circular sequence. One can verify that f_{⟲}^{p} is clockwise oriented.

One can prove the following proposition

(38) Let f be a non constant standard special circular sequence. Then f is clockwise oriented or Rev(f) is clockwise oriented.

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

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

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

[7] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.

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

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

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

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

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

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

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

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

20 andrzej trybulec

[16] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

[17] Yatsuka Nakamura and Adam Grabowski. Bounding boxes for special sequences in E^{2}. Formalized Mathematics, 7(1):115–121, 1998.

[18] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

[19] Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1–13, 1999.

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

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

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

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

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

[25] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

[26] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5(3):317–322, 1996.

[27] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

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

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

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

On the Components of the Complement of a Special Polygonal Curve

Andrzej Trybulec^{1} University of Bialystok Yatsuka Nakamura Shinshu University Nagano

Summary. By the special polygonal curve we meana simple closed curve, that is a polygone and moreover has edges parallel to axes. We continue the formalization of the Takeuti-Nakamura proof [21] of the Jordan curve theorem. In the paper we prove that the complement of the special polygonal curve consists of at least two components. With the theorem which has at most two components we completed the theorem that a special polygonal curve cuts the plane into exactly two components.

MML Identifier: SPRECT 4.

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

In this paper j denotes a natural number. One can prove the following propositions:

(1) Let f be a S-sequence in ℝ^{2} and Q be a non empty compact subset of E_{T}^{2}. If L̃(f) meets Q and π_{1}f ∉ Q, then L̃(⇂ f, FPoint(L̃(f), π_{1}f, π_{len f}f, Q)) ∩ Q = {FPoint(L̃(f), π_{1}f, π_{len f}f, Q)}.

(2) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence and p = π_{len f}f, then L̃(⇃ p, f) = {p}.

(3) Let f be a finite sequence of elements of E_{T}^{2} and p be a point of E_{T}^{2}. If f is a special sequence and p ∈ L̃(f), then L̃(⇃ p, f) ⊆ L̃(f).

(4) Let f be a S-sequence in ℝ^{2}, p be a point of E_{T}^{2}, and given j. If 1 j and j < lenf and p ∈ L̃(mid(f, j, lenf)), then LE π_{j}f, p, L̃(f), π_{1}f, π_{len f}f.

1The work had been done when the first author visited Nagano in fall of 1998.

22 andrzej trybulec and yatsuka nakamura

(5) Let f be a S-sequence in ℝ^{2}, p, q be points of E_{T}^{2}, and given j. If 1 j and j < lenf and p ∈ L(f, j) and q ∈ L(p, π_{j+1}f), then LE p, q, L̃(f), π_{1}f, π_{len f}f.

(6) Let f be a S-sequence in ℝ^{2} and Q be a non empty compact subset of E_{T}^{2}. If L̃(f) meets Q and π_{len f}f ∉ Q, then L̃(⇃ LPoint(L̃(f), π_{1}f, π_{len f}f, Q), f) ∩ Q = {LPoint(L̃(f), π_{1}f, π_{len f}f, Q)}.

(7) For every non constant standard special circular sequence f holds LeftComp(f) ≠ RightComp(f).

References

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

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

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

[4] Czesław Byliński. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241–245, 1996.

[5] Czesław Byliński and Yatsuka Nakamura. Special polygons. Formalized Mathematics, 5(2):247–252, 1996.

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

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

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

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

[10] Adam Grabowski and Yatsuka Nakamura. The ordering of points on a curve. Part II. Formalized Mathematics, 6(4):467–473, 1997.

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

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

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

[14] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

[15] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[16] Yatsuka Nakamura and Jarosław Kotowicz. Connectedness conditions using polygonal arcs. Formalized Mathematics, 3(1):101–106, 1992.

[17] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Formalized Mathematics, 6(2):255–263, 1997.

[18] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

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

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

[21] Yukio Takeuchi and Yatsuka Nakamura. On the Jordan curve theorem. Technical Report 19804, Dept. of Information Eng., Shinshu University, 500 Wakasato, Nagano city, Japan, April 1980.

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

on the components of the complement of a … 23

[23] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Formalized Mathematics, 5(4):465–468, 1996.

[24] Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5(3):317–322, 1996.

[25] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Formalized Mathematics, 6(4):541–548, 1997.

[26] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Formalized Mathematics, 6(4):531–539, 1997.

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

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

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

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

24 andrzej trybulec and yatsuka nakamura

Gauges

Czesław Byliński University of Białystok

MML Identifier: JORDAN8.

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

We follow the rules: i, i_{1}, i_{2}, j, j_{1}, j_{2}, k, m, n are natural numbers, D is a non empty set, and f is a finite sequence of elements of D.

We now state two propositions:

(1) If lenf 2, then f↾2 = ⟨π_{1}f, π_{2}f⟩.

(2) If k + 1 lenf, then f↾(k + 1) = (f↾k)^{⌢} ⟨π_{k+1}f⟩. In the sequel f denotes a finite sequence of elements of E_{T}^{2}, G denotes a

Go-board, and p denotes a point of E_{T}^{2}.

The following propositions are true:

(3) ε_{(the carrier of E_{T}^{2})} is a sequence which elements belong to G.

(4) If f is a sequence which elements belong to G, then f↾m is a sequence which elements belong to G.

(5) If f is a sequence which elements belong to G, then f_{⇂m} is a sequence which elements belong to G.

(6) Suppose 1 k and k + 1 lenf and f is a sequence which elements belong to G. Then there exist natural numbers i_{1}, j_{1}, i_{2}, j_{2} such that

(i) ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G, (ii) π_{k}f = G_{i_{1},j_{1}}, (iii) ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G, (iv) π_{k+1}f = G_{i_{2},j_{2}}, and

(v) i_{1} = i_{2} and j_{1} + 1 = j_{2} or i_{1} + 1 = i_{2} and j_{1} = j_{2} or i_{1} = i_{2} + 1 and j_{1} = j_{2} or i_{1} = i_{2} and j_{1} = j_{2} + 1.

(7) Let f be a non empty finite sequence of elements of E_{T}^{2}. Suppose f is a sequence which elements belong to G. Then f is standard and special.

(8) Let f be a non empty finite sequence of elements of E_{T}^{2}. Suppose lenf 2 and f is a sequence which elements belong to G. Then f is non constant.

(9) Let f be a non empty finite sequence of elements of E_{T}^{2}. Suppose that (i) f is a sequence which elements belong to G, (ii) there exist i, j such that ⟨⟨i, j⟩⟩∈ the indices of G and p = G_{i,j}, and (iii) for all i_{1}, j_{1}, i_{2}, j_{2} such that ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G and ⟨⟨i_{2}, j_{2}⟩⟩∈ the

indices of G and π_{len f}f = G_{i_{1},j_{1}} and p = G_{i_{2},j_{2}} holds |i_{2}−i_{1}|+|j_{2}−j_{1}| = 1. Then f ^{⌢} ⟨p⟩ is a sequence which elements belong to G.

(10) If i + k < lenG and 1 j and j < widthG and cell(G, i, j) meets cell(G, i + k, j), then k 1.

(11) For every non empty compact subset C of E_{T}^{2} holds C is vertical iff E-boundC W-boundC.

(12) For every non empty compact subset C of E_{T}^{2} holds C is horizontal iff

N-boundC S-boundC. Let C be a non empty subset of E_{T}^{2} and let n be a natural number. The func-

tor Gauge(C, n) yielding a matrix over E_{T}^{2} is defined by the conditions (Def. 1). (Def. 1)(i) lenGauge(C, n) = 2^{n} + 3,

(ii) lenGauge(C, n) = widthGauge(C, n), and (iii) for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, n) holds

(Gauge(C, n))_{i,j} = [W-boundC + \frac{E-bound C−W-bound C}{2^{n}} ·(i−2), S-boundC + \frac{N-bound C−S-bound C}{2^{n}} · (j − 2)].

Let C be a compact non empty subset of E_{T}^{2} and let n be a natural number. Note that Gauge(C, n) is non trivial line X-constant and column Y-constant.

In the sequel C is a compact non vertical non horizontal non empty subset of E_{T}^{2}.

Let us consider C, n. Observe that Gauge(C, n) is line Y-increasing and column X-increasing.

The following propositions are true:

(13) lenGauge(C, n) 4.

(14) If 1 j and j lenGauge(C, n), then ((Gauge(C, n))_{2,j})_{1} = W-boundC.

(15) If 1 j and j lenGauge(C, n), then ((Gauge(C, n))_{len Gauge(C,n)− ′1,j})_{1} = E-boundC.

(16) If 1 i and i lenGauge(C, n), then ((Gauge(C, n))_{i,2})_{2} = S-boundC.

(17) If 1 i and i lenGauge(C, n), then ((Gauge(C, n))_{i,len Gauge(C,n)− ′1})_{2} = N-boundC.

(18) If i lenGauge(C, n), then cell(Gauge(C, n), i, lenGauge(C, n))∩C = ∅.

(19) If j lenGauge(C, n), then cell(Gauge(C, n), lenGauge(C, n), j) ∩ C = ∅.

(20) If i lenGauge(C, n), then cell(Gauge(C, n), i, 0) ∩ C = ∅.

gauges 27

(21) If j lenGauge(C, n), then cell(Gauge(C, n), 0, j) ∩ C = ∅.

References

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

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

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

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

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

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

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

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

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

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

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

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

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

[14] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

[15] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[16] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

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

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

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

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

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

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

28 czesław byliński

The Ring of Integers, Euclidean Rings and Modulo Integers

Christoph Schwarzweller University of Tübingen

Summary. In this article we introduce the ring of Integers, Euclidean rings and Integers modulo p. In particular we prove that the Ring of Integers is an Euclidean ring and that the Integers modulo p constitutes a field if and only if p is a prime.

MML Identifier: INT 3.

The notation and terminology used here are introduced in the following papers: [16], [21], [20], [17], [22], [4], [5], [14], [10], [12], [13], [3], [8], [7], [15], [18], [2], [6], [11], [9], [1], and [19].

1. The Ring of Integers

The binary operation multint on ℤ is defined as follows: (Def. 1) For all elements a, b of ℤ holds (multint)(a, b) = ·_{ℝ}(a, b).

The unary operation compint on ℤ is defined as follows: (Def. 2) For every element a of ℤ holds (compint)(a) = −_{ℝ}(a).

The double loop structure INT.Ring is defined by: (Def. 3) INT.Ring = ⟨ℤ, +_{ℤ}, multint, 1(∈ ℤ), 0(∈ ℤ)⟩.

Let us mention that INT.Ring is strict and non empty.

Let us mention that INT.Ring is Abelian add-associative right zeroed right complementable well unital distributive commutative associative integral domain-like and non degenerated.

Let a, b be elements of the carrier of INT.Ring. The predicate a b is defined by:

(Def. 4) There exist integers a′, b′ such that a′= a and b′= b and a′ b′. Let us notice that the predicate a b is reflexive and connected. We introduce b a as a synonym of a b. We introduce b < a and a > b as antonyms of a b.

Let a be an element of the carrier of INT.Ring. The functor |a| yields an element of the carrier of INT.Ring and is defined as follows:

(Def. 5) |a| = \{ \begin{matrix}a, if a 0_{INT.Ring} , \\ − a, otherwise. \end{matrix}

The function absint from the carrier of INT.Ring into ℕ is defined as follows: (Def. 6) For every element a of the carrier of INT.Ring holds (absint)(a) =

|□|_{ℝ}(a). One can prove the following two propositions:

(1) For every element a of the carrier of INT.Ring holds (absint)(a) = |a|.

(2) Let a, b, q_{1}, q_{2}, r_{1}, r_{2} be elements of the carrier of INT.Ring. Suppose b ≠ 0_{INT.Ring} and a = q_{1} · b + r_{1} and 0_{INT.Ring} r_{1} and r_{1} < |b| and a = q_{2} · b + r_{2} and 0_{INT.Ring} r_{2} and r_{2} < |b|. Then q_{1} = q_{2} and r_{1} = r_{2}.

Let a, b be elements of the carrier of INT.Ring. Let us assume that b ≠ 0_{INT.Ring} . The functor a ÷ b yields an element of the carrier of INT.Ring and is defined by:

(Def. 7) There exists an element r of the carrier of INT.Ring such that a =

(a ÷ b) · b + r and 0_{INT.Ring} r and r < |b|. Let a, b be elements of the carrier of INT.Ring. Let us assume that b ≠

0_{INT.Ring} . The functor a modb yields an element of the carrier of INT.Ring and is defined as follows:

(Def. 8) There exists an element q of the carrier of INT.Ring such that a =

q · b + (a mod b) and 0_{INT.Ring} a mod b and a mod b < |b|. Next we state the proposition

(3) For all elements a, b of the carrier of INT.Ring such that b ≠ 0_{INT.Ring} holds a = (a ÷ b) · b + (a mod b).

2. Euclidean Rings

Let I be a non empty double loop structure. We say that I is Euclidian if and only if the condition (Def. 9) is satisfied.

(Def. 9) There exists a function f from the carrier of I into ℕ such that for all elements a, b of the carrier of I if b ≠ 0_{I}, then there exist elements q, r of the carrier of I such that a = q · b + r but r = 0_{I} or f(r) < f(b).

One can check that INT.Ring is Euclidian. Let us observe that there exists a ring which is strict, Euclidian, integral

domain-like, non degenerated, well unital, and distributive.

A EuclidianRing is a Euclidian integral domain-like non degenerated well unital distributive ring.

Let us mention that there exists a EuclidianRing which is strict. Let E be a Euclidian non empty double loop structure. A function from the

carrier of E into ℕ is said to be a DegreeFunction of E if it satisfies the condition (Def. 10).

(Def. 10) Let a, b be elements of the carrier of E. Suppose b ≠ 0_{E}. Then there exist elements q, r of the carrier of E such that a = q · b + r but r = 0_{E} or it(r) < it(b).

Next we state the proposition

(4) Every EuclidianRing is a gcdDomain. Let us note that every integral domain-like non degenerated Abelian add-

associative right zeroed right complementable associative commutative right unital right-distributive non empty double loop structure which is Euclidian is also gcd-like.

absint is a DegreeFunction of INT.Ring. One can prove the following proposition

(5) Every commutative associative left unital field-like right zeroed non empty double loop structure is Euclidian.

Let us observe that every non empty double loop structure which is commutative, associative, left unital, field-like, right zeroed, and field-like is also Euclidian.

One can prove the following proposition

(6) Let F be a commutative associative left unital field-like right zeroed non empty double loop structure. Then every function from the carrier of F into ℕ is a DegreeFunction of F.

3. Some Theorems about Div and Mod

The following propositions are true:

(7) Let n be a natural number. Suppose n > 0. Let a be an integer and a′be a natural number. If a′= a, then a ÷ n = a′÷ n and a mod n = a′mod n.

(8) For every natural number n such that n > 0 and for all integers a, k holds (a + n · k) ÷ n = (a ÷ n) + k and (a + n · k) mod n = a mod n.

(9) For every natural number n such that n > 0 and for every integer a holds a mod n 0 and a mod n < n.

(10) Let n be a natural number. Suppose n > 0. Let a be an integer. Then (i) if 0 a and a < n, then a mod n = a, and (ii) if 0 > a and a − n, then a mod n = n + a.

(11) For every natural number n such that n > 0 and for every integer a holds a mod n = 0 iff n | a.

(12) For every natural number n such that n > 0 and for all integers a, b holds a mod n = b mod n iff a ≡ b(modn).

(13) For every natural number n such that n > 0 and for every integer a holds a mod n mod n = a mod n.

(14) For every natural number n such that n > 0 and for all integers a, b holds (a + b) mod n = ((a mod n) + (b mod n)) mod n.

(15) For every natural number n such that n > 0 and for all integers a, b holds a · b mod n = (a mod n) · (b mod n) mod n.

(16) For all integers a, b there exist integers s, t such that a gcdb = s·a+t·b.

4. Modulo Integers

Let n be a natural number. Let us assume that n > 0. The functor multintn yielding a binary operation on ℤ_{n} is defined as follows:

(Def. 11) For all elements k, l of ℤ_{n} holds (multintn)(k, l) = k · l mod n.

Let n be a natural number. Let us assume that n > 0. The functor compintn yielding a unary operation on ℤ_{n} is defined by:

(Def. 12) For every element k of ℤ_{n} holds (compintn)(k) = (n − k) mod n.

Next we state three propositions:

(17) Let n be a natural number. Suppose n > 0. Let a, b be elements of ℤ_{n}.

Then

(i) a + b < n iff +_{n}(a, b) = a + b, and (ii) a + b n iff +_{n}(a, b) = (a + b) − n.

(18) Let n be a natural number. Suppose n > 0. Let a, b be elements of ℤ_{n} and k be a natural number. Then k · n a · b and a · b < (k + 1) · n if and only if (multintn)(a, b) = a · b − k · n.

(19) Let n be a natural number. Suppose n > 0. Let a be an element of ℤ_{n}.

Then

(i) a = 0 iff (compintn)(a) = 0, and (ii) a ≠ 0 iff (compintn)(a) = n − a. Let n be a natural number. The functor INT.Ringn yields a double loop

structure and is defined by: (Def. 13) INT.Ringn = ⟨ℤ_{n}, +_{n}, multintn, 1(∈ ℤ_{n}), 0(∈ ℤ_{n})⟩.

the ring of integers, euclidean rings and … 33

Let n be a natural number. Observe that INT.Ringn is strict and non empty. We now state the proposition

(20) INT.Ring1 is degenerated and INT.Ring1 is a ring and INT.Ring1 is field-like, well unital, and distributive.

Let us note that there exists a ring which is strict, degenerated, well unital, distributive, and field-like.

One can prove the following propositions:

(21) For every natural number n such that n > 1 holds INT.Ringn is non degenerated and INT.Ringn is a well unital distributive ring.

(22) Let p be a natural number. Suppose p > 1. Then INT.Ringp is an add-associative right zeroed right complementable Abelian commutative associative left unital distributive field-like non degenerated non empty double loop structure if and only if p is a prime number.

Let p be a prime number. Observe that INT.Ringp is add-associative right zeroed right complementable Abelian commutative associative left unital distributive field-like and non degenerated.

References

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

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

[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] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

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

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

[9] Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321–328, 1990.

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

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

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

[13] Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555–561, 1990.

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34 christoph schwarzweller

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Logic Gates and Logical Equivalence of Adders

Yatsuka Nakamura Shinshu University Nagano

Summary. This is an experimental article which shows that logical correctness of logic circuits can be easily proven by the Mizar system. First, we define the notion of logic gates. Then we prove that an MSB carry of ’4 Bit Carry Skip Adder’ is equivalent to an MSB carry of a normal 4 bit adder. In the last theorem, we show that outputs of the ’4 Bit Carry Look Ahead Adder’ are equivalent to the corresponding outputs of the normal 4 bits adder. The policy here is as follows: when the functional (semantic) correctness of a system is already proven, and the correspondence of the system to a (normal) logic circuit is given, it is enough to prove the correctness of the new circuit if we only prove the logical equivalence between them. Although the article is very fundamental (it contains few environment files), it can be applied to real problems. The key of the method introduced here is to put the specification of the logic circuit into the Mizar propositional formulae, and to use the strong inference ability of the Mizar checker. The proof is done formally so that the automation of the proof writing is possible. Even in the 5.3.07 version of Mizar, it can handle a formulae of more than 100 lines, and a formula which contains more than 100 variables. This means that the Mizar system is enough to prove logical correctness of middle scaled logic circuits.

MML Identifier: GATE 1.

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

1. Definition of Logical Values and Logic Gates

Let a be a set. We introduce NE a as an antonym of a is empty. We now state three propositions:

(1) For every set a such that a = {∅} holds NE a.

(2) There exists a set a such that NE a.

(3) NE ∅ iff contradiction. let a be a set. The functor NOT1a yielding a set is defined by:

(Def. 1) NOT1a = \{ \begin{matrix}∅, if NE a, \\ \{∅\}, otherwise. \end{matrix}

The following proposition is true

(4) For every set a holds NE NOT1a iff not NE a. In the sequel a, b are sets. We now state the proposition

(5) NE NOT1∅. Let a, b be sets. The functor AND2(a, b) yields a set and is defined by:

(Def. 2) AND2(a, b) = \{ \begin{matrix}NOT1∅, if NE a and NE b, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(6) For all sets a, b holds NE AND2(a, b) iff NE a and NE b. Let a, b be sets. The functor OR2(a, b) yielding a set is defined as follows:

(Def. 3) OR2(a, b) = \{ \begin{matrix}NOT1∅, if NE a or NE b, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(7) For all sets a, b holds NE OR2(a, b) iff NE a or NE b. Let a, b be sets. The functor XOR2(a, b) yields a set and is defined by:

(Def. 4) XOR2(a, b) = \{ \begin{matrix}NOT1∅, if NE a and not NE b or not NE a and NE b, \\ ∅, otherwise. \end{matrix}

The following four propositions are true:

(8) For all sets a, b holds NE XOR2(a, b) iff NE a and not NE b or not NE a and NE b.

(9) NE XOR2(a, a) iff contradiction.

(10) NE XOR2(a, ∅) iff NE a.

(11) NE XOR2(a, b) iff NE XOR2(b, a).

Let a, b be sets. The functor EQV2(a, b) yielding a set is defined by:

(Def. 5) EQV2(a, b) = \{ \begin{matrix}NOT1∅, if NE a iff NE b, \\ ∅, otherwise. \end{matrix}

We now state two propositions:

(12) For all sets a, b holds NE EQV2(a, b) iff NE a iff NE b.

(13) NE EQV2(a, b) iff not NE XOR2(a, b).

Let a, b be sets. The functor NAND2(a, b) yielding a set is defined by:

(Def. 6) NAND2(a, b) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(14) For all sets a, b holds NE NAND2(a, b) iff not NE a or not NE b. Let a, b be sets. The functor NOR2(a, b) yielding a set is defined as follows:

(Def. 7) NOR2(a, b) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b, \\ ∅, otherwise. \end{matrix}

We now state the proposition

(15) For all sets a, b holds NE NOR2(a, b) iff not NE a and not NE b. Let a, b, c be sets. The functor AND3(a, b, c) yields a set and is defined by:

(Def. 8) AND3(a, b, c) = \{ \begin{matrix}NOT1∅, if NE a and NE b and NE c, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(16) For all sets a, b, c holds NE AND3(a, b, c) iff NE a and NE b and NE c. Let a, b, c be sets. The functor OR3(a, b, c) yielding a set is defined by:

(Def. 9) OR3(a, b, c) = \{ \begin{matrix}NOT1∅, if NE a or NE b or NE c, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(17) For all sets a, b, c holds NE OR3(a, b, c) iff NE a or NE b or NE c. Let a, b, c be sets. The functor XOR3(a, b, c) yielding a set is defined by:

(Def. 10) XOR3(a, b, c) = \{ \begin{matrix}NOT1∅, if NE a and not NE b or not NE a and NE \\ b but not NE c or not NE a or not NE b but not \\ NE a or not NE b and NE c, \\ ∅, otherwise. \end{matrix}

We now state the proposition

(18) Let a, b, c be sets. Then NE XOR3(a, b, c) if and only if one of the

following conditions is satisfied:

(i) NE a and not NE b or not NE a and NE b but not NE c, or (ii) not NE a or not NE b but not NE a or not NE b and NE c. Let a, b, c be sets. The functor MAJ3(a, b, c) yields a set and is defined as

follows:

(Def. 11) MAJ3(a, b, c) = \{ \begin{matrix}NOT1∅, if NE a and NE b or NE b and NE c or NE \\ c and NE a, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(19) For all sets a, b, c holds NE MAJ3(a, b, c) iff NE a and NE b or NE b

and NE c or NE c and NE a. Let a, b, c be sets. The functor NAND3(a, b, c) yielding a set is defined by:

(Def. 12) NAND3(a, b, c) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b or not NE c, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(20) For all sets a, b, c holds NE NAND3(a, b, c) iff not NE a or not NE b or

not NE c. Let a, b, c be sets. The functor NOR3(a, b, c) yields a set and is defined by:

(Def. 13) NOR3(a, b, c) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b and not NE c, \\ ∅, otherwise. \end{matrix}

We now state the proposition

(21) For all sets a, b, c holds NE NOR3(a, b, c) iff not NE a and not NE b and

not NE c. Let a, b, c, d be sets. The functor AND4(a, b, c, d) yields a set and is defined

by:

(Def. 14) AND4(a, b, c, d) = \{ \begin{matrix}NOT1∅, if NE a and NE b and NE c and NE d, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(22) For all sets a, b, c, d holds NE AND4(a, b, c, d) iff NE a and NE b and

NE c and NE d. Let a, b, c, d be sets. The functor OR4(a, b, c, d) yielding a set is defined as

follows:

(Def. 15) OR4(a, b, c, d) = \{ \begin{matrix}NOT1∅, if NE a or NE b or NE c or NE d, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(23) For all sets a, b, c, d holds NE OR4(a, b, c, d) iff NE a or NE b or NE c

or NE d. Let a, b, c, d be sets. The functor NAND4(a, b, c, d) yielding a set is defined

by:

(Def. 16) NAND4(a, b, c, d) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b or not NE c or \\ not NE d, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(24) For all sets a, b, c, d holds NE NAND4(a, b, c, d) iff not NE a or not NE

b or not NE c or not NE d. Let a, b, c, d be sets. The functor NOR4(a, b, c, d) yielding a set is defined

by:

(Def. 17) NOR4(a, b, c, d) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b and not NE \\ c and not NE d, \\ ∅, otherwise. \end{matrix}

The following proposition is true (25) For all sets a, b, c, d holds NE NOR4(a, b, c, d) iff not NE a and not NE

b and not NE c and not NE d. Let a, b, c, d, e be sets. The functor AND5(a, b, c, d, e) yielding a set is defined

as follows:

(Def. 18) AND5(a, b, c, d, e) = \{ \begin{matrix}NOT1∅, if NE a and NE b and NE c and NE d \\ and NE e, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(26) For all sets a, b, c, d, e holds NE AND5(a, b, c, d, e) iff NE a and NE b

and NE c and NE d and NE e. Let a, b, c, d, e be sets. The functor OR5(a, b, c, d, e) yields a set and is

defined by:

(Def. 19) OR5(a, b, c, d, e) = \{ \begin{matrix}NOT1∅, if NE a or NE b or NE c or NE d or NE e, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(27) For all sets a, b, c, d, e holds NE OR5(a, b, c, d, e) iff NE a or NE b or

NE c or NE d or NE e. Let a, b, c, d, e be sets. The functor NAND5(a, b, c, d, e) yields a set and is

defined as follows:

(Def. 20) NAND5(a, b, c, d, e) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b or not NE c \\ or not NE d or not NE e, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(28) For all sets a, b, c, d, e holds NE NAND5(a, b, c, d, e) iff not NE a or not

NE b or not NE c or not NE d or not NE e. Let a, b, c, d, e be sets. The functor NOR5(a, b, c, d, e) yielding a set is defined

as follows:

(Def. 21) NOR5(a, b, c, d, e) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b and not NE c \\ and not NE d and not NE e, \\ ∅, otherwise. \end{matrix}

We now state the proposition

(29) For all sets a, b, c, d, e holds NE NOR5(a, b, c, d, e) iff not NE a and not

NE b and not NE c and not NE d and not NE e. Let a, b, c, d, e, f be sets. The functor AND6(a, b, c, d, e, f) yielding a set is

defined by:

(Def. 22) AND6(a, b, c, d, e, f) = \{ \begin{matrix}NOT1∅, if NE a and NE b and NE c and NE d \\ and NE e and NE f, \\ ∅, otherwise. \end{matrix} (30) Let a, b, c, d, e, f be sets. Then NE AND6(a, b, c, d, e, f) if and only if

the following conditions are satisfied:

(i) NE a, (ii) NE b, (iii) NE c, (iv) NE d,

(v) NE e, and (vi) NE f. Let a, b, c, d, e, f be sets. The functor OR6(a, b, c, d, e, f) yielding a set is

defined by:

(Def. 23) OR6(a, b, c, d, e, f) = \{ \begin{matrix}NOT1∅, if NE a or NE b or NE c or NE d or \\ NE e or NE f, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(31) Let a, b, c, d, e, f be sets. Then NE OR6(a, b, c, d, e, f) if and only if one

of the following conditions is satisfied:

(i) NE a, or (ii) NE b, or (iii) NE c, or (iv) NE d, or

(v) NE e, or (vi) NE f. Let a, b, c, d, e, f be sets. The functor NAND6(a, b, c, d, e, f) yields a set

and is defined by:

(Def. 24) NAND6(a, b, c, d, e, f) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b or not NE \\ c or not NE d or not NE e or not NE f, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(32) Let a, b, c, d, e, f be sets. Then NE NAND6(a, b, c, d, e, f) if and only if

one of the following conditions is satisfied:

(i) not NE a, or (ii) not NE b, or (iii) not NE c, or (iv) not NE d, or

(v) not NE e, or (vi) not NE f. Let a, b, c, d, e, f be sets. The functor NOR6(a, b, c, d, e, f) yields a set and

is defined as follows:

(Def. 25) NOR6(a, b, c, d, e, f) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b and not NE \\ c and not NE d and not NE e and not NE f, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(33) Let a, b, c, d, e, f be sets. Then NE NOR6(a, b, c, d, e, f) if and only if

the following conditions are satisfied:

(i) not NE a, (ii) not NE b, (iii) not NE c, (iv) not NE d,

(v) not NE e, and (vi) not NE f. Let a, b, c, d, e, f, g be sets. The functor AND7(a, b, c, d, e, f, g) yields a set

and is defined by:

(Def. 26) AND7(a, b, c, d, e, f, g) = \{ \begin{matrix}NOT1∅, if NE a and NE b and NE c and \\ NE d and NE e and NE f and NE g, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(34) Let a, b, c, d, e, f, g be sets. Then NE AND7(a, b, c, d, e, f, g) if and only if the following conditions are satisfied: NE a and NE b and NE c and NE d and NE e and NE f and NE g.

Let a, b, c, d, e, f, g be sets. The functor OR7(a, b, c, d, e, f, g) yielding a set is defined as follows:

(Def. 27) OR7(a, b, c, d, e, f, g) = \{ \begin{matrix}NOT1∅, if NE a or NE b or NE c or NE d or \\ NE e or NE f or NE g, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(35) Let a, b, c, d, e, f, g be sets. Then NE OR7(a, b, c, d, e, f, g) if and only if one of the following conditions is satisfied: NE a or NE b or NE c or NE d or NE e or NE f or NE g.

Let a, b, c, d, e, f, g be sets. The functor NAND7(a, b, c, d, e, f, g) yielding a set is defined as follows:

(Def. 28) NAND7(a, b, c, d, e, f, g) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b or \\ not NE c or not NE d or not NE e or not \\ NE f or not NE g, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(36) Let a, b, c, d, e, f, g be sets. Then NE NAND7(a, b, c, d, e, f, g) if and only if one of the following conditions is satisfied: not NE a or not NE b or not NE c or not NE d or not NE e or not NE f or not NE g.

Let a, b, c, d, e, f, g be sets. The functor NOR7(a, b, c, d, e, f, g) yielding a set is defined as follows:

(Def. 29) NOR7(a, b, c, d, e, f, g) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b and \\ not NE c and not NE d and not NE e and \\ not NE f and not NE g, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(37) Let a, b, c, d, e, f, g be sets. Then NE NOR7(a, b, c, d, e, f, g) if and only if the following conditions are satisfied: not NE a and not NE b and not NE c and not NE d and not NE e and not NE f and not NE g.

Let a, b, c, d, e, f, g, h be sets. The functor AND8(a, b, c, d, e, f, g, h) yields a set and is defined by:

(Def. 30) AND8(a, b, c, d, e, f, g, h) = \{ \begin{matrix}NOT1∅, if NE a and NE b and NE c and \\ NE d and NE e and NE f and NE g and \\ NE h, \\ ∅, otherwise. \end{matrix}

The following proposition is true

(38) Let a, b, c, d, e, f, g, h be sets. Then NE AND8(a, b, c, d, e, f, g, h) if and only if the following conditions are satisfied: NE a and NE b and NE c and NE d and NE e and NE f and NE g and NE h.

Let a, b, c, d, e, f, g, h be sets. The functor OR8(a, b, c, d, e, f, g, h) yielding a set is defined as follows:

(Def. 31) OR8(a, b, c, d, e, f, g, h) = \{ \begin{matrix}NOT1∅, if NE a or NE b or NE c or NE d \\ or NE e or NE f or NE g or NE h, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(39) Let a, b, c, d, e, f, g, h be sets. Then NE OR8(a, b, c, d, e, f, g, h) if and only if one of the following conditions is satisfied: NE a or NE b or NE c or NE d or NE e or NE f or NE g or NE h.

Let a, b, c, d, e, f, g, h be sets. The functor NAND8(a, b, c, d, e, f, g, h) yielding a set is defined as follows:

(Def. 32) NAND8(a, b, c, d, e, f, g, h) = \{ \begin{matrix}NOT1∅, if not NE a or not NE b or \\ not NE c or not NE d or not NE e or \\ not NE f or not NE g or not NE h, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(40) Let a, b, c, d, e, f, g, h be sets. Then NE NAND8(a, b, c, d, e, f, g, h) if and only if one of the following conditions is satisfied: not NE a or not NE b or not NE c or not NE d or not NE e or not NE f or not NE g or not NE h.

Let a, b, c, d, e, f, g, h be sets. The functor NOR8(a, b, c, d, e, f, g, h) yielding a set is defined as follows:

(Def. 33) NOR8(a, b, c, d, e, f, g, h) = \{ \begin{matrix}NOT1∅, if not NE a and not NE b and \\ not NE c and not NE d and not NE e \\ and not NE f and not NE g and not \\ NE h, \\ ∅, otherwise. \end{matrix}

One can prove the following proposition

(41) Let a, b, c, d, e, f, g, h be sets. Then NE NOR8(a, b, c, d, e, f, g, h) if and only if the following conditions are satisfied: not NE a and not NE b and not NE c and not NE d and not NE e and not NE f and not NE g and not NE h.

2. Logical Equivalence of 4 Bits Adders

We now state the proposition

(42) Let c_{1}, x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}, y_{3}, y_{4}, c_{2}, c_{3}, c_{4}, c_{5}, n_{1}, n_{2}, n_{3}, n_{4}, n, c_{6} be sets. Suppose that NE c_{2} iff NE MAJ3(x_{1}, y_{1}, c_{1}) and NE c_{3} iff NE MAJ3(x_{2}, y_{2}, c_{2}) and NE c_{4} iff NE MAJ3(x_{3}, y_{3}, c_{3}) and NE c_{5} iff NE MAJ3(x_{4}, y_{4}, c_{4}) and NE n_{1} iff NE OR2(x_{1}, y_{1}) and NE n_{2} iff NE OR2(x_{2}, y_{2}) and NE n_{3} iff NE OR2(x_{3}, y_{3}) and NE n_{4} iff NE OR2(x_{4}, y_{4}) and NE n iff NE AND5(c_{1}, n_{1}, n_{2}, n_{3}, n_{4}) and NE c_{6} iff NE OR2(c_{5}, n). Then NE c_{5} if and only if NE c_{6}.

Let a, b be sets. The functor MODADD2(a, b) yields a set and is defined as follows:

(Def. 34) MODADD2(a, b) = \{ \begin{matrix}NOT1∅, if NE a or NE b but NE a but NE b, \\ ∅, otherwise. \end{matrix}

Next we state the proposition

(43) For all sets a, b holds NE MODADD2(a, b) iff NE a or NE b but NE a

but NE b. Let a, b, c be sets. The functor ADD1(a, b, c) yields a set and is defined by:

(Def. 35) ADD1(a, b, c) = XOR3(a, b, c).

Let a, b, c be sets. The functor CARR1(a, b, c) yielding a set is defined by: (Def. 36) CARR1(a, b, c) = MAJ3(a, b, c).

Let a_{1}, b_{1}, a_{2}, b_{2}, c be sets. The functor ADD2(a_{2}, b_{2}, a_{1}, b_{1}, c) yielding a set is defined as follows:

(Def. 37) ADD2(a_{2}, b_{2}, a_{1}, b_{1}, c) = XOR3(a_{2}, b_{2}, CARR1(a_{1}, b_{1}, c)).

Let a_{1}, b_{1}, a_{2}, b_{2}, c be sets. The functor CARR2(a_{2}, b_{2}, a_{1}, b_{1}, c) yields a set and is defined as follows:

(Def. 38) CARR2(a_{2}, b_{2}, a_{1}, b_{1}, c) = MAJ3(a_{2}, b_{2}, CARR1(a_{1}, b_{1}, c)).

Let a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, c be sets. The functor ADD3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yields a set and is defined by:

(Def. 39) ADD3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) = XOR3(a_{3}, b_{3}, CARR2(a_{2}, b_{2}, a_{1}, b_{1}, c)).

Let a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, c be sets. The functor CARR3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yields a set and is defined as follows:

(Def. 40) CARR3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) = MAJ3(a_{3}, b_{3}, CARR2(a_{2}, b_{2}, a_{1}, b_{1}, c)). Let a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, a_{4}, b_{4}, c be sets. The functor ADD4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yielding a set is defined by:

(Def. 41) ADD4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) =

XOR3(a_{4}, b_{4}, CARR3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c)). Let a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}, a_{4}, b_{4}, c be sets. The functor CARR4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) yields a set and is defined

as follows: (Def. 42) CARR4(a_{4}, b_{4}, a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c) =

MAJ3(a_{4}, b_{4}, CARR3(a_{3}, b_{3}, a_{2}, b_{2}, a_{1}, b_{1}, c)). One can prove the following proposition

(44) Let c_{1}, x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}, x_{4}, y_{4}, c_{4}, q_{1}, p_{1}, s_{1}, q_{2}, p_{2}, s_{2}, q_{3}, p_{3}, s_{3}, q_{4}, p_{4}, s_{4}, c_{7}, c_{8}, l_{2}, t_{2}, l_{3}, m_{3}, t_{3}, l_{4}, m_{4}, n_{4}, t_{4}, l_{5}, m_{5}, n_{5}, o_{5}, s_{5}, s_{6}, s_{7}, s_{8} be sets such that NE q_{1} iff NE NOR2(x_{1}, y_{1}) and NE p_{1} iff NE NAND2(x_{1}, y_{1}) and NE s_{1} iff NE MODADD2(x_{1}, y_{1}) and NE q_{2} iff NE NOR2(x_{2}, y_{2}) and NE p_{2} iff NE NAND2(x_{2}, y_{2}) and NE s_{2} iff NE MODADD2(x_{2}, y_{2}) and NE q_{3} iff NE NOR2(x_{3}, y_{3}) and NE p_{3} iff NE NAND2(x_{3}, y_{3}) and NE s_{3} iff NE MODADD2(x_{3}, y_{3}) and NE q_{4} iff NE NOR2(x_{4}, y_{4}) and NE p_{4} iff NE NAND2(x_{4}, y_{4}) and NE s_{4} iff NE MODADD2(x_{4}, y_{4}) and NE c_{7} iff NE NOT1c_{1} and NE c_{8} iff NE NOT1c_{7} and NE s_{5} iff NE XOR2(c_{8}, s_{1}) and NE l_{2} iff NE AND2(c_{7}, p_{1}) and NE t_{2} iff NE NOR2(l_{2}, q_{1}) and NE s_{6} iff NE XOR2(t_{2}, s_{2}) and NE l_{3} iff NE AND2(q_{1}, p_{2}) and NE m_{3} iff NE AND3(p_{2}, p_{1}, c_{7}) and NE t_{3} iff NE NOR3(l_{3}, m_{3}, q_{2}) and NE s_{7} iff NE XOR2(t_{3}, s_{3}) and NE l_{4} iff NE AND2(q_{2}, p_{3}) and NE m_{4} iff NE AND3(q_{1}, p_{3}, p_{2}) and NE n_{4} iff NE AND4(p_{3}, p_{2}, p_{1}, c_{7}) and NE t_{4} iff NE NOR4(l_{4}, m_{4}, n_{4}, q_{3}) and NE s_{8} iff NE XOR2(t_{4}, s_{4}) and NE l_{5} iff NE AND2(q_{3}, p_{4}) and NE m_{5} iff NE AND3(q_{2}, p_{4}, p_{3}) and NE n_{5} iff NE AND4(q_{1}, p_{4}, p_{3}, p_{2}) and NE o_{5} iff NE AND5(p_{4}, p_{3}, p_{2}, p_{1}, c_{7}) and NE c_{4} iff NE NOR5(q_{4}, l_{5}, m_{5}, n_{5}, o_{5}). Then

(i) NE s_{5} iff NE ADD1(x_{1}, y_{1}, c_{1}), (ii) NE s_{6} iff NE ADD2(x_{2}, y_{2}, x_{1}, y_{1}, c_{1}), (iii) NE s_{7} iff NE ADD3(x_{3}, y_{3}, x_{2}, y_{2}, x_{1}, y_{1}, c_{1}),

logic gates and logical equivalence of adders 45

(iv) NE s_{8} iff NE ADD4(x_{4}, y_{4}, x_{3}, y_{3}, x_{2}, y_{2}, x_{1}, y_{1}, c_{1}), and

(v) NE c_{4} iff NE CARR4(x_{4}, y_{4}, x_{3}, y_{3}, x_{2}, y_{2}, x_{1}, y_{1}, c_{1}).

References

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

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

46 yatsuka nakamura

The Sequential Closure Operator in Sequential and Frechet Spaces

Bartłomiej Skorulski University of Białystok

MML Identifier: FRECHET2.

The articles [26], [30], [2], [21], [10], [3], [11], [29], [9], [31], [6], [7], [23], [8], [4], [13], [1], [20], [19], [24], [18], [17], [14], [16], [5], [12], [22], [28], [15], [27], and [25] provide the notation and terminology for this paper.

1. The Properties of Sequences and Subsequences

Let T be a non empty 1-sorted structure, let f be a function from ℕ into ℕ, and let S be a sequence of T. Then S · f is a sequence of T.

One can prove the following two propositions:

(1) Let T be a non empty 1-sorted structure, S be a sequence of T, and N_{1} be an increasing sequence of naturals. Then S · N_{1} is a sequence of T.

(2) For every sequence R_{1} of real numbers such that R_{1} = id_{ℕ} holds R_{1} is an increasing sequence of naturals.

Let T be a non empty 1-sorted structure and let S be a sequence of T. A sequence of T is called a subsequence of S if:

(Def. 1) There exists an increasing sequence N_{1} of naturals such that it = S · N_{1}. The following two propositions are true:

(3) For every non empty 1-sorted structure T holds every sequence S of T is a subsequence of S.

(4) For every non empty 1-sorted structure T and for every sequence S of T and for every subsequence S_{1} of S holds rngS_{1} ⊆ rngS.

Let T be a non empty 1-sorted structure, let N_{1} be an increasing sequence of naturals, and let S be a sequence of T. Then S · N_{1} is a subsequence of S.

One can prove the following proposition

(5) Let T be a non empty 1-sorted structure, S_{1} be a sequence of T, and S_{2} be a subsequence of S_{1}. Then every subsequence of S_{2} is a subsequence of S_{1}.

In this article we present several logical schemes. The scheme SubSeqChoice deals with a non empty 1-sorted structure A, a sequence B of A, and and states that:

There exists a subsequence S_{1} of B such that for every natural number n holds P[S_{1}(n)]

provided the following requirement is met:

• For every natural number n there exists a natural number m and there exists a point x of A such that n m and x = B(m) and P[x].

The scheme SubSeqChoice1 deals with a non empty topological structure A, a sequence B of A, and and states that:

There exists a subsequence S_{1} of B such that for every natural number n holds P[S_{1}(n)]

provided the parameters have the following property:

• For every natural number n there exists a natural number m and there exists a point x of A such that n m and x = B(m) and P[x].

One can prove the following propositions:

(6) Let T be a non empty 1-sorted structure, S be a sequence of T, and A be a subset of the carrier of T. Suppose that for every subsequence S_{1} of S holds rngS_{1} ⊈ A. Then there exists a natural number n such that for every natural number m such that n m holds S(m) ∉ A.

(7) Let T be a non empty 1-sorted structure, S be a sequence of T, and A, B be subsets of the carrier of T. If rngS ⊆ A ∪ B, then there exists a subsequence S_{1} of S such that rngS_{1} ⊆ A or rngS_{1} ⊆ B.

(8) Let T be a non empty topological space. Suppose that for every sequence S of T and for all points x_{1}, x_{2} of T such that x_{1} ∈ LimS and x_{2} ∈ LimS holds x_{1} = x_{2}. Then T is a T_{1} space.

(9) Let T be a non empty topological space. Suppose T is a T_{2} space. Let S be a sequence of T and x_{1}, x_{2} be points of T. If x_{1} ∈ LimS and x_{2} ∈ LimS, then x_{1} = x_{2}.

(10) Let T be a non empty topological space. Suppose T is first-countable. Then T is a T_{2} space if and only if for every sequence S of T and for all points x_{1}, x_{2} of T such that x_{1} ∈ LimS and x_{2} ∈ LimS holds x_{1} = x_{2}.

(11) For every non empty topological structure T and for every sequence S of T such that S is not convergent holds LimS = ∅.

(12) Let T be a non empty topological space and A be a subset of T. If A is closed, then for every sequence S of T such that rngS ⊆ A holds LimS ⊆ A.

(13) Let T be a non empty topological structure, S be a sequence of T, and x be a point of T. Suppose S is not convergent to x. Then there exists a subsequence S_{1} of S such that every subsequence of S_{1} is not convergent to x.

2. The Continuous Maps

One can prove the following two propositions:

(14) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into T_{2}. Suppose f is continuous. Let S_{1} be a sequence of T_{1} and S_{2} be a sequence of T_{2}. If S_{2} = f · S_{1}, then f^{∘} LimS_{1} ⊆ LimS_{2}.

(15) Let T_{1}, T_{2} be non empty topological spaces and f be a map from T_{1} into T_{2}. Suppose T_{1} is sequential. Then f is continuous if and only if for every sequence S_{1} of T_{1} and for every sequence S_{2} of T_{2} such that S_{2} = f · S_{1} holds f^{∘} LimS_{1} ⊆ LimS_{2}.

3. The Sequential Closure Operator

Let T be a non empty topological structure and let A be a subset of the carrier of T. The functor Cl_{Seq} A yielding a subset of T is defined by:

(Def. 2) For every point x of T holds x ∈ Cl_{Seq} A iff there exists a sequence S of

T such that rngS ⊆ A and x ∈ LimS. The following propositions are true:

(16) Let T be a non empty topological structure, A be a subset of T, S be a sequence of T, and x be a point of T. If rngS ⊆ A and x ∈ LimS, then x ∈ \overline{A}.

(17) For every non empty topological structure T and for every subset A of T holds Cl_{Seq} A ⊆ \overline{A}.

(18) Let T be a non empty topological structure, S be a sequence of T, S_{1} be a subsequence of S, and x be a point of T. If S is convergent to x, then S_{1} is convergent to x.

(19) Let T be a non empty topological structure, S be a sequence of T, and S_{1} be a subsequence of S. Then LimS ⊆ LimS_{1}.

(20) For every non empty topological structure T holds Cl_{Seq}(∅_{T}) = ∅.

(21) For every non empty topological structure T and for every subset A of T holds A ⊆ Cl_{Seq} A.

(22) For every non empty topological structure T and for all subsets A, B of T holds Cl_{Seq} A ∪ Cl_{Seq} B = Cl_{Seq}(A ∪ B).

(23) Let T be a non empty topological structure. Then T is Frechet if and only if for every subset A of the carrier of T holds \overline{A} = Cl_{Seq} A.

(24) Let T be a non empty topological space. Suppose T is Frechet. Let A, B be subsets of T. Then Cl_{Seq}(∅_{T}) = ∅ and A ⊆ Cl_{Seq} A and Cl_{Seq}(A ∪ B) = Cl_{Seq} A ∪ Cl_{Seq} B and Cl_{Seq} Cl_{Seq} A = Cl_{Seq} A.

(25) Let T be a non empty topological space. Suppose T is sequential. If for every subset A of T holds Cl_{Seq} Cl_{Seq} A = Cl_{Seq} A, then T is Frechet.

(26) Let T be a non empty topological space. Suppose T is sequential. Then T is Frechet if and only if for all subsets A, B of T holds Cl_{Seq}(∅_{T}) = ∅ and A ⊆ Cl_{Seq} A and Cl_{Seq}(A∪B) = Cl_{Seq} A∪Cl_{Seq} B and Cl_{Seq} Cl_{Seq} A = Cl_{Seq} A.

4. The Limit

Let T be a non empty topological space and let S be a sequence of T. Let us assume that there exists a point x of T such that LimS = {x}. The functor limS yields a point of T and is defined as follows:

(Def. 3) S is convergent to limS.

The following propositions are true:

(27) Let T be a non empty topological space. Suppose T is a T_{2} space. Let S be a sequence of T. If S is convergent, then there exists a point x of T such that LimS = {x}.

(28) Let T be a non empty topological space. Suppose T is a T_{2} space. Let S be a sequence of T and x be a point of T. Then S is convergent to x if and only if S is convergent and x = limS.

(29) For every metric structure M holds every sequence of M is a sequence of M_{top}.

(30) For every non empty metric structure M holds every sequence of M_{top} is a sequence of M.

(31) Let M be a non empty metric space, S be a sequence of M, x be a point of M, S′be a sequence of M_{top}, and x′be a point of M_{top}. Suppose S = S′ and x = x′. Then S is convergent to x if and only if S′is convergent to x′.

(32) Let M be a non empty metric space, S_{3} be a sequence of M, and S_{4} be a sequence of M_{top}. If S_{3} = S_{4}, then S_{3} is convergent iff S_{4} is convergent.

(33) Let M be a non empty metric space, S_{3} be a sequence of M, and S_{4} be a sequence of M_{top}. If S_{3} = S_{4} and S_{3} is convergent, then limS_{3} = limS_{4}.

5. The Cluster Points

Let T be a topological structure, let S be a sequence of T, and let x be a point of T. We say that x is a cluster point of S if and only if the condition (Def. 4) is satisfied.

(Def. 4) Let O be a subset of T and n be a natural number. Suppose O is open and x ∈ O. Then there exists a natural number m such that n m and S(m) ∈ O.

Next we state several propositions:

(34) Let T be a non empty topological structure, S be a sequence of T, and x be a point of T. If there exists a subsequence of S which is convergent to x, then x is a cluster point of S.

(35) Let T be a non empty topological structure, S be a sequence of T, and x be a point of T. If S is convergent to x, then x is a cluster point of S.

(36) Let T be a non empty topological structure, S be a sequence of T, x be a point of T, and Y be a subset of the carrier of T. If Y = {y; y ranges over points of T: x ∈ \overline{\{y\}}\} and rngS ⊆ Y, then S is convergent to x.

(37) Let T be a non empty topological structure, S be a sequence of T, and x, y be points of T. Suppose that for every natural number n holds S(n) = y and S is convergent to x. Then x ∈ \overline{\{y\}}.

(38) Let T be a non empty topological structure, x be a point of T, Y be a subset of the carrier of T, and S be a sequence of T. Suppose Y = {y; y ranges over points of T: x ∈ \overline{\{y\}}\} and rngS ∩ Y = ∅ and S is convergent to x. Then there exists a subsequence of S which is one-to-one.

(39) Let T be a non empty topological structure and S_{1}, S_{2} be sequences of T. Suppose rngS_{2} ⊆ rngS_{1} and S_{2} is one-to-one. Then there exists a permutation P of ℕ such that S_{2} · P is a subsequence of S_{1}.

Now we present two schemes. The scheme PermSeq deals with a non empty 1-sorted structure A, a sequence B of A, a permutation C of ℕ, and and states that:

There exists a natural number n such that for every natural number m such that n m holds P[(B · C)(m)]

provided the following condition is satisfied:

• There exists a natural number n such that for every natural number m and for every point x of A if n m and x = B(m), then P[x].

The scheme PermSeq2 deals with a non empty topological structure A, a sequence B of A, a permutation C of ℕ, and and states that:

There exists a natural number n such that for every natural number m such that n m holds P[(B · C)(m)]

provided the parameters meet the following condition:

• There exists a natural number n such that for every natural number m and for every point x of A if n m and x = B(m), then P[x].

We now state several propositions:

(40) Let T be a non empty topological structure, S be a sequence of T, P be a permutation of ℕ, and x be a point of T. If S is convergent to x, then S · P is convergent to x.

(41) Let n_{0} be a natural number. Then there exists an increasing sequence N_{1} of naturals such that for every natural number n holds N_{1}(n) = n+n_{0}.

(42) Let T be a non empty 1-sorted structure, S be a sequence of T, and n_{0} be a natural number. Then there exists a subsequence S_{1} of S such that for every natural number n holds S_{1}(n) = S(n + n_{0}).

(43) Let T be a non empty topological structure, S be a sequence of T, x be a point of T, and S_{1} be a subsequence of S. Suppose x is a cluster point of S and there exists a natural number n_{0} such that for every natural number n holds S_{1}(n) = S(n + n_{0}). Then x is a cluster point of S_{1}.

(44) Let T be a non empty topological structure, S be a sequence of T, and x be a point of T. If x is a cluster point of S, then x ∈ \overline{rngS}.

(45) Let T be a non empty topological structure. Suppose T is Frechet. Let S be a sequence of T and x be a point of T. If x is a cluster point of S, then there exists a subsequence of S which is convergent to x.

6. Auxiliary Theorems

We now state several propositions:

(46) Let T be a non empty topological space. Suppose T is first-countable. Let x be a point of T. Then there exists a basis B of x and there exists a function S such that domS = ℕ and rngS = B and for all natural numbers n, m such that m n holds S(m) ⊆ S(n).

(47) For every non empty topological space T holds T is a T_{1} space iff for every point p of T holds \overline{\{p\}} = \{p\}.

(48) For every non empty topological space T such that T is a T_{2} space holds T is a T_{1} space.

the sequential closure operator in … 53

(49) Let T be a non empty topological space. Suppose T is not a T_{1} space. Then there exist points x_{1}, x_{2} of T and there exists a sequence S of T such that S = ℕ ⟼→ x_{1} and x_{1} ≠ x_{2} and S is convergent to x_{2}.

(50) For every function f such that domf is infinite and f is one-to-one holds rngf is infinite.

(51) For every non empty finite subset X of ℕ and for every natural number x such that x ∈ X holds x maxX.

References

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

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

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

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

[11] Agata Darmochwał and Andrzej Trybulec. Similarity of formulae. Formalized Mathematics, 2(5):635–642, 1991.

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54 bartłomiej skorulski

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

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

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Properties of the Product of Compact Topological Spaces

Adam Grabowski University of Białystok

MML Identifier: BORSUK 3.

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

1. Preliminaries

One can prove the following proposition

(1) For all topological spaces S, T holds Ω_{[: S, T :]} = [: Ω_{S}, Ω_{T} :]. Let X be a set and let Y be an empty set. Note that [: X, Y :] is empty. Let X be an empty set and let Y be a set. Observe that [: X, Y :] is empty. We now state the proposition

(2) Let X, Y be non empty topological spaces and x be a point of X. Then Y ⟼→ x is a continuous map from Y into X↾{x}.

Let T be a non empty topological structure. One can verify that id_{T} is homeomorphism.

Let S, T be non empty topological structures. Let us notice that the predicate S and T are homeomorphic is reflexive and symmetric.

The following proposition is true

(3) Let S, T, V be non empty topological spaces. Suppose S and T are homeomorphic and T and V are homeomorphic. Then S and V are homeomorphic.

2. On the Projections and Empty Topological Spaces

Let T be a topological structure and let P be an empty subset of the carrier of T. One can verify that T ↾P is empty.

One can check that there exists a topological space which is strict and empty. One can prove the following propositions:

(4) For every topological space T_{1} and for every empty topological space T_{2} holds [: T_{1}, T_{2} :] is empty and [: T_{2}, T_{1} :] is empty.

(5) Every empty topological space is compact. Let us note that every topological space which is empty is also compact. Let T_{1} be a topological space and let T_{2} be an empty topological space.

Observe that [: T_{1}, T_{2} :] is empty.

One can prove the following propositions:

(6) Let X, Y be non empty topological spaces, x be a point of X, and f be a map from [: Y, X↾{x}:] into Y. If f = π_{1}((the carrier of Y) × {x}), then f is one-to-one.

(7) Let X, Y be non empty topological spaces, x be a point of X, and f be a map from [: X↾{x}, Y :] into Y. If f = π_{2}({x} × the carrier of Y), then f is one-to-one.

(8) Let X, Y be non empty topological spaces, x be a point of X, and f be a map from [: Y, X↾{x}:] into Y. If f = π_{1}((the carrier of Y) × {x}), then f^{−1} = ⟨id_{Y} , Y ⟼→ x⟩.

(9) Let X, Y be non empty topological spaces, x be a point of X, and f be a map from [: X↾{x}, Y :] into Y. If f = π_{2}({x} × the carrier of Y), then f^{−1} = ⟨Y ⟼→ x, id_{Y} ⟩.

(10) Let X, Y be non empty topological spaces, x be a point of X, and f be a map from [: Y, X↾{x}:] into Y. If f = π_{1}((the carrier of Y) × {x}), then f is a homeomorphism.

(11) Let X, Y be non empty topological spaces, x be a point of X, and f be a map from [: X↾{x}, Y :] into Y. If f = π_{2}({x} × the carrier of Y), then f is a homeomorphism.

3. On the Product of Compact Spaces

One can prove the following propositions:

(12) Let X be a non empty topological space, Y be a compact non empty topological space, G be an open subset of [: X, Y :], and x be a set. Suppose x ∈ {x′; x′ ranges over points of X: [: {x′}, the carrier of Y :] ⊆ G}. Then there exists a many sorted set f indexed by the carrier of Y such that for every set i if i ∈ the carrier of Y, then there exists a subset G_{1} of X and there exists a subset H_{1} of Y such that f(i) = ⟨⟨G_{1}, H_{1}⟩⟩and ⟨⟨x, i⟩⟩∈ [: G_{1}, H_{1} :] and G_{1} is open and H_{1} is open and [: G_{1}, H_{1} :] ⊆ G.

(13) Let X be a non empty topological space, Y be a compact non empty topological space, G be an open subset of [: Y, X:], and x be a set. Suppose x ∈ {y; y ranges over points of X: [: Ω_{Y} , {y}:] ⊆ G}. Then there exists an open subset R of X such that x ∈ R and R ⊆ {y; y ranges over points of X: [: Ω_{Y} , {y}:] ⊆ G}.

(14) Let X be a non empty topological space, Y be a compact non empty topological space, and G be an open subset of [: Y, X:]. Then {x; x ranges over points of X: [: Ω_{Y} , {x}:] ⊆ G} ∈ the topology of X.

(15) For all non empty topological spaces X, Y and for every point x of X holds [: X↾{x}, Y :] and Y are homeomorphic.

(16) For all non empty topological spaces S, T such that S and T are homeomorphic and S is compact holds T is compact.

(17) For all topological spaces X, Y and for every subspace X_{1} of X holds [: Y, X_{1} :] is a subspace of [: Y, X:].

(18) Let X be a non empty topological space, Y be a compact non empty topological space, x be a point of X, and Z be a subset of [: Y, X:]. If Z = [: Ω_{Y} , {x}:], then Z is compact.

(19) Let X be a non empty topological space, Y be a compact non empty topological space, and x be a point of X. Then [: Y, X↾{x}:] is compact.

(20) Let X, Y be compact non empty topological spaces and R be a family of subsets of X. Suppose R = {Q; Q ranges over open subsets of X: [: Ω_{Y} , Q:] ⊆ ⋃ BaseAppr(Ω_{[: Y, X :]})}. Then R is open and a cover of Ω_{X}.

(21) Let X, Y be compact non empty topological spaces, R be a family of

subsets of X, and F be a family of subsets of [: Y, X:]. Suppose that

(i) F is a cover of [: Y, X:] and open, and (ii) R = {Q; Q ranges over open subsets of X: ⋁_{F_{1} : family of subsets of [: Y, X :]}

(F_{1} ⊆ F ∧ F_{1} is finite ∧ [: Ω_{Y} , Q:] ⊆ ⋃ F_{1})}. Then R is open and a cover of X.

(22) Let X, Y be compact non empty topological spaces, R be a family of

subsets of X, and F be a family of subsets of [: Y, X:]. Suppose that

(i) F is a cover of [: Y, X:] and open, and (ii) R = {Q; Q ranges over open subsets of X: ⋁_{F_{1} : family of subsets of [: Y, X :]}

(F_{1} ⊆ F ∧ F_{1} is finite ∧ [: Ω_{Y} , Q:] ⊆ ⋃ F_{1})}. Then there exists a family C of subsets of X such that C ⊆ R and C is finite and a cover of X.

(23) Let X, Y be compact non empty topological spaces and F be a family of 58 adam grabowski

subsets of [: Y, X:]. Suppose F is a cover of [: Y, X:] and open. Then there exists a family G of subsets of [: Y, X:] such that G ⊆ F and G is a cover of [: Y, X:] and finite.

(24) For all topological spaces T_{1}, T_{2} such that T_{1} is compact and T_{2} is com-

pact holds [: T_{1}, T_{2} :] is compact. Let T_{1}, T_{2} be compact topological spaces. Observe that [: T_{1}, T_{2} :] is compact. Next we state two propositions:

(25) Let X, Y be non empty topological spaces, X_{1} be a non empty subspace of X, and Y_{1} be a non empty subspace of Y. Then [: X_{1}, Y_{1} :] is a subspace of [: X, Y :].

(26) Let X, Y be non empty topological spaces, Z be a non empty subset of [: Y, X:], V be a non empty subset of X, and W be a non empty subset of Y. Suppose Z = [: W, V :]. Then the topological structure of [: Y ↾W, X↾V :] = the topological structure of [: Y, X:]↾Z.

Let T be a topological space. Observe that there exists a subset of T which is compact.

Let T be a topological space and let P be a compact subset of T. Note that T ↾P is compact.

We now state the proposition

(27) Let T_{1}, T_{2} be topological spaces, S_{1} be a subset of T_{1}, and S_{2} be a subset of T_{2}. If S_{1} is compact and S_{2} is compact, then [: S_{1}, S_{2} :] is a compact subset of [: T_{1}, T_{2} :].

References

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

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

[3] Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1(2):409–420, 1990.

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

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

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

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

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

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

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

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

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

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

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

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

properties of the product of compact … 59

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

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

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

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

60 adam grabowski

Compactness of the Bounded Closed Subsets of E_{T}^{2}

Artur Korniłowicz^{1} University of Bialystok

Summary. This paper contains theorems which describe the correspondence between topological properties of real numbers subsets introduced in [40] and introduced in [38], [16]. We also show the homeomorphism between the cartesian product of two R^{1} and E_{T}^{2}. The compactness of the bounded closed subset of E_{T}^{2} is proven.

MML Identifier: TOPREAL6.

The articles [41], [48], [12], [49], [10], [11], [6], [47], [7], [18], [24], [43], [1], [39], [35], [8], [14], [28], [27], [26], [45], [25], [23], [3], [9], [13], [29], [2], [46], [40], [38], [50], [17], [36], [37], [16], [42], [5], [19], [4], [20], [21], [22], [51], [33], [32], [15], [31], [30], [44], and [34] provide the notation and terminology for this paper.

1. Real Numbers

For simplicity, we use the following convention: a, b are real numbers, r is a real number, i, j, n are natural numbers, M is a non empty metric space, p, q, s are points of E_{T}^{2}, e is a point of E^{2}, w is a point of E^{n}, z is a point of M, A, B are subsets of E_{T}^{n}, P is a subset of E_{T}^{2}, and D is a non empty subset of E_{T}^{2}.

One can prove the following propositions:

(2)^{2} a − 2 · a = − a.

(3) − a + 2 · a = a.

1This paper was written while the author visited Shinshu University, winter 1999.

2The proposition (1) has been removed.

(4) a − \frac{a}{2} = \frac{a}{2}.

(5) If a ≠ 0 and b ≠ 0, then \frac{a}{\frac{a}{b}} = b.

(6) For all real numbers a, b such that 0 a and 0 b holds \sqrt{a + b} \sqrt{a} + \sqrt{b}.

(7) If 0 a and a b, then |a| |b|.

(8) If b a and a 0, then |a| |b|.

(9) ∏(0 ↦ r) = 1.

(10) ∏(1 ↦ r) = r.

(11) ∏(2 ↦ r) = r · r.

(12) ∏((n + 1) ↦ r) = ∏(n ↦ r) · r.

(13) j ≠ 0 and r = 0 iff ∏(j ↦ r) = 0.

(14) If r ≠ 0 and j i, then ∏((i − ′j) ↦ r) = ^{\frac{∏(i↦r)}{∏(j↦r)}}.

(15) If r ≠ 0 and j i, then r^{i− ′j} = \frac{r^{i}}{r^{j}} .

In the sequel a, b denote real numbers.

The following propositions are true:

(16) ^{2}⟨a, b⟩ = ⟨a^{2}, b^{2}⟩.

(17) For every finite sequence F of elements of ℝ such that i ∈ dom|F | and a = F (i) holds |F |(i) = |a|.

(18) |⟨a, b⟩| = ⟨|a|, |b|⟩.

(19) For all real numbers a, b, c, d such that a b and c d holds |b − a| + |d − c| = (b − a) + (d − c).

(20) If r > 0, then a ∈ ]a − r, a + r[.

(21) If r 0, then a ∈ [a − r, a + r].

(22) If a < b, then inf]a, b[ = a and sup]a, b[ = b.

(23) ]a, b[ ⊆ [a, b].

(24) For every bounded subset A of ℝ holds A ⊆ [inf A, supA].

2. Topological Preliminaries

Let T be a topological structure and let A be a finite subset of the carrier of T. One can verify that T ↾A is finite.

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

Let T be a topological structure. Note that every subset of T which is empty is also connected.

Let T be a topological space. Observe that every subset of T which is finite is also compact.

Let T be T_{2} non empty topological space. Observe that every subset of T which is compact is also closed.

The following two propositions are true:

(25) For all topological spaces S, T such that S and T are homeomorphic and S is connected holds T is connected.

(26) Let T be a topological space and F be a finite family of subsets of T. Suppose that for every subset X of T such that X ∈ F holds X is compact. Then ⋃ F is compact.

3. Points and Subsets in E_{T}^{2}

The following propositions are true:

(27) For every non empty set X and for every set Y such that X ⊆ Y holds X meets Y.

(28) For all sets A, B, C, D, X such that A ∪ B = X and C ∪ D = X and A ∩ B = ∅ and C ∩ D = ∅ and B = D holds A = C.

(29) For all sets A, B, C, D, a, b such that A ⊆ B and C ⊆ D holds ∏[a ⟼→ A, b ⟼→ C] ⊆ ∏[a ⟼→ B, b ⟼→ D].

(30) For all subsets A, B of ℝ holds ∏[1 ⟼→ A, 2 ⟼→ B] is a subset of E_{T}^{2}.

(31) |[0, a]| = |a| and |[a, 0]| = |a|.

(32) For every point p of E^{2} and for every point q of E_{T}^{2} such that p = 0_{E_{T}^{2}} and p = q holds q = ⟨0, 0⟩ and q_{1} = 0 and q_{2} = 0.

(33) For all points p, q of E^{2} and for every point z of E_{T}^{2} such that p = 0_{E_{T}^{2}} and q = z holds ρ(p, q) = |z|.

(34) r · p = [r · p_{1}, r · p_{2}].

(35) If s = (1 − r) · p + r · q and s ≠ p and 0 r, then 0 < r.

(36) If s = (1 − r) · p + r · q and s ≠ q and r 1, then r < 1.

(37) If s ∈ L(p, q) and s ≠ p and s ≠ q and p_{1} < q_{1}, then p_{1} < s_{1} and s_{1} < q_{1}.

(38) If s ∈ L(p, q) and s ≠ p and s ≠ q and p_{2} < q_{2}, then p_{2} < s_{2} and s_{2} < q_{2}.

(39) For every point p of E_{T}^{2} there exists a point q of E_{T}^{2} such that q_{1} < W-boundD and p ≠ q.

(40) For every point p of E_{T}^{2} there exists a point q of E_{T}^{2} such that q_{1} > E-boundD and p ≠ q.

(41) For every point p of E_{T}^{2} there exists a point q of E_{T}^{2} such that q_{2} > N-boundD and p ≠ q.

(42) For every point p of E_{T}^{2} there exists a point q of E_{T}^{2} such that q_{2} <

S-boundD and p ≠ q. One can verify the following observations:

∗ every subset of E_{T}^{2} which is convex and non empty is also connected,

∗ every subset of E_{T}^{2} which is non horizontal is also non empty,

∗ every subset of E_{T}^{2} which is non vertical is also non empty,

∗ every subset of E_{T}^{2} which is region is also open and connected, and

∗ every subset of E_{T}^{2} which is open and connected is also region.

Let us observe that every subset of E_{T}^{2} which is empty is also horizontal and every subset of E_{T}^{2} which is empty is also vertical.

Let us mention that there exists a subset of E_{T}^{2} which is non empty and convex.

Let a, b be points of E_{T}^{2}. Observe that L(a, b) is convex and connected.

Let us mention that □_{E^{2}} is connected.

Let us observe that every subset of E_{T}^{2} which is simple closed curve is also connected and compact.

One can prove the following propositions:

(43) L(NE-cornerP, SE-cornerP) ⊆ L̃(SpStSeqP).

(44) L(SW-cornerP, SE-cornerP) ⊆ L̃(SpStSeqP).

(45) L(SW-cornerP, NW-cornerP) ⊆ L̃(SpStSeqP).

(46) For every subset C of E_{T}^{2} holds {p; p ranges over points of E_{T}^{2}: p_{1} < W-boundC} is a non empty convex connected subset of E_{T}^{2}.

4. Balls as subsets of E_{T}^{n}

We now state a number of propositions:

(47) If e = q and p ∈ Ball(e, r), then q_{1} − r < p_{1} and p_{1} < q_{1} + r.

(48) If e = q and p ∈ Ball(e, r), then q_{2} − r < p_{2} and p_{2} < q_{2} + r.

(49) If p = e, then ∏[1 ⟼→ ]p_{1} − \frac{r}{\sqrt{2}}, p_{1} + \frac{r}{\sqrt{2}}[, 2 ⟼→ ]p_{2} − \frac{r}{\sqrt{2}}, p_{2} + \frac{r}{\sqrt{2}}[] ⊆ Ball(e, r).

(50) If p = e, then Ball(e, r) ⊆ ∏[1 ⟼→ ]p_{1} − r, p_{1} + r[, 2 ⟼→ ]p_{2} − r, p_{2} + r[].

(51) If P = Ball(e, r) and p = e, then (proj1)^{∘}P = ]p_{1} − r, p_{1} + r[.

(52) If P = Ball(e, r) and p = e, then (proj2)^{∘}P = ]p_{2} − r, p_{2} + r[.

(53) If D = Ball(e, r) and p = e, then W-boundD = p_{1} − r.

(54) If D = Ball(e, r) and p = e, then E-boundD = p_{1} + r.

(55) If D = Ball(e, r) and p = e, then S-boundD = p_{2} − r.

(56) If D = Ball(e, r) and p = e, then N-boundD = p_{2} + r.

(57) If D = Ball(e, r), then D is non horizontal.

(58) If D = Ball(e, r), then D is non vertical.

(59) For every point f of E^{2} and for every point x of E_{T}^{2} such that x ∈ Ball(f, a) holds [x_{1} − 2 · a, x_{2}] ∉ Ball(f, a).

(60) Let X be a non empty compact subset of E_{T}^{2} and p be a point of E^{2}. If p = 0_{E_{T}^{2}} and a > 0, then X ⊆ Ball(p, | E-boundX| + | N-boundX| + | W-boundX| + | S-boundX| + a).

(61) Let M be a Reflexive symmetric triangle non empty metric structure and z be a point of M. If r < 0, then Sphere(z, r) = ∅.

(62) For every Reflexive discernible non empty metric structure M and for every point z of M holds Sphere(z, 0) = {z}.

(63) Let M be a Reflexive symmetric triangle non empty metric structure and z be a point of M. If r < 0, then \overline{Ball}(z, r) = ∅.

(64) \overline{Ball}(z, 0) = \{z\}.

(65) For every subset A of M_{top} such that A = \overline{Ball}(z, r) holds A is closed.

(66) If A = \overline{Ball}(w, r), then A is closed.

(67) \overline{Ball}(z, r) is bounded.

(68) For every subset A of M_{top} such that A = Sphere(z, r) holds A is closed.

(69) If A = Sphere(w, r), then A is closed.

(70) Sphere(z, r) is bounded.

(71) If A is Bounded, then \overline{A} is Bounded.

(72) For every non empty metric structure M holds M is bounded iff every subset of the carrier of M is bounded.

(73) Let M be a Reflexive symmetric triangle non empty metric structure and X, Y be subsets of the carrier of M. Suppose the carrier of M = X ∪ Y and M is non bounded and X is bounded. Then Y is non bounded.

(74) For all subsets X, Y of E_{T}^{n} such that n 1 and the carrier of E_{T}^{n} = X ∪Y and X is Bounded holds Y is non Bounded.

(76)^{3} If A is Bounded and B is Bounded, then A ∪ B is Bounded.

5. Topological Properties of Real Numbers Subsets

Let X be a non empty subset of ℝ. Observe that \overline{X} is non empty.

Let D be a lower bounded subset of ℝ. One can verify that \overline{D} is lower bounded.

3The proposition (75) has been removed.

Let D be an upper bounded subset of ℝ. One can verify that \overline{D} is upper bounded.

We now state two propositions:

(77) For every non empty lower bounded subset D of ℝ holds inf D = inf \overline{D}.

(78) For every non empty upper bounded subset D of ℝ holds supD = sup \overline{D}. Let us observe that ℝ^{1} is T_{2}. The following three propositions are true:

(79) For every subset A of ℝ and for every subset B of ℝ^{1} such that A = B holds A is closed iff B is closed.

(80) For every subset A of ℝ and for every subset B of ℝ^{1} such that A = B holds \overline{A} = \overline{B}.

(81) For every subset A of ℝ and for every subset B of ℝ^{1} such that A = B

holds A is compact iff B is compact. One can check that every subset of ℝ which is finite is also compact. Let a, b be real numbers. Note that [a, b] is compact. Next we state the proposition

(82) a ≠ b iff \overline{]a, b[} = [a, b].

Let us observe that there exists a subset of ℝ which is non empty, finite, and bounded.

The following propositions are true:

(83) Let T be a topological structure, f be a real map of T, and g be a map from T into ℝ^{1}. If f = g, then f is continuous iff g is continuous.

(84) Let A, B be subsets of ℝ and f be a map from [: ℝ^{1}, ℝ^{1} :] into E_{T}^{2}. If for all real numbers x, y holds f(⟨⟨x, y⟩⟩) = ⟨x, y⟩, then f^{∘}[: A, B:] = ∏[1 ⟼→ A, 2 ⟼→ B].

(85) For every map f from [: ℝ^{1}, ℝ^{1} :] into E_{T}^{2} such that for all real numbers x, y holds f(⟨⟨x, y⟩⟩) = ⟨x, y⟩ holds f is a homeomorphism.

(86) [: ℝ^{1}, ℝ^{1} :] and E_{T}^{2} are homeomorphic.

6. Bounded Subsets

One can prove the following propositions:

(87) For all compact subsets A, B of ℝ holds ∏[1 ⟼→ A, 2 ⟼→ B] is a compact subset of E_{T}^{2}.

(88) If P is Bounded and closed, then P is compact.

(89) If P is Bounded, then for every continuous real map g of E_{T}^{2} holds \overline{g^{∘}P} ⊆ g^{∘}\overline{P} .

(90) (proj1)^{∘}\overline{P} ⊆ \overline{(proj1)^{∘}P} .

compactness of the bounded closed subsets of … 67

(91) (proj2)^{∘}\overline{P} ⊆ \overline{(proj2)^{∘}P} .

(92) If P is Bounded, then \overline{(proj1)^{∘}P} = (proj1)^{∘}\overline{P} .

(93) If P is Bounded, then \overline{(proj2)^{∘}P} = (proj2)^{∘}\overline{P} .

(94) If D is Bounded, then W-boundD = W-bound \overline{D}.

(95) If D is Bounded, then E-boundD = E-bound \overline{D}.

(96) If D is Bounded, then N-boundD = N-bound \overline{D}.

(97) If D is Bounded, then S-boundD = S-bound \overline{D}.

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in the preparation of the article.

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[44] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Formalized Mathematics, 6(4):531–539, 1997.

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

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

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

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

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

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

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

Hilbert Positive Propositional Calculus

Adam Grabowski University of Białystok

MML Identifier: HILBERT1.

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

1. Definition of the Language

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

Let D be a set. We say that D has implication if and only if: (Def. 2) For all finite sequences p, q such that p ∈ D and q ∈ D holds ⟨1⟩^{⌢}p^{⌢}q ∈

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

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

D. Let D be a set. We say that D has propositional variables if and only if:

(Def. 4) For every natural number n holds ⟨3 + n⟩ ∈ D.

Let D be a set. We say that D is HP-closed if and only if: (Def. 5) D ⊆ ℕ^{∗} and D has VERUM, implication, conjunction, and propositional

variables. Let us note that every set which is HP-closed is also non empty and has

VERUM, implication, conjunction, and propositional variables and every subset of ℕ^{∗} which has VERUM, implication, conjunction, and propositional variables is HP-closed.

The set HP-WFF is defined as follows:

(Def. 6) HP-WFF is HP-closed and for every set D such that D is HP-closed

holds HP-WFF ⊆ D. Let us note that HP-WFF is HP-closed. Let us mention that there exists a set which is HP-closed and non empty. One can verify that every element of HP-WFF is relation-like and function-

like.

Let us mention that every element of HP-WFF is finite sequence-like. A HP-formula is an element of HP-WFF. The HP-formula VERUM is defined by:

(Def. 7) VERUM = ⟨0⟩.

Let p, q be elements of HP-WFF. The functor p ⇒ q yielding a HP-formula is defined by:

(Def. 8) p ⇒ q = ⟨1⟩^{⌢} p^{⌢} q.

The functor p ∧ q yielding a HP-formula is defined as follows: (Def. 9) p ∧ q = ⟨2⟩^{⌢} p^{⌢} q.

We follow the rules: T, X, Y denote subsets of HP-WFF and p, q, r, s denote elements of HP-WFF.

Let T be a subset of HP-WFF. We say that T is Hilbert theory if and only if the conditions (Def. 10) are satisfied.

(Def. 10)(i) VERUM ∈ T, and

(ii) for all elements p, q, r of HP-WFF holds 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 if p ∈ T and p ⇒ q ∈ T, then q ∈ T.

Let us consider X. The functor CnPosX yields a subset of HP-WFF and is defined by:

(Def. 11) r ∈ CnPosX iff for every T such that T is Hilbert theory and X ⊆ T

holds r ∈ T. The subset HP TAUT of HP-WFF is defined by:

(Def. 12) HP TAUT = CnPos∅_{HP-WFF} .

The following propositions are true:

(1) VERUM ∈ CnPosX.

(2) p ⇒ (q ⇒ p ∧ q) ∈ CnPosX.

(3) (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ CnPosX.

(4) p ⇒ (q ⇒ p) ∈ CnPosX.

(5) p ∧ q ⇒ p ∈ CnPosX.

(6) p ∧ q ⇒ q ∈ CnPosX.

(7) If p ∈ CnPosX and p ⇒ q ∈ CnPosX, then q ∈ CnPosX.

(8) If T is Hilbert theory and X ⊆ T, then CnPosX ⊆ T.

(9) X ⊆ CnPosX.

(10) If X ⊆ Y, then CnPosX ⊆ CnPosY.

(11) CnPosCnPosX = CnPosX.

Let X be a subset of HP-WFF. One can verify that CnPosX is Hilbert theory.

We now state two propositions:

(12) T is Hilbert theory iff CnPosT = T.

(13) If T is Hilbert theory, then HP TAUT ⊆ T.

Let us mention that HP TAUT is Hilbert theory.

2. The Tautologies of the Hilbert Calculus - Implicational Part

We now state a number of propositions:

(14) p ⇒ p ∈ HP TAUT.

(15) If q ∈ HP TAUT, then p ⇒ q ∈ HP TAUT.

(16) p ⇒ VERUM ∈ HP TAUT.

(17) (p ⇒ q) ⇒ (p ⇒ p) ∈ HP TAUT.

(18) (q ⇒ p) ⇒ (p ⇒ p) ∈ HP TAUT.

(19) (q ⇒ r) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ HP TAUT.

(20) If p ⇒ (q ⇒ r) ∈ HP TAUT, then q ⇒ (p ⇒ r) ∈ HP TAUT.

(21) (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒ r)) ∈ HP TAUT.

(22) If p ⇒ q ∈ HP TAUT, then (q ⇒ r) ⇒ (p ⇒ r) ∈ HP TAUT.

(23) If p ⇒ q ∈ HP TAUT and q ⇒ r ∈ HP TAUT, then p ⇒ r ∈ HP TAUT.

(24) (p ⇒ (q ⇒ r)) ⇒ ((s ⇒ q) ⇒ (p ⇒ (s ⇒ r))) ∈ HP TAUT.

(25) ((p ⇒ q) ⇒ r) ⇒ (q ⇒ r) ∈ HP TAUT.

(26) (p ⇒ (q ⇒ r)) ⇒ (q ⇒ (p ⇒ r)) ∈ HP TAUT.

(27) (p ⇒ (p ⇒ q)) ⇒ (p ⇒ q) ∈ HP TAUT.

(28) q ⇒ ((q ⇒ p) ⇒ p) ∈ HP TAUT.

(29) If s ⇒ (q ⇒ p) ∈ HP TAUT and q ∈ HP TAUT, then s ⇒ p ∈ HP TAUT.

3. Conjunctional Part of the Calculus

The following propositions are true:

(30) p ⇒ p ∧ p ∈ HP TAUT.

(31) p ∧ q ∈ HP TAUT iff p ∈ HP TAUT and q ∈ HP TAUT.

(32) p ∧ q ∈ HP TAUT iff q ∧ p ∈ HP TAUT.

(33) (p ∧ q ⇒ r) ⇒ (p ⇒ (q ⇒ r)) ∈ HP TAUT.

(34) (p ⇒ (q ⇒ r)) ⇒ (p ∧ q ⇒ r) ∈ HP TAUT.

(35) (r ⇒ p) ⇒ ((r ⇒ q) ⇒ (r ⇒ p ∧ q)) ∈ HP TAUT.

(36) (p ⇒ q) ∧ p ⇒ q ∈ HP TAUT.

(37) (p ⇒ q) ∧ p ∧ s ⇒ q ∈ HP TAUT.

(38) (q ⇒ s) ⇒ (p ∧ q ⇒ s) ∈ HP TAUT.

(39) (q ⇒ s) ⇒ (q ∧ p ⇒ s) ∈ HP TAUT.

(40) (p ∧ s ⇒ q) ⇒ (p ∧ s ⇒ q ∧ s) ∈ HP TAUT.

(41) (p ⇒ q) ⇒ (p ∧ s ⇒ q ∧ s) ∈ HP TAUT.

(42) (p ⇒ q) ∧ (p ∧ s) ⇒ q ∧ s ∈ HP TAUT.

(43) p ∧ q ⇒ q ∧ p ∈ HP TAUT.

(44) (p ⇒ q) ∧ (p ∧ s) ⇒ s ∧ q ∈ HP TAUT.

(45) (p ⇒ q) ⇒ (p ∧ s ⇒ s ∧ q) ∈ HP TAUT.

(46) (p ⇒ q) ⇒ (s ∧ p ⇒ s ∧ q) ∈ HP TAUT.

(47) p ∧ (s ∧ q) ⇒ p ∧ (q ∧ s) ∈ HP TAUT.

(48) (p ⇒ q) ∧ (p ⇒ s) ⇒ (p ⇒ q ∧ s) ∈ HP TAUT.

(49) p ∧ q ∧ s ⇒ p ∧ (q ∧ s) ∈ HP TAUT.

(50) p ∧ (q ∧ s) ⇒ p ∧ q ∧ s ∈ HP 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] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

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

Homeomorphism between [:E_{T}^{i}, E_{T}^{j}:] and E_{T}^{i+j}

Artur Korniłowicz^{1} University of Bialystok

Summary. In this paper we introduce the cartesian product of two metric spaces. As the distance between two points in the product we take maximal distance between coordinates of these points. In the main theorem we show the homeomorphism between [:E_{T}^{i}, E_{T}^{j}:] and E_{T}^{i+j}.

MML Identifier: TOPREAL7.

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

We use the following convention: i, j, n denote natural numbers, f, g, h, k denote finite sequences of elements of ℝ, and M, N denote non empty metric spaces.

We now state a number of propositions:

(1) For all real numbers a, b such that max(a, b) a holds max(a, b) = a.

(2) For all real numbers a, b, c, d holds max(a + c, b + d) max(a, b) + max(c, d).

(3) For all real numbers a, b, c, d, e, f such that a b + c and d e + f holds max(a, d) max(b, e) + max(c, f).

(4) For all finite sequences f, g holds domg ⊆ dom(f ^{⌢} g).

(5) For all finite sequences f, g such that lenf < i and i lenf + leng holds i − lenf ∈ domg.

(6) For all finite sequences f, g, h, k such that f^{⌢}g = h^{⌢}k and lenf = lenh and leng = lenk holds f = h and g = k.

(7) If lenf = leng or domf = domg, then len(f + g) = lenf and dom(f + g) = domf.

1This paper was written while the author visited Shinshu University, winter 1999.

(8) If lenf = leng or domf = domg, then len(f − g) = lenf and dom(f − g) = domf.

(9) lenf = len^{2}f and domf = dom^{2}f.

(10) lenf = len|f| and domf = dom|f|.

(11) ^{2}(f ^{⌢} g) = (^{2}f)^{⌢} (^{2}g).

(12) |f ^{⌢} g| = |f|^{⌢} |g|.

(13) If lenf = lenh and leng = lenk, then ^{2}(f ^{⌢} g + h^{⌢} k) = (^{2}(f + h))^{⌢} (^{2}(g + k)).

(14) If lenf = lenh and leng = lenk, then |f ^{⌢} g + h^{⌢} k| = |f + h|^{⌢} |g + k|.

(15) If lenf = lenh and leng = lenk, then ^{2}(f ^{⌢} g − h^{⌢} k) = (^{2}(f − h))^{⌢} (^{2}(g − k)).

(16) If lenf = lenh and leng = lenk, then |f ^{⌢} g − h^{⌢} k| = |f − h|^{⌢} |g − k|.

(17) If lenf = n, then f ∈ the carrier of E^{n}.

(18) If lenf = n, then f ∈ the carrier of E_{T}^{n}.

(19) For every finite sequence f such that f ∈ the carrier of E^{n} holds lenf =

n. Let M, N be non empty metric structures. The functor max-Prod2(M, N)

yielding a strict metric structure is defined by the conditions (Def. 1). (Def. 1)(i) The carrier of max-Prod2(M, N) = [: the carrier of M, the carrier of

N :], and (ii) for all points x, y of max-Prod2(M, N) there exist points x_{1}, y_{1} of M

and there exist points x_{2}, y_{2} of N such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ and (the distance of max-Prod2(M, N))(x, y) = max((the distance of M)(x_{1}, y_{1}), (the distance of N)(x_{2}, y_{2})).

Let M, N be non empty metric structures. One can verify that max-Prod2(M, N) is non empty.

Let M, N be non empty metric structures, let x be a point of M, and let y be a point of N. Then ⟨⟨x, y⟩⟩ is an element of max-Prod2(M, N).

Let M, N be non empty metric structures and let x be a point of max-Prod2(M, N). Then x_{1} is an element of M. Then x_{2} is an element of N.

The following three propositions are true:

(20) Let M, N be non empty metric structures, m_{1}, m_{2} be points of M, and n_{1}, n_{2} be points of N. Then ρ(⟨⟨m_{1}, n_{1}⟩⟩, ⟨⟨m_{2}, n_{2}⟩⟩) = max(ρ(m_{1}, m_{2}), ρ(n_{1}, n_{2})).

(21) For all non empty metric structures M, N and for all points m, n of max-Prod2(M, N) holds ρ(m, n) = max(ρ(m_{1}, n_{1}), ρ(m_{2}, n_{2})).

(22) For all Reflexive non empty metric structures M, N holds max-Prod2(M, N) is Reflexive.

homeomorphism between [:⌉_{t}^{i}, … 75

Let M, N be Reflexive non empty metric structures. Observe that max-Prod2(M, N) is Reflexive.

Next we state the proposition

(23) For all symmetric non empty metric structures M, N holds

max-Prod2(M, N) is symmetric. Let M, N be symmetric non empty metric structures. Note that

max-Prod2(M, N) is symmetric.

Next we state the proposition

(24) For all triangle non empty metric structures M, N holds

max-Prod2(M, N) is triangle. Let M, N be triangle non empty metric structures. One can check that

max-Prod2(M, N) is triangle.

Let M, N be non empty metric spaces. Note that max-Prod2(M, N) is discernible.

The following three propositions are true:

(25) [: M_{top}, N_{top} :] = (max-Prod2(M, N))_{top}.

(26) Suppose that (i) the carrier of M = the carrier of N, (ii) for every point m of M and for every point n of N and for every real

number r such that r > 0 and m = n there exists a real number r_{1} such that r_{1} > 0 and Ball(n, r_{1}) ⊆ Ball(m, r), and

(iii) for every point m of M and for every point n of N and for every real number r such that r > 0 and m = n there exists a real number r_{1} such that r_{1} > 0 and Ball(m, r_{1}) ⊆ Ball(n, r). Then M_{top} = N_{top}.

(27) [: E_{T}^{i}, E_{T}^{j}:] and E_{T}^{i+j} are homeomorphic.

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in the preparation of the article.

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

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

[5] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Full Subtracter Circuit. Part I

Katsumi Wasaki Shinshu University Nagano

Noboru Endou Shinshu University Nagano

Summary. We formalize the concept of the full subtracter circuit, define the structures of bit subtract/borrow units for binary operations, and prove the stability of the circuit.

MML Identifier: FSCIRC 1.

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

1. Bit Subtract and Borrow Circuit

In this paper x, y, c are sets.

Let x, y, c be sets. The functor BitSubtracterOutput(x, y, c) yields an element of InnerVertices(2GatesCircStr(x, y, c, xor)) and is defined as follows:

(Def. 1) BitSubtracterOutput(x, y, c) = 2GatesCircOutput(x, y, c, xor).

Let x, y, c be sets. The functor BitSubtracterCirc(x, y, c) yields a strict Boolean circuit of 2GatesCircStr(x, y, c, xor) with denotation held in gates and is defined as follows:

(Def. 2) BitSubtracterCirc(x, y, c) = 2GatesCircuit(x, y, c, xor).

Let x, y, c be sets. The functor BorrowIStr(x, y, c) 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:

(Def. 3) BorrowIStr(x, y, c) = 1GateCircStr(⟨x, y⟩, and_{2a})+·1GateCircStr(⟨y, c⟩, and_{2})+·1GateCircStr(⟨x, c⟩, and_{2a}).

Let x, y, c be sets. The functor BorrowStr(x, y, c) yielding an unsplit non void strict non empty many sorted signature with arity held in gates and Boolean denotation held in gates is defined by:

(Def. 4) BorrowStr(x, y, c) = BorrowIStr(x, y, c)+·1GateCircStr(⟨⟨⟨⟨x, y⟩, and_{2a} ⟩⟩,

⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩⟩, or_{3}). Let x, y, c be sets. The functor BorrowICirc(x, y, c) yielding a strict Boolean

circuit of BorrowIStr(x, y, c) with denotation held in gates is defined by: (Def. 5) BorrowICirc(x, y, c) = 1GateCircuit(x, y, and_{2a})+·1GateCircuit(y, c, and_{2})

+·1GateCircuit(x, c, and_{2a}). The following propositions are true:

(1) InnerVertices(BorrowStr(x, y, c)) is a binary relation.

(2) For all non pair sets x, y, c holds InputVertices(BorrowStr(x, y, c)) has no pairs.

(3) For every state s of BorrowICirc(x, y, c) and for all elements a, b of Boolean such that a = s(x) and b = s(y) holds (Following(s))(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) = ¬a ∧ b.

(4) For every state s of BorrowICirc(x, y, c) and for all elements a, b of Boolean such that a = s(y) and b = s(c) holds (Following(s))(⟨⟨⟨y, c⟩, and_{2} ⟩⟩) = a ∧ b.

(5) For every state s of BorrowICirc(x, y, c) and for all elements a, b of Boolean such that a = s(x) and b = s(c) holds (Following(s))(⟨⟨⟨x, c⟩, and_{2a} ⟩⟩) = ¬a ∧ b.

Let x, y, c be sets. The functor BorrowOutput(x, y, c) yields an element of InnerVertices(BorrowStr(x, y, c)) and is defined by:

(Def. 6) BorrowOutput(x, y, c) = ⟨⟨⟨⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩⟩,

or_{3} ⟩⟩. Let x, y, c be sets. The functor BorrowCirc(x, y, c) yielding a strict Boolean

circuit of BorrowStr(x, y, c) with denotation held in gates is defined by: (Def. 7) BorrowCirc(x, y, c) = BorrowICirc(x, y, c)+·1GateCircuit(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩,

⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩, or_{3}). Next we state a number of propositions:

(6) x ∈ the carrier of BorrowStr(x, y, c) and y ∈ the carrier of BorrowStr(x, y, c) and c ∈ the carrier of BorrowStr(x, y, c).

(7) ⟨⟨⟨x, y⟩, and_{2a} ⟩⟩ ∈ InnerVertices(BorrowStr(x, y, c)) and ⟨⟨⟨y, c⟩, and_{2} ⟩⟩ ∈ InnerVertices(BorrowStr(x, y, c)) and ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩ ∈ InnerVertices(BorrowStr(x, y, c)).

(8) For all non pair sets x, y, c holds x ∈ InputVertices(BorrowStr(x, y, c)) and y ∈ InputVertices(BorrowStr(x, y, c)) and c ∈ InputVertices(BorrowStr(x, y, c)).

(9) For all non pair sets x, y, c holds InputVertices(BorrowStr(x, y, c)) = {x, y, c} and InnerVertices(BorrowStr(x, y, c)) = {⟨⟨⟨x, y⟩, and_{2a} ⟩⟩, ⟨⟨⟨y, c⟩, and_{2} ⟩⟩, ⟨⟨⟨x, c⟩, and_{2a} ⟩⟩} ∪ {BorrowOutput(x, y, c)}.

(10) Let x, y, c be non pair sets, s be a state of BorrowCirc(x, y, c), and a_{1}, a_{2} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y), then (Following(s))(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) = ¬a_{1} ∧ a_{2}.

(11) Let x, y, c be non pair sets, s be a state of BorrowCirc(x, y, c), and a_{2}, a_{3} be elements of Boolean. If a_{2} = s(y) and a_{3} = s(c), then (Following(s))(⟨⟨⟨y, c⟩, and_{2} ⟩⟩) = a_{2} ∧ a_{3}.

(12) Let x, y, c be non pair sets, s be a state of BorrowCirc(x, y, c), and a_{1}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{3} = s(c), then (Following(s))(⟨⟨⟨x, c⟩, and_{2a} ⟩⟩) = ¬a_{1} ∧ a_{3}.

(13) Let x, y, c be non pair 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}.

(14) Let x, y, c be non pair sets, s be a state of BorrowCirc(x, y, c), and a_{1}, a_{2} be elements of Boolean. If a_{1} = s(x) and a_{2} = s(y), then (Following(s, 2))(⟨⟨⟨x, y⟩, and_{2a} ⟩⟩) = ¬a_{1} ∧ a_{2}.

(15) Let x, y, c be non pair sets, s be a state of BorrowCirc(x, y, c), and a_{2}, a_{3} be elements of Boolean. If a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(⟨⟨⟨y, c⟩, and_{2} ⟩⟩) = a_{2} ∧ a_{3}.

(16) Let x, y, c be non pair sets, s be a state of BorrowCirc(x, y, c), and a_{1}, a_{3} be elements of Boolean. If a_{1} = s(x) and a_{3} = s(c), then (Following(s, 2))(⟨⟨⟨x, c⟩, and_{2a} ⟩⟩) = ¬a_{1} ∧ a_{3}.

(17) Let x, y, c be non pair 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) and a_{2} = s(y) and a_{3} = s(c), then (Following(s, 2))(BorrowOutput(x, y, c)) = ¬a_{1} ∧ a_{2} ∨ a_{2} ∧ a_{3} ∨ ¬a_{1} ∧ a_{3}.

(18) For all non pair sets x, y, c and for every state s of BorrowCirc(x, y, c) holds Following(s, 2) is stable.

2. Bit Subtracter with Borrow Circuit

Let x, y, c be sets. The functor BitSubtracterWithBorrowStr(x, y, c) 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:

(Def. 8) BitSubtracterWithBorrowStr(x, y, c) = 2GatesCircStr(x, y, c, xor)

+·BorrowStr(x, y, c). The following propositions are true:

(19) For all non pair sets x, y, c holds InputVertices(BitSubtracterWithBorrowStr(x, y, c)) = {x, y, c}.

(20) For all non pair 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)}.

(21) Let S be a non empty many sorted signature. Suppose S = BitSubtracterWithBorrowStr(x, y, c). Then x ∈ the carrier of S and y ∈ the carrier of S and c ∈ the carrier of S.

Let x, y, c be sets. The functor BitSubtracterWithBorrowCirc(x, y, c) yields a strict Boolean circuit of BitSubtracterWithBorrowStr(x, y, c) with denotation held in gates and is defined as follows:

(Def. 9) BitSubtracterWithBorrowCirc(x, y, c) = BitSubtracterCirc(x, y, c)

+·BorrowCirc(x, y, c). We now state several propositions:

(22) InnerVertices(BitSubtracterWithBorrowStr(x, y, c)) is a binary relation.

(23) For all non pair sets x, y, c holds InputVertices(BitSubtracterWithBorrowStr(x, y, c)) has no pairs.

(24) BitSubtracterOutput(x, y, c) ∈ InnerVertices(BitSubtracterWithBorrowStr(x, y, c)) and BorrowOutput (x, y, c) ∈ InnerVertices(BitSubtracterWithBorrowStr(x, y, c)).

(25) Let x, y, c be non pair sets, 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}.

(26) For all non pair sets x, y, c and for every state s of BitSubtracterWithBorrowCirc(x, y, c) holds Following(s, 2) is stable.

References

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

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

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

[6] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Introduction to circuits, I. Formalized Mathematics, 5(2):227–232, 1996.

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

full subtracter circuit. part i 81

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

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

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

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

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

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

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

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

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

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

82 katsumi wasaki and noboru endou

Correctness of Binary Counter Circuits

Yuguang Yang Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Yatsuka Nakamura Shinshu University Nagano

Summary. This article introduces the verification of the correctness for the operations and the specification of the 3-bit counter. Both cases: without reset input and with reset input are considered. The proof was proposed by Y. Nakamura in [1].

MML Identifier: GATE 2.

The paper [1] provides the terminology and notation for this paper. In this paper a, b, c, d denote sets. Next we state four propositions:

(1) Let s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, n_{0}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}, n_{7}, q_{1}, q_{2}, q_{3}, n_{8}, n_{9}, n_{10} be sets such that NE s_{0} iff NE AND3(NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND3(NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND3(NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND3(NOT1q_{3}, q_{2}, q_{1}) and NE s_{4} iff NE AND3(q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{5} iff NE AND3(q_{3}, NOT1q_{2}, q_{1}) and NE s_{6} iff NE AND3(q_{3}, q_{2}, NOT1q_{1}) and NE s_{7} iff NE AND3(q_{3}, q_{2}, q_{1}) and NE n_{0} iff NE AND3(NOT1n_{10}, NOT1n_{9}, NOT1n_{8}) and NE n_{1} iff NE AND3(NOT1n_{10}, NOT1n_{9}, n_{8}) and NE n_{2} iff NE AND3(NOT1n_{10}, n_{9}, NOT1n_{8}) and NE n_{3} iff NE AND3(NOT1n_{10}, n_{9}, n_{8}) and NE n_{4} iff NE AND3(n_{10}, NOT1n_{9}, NOT1n_{8}) and NE n_{5} iff NE AND3(n_{10}, NOT1n_{9}, n_{8}) and NE n_{6} iff NE AND3(n_{10}, n_{9}, NOT1n_{8}) and NE n_{7} iff NE AND3(n_{10}, n_{9}, n_{8}) and NE n_{8} iff NE NOT1q_{1} and NE n_{9} iff NE XOR2(q_{1}, q_{2}) and NE n_{10} iff NE OR2(AND2(q_{3}, NOT1q_{1}), AND2(q_{1}, XOR2(q_{2}, q_{3}))). Then (i) NE n_{1} iff NE s_{0}, (ii) NE n_{2} iff NE s_{1}, (iii) NE n_{3} iff NE s_{2}, (iv) NE n_{4} iff NE s_{3},

(v) NE n_{5} iff NE s_{4}, (vi) NE n_{6} iff NE s_{5}, (vii) NE n_{7} iff NE s_{6}, and (viii) NE n_{0} iff NE s_{7}.

(2) NE AND3(AND2(a, d), AND2(b, d), AND2(c, d))

iff NE AND2(AND3(a, b, c), d). (3)(i) Not NE AND2(a, b) iff NE OR2(NOT1a, NOT1b), (ii) NE OR2(a, b) and NE OR2(c, b) iff NE OR2(AND2(a, c), b), (iii) NE OR2(a, b) and NE OR2(c, b) and NE OR2(d, b) iff NE

OR2(AND3(a, c, d), b), and (iv) if NE OR2(a, b) and NE a iff NE c, then NE OR2(c, b).

(4) Let s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, n_{0}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}, n_{7}, q_{1}, q_{2}, q_{3}, n_{8}, n_{9}, n_{10}, R be sets such that NE s_{0} iff NE AND3(NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND3(NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND3(NOT1q_{3}, q_{2}, NOT1 q_{1}) and NE s_{3} iff NE AND3(NOT1q_{3}, q_{2}, q_{1}) and NE s_{4} iff NE AND3(q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{5} iff NE AND3(q_{3}, NOT1q_{2}, q_{1}) and NE s_{6} iff NE AND3(q_{3}, q_{2}, NOT1q_{1}) and NE s_{7} iff NE AND3(q_{3}, q_{2}, q_{1}) and NE n_{0} iff NE AND3(NOT1n_{10}, NOT1n_{9}, NOT1n_{8}) and NE n_{1} iff NE AND3(NOT1n_{10}, NOT1n_{9}, n_{8}) and NE n_{2} iff NE AND3(NOT1n_{10}, n_{9}, NOT1n_{8}) and NE n_{3} iff NE AND3(NOT1n_{10}, n_{9}, n_{8}) and NE n_{4} iff NE AND3(n_{10}, NOT1n_{9}, NOT1n_{8}) and NE n_{5} iff NE AND3(n_{10}, NOT1n_{9}, n_{8}) and NE n_{6} iff NE AND3(n_{10}, n_{9}, NOT1n_{8}) and NE n_{7} iff NE AND3(n_{10}, n_{9}, n_{8}) and NE n_{8} iff NE AND2(NOT1q_{1}, R) and NE n_{9} iff NE AND2(XOR2(q_{1}, q_{2}), R) and NE n_{10} iff NE AND2(OR2(AND2(q_{3}, NOT1q_{1}), AND2(q_{1}, XOR2(q_{2}, q_{3}))), R). Then

(i) NE n_{1} iff NE AND2(s_{0}, R), (ii) NE n_{2} iff NE AND2(s_{1}, R), (iii) NE n_{3} iff NE AND2(s_{2}, R), (iv) NE n_{4} iff NE AND2(s_{3}, R),

(v) NE n_{5} iff NE AND2(s_{4}, R), (vi) NE n_{6} iff NE AND2(s_{5}, R), (vii) NE n_{7} iff NE AND2(s_{6}, R), and (viii) NE n_{0} iff NE OR2(s_{7}, NOT1R). correctness of binary counter circuits 85

References

[1] Yatsuka Nakamura. Logic gates and logical equivalence of adders. Formalized Mathematics,

8(1):35–45, 1999.

Received March 13, 1999 86 yuguang yang et al.

Correctness of Johnson Counter Circuits

Yuguang Yang Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Yatsuka Nakamura Shinshu University Nagano

Summary. This article introduces the verification of the correctness for the operations and the specification of the Johnson counter. We formalize the concepts of 2-bit, 3-bit and 4-bit Johnson counter circuits with a reset input, and define the specification of the state transitions without the minor loop.

MML Identifier: GATE 3.

The notation and terminology used here are introduced in the paper [1]. The following propositions are true:

(1) Let s_{0}, s_{1}, s_{2}, s_{3}, n_{0}, n_{1}, n_{2}, n_{3}, q_{1}, q_{2}, n_{4}, n_{5} be sets such that NE s_{0} iff NE AND2(NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND2(NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND2(q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND2(q_{2}, q_{1}) and NE n_{0} iff NE AND2(NOT1n_{5}, NOT1n_{4}) and NE n_{1} iff NE AND2(NOT1n_{5}, n_{4}) and NE n_{2} iff NE AND2(n_{5}, NOT1n_{4}) and NE n_{3} iff NE AND2(n_{5}, n_{4}) and NE n_{4} iff NE NOT1q_{2} and NE n_{5} iff NE q_{1}. Then

(i) NE n_{1} iff NE s_{0}, (ii) NE n_{3} iff NE s_{1}, (iii) NE n_{2} iff NE s_{3}, and (iv) NE n_{0} iff NE s_{2}.

(2) Let s_{0}, s_{1}, s_{2}, s_{3}, n_{0}, n_{1}, n_{2}, n_{3}, q_{1}, q_{2}, n_{4}, n_{5}, R be sets such that NE s_{0} iff NE AND2(NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND2(NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND2(q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND2(q_{2}, q_{1}) and NE n_{0} iff NE AND2(NOT1n_{5}, NOT1n_{4}) and NE n_{1} iff NE AND2(NOT1n_{5}, n_{4}) and NE n_{2} iff NE AND2(n_{5}, NOT1n_{4}) and NE n_{3} iff NE AND2(n_{5}, n_{4}) and NE n_{4} iff NE AND2(NOT1q_{2}, R) and NE n_{5} iff NE AND2(q_{1}, R). Then

(i) NE n_{1} iff NE AND2(s_{0}, R), (ii) NE n_{3} iff NE AND2(s_{1}, R), (iii) NE n_{2} iff NE AND2(s_{3}, R), and (iv) NE n_{0} iff NE OR2(AND2(s_{2}, R), NOT1R).

(3) Let s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, n_{0}, n_{1}, n_{2}, n_{3}, n_{6}, n_{7}, n_{8}, n_{9}, q_{1}, q_{2}, q_{3}, n_{4}, n_{5}, n_{10} be sets such that NE s_{0} iff NE AND3(NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND3(NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND3(NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND3(NOT1q_{3}, q_{2}, q_{1}) and NE s_{4} iff NE AND3(q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{5} iff NE AND3(q_{3}, NOT1q_{2}, q_{1}) and NE s_{6} iff NE AND3(q_{3}, q_{2}, NOT1q_{1}) and NE s_{7} iff NE AND3(q_{3}, q_{2}, q_{1}) and NE n_{0} iff NE AND3(NOT1n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{1} iff NE AND3(NOT1n_{10}, NOT1n_{5}, n_{4}) and NE n_{2} iff NE AND3(NOT1n_{10}, n_{5}, NOT1n_{4}) and NE n_{3} iff NE AND3(NOT1n_{10}, n_{5}, n_{4}) and NE n_{6} iff NE AND3(n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{7} iff NE AND3(n_{10}, NOT1n_{5}, n_{4}) and NE n_{8} iff NE AND3(n_{10}, n_{5}, NOT1n_{4}) and NE n_{9} iff NE AND3(n_{10}, n_{5}, n_{4}) and NE n_{4} iff NE NOT1q_{3} and NE n_{5} iff NE q_{1} and NE n_{10} iff NE q_{2}. Then

(i) NE n_{1} iff NE s_{0}, (ii) NE n_{3} iff NE s_{1}, (iii) NE n_{9} iff NE s_{3}, (iv) NE n_{8} iff NE s_{7},

(v) NE n_{6} iff NE s_{6}, (vi) NE n_{0} iff NE s_{4}, (vii) NE n_{2} iff NE s_{5}, and (viii) NE n_{7} iff NE s_{2}.

(4) Let s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, n_{0}, n_{1}, n_{2}, n_{3}, n_{6}, n_{7}, n_{8}, n_{9}, q_{1}, q_{2}, q_{3}, n_{4}, n_{5}, n_{10}, R be sets such that NE s_{0} iff NE AND3(NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND3(NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND3(NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND3(NOT1q_{3}, q_{2}, q_{1}) and NE s_{4} iff NE AND3(q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{5} iff NE AND3(q_{3}, NOT1q_{2}, q_{1}) and NE s_{6} iff NE AND3(q_{3}, q_{2}, NOT1q_{1}) and NE s_{7} iff NE AND3(q_{3}, q_{2}, q_{1}) and NE n_{0} iff NE AND3(NOT1n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{1} iff NE AND3(NOT1n_{10}, NOT1n_{5}, n_{4}) and NE n_{2} iff NE AND3(NOT1n_{10}, n_{5}, NOT1n_{4}) and NE n_{3} iff NE AND3(NOT1n_{10}, n_{5}, n_{4}) and NE n_{6} iff NE AND3(n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{7} iff NE AND3(n_{10}, NOT1n_{5}, n_{4}) and NE n_{8} iff NE AND3(n_{10}, n_{5}, NOT1n_{4}) and NE n_{9} iff NE AND3(n_{10}, n_{5}, n_{4}) and NE n_{4} iff NE AND2(NOT1q_{3}, R) and NE n_{5} iff NE AND2(q_{1}, R) and NE n_{10} iff NE AND2(q_{2}, R). Then (i) NE n_{1} iff NE AND2(s_{0}, R), (ii) NE n_{3} iff NE AND2(s_{1}, R), (iii) NE n_{9} iff NE AND2(s_{3}, R), (iv) NE n_{8} iff NE AND2(s_{7}, R),

(v) NE n_{6} iff NE AND2(s_{6}, R), (vi) NE n_{0} iff NE OR2(AND2(s_{4}, R), NOT1R), (vii) NE n_{2} iff NE AND2(s_{5}, R), and (viii) NE n_{7} iff NE AND2(s_{2}, R).

(5) Let s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}, s_{9}, s_{10}, s_{11}, s_{12}, s_{13}, s_{14}, s_{15}, n_{0}, n_{1}, n_{2}, n_{3}, n_{6}, n_{7}, n_{8}, n_{9}, n_{11}, n_{12}, n_{13}, n_{14}, n_{15}, n_{16}, n_{17}, n_{18}, q_{1}, q_{2}, q_{3}, q_{4}, n_{4}, n_{5}, n_{10}, n_{19} be sets such that NE s_{0} iff NE AND4(NOT1q_{4}, NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND4(NOT1q_{4}, NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND4(NOT1q_{4}, NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND4(NOT1q_{4}, NOT1q_{3}, q_{2}, q_{1}) and NE s_{4} iff NE AND4(NOT1q_{4}, q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{5} iff NE AND4(NOT1q_{4}, q_{3}, NOT1q_{2}, q_{1}) and NE s_{6} iff NE AND4(NOT1q_{4}, q_{3}, q_{2}, NOT1q_{1}) and NE s_{7} iff NE AND4(NOT1q_{4}, q_{3}, q_{2}, q_{1}) and NE s_{8} iff NE AND4(q_{4}, NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{9} iff NE AND4(q_{4}, NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{10} iff NE AND4(q_{4}, NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{11} iff NE AND4(q_{4}, NOT1q_{3}, q_{2}, q_{1}) and NE s_{12} iff NE AND4(q_{4}, q_{3}, NOT1q_{2}, NOT1 q_{1}) and NE s_{13} iff NE AND4(q_{4}, q_{3}, NOT1q_{2}, q_{1}) and NE s_{14} iff NE AND4(q_{4}, q_{3}, q_{2}, NOT1q_{1}) and NE s_{15} iff NE AND4(q_{4}, q_{3}, q_{2}, q_{1}) and NE n_{0} iff NE AND4(NOT1n_{19}, NOT1n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{1} iff NE AND4(NOT1n_{19}, NOT1n_{10}, NOT1n_{5}, n_{4}) and NE n_{2} iff NE AND4(NOT1n_{19}, NOT1n_{10}, n_{5}, NOT1n_{4}) and NE n_{3} iff NE AND4(NOT1n_{19}, NOT1n_{10}, n_{5}, n_{4}) and NE n_{6} iff NE AND4(NOT1n_{19}, n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{7} iff NE AND4(NOT1 n_{19}, n_{10}, NOT1n_{5}, n_{4}) and NE n_{8} iff NE AND4(NOT1n_{19}, n_{10}, n_{5}, NOT1n_{4}) and NE n_{9} iff NE AND4(NOT1n_{19}, n_{10}, n_{5}, n_{4}) and NE n_{11} iff NE AND4(n_{19}, NOT1n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{12} iff NE AND4(n_{19}, NOT1n_{10}, NOT1n_{5}, n_{4}) and NE n_{13} iff NE AND4(n_{19}, NOT1 n_{10}, n_{5}, NOT1n_{4}) and NE n_{14} iff NE AND4(n_{19}, NOT1n_{10}, n_{5}, n_{4}) and NE n_{15} iff NE AND4(n_{19}, n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{16} iff NE AND4(n_{19}, n_{10}, NOT1n_{5}, n_{4}) and NE n_{17} iff NE AND4(n_{19}, n_{10}, n_{5}, NOT1n_{4}) and NE n_{18} iff NE AND4(n_{19}, n_{10}, n_{5}, n_{4}) and NE n_{4} iff NE NOT1q_{4} and NE n_{5} iff NE q_{1} and NE n_{10} iff NE q_{2} and NE n_{19} iff NE q_{3}. Then

(i) NE n_{1} iff NE s_{0}, (ii) NE n_{3} iff NE s_{1}, (iii) NE n_{9} iff NE s_{3}, (iv) NE n_{18} iff NE s_{7},

(v) NE n_{17} iff NE s_{15}, (vi) NE n_{15} iff NE s_{14}, (vii) NE n_{11} iff NE s_{12}, (viii) NE n_{0} iff NE s_{8}, (ix) NE n_{7} iff NE s_{2},

(x) NE n_{14} iff NE s_{5}, (xi) NE n_{8} iff NE s_{11}, (xii) NE n_{16} iff NE s_{6}, (xiii) NE n_{13} iff NE s_{13}, (xiv) NE n_{6} iff NE s_{10}, (xv) NE n_{12} iff NE s_{4}, and (xvi) NE n_{2} iff NE s_{9}.

(6) Let s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8}, s_{9}, s_{10}, s_{11}, s_{12}, s_{13}, s_{14}, s_{15}, n_{0}, n_{1}, n_{2}, n_{3}, n_{6}, n_{7}, n_{8}, n_{9}, n_{11}, n_{12}, n_{13}, n_{14}, n_{15}, n_{16}, n_{17}, n_{18}, q_{1}, q_{2}, q_{3}, q_{4}, n_{4}, n_{5}, n_{10}, n_{19}, R be sets such that NE s_{0} iff NE AND4(NOT1q_{4}, NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{1} iff NE AND4(NOT1q_{4}, NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{2} iff NE AND4(NOT1q_{4}, NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{3} iff NE AND4(NOT1q_{4}, NOT1q_{3}, q_{2}, q_{1}) and NE s_{4} iff NE AND4(NOT1q_{4}, q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{5} iff NE AND4(NOT1q_{4}, q_{3}, NOT1q_{2}, q_{1}) and NE s_{6} iff NE AND4(NOT1q_{4}, q_{3}, q_{2}, NOT1q_{1}) and NE s_{7} iff NE AND4(NOT1q_{4}, q_{3}, q_{2}, q_{1}) and NE s_{8} iff NE AND4(q_{4}, NOT1q_{3}, NOT1q_{2}, NOT1q_{1}) and NE s_{9} iff NE AND4(q_{4}, NOT1q_{3}, NOT1q_{2}, q_{1}) and NE s_{10} iff NE AND4(q_{4}, NOT1q_{3}, q_{2}, NOT1q_{1}) and NE s_{11} iff NE AND4(q_{4}, NOT1q_{3}, q_{2}, q_{1}) and NE s_{12} iff NE AND4(q_{4}, q_{3}, NOT1q_{2}, NOT1 q_{1}) and NE s_{13} iff NE AND4(q_{4}, q_{3}, NOT1q_{2}, q_{1}) and NE s_{14} iff NE AND4(q_{4}, q_{3}, q_{2}, NOT1q_{1}) and NE s_{15} iff NE AND4(q_{4}, q_{3}, q_{2}, q_{1}) and NE n_{0} iff NE AND4(NOT1n_{19}, NOT1n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{1} iff NE AND4(NOT1n_{19}, NOT1n_{10}, NOT1n_{5}, n_{4}) and NE n_{2} iff NE AND4(NOT1n_{19}, NOT1n_{10}, n_{5}, NOT1n_{4}) and NE n_{3} iff NE AND4(NOT1n_{19}, NOT1n_{10}, n_{5}, n_{4}) and NE n_{6} iff NE AND4(NOT1n_{19}, n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{7} iff NE AND4(NOT1 n_{19}, n_{10}, NOT1n_{5}, n_{4}) and NE n_{8} iff NE AND4(NOT1n_{19}, n_{10}, n_{5}, NOT1n_{4}) and NE n_{9} iff NE AND4(NOT1n_{19}, n_{10}, n_{5}, n_{4}) and NE n_{11} iff NE AND4(n_{19}, NOT1n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{12} iff NE AND4(n_{19}, NOT1n_{10}, NOT1n_{5}, n_{4}) and NE n_{13} iff NE AND4(n_{19}, NOT1 n_{10}, n_{5}, NOT1n_{4}) and NE n_{14} iff NE AND4(n_{19}, NOT1n_{10}, n_{5}, n_{4}) and NE n_{15} iff NE AND4(n_{19}, n_{10}, NOT1n_{5}, NOT1n_{4}) and NE n_{16} iff NE AND4(n_{19}, n_{10}, NOT1n_{5}, n_{4}) and NE n_{17} iff NE AND4(n_{19}, n_{10}, n_{5}, NOT1 n_{4}) and NE n_{18} iff NE AND4(n_{19}, n_{10}, n_{5}, n_{4}) and NE n_{4} iff NE AND2(NOT1q_{4}, R) and NE n_{5} iff NE AND2(q_{1}, R) and NE n_{10} iff NE AND2(q_{2}, R) and NE n_{19} iff NE AND2(q_{3}, R). Then

(i) NE n_{1} iff NE AND2(s_{0}, R), (ii) NE n_{3} iff NE AND2(s_{1}, R), (iii) NE n_{9} iff NE AND2(s_{3}, R), (iv) NE n_{18} iff NE AND2(s_{7}, R),

(v) NE n_{17} iff NE AND2(s_{15}, R), (vi) NE n_{15} iff NE AND2(s_{14}, R), (vii) NE n_{11} iff NE AND2(s_{12}, R), (viii) NE n_{0} iff NE OR2(AND2(s_{8}, R), NOT1R), (ix) NE n_{7} iff NE AND2(s_{2}, R),

(x) NE n_{14} iff NE AND2(s_{5}, R), (xi) NE n_{8} iff NE AND2(s_{11}, R), (xii) NE n_{16} iff NE AND2(s_{6}, R), (xiii) NE n_{13} iff NE AND2(s_{13}, R), (xiv) NE n_{6} iff NE AND2(s_{10}, R), (xv) NE n_{12} iff NE AND2(s_{4}, R), and (xvi) NE n_{2} iff NE AND2(s_{9}, R).

References

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

92 yuguang yang et al.

The Definition of the Riemann Definite Integral and some Related Lemmas

Noboru Endou Shinshu University Nagano

Artur Korniłowicz^{1} University of Bialystok

Summary. This article introduces the Riemann definite integral on the closed interval of real. We present the definitions and related lemmas of the closed interval. We formalize the concept of the Riemann definite integral and the division of the closed interval of real, and prove the additivity of the integral.

MML Identifier: INTEGRA1.

The notation and terminology used in this paper are introduced in the following papers: [28], [31], [8], [14], [2], [5], [6], [30], [22], [32], [18], [15], [7], [20], [26], [10], [12], [3], [27], [21], [4], [29], [16], [17], [24], [9], [11], [19], [25], [13], [23], and [1].

1. Definition of Closed Interval and its Properties

For simplicity, we adopt the following rules: a, a_{1}, a_{2}, b, b_{1}, b_{2} are real numbers, p is a finite sequence, F, G, H are finite sequences of elements of ℝ, i, j, k are natural numbers, f is a function from ℝ into ℝ, and x_{1} is a set.

Let I_{1} be a subset of ℝ. We say that I_{1} is closed-interval if and only if: (Def. 1) There exist real numbers a, b such that a b and I_{1} = [a, b].

Let us mention that there exists a subset of ℝ which is closed-interval.

In the sequel A, A_{1}, A_{2} are closed-interval subsets of ℝ.

The following propositions are true:

(1) Every closed-interval subset of ℝ is compact.

1This paper was written while the second author visited Shinshu University, winter 1999.

(2) If A is a closed-interval subset of ℝ, then A is non empty. Let us observe that every subset of ℝ which is closed-interval is also non

empty and compact.

The following proposition is true

(3) If A is a closed-interval subset of ℝ, then A is lower bounded and upper bounded.

Let us observe that every subset of ℝ which is closed-interval is also bounded. One can verify that there exists a subset of ℝ which is closed-interval. Next we state three propositions:

(4) If A is a closed-interval subset of ℝ, then there exist a, b such that a b and a = inf A and b = supA.

(5) If A is a closed-interval subset of ℝ, then A = [inf A, supA].

(6) If A = [a_{1}, b_{1}] and A = [a_{2}, b_{2}], then a_{1} = a_{2} and b_{1} = b_{2}.

2. Definition of Division of Closed Interval and its Properties

Let A be a closed-interval subset of ℝ. A non empty increasing finite sequence of elements of ℝ is said to be a DivisionPoint of A if:

(Def. 2) rngit ⊆ A and it(lenit) = supA.

Let A be a closed-interval subset of ℝ. The functor divsA yielding a set is defined by:

(Def. 3) x_{1} ∈ divsA iff x_{1} is a DivisionPoint of A.

Let A be a closed-interval subset of ℝ. One can check that divsA is non empty.

Let A be a closed-interval subset of ℝ. A non empty set is called a Division of A if:

(Def. 4) x_{1} ∈ it iff x_{1} is a DivisionPoint of A.

Let A be a closed-interval subset of ℝ. Observe that there exists a Division of A which is non empty.

The following proposition is true

(7) For every closed-interval subset A of ℝ and for every non empty Division S of A holds every element of S is a DivisionPoint of A.

Let A be a closed-interval subset of ℝ and let S be a non empty Division of A. We see that the element of S is a DivisionPoint of A.

In the sequel S denotes a non empty Division of A and D, D_{1}, D_{2} denote elements of S.

Next we state two propositions:

(8) If i ∈ domD, then D(i) ∈ A.

(9) If i ∈ domD and i ≠ 1, then i − 1 ∈ domD and D(i − 1) ∈ A and i − 1 ∈ ℕ.

Let A be a closed-interval subset of ℝ, let S be a non empty Division of A, let D be an element of S, and let i be a natural number. Let us assume that i ∈ domD. The functor divset(D, i) yielding a closed-interval subset of ℝ is defined as follows:

(Def. 5)(i) inf divset(D, i) = inf A and supdivset(D, i) = D(i) if i = 1, (ii) inf divset(D, i) = D(i − 1) and supdivset(D, i) = D(i), otherwise. Next we state the proposition

(10) If i ∈ domD, then divset(D, i) ⊆ A.

Let A be a subset of ℝ. The functor vol(A) yielding a real number is defined by:

(Def. 6) vol(A) = supA − inf A.

One can prove the following proposition

(11) For every closed-interval subset A of ℝ holds 0 vol(A).

3. Definitions of Integrability and Related Topics

Let A be a closed-interval subset of ℝ, let f be a partial function from A to ℝ, let S be a non empty Division of A, and let D be an element of S. The functor upper volume(f, D) yielding a finite sequence of elements of ℝ is defined as follows:

(Def. 7) lenupper volume(f, D) = lenD and for every i such that i ∈ SeglenD holds (upper volume(f, D))(i) = suprng(f↾ divset(D, i)) · vol(divset(D, i)).

The functor lower volume(f, D) yielding a finite sequence of elements of ℝ is defined by:

(Def. 8) lenlower volume(f, D) = lenD and for every i such that i ∈ SeglenD

holds (lower volume(f, D))(i) = inf rng(f↾ divset(D, i)) · vol(divset(D, i)). Let A be a closed-interval subset of ℝ, let f be a partial function from A

to ℝ, let S be a non empty Division of A, and let D be an element of S. The functor upper sum(f, D) yields a real number and is defined by:

(Def. 9) upper sum(f, D) = ∑ upper volume(f, D).

The functor lower sum(f, D) yields a real number and is defined by: (Def. 10) lower sum(f, D) = ∑ lower volume(f, D).

Let A be a closed-interval subset of ℝ. Then divsA is a Division of A.

Let A be a closed-interval subset of ℝ and let f be a partial function from A to ℝ. The functor upper sum setf yielding a partial function from divsA to ℝ is defined as follows:

(Def. 11) domupper sum setf = divsA and for every element D of divsA such that D ∈ domupper sum setf holds (upper sum setf)(D) = upper sum(f, D).

The functor lower sum setf yields a partial function from divsA to ℝ and is defined as follows:

(Def. 12) domlower sum setf = divsA and for every element D of divsA such that D ∈ domlower sum setf holds (lower sum setf)(D) = lower sum(f, D).

Let A be a closed-interval subset of ℝ and let f be a partial function from A to ℝ. We say that f is upper integrable on A if and only if:

(Def. 13) rngupper sum setf is lower bounded.

We say that f is lower integrable on A if and only if: (Def. 14) rnglower sum setf is upper bounded.

Let A be a closed-interval subset of ℝ and let f be a partial function from A to ℝ. The functor upper integralf yielding a real number is defined by:

(Def. 15) upper integralf = inf rngupper sum setf.

Let A be a closed-interval subset of ℝ and let f be a partial function from A to ℝ. The functor lower integralf yields a real number and is defined as follows:

(Def. 16) lower integralf = suprnglower sum setf.

Let A be a closed-interval subset of ℝ and let f be a partial function from A to ℝ. We say that f is integrable on A if and only if:

(Def. 17) f is upper integrable on A and f is lower integrable on A and

upper integralf = lower integralf. Let A be a closed-interval subset of ℝ and let f be a partial function from

A to ℝ. The functor integralf yields a real number and is defined by: (Def. 18) integralf = upper integralf.

4. Real Function’s Properties

Next we state several propositions:

(12) For every non empty set X and for all partial functions f, g from X to ℝ holds rng(f + g) ⊆ rngf + rngg.

(13) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is lower bounded on A, then rngf is lower bounded.

(14) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If rngf is lower bounded, then f is lower bounded on A.

(15) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is upper bounded on A, then rngf is upper bounded.

(16) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If rngf is upper bounded, then f is upper bounded on A.

(17) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is bounded on A, then rngf is bounded.

5. Characteristic Function’s Properties

The following propositions are true:

(18) For every closed-interval subset A of ℝ holds χ_{A,A} is a constant on A.

(19) For every closed-interval subset A of ℝ holds rng(χ_{A,A}) = {1}.

(20) For every closed-interval subset A of ℝ and for every set B such that B ∩ dom(χ_{A,A}) ≠ ∅ holds rng(χ_{A,A}↾B) = {1}.

(21) If i ∈ SeglenD, then vol(divset(D, i)) = (lower volume(χ_{A,A}, D))(i).

(22) If i ∈ SeglenD, then vol(divset(D, i)) = (upper volume(χ_{A,A}, D))(i).

(23) If lenF = lenG and lenF = lenH and for every k such that k ∈ domF holds H(k) = F_{k} + G_{k}, then ∑ H = ∑ F + ∑ G.

(24) If lenF = lenG and lenF = lenH and for every k such that k ∈ domF holds H(k) = F_{k} − G_{k}, then ∑ H = ∑ F − ∑ G.

(25) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D be an element of S. Then ∑ lower volume(χ_{A,A}, D) = vol(A).

(26) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D be an element of S. Then ∑ upper volume(χ_{A,A}, D) = vol(A).

6. Some Properties of Darboux Sum

Let A be a closed-interval subset of ℝ, let f be a partial function from A to ℝ, let S be a non empty Division of A, and let D be an element of S. Then upper volume(f, D) is a non empty finite sequence of elements of ℝ.

Let A be a closed-interval subset of ℝ, let f be a partial function from A to ℝ, let S be a non empty Division of A, and let D be an element of S. Then lower volume(f, D) is a non empty finite sequence of elements of ℝ.

One can prove the following propositions:

(27) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is total and lower bounded on A, then inf rngf · vol(A) lower sum(f, D).

(28) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and i be a natural number. Suppose f is total and upper bounded on A and i ∈ SeglenD. Then suprngf ·vol(divset(D, i)) suprng(f↾ divset(D, i))· vol(divset(D, i)).

(29) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is total and upper bounded on A, then upper sum(f, D) suprngf · vol(A).

(30) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is total and bounded on A, then lower sum(f, D) upper sum(f, D).

Let x be a non empty finite sequence of elements of ℝ. Then rngx is a finite non empty subset of ℝ.

Let A be a closed-interval subset of ℝ and let D be an element of divsA. The functor δ_{D} yielding a real number is defined by:

(Def. 19) δ_{D} = maxrngupper volume(χ_{A,A}, D).

Let A be a closed-interval subset of ℝ, let S be a non empty Division of A, and let D_{1}, D_{2} be elements of S. The predicate D_{1} D_{2} is defined as follows:

(Def. 20) lenD_{1} lenD_{2} and rngD_{1} ⊆ rngD_{2}.

We introduce D_{2} D_{1} as a synonym of D_{1} D_{2}.

One can prove the following propositions:

(31) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If lenD_{1} = 1, then D_{1} D_{2}.

(32) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If f is total and upper bounded on A and lenD_{1} = 1, then upper sum(f, D_{1}) upper sum(f, D_{2}).

(33) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If f is total and lower bounded on A and lenD_{1} = 1, then lower sum(f, D_{1}) lower sum(f, D_{2}).

(34) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D be an element of S. If i ∈ domD, then there exist A_{1}, A_{2} such that A_{1} = [inf A, D(i)] and A_{2} = [D(i), supA] and A = A_{1} ∪ A_{2}.

(35) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If i ∈ domD_{1}, then if D_{1} D_{2}, then there exists j such that j ∈ domD_{2} and D_{1}(i) = D_{2}(j).

Let A be a closed-interval subset of ℝ, let S be a non empty Division of A, let D_{1}, D_{2} be elements of S, and let i be a natural number. Let us assume that D_{1} D_{2}. The functor indx(D_{2}, D_{1}, i) yields a natural number and is defined as follows:

(Def. 21)(i) indx(D_{2}, D_{1}, i) ∈ domD_{2} and D_{1}(i) = D_{2}(indx(D_{2}, D_{1}, i)) if i ∈

domD_{1}, (ii) indx(D_{2}, D_{1}, i) = 0, otherwise. Next we state four propositions:

(36) Let p be an increasing finite sequence of elements of ℝ and n be a natural number. Suppose n lenp. Then p_{⇂n} is an increasing finite sequence of elements of ℝ.

(37) Let p be an increasing finite sequence of elements of ℝ and i, j be natural numbers. Suppose j ∈ domp and i j. Then mid(p, i, j) is an increasing finite sequence of elements of ℝ.

(38) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, D be an element of S, and i, j be natural numbers. Suppose i ∈ domD and j ∈ domD and i j. Then there exists a closed-interval subset B of ℝ such that inf B = (mid(D, i, j))(1) and supB = (mid(D, i, j))(lenmid(D, i, j)) and lenmid(D, i, j) = (j − i) + 1 and mid(D, i, j) is a DivisionPoint of B.

(39) Let A, B be closed-interval subsets of ℝ, S be a non empty Division of A, S_{1} be a non empty Division of B, D be an element of S, and i, j be natural numbers. Suppose i ∈ domD and j ∈ domD and i j and D(i) inf B and D(j) = supB. Then mid(D, i, j) is an element of S_{1}.

Let p be a finite sequence of elements of ℝ. The functor PartSumsp yielding a finite sequence of elements of ℝ is defined by:

(Def. 22) lenPartSumsp = lenp and for every i such that i ∈ Seglenp holds

(PartSumsp)(i) = ∑(p↾i). We now state a number of propositions:

(40) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. Suppose D_{1} D_{2} and f is total and upper bounded on A. Let i be a non empty natural number. If i ∈ domD_{1}, then ∑(upper volume(f, D_{1})↾i) ∑(upper volume(f, D_{2})↾ indx(D_{2}, D_{1}, i)).

(41) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. Suppose D_{1} D_{2} and f is total and lower bounded on A. Let i be a non empty natural number. If i ∈ domD_{1}, then ∑(lower volume(f, D_{1})↾i) ∑(lower volume(f, D_{2})↾ indx(D_{2}, D_{1}, i)).

(42) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D_{1}, D_{2} be elements of S, and i be a natural number. Suppose D_{1} D_{2} and i ∈ domD_{1} and f is total and upper bounded on A. Then (PartSumsupper volume(f, D_{1}))(i) (PartSumsupper volume(f, D_{2}))(indx(D_{2}, D_{1}, i)).

(43) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D_{1}, D_{2} be elements of S, and i be a natural number. Suppose D_{1} D_{2} and i ∈ domD_{1} and f is total and lower bounded on A. Then (PartSumslower volume(f, D_{1}))(i) (PartSumslower volume(f, D_{2}))(indx(D_{2}, D_{1}, i)).

(44) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. Then (PartSumsupper volume(f, D))(lenD) = upper sum(f, D).

(45) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. Then (PartSumslower volume(f, D))(lenD) = lower sum(f, D).

(46) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If D_{1} D_{2}, then indx(D_{2}, D_{1}, lenD_{1}) = lenD_{2}.

(47) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If D_{1} D_{2} and f is total and upper bounded on A, then upper sum(f, D_{2}) upper sum(f, D_{1}).

(48) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If D_{1} D_{2} and f is total and lower bounded on A, then lower sum(f, D_{2}) lower sum(f, D_{1}).

(49) Let A be a closed-interval subset of ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. Then there exists an element D of S such that D_{1} D and D_{2} D.

(50) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D_{1}, D_{2} be elements of S. If f is total and bounded on A, then lower sum(f, D_{1}) upper sum(f, D_{2}).

7. Additivity of Integral

One can prove the following propositions:

(51) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. Suppose f is upper integrable on A and f is lower integrable on A and f is total and bounded on A. Then upper integralf lower integralf.

(52) For all subsets X, Y of ℝ holds − X + − Y = − (X + Y).

(53) For all subsets X, Y of ℝ such that X is upper bounded and Y ≠ ∅ and Y is upper bounded holds X + Y is upper bounded.

(54) For all non empty subsets X, Y of ℝ such that X is upper bounded and Y is upper bounded holds sup(X + Y) = supX + supY.

(55) Let A be a closed-interval subset of ℝ, f, g be partial functions from A to ℝ, S be a non empty Division of A, and D be an element of S. Suppose i ∈ SeglenD and f is upper bounded on A and total and g is upper bounded on A and total. Then (upper volume(f + g, D))(i) (upper volume(f, D))(i) + (upper volume(g, D))(i).

(56) Let A be a closed-interval subset of ℝ, f, g be partial functions from A to ℝ, S be a non empty Division of A, and D be an element of S. Suppose i ∈ SeglenD and f is lower bounded on A and total and g is lower bounded on A and total. Then (lower volume(f, D))(i)+(lower volume(g, D))(i) (lower volume(f + g, D))(i).

(57) Let A be a closed-interval subset of ℝ, f, g be partial functions from A to ℝ, S be a non empty Division of A, and D be an element of S. Suppose f is upper bounded on A and total and g is upper bounded on A and total. Then upper sum(f + g, D) upper sum(f, D) + upper sum(g, D).

(58) Let A be a closed-interval subset of ℝ, f, g be partial functions from A to ℝ, S be a non empty Division of A, and D be an element of S. Suppose f is lower bounded on A and total and g is lower bounded on A and total. Then lower sum(f, D) + lower sum(g, D) lower sum(f + g, D).

(59) Let X be a non empty set and f be a partial function from X to ℝ. If f is upper bounded on X and total, then rngf is upper bounded.

(60) Let X be a non empty set and f be a partial function from X to ℝ. If rngf is upper bounded and f is total, then f is upper bounded on X.

(61) Let X be a non empty set and f be a partial function from X to ℝ. If f is lower bounded on X and total, then rngf is lower bounded.

(62) Let X be a non empty set and f be a partial function from X to ℝ. If rngf is lower bounded and f is total, then f is lower bounded on X.

(63) Let A be a closed-interval subset of ℝ and f, g be partial functions from

A to ℝ. Suppose that

(i) f is total and bounded on A, (ii) g is total and bounded on A, (iii) f is integrable on A, and (iv) g is integrable on A.

Then f + g is integrable on A and integralf + g = integralf + integralg. 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.

102 noboru endou and artur korniłowicz

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

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

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

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

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

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

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

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

[11] Czesław Byliński and Andrzej Trybulec. Complex spaces. Formalized Mathematics, 2(1):151–158, 1991.

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

[13] 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.

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

[15] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[16] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[17] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

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

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

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

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

[22] Robert Milewski. Natural numbers. Formalized Mathematics, 7(1):19–22, 1998.

[23] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Formalized Mathematics, 6(2):255–263, 1997.

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

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

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

[27] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

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

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

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

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

Properties of the Trigonometric Function

Takashi Mitsuishi Shinshu University Nagano

Yuguang Yang Shinshu University Nagano

Summary. This article introduces the monotone increasing and the monotone decreasing of sinus and cosine, and definitions of hyperbolic sinus, hyperbolic cosine and hyperbolic tangent, and some related formulas about them.

MML Identifier: SIN COS2.

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

1. Monotone Increasing and Monotone Decreasing of Sinus and Cosine

We adopt the following rules: p, q, r, t_{1} are elements of ℝ and n is a natural number.

Next we state a number of propositions:

(1) If p 0 and r 0, then p + r 2 · \sqrt{p · r}.

(2) sin is increasing on ]0, \frac{Pai}{2} [.

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

(4) cos is decreasing on ]0, \frac{Pai}{2} [.

(5) cos is decreasing on ]\frac{Pai}{2} , Pai[.

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

(7) sin is increasing on ]\frac{3}{2} · Pai, 2 · Pai[.

(8) cos is increasing on ]Pai, \frac{3}{2} · Pai[.

(9) cos is increasing on ]\frac{3}{2} · Pai, 2 · Pai[.

(10) (sin)(t_{1}) = (sin)(2 · Pai·n + t_{1}).

(11) (cos)(t_{1}) = (cos)(2 · Pai·n + t_{1}).

2. Hyperbolic Sinus, Hyperbolic Cosine and Hyperbolic Tangent

The partial function sinh from ℝ to ℝ is defined as follows: (Def. 1) domsinh = ℝ and for every real number d holds (sinh)(d) =

\frac{(exp)(d)−(exp)(− d)}{2} . Let d be a real number. The functor sinhd yielding an element of ℝ is defined

by: (Def. 2) sinhd = (sinh)(d).

The partial function cosh from ℝ to ℝ is defined as follows: (Def. 3) domcosh = ℝ and for every real number d holds (cosh)(d) =

\frac{(exp)(d)+(exp)(− d)}{2}. Let d be a real number. The functor coshd yields an element of ℝ and is

defined as follows: (Def. 4) coshd = (cosh)(d).

The partial function tanh from ℝ to ℝ is defined as follows: (Def. 5) domtanh = ℝ and for every real number d holds (tanh)(d) =

\frac{(exp)(d)−(exp)(− d)}{(exp)(d)+(exp)(− d)}. Let d be a real number. The functor tanhd yields an element of ℝ and is

defined as follows: (Def. 6) tanhd = (tanh)(d).

We now state a number of propositions:

(12) (exp)(p + q) = (exp)(p) · (exp)(q).

(13) (exp)(0) = 1.

(14) (cosh)(p)^{2} − (sinh)(p)^{2} = 1 and (cosh)(p) · (cosh)(p) − (sinh)(p) · (sinh)(p) = 1.

(15) (cosh)(p) ≠ 0 and (cosh)(p) > 0 and (cosh)(0) = 1.

(16) (sinh)(0) = 0.

(17) (tanh)(p) = \frac{(sinh)(p)}{(cosh)(p)}.

(18) (sinh)(p)^{2} = \frac{1}{2} · ((cosh)(2 · p) − 1) and (cosh)(p)^{2} = \frac{1}{2} · ((cosh)(2 · p) + 1).

(19) (cosh)(− p) = (cosh)(p) and (sinh)(− p) = − (sinh)(p) and (tanh)(− p) = − (tanh)(p).

(20) (cosh)(p + r) = (cosh)(p) · (cosh)(r) + (sinh)(p) · (sinh)(r) and (cosh)(p − r) = (cosh)(p) · (cosh)(r) − (sinh)(p) · (sinh)(r).

(21) (sinh)(p + r) = (sinh)(p) · (cosh)(r) + (cosh)(p) · (sinh)(r) and (sinh)(p − r) = (sinh)(p) · (cosh)(r) − (cosh)(p) · (sinh)(r).

(22) (tanh)(p+r) = \frac{(tanh)(p)+(tanh)(r)}{1+(tanh)(p)·(tanh)(r)} and (tanh)(p−r) = \frac{(tanh)(p)−(tanh)(r)}{1−(tanh)(p)·(tanh)(r)}.

(23) (sinh)(2 · p) = 2 · (sinh)(p) · (cosh)(p) and (cosh)(2 · p) = 2 · (cosh)(p)^{2} − 1 and (tanh)(2 · p) = \frac{2·(tanh)(p)}{1+(tanh)(p)^{2}} .

(24) (sinh)(p)^{2} − (sinh)(q)^{2} = (sinh)(p + q) · (sinh)(p − q) and (sinh)(p + q) · (sinh)(p − q) = (cosh)(p)^{2} − (cosh)(q)^{2} and (sinh)(p)^{2} − (sinh)(q)^{2} = (cosh)(p)^{2} − (cosh)(q)^{2}.

(25) (sinh)(p)^{2} + (cosh)(q)^{2} = (cosh)(p + q) · (cosh)(p − q) and (cosh)(p + q) · (cosh)(p − q) = (cosh)(p)^{2} + (sinh)(q)^{2} and (sinh)(p)^{2} + (cosh)(q)^{2} = (cosh)(p)^{2} + (sinh)(q)^{2}.

(26) (sinh)(p) + (sinh)(r) = 2 · (sinh)(\frac{p}{2} + \frac{r}{2}) · (cosh)(\frac{p}{2} − \frac{r}{2}) and (sinh)(p) − (sinh)(r) = 2 · (sinh)(\frac{p}{2} − \frac{r}{2}) · (cosh)(\frac{p}{2} + \frac{r}{2}).

(27) (cosh)(p) + (cosh)(r) = 2 · (cosh)(\frac{p}{2} + \frac{r}{2}) · (cosh)(\frac{p}{2} − \frac{r}{2}) and (cosh)(p) − (cosh)(r) = 2 · (sinh)(\frac{p}{2} − \frac{r}{2}) · (sinh)(\frac{p}{2} + \frac{r}{2}).

(28) (tanh)(p) + (tanh)(r) = \frac{(sinh)(p+r)}{(cosh)(p)·(cosh)(r)} and (tanh)(p) − (tanh)(r) = \frac{(sinh)(p−r)}{(cosh)(p)·(cosh)(r)}.

(29) ((cosh)(p) + (sinh)(p))_{ℕ}^{n} = (cosh)(n · p) + (sinh)(n · p).

One can check the following observations:

∗ sinh is total,

∗ cosh is total, and

∗ tanh is total.

One can prove the following propositions:

(30) domsinh = ℝ and domcosh = ℝ and domtanh = ℝ.

(31) sinh is differentiable in p and (sinh)′(p) = (cosh)(p).

(32) cosh is differentiable in p and (cosh)′(p) = (sinh)(p).

(33) tanh is differentiable in p and (tanh)′(p) = \frac{1}{(cosh)(p)^{2}} .

(34) sinh is differentiable on ℝ and (sinh)′(p) = (cosh)(p).

(35) cosh is differentiable on ℝ and (cosh)′(p) = (sinh)(p).

(36) tanh is differentiable on ℝ and (tanh)′(p) = \frac{1}{(cosh)(p)^{2}} .

(37) (cosh)(p) 1.

(38) sinh is continuous in p.

(39) cosh is continuous in p.

(40) tanh is continuous in p.

(41) sinh is continuous on ℝ.

(42) cosh is continuous on ℝ.

(43) tanh is continuous on ℝ.

106 takashi mitsuishi and yuguang yang

(44) (tanh)(p) < 1 and (tanh)(p) > − 1.

References

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

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

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

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

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

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

[9] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

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

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

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

[13] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

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

[15] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

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

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

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

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

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

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

Predicate Calculus for Boolean Valued Functions. Part II

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we have proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC 4.

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

1. Preliminaries

In this paper Y denotes a non empty set. Next we state a number of propositions:

(1) For all elements a, b, c of BVF(Y) such that a ⋐ b ⇒ c holds a ∧ b ⋐ c.

(2) For all elements a, b, c of BVF(Y) such that a ∧ b ⋐ c holds a ⋐ b ⇒ c.

(3) For all elements a, b of BVF(Y) holds a ∨ a ∧ b = a.

(4) For all elements a, b of BVF(Y) holds a ∧ (a ∨ b) = a.

(5) For every element a of BVF(Y) holds a ∧ ¬a = false(Y).

(6) For every element a of BVF(Y) holds a ∨ ¬a = true(Y).

(7) For all elements a, b of BVF(Y) holds a ⇔ b = (a ⇒ b) ∧ (b ⇒ a).

(8) For all elements a, b of BVF(Y) holds a ⇒ b = ¬a ∨ b.

(9) For all elements a, b of BVF(Y) holds a ⊕ b = ¬a ∧ b ∨ a ∧ ¬b.

(10) For all elements a, b of BVF(Y) holds a ⇔ b = true(Y) iff a ⇒ b = true(Y) and b ⇒ a = true(Y).

(11) For all elements a, b, c of BVF(Y) such that a ⇔ b = true(Y) and b ⇔ c = true(Y) holds a ⇔ c = true(Y).

(12) For all elements a, b of BVF(Y) such that a ⇔ b = true(Y) holds ¬a ⇔ ¬b = true(Y).

(13) For all elements a, b, c, d of BVF(Y) such that a ⇔ b = true(Y) and c ⇔ d = true(Y) holds a ∧ c ⇔ b ∧ d = true(Y).

(14) For all elements a, b, c, d of BVF(Y) such that a ⇔ b = true(Y) and c ⇔ d = true(Y) holds a ⇒ c ⇔ b ⇒ d = true(Y).

(15) For all elements a, b, c, d of BVF(Y) such that a ⇔ b = true(Y) and c ⇔ d = true(Y) holds a ∨ c ⇔ b ∨ d = true(Y).

(16) For all elements a, b, c, d of BVF(Y) such that a ⇔ b = true(Y) and c ⇔ d = true(Y) holds a ⇔ c ⇔ b ⇔ d = true(Y).

2. Predicate Calculus

Next we state a number of propositions:

(17) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a⇔b,P_{1}} G = ∀_{a⇒b,P_{1}} G ∧ ∀_{b⇒a,P_{1}} G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1}, P_{2} be partitions of Y. Suppose G is a coordinate and P_{1} ∈ G and P_{2} ∈ G. Then ∀_{a,P_{1}}G ⋐ ∃_{a,P_{1}}G and ∀_{a,P_{1}}G ⋐ ∃_{a,P_{2}}G.

(19) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. If a ⇒ u = true(Y), then ∀_{a,P_{1}}G ⇒ u = true(Y).

(20) Let u be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∃_{u,P_{1}}G ⋐ u.

(21) Let u be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then u ⋐ ∀_{u,P_{1}}G.

(22) Let u be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1}, P_{2} be partitions of Y. Suppose G is a coordinate and P_{1} ∈ G and P_{2} ∈ G and u is independent of P_{2}, G. Then ∀_{u,P_{1}}G ⋐ ∀_{u,P_{2}}G.

(23) Let u be an element of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1}, P_{2} be partitions of Y. Suppose G is a coordinate and P_{1} ∈ G and P_{2} ∈ G and u is independent of P_{1}, G. Then ∃_{u,P_{1}}G ⋐ ∃_{u,P_{2}}G.

predicate calculus for boolean valued … 109

(24) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a⇔b,P_{1}} G ⋐ ∀_{a,P_{1}}G ⇔ ∀_{b,P_{1}}G.

(25) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. If G is a coordinate and P_{1} ∈ G, then ∀_{a∧b,P_{1}} G ⋐ a ∧ ∀_{b,P_{1}}G.

(26) Let a, u be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G and u is independent of P_{1}, G. Then ∀_{a,P_{1}}G ⇒ u ⋐ ∃_{a⇒u,P_{1}} G.

(27) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G. If a ⇔ b = true(Y), then ∀_{a,P_{1}}G ⇔ ∀_{b,P_{1}}G = true(Y).

(28) Let a, b be elements of BVF(Y), G be a subset of PARTITIONS(Y), and P_{1} be a partition of Y. Suppose G is a coordinate and P_{1} ∈ G. If a ⇔ b = true(Y), then ∃_{a,P_{1}}G ⇔ ∃_{b,P_{1}}G = true(Y).

References

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

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

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

[4] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

[5] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

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

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

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

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

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

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

110 shunichi kobayashi and yatsuka nakamura

Propositional Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC 5.

The terminology and notation used in this paper have been introduced in the following articles: [6], [8], [9], [2], [3], [5], [1], [7], and [4].

In this paper Y is a non empty set. Next we state a number of propositions:

(1) For all elements a, b of BVF(Y) holds a = true(Y) and b = true(Y) iff a ∧ b = true(Y).

(2) For all elements a, b of BVF(Y) such that a = true(Y) and a ⇒ b = true(Y) holds b = true(Y).

(3) For all elements a, b of BVF(Y) such that a = true(Y) holds a ∨ b = true(Y).

(5)^{1} For all elements a, b of BVF(Y) such that b = true(Y) holds a ⇒ b =

true(Y).

(6) For all elements a, b of BVF(Y) such that ¬a = true(Y) holds a ⇒ b = true(Y).

(7) For every element a of BVF(Y) holds ¬(a ∧ ¬a) = true(Y).

(8) For every element a of BVF(Y) holds a ⇒ a = true(Y).

(9) For all elements a, b of BVF(Y) holds a ⇒ b = true(Y) iff ¬b ⇒ ¬a = true(Y).

1The proposition (4) has been removed.

(10) For all elements a, b, c of BVF(Y) such that a ⇒ b = true(Y) and b ⇒ c = true(Y) holds a ⇒ c = true(Y).

(11) For all elements a, b of BVF(Y) such that a ⇒ b = true(Y) and a ⇒ ¬b = true(Y) holds ¬a = true(Y).

(12) For every element a of BVF(Y) holds ¬a ⇒ a ⇒ a = true(Y).

(13) For all elements a, b, c of BVF(Y) holds a ⇒ b ⇒ ¬(b ∧ c) ⇒ ¬(a ∧ c) = true(Y).

(14) For all elements a, b, c of BVF(Y) holds a ⇒ b ⇒ b ⇒ c ⇒ a ⇒ c = true(Y).

(15) For all elements a, b, c of BVF(Y) such that a ⇒ b = true(Y) holds b ⇒ c ⇒ a ⇒ c = true(Y).

(16) For all elements a, b of BVF(Y) holds b ⇒ a ⇒ b = true(Y).

(17) For all elements a, b, c of BVF(Y) holds a ⇒ b ⇒ c ⇒ b ⇒ c = true(Y).

(18) For all elements a, b of BVF(Y) holds b ⇒ b ⇒ a ⇒ a = true(Y).

(19) For all elements a, b, c of BVF(Y) holds c ⇒ b ⇒ a ⇒ b ⇒ c ⇒ a = true(Y).

(20) For all elements a, b, c of BVF(Y) holds b ⇒ c ⇒ a ⇒ b ⇒ a ⇒ c = true(Y).

(21) For all elements a, b, c of BVF(Y) holds b ⇒ b ⇒ c ⇒ b ⇒ c = true(Y).

(22) For all elements a, b, c of BVF(Y) holds a ⇒ b ⇒ c ⇒ a ⇒ b ⇒ a ⇒ c = true(Y).

(23) For all elements a, b of BVF(Y) such that a = true(Y) holds a ⇒ b ⇒ b = true(Y).

(24) For all elements a, b, c of BVF(Y) such that c ⇒ b ⇒ a = true(Y) holds b ⇒ c ⇒ a = true(Y).

(25) For all elements a, b, c of BVF(Y) such that c ⇒ b ⇒ a = true(Y) and b = true(Y) holds c ⇒ a = true(Y).

(26) For all elements a, b, c of BVF(Y) such that c ⇒ b ⇒ a = true(Y) and b = true(Y) and c = true(Y) holds a = true(Y).

(27) For all elements b, c of BVF(Y) such that b ⇒ b ⇒ c = true(Y) holds b ⇒ c = true(Y).

(28) For all elements a, b, c of BVF(Y) such that a ⇒ b ⇒ c = true(Y) holds a ⇒ b ⇒ a ⇒ c = true(Y).

(29) For all elements a, b, c of BVF(Y) such that a ⇒ b ⇒ c = true(Y) and a ⇒ b = true(Y) holds a ⇒ c = true(Y).

(30) For all elements a, b, c of BVF(Y) such that a ⇒ b ⇒ c = true(Y) and a ⇒ b = true(Y) and a = true(Y) holds c = true(Y).

(31) For all elements a, b, c, d of BVF(Y) such that a ⇒ b ⇒ c = true(Y) and a ⇒ c ⇒ d = true(Y) holds a ⇒ b ⇒ d = true(Y).

propositional calculus for boolean valued … 113

(32) For all elements a, b of BVF(Y) holds ¬a ⇒ ¬b ⇒ b ⇒ a = true(Y).

(33) For all elements a, b of BVF(Y) holds a ⇒ b ⇒ ¬b ⇒ ¬a = true(Y).

(34) For all elements a, b of BVF(Y) holds a ⇒ ¬b ⇒ b ⇒ ¬a = true(Y).

(35) For all elements a, b of BVF(Y) holds ¬a ⇒ b ⇒ ¬b ⇒ a = true(Y).

(36) For every element a of BVF(Y) holds a ⇒ ¬a ⇒ ¬a = true(Y).

(37) For all elements a, b of BVF(Y) holds ¬a ⇒ a ⇒ b = true(Y).

(38) For all elements a, b, c of BVF(Y) holds ¬(a ∧ b ∧ c) = ¬a ∨ ¬b ∨ ¬c.

(39) For all elements a, b, c of BVF(Y) holds ¬(a ∨ b ∨ c) = ¬a ∧ ¬b ∧ ¬c.

(40) For all elements a, b, c, d of BVF(Y) holds a ∨ b ∧ c ∧ d = (a ∨ b) ∧ (a ∨ c) ∧ (a ∨ d).

(41) For all elements a, b, c, d of BVF(Y) holds a∧(b∨c∨d) = a∧b∨a∧c∨a∧d.

References

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

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

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

[4] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

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

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

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

114 shunichi kobayashi and yatsuka nakamura

Propositional Calculus for Boolean Valued Functions. Part II

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC 6.

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

In this paper Y denotes a non empty set. The following propositions are true:

(1) For all elements a, b of BVF(Y) holds a ⇒ b ⇒ a ∧ b = true(Y).

(2) For all elements a, b of BVF(Y) holds a ⇒ b ⇒ b ⇒ a ⇒ a ⇔ b = true(Y).

(3) For all elements a, b of BVF(Y) holds a ∨ b ⇔ b ∨ a = true(Y).

(4) For all elements a, b, c of BVF(Y) holds a ∧ b ⇒ c ⇒ a ⇒ b ⇒ c = true(Y).

(5) For all elements a, b, c of BVF(Y) holds a ⇒ b ⇒ c ⇒ a ∧ b ⇒ c = true(Y).

(6) For all elements a, b, c of BVF(Y) holds c ⇒ a ⇒ c ⇒ b ⇒ c ⇒ a ∧ b = true(Y).

(7) For all elements a, b, c of BVF(Y) holds a ∨ b ⇒ c ⇒ (a ⇒ c) ∨ (b ⇒ c) = true(Y).

(8) For all elements a, b, c of BVF(Y) holds a ⇒ c ⇒ b ⇒ c ⇒ a ∨ b ⇒ c = true(Y).

(9) For all elements a, b, c of BVF(Y) holds (a ⇒ c) ∧ (b ⇒ c) ⇒ a ∨ b ⇒ c = true(Y).

(10) For all elements a, b of BVF(Y) holds a ⇒ b ∧ ¬b ⇒ ¬a = true(Y).

(11) For all elements a, b, c of BVF(Y) holds (a ∨ b) ∧ (a ∨ c) ⇒ a ∨ b ∧ c = true(Y).

(12) For all elements a, b, c of BVF(Y) holds a∧(b∨c) ⇒ a∧b∨a∧c = true(Y).

(13) For all elements a, b, c of BVF(Y) holds (a ∨ c) ∧ (b ∨ c) ⇒ a ∧ b ∨ c = true(Y).

(14) For all elements a, b, c of BVF(Y) holds (a∨b)∧c ⇒ a∧c∨b∧c = true(Y).

(15) For all elements a, b of BVF(Y) such that a ∧ b = true(Y) holds a ∨ b = true(Y).

(16) For all elements a, b, c of BVF(Y) such that a ⇒ b = true(Y) holds a ∨ c ⇒ b ∨ c = true(Y).

(17) For all elements a, b, c of BVF(Y) such that a ⇒ b = true(Y) holds a ∧ c ⇒ b ∧ c = true(Y).

(18) For all elements a, b, c of BVF(Y) such that c ⇒ a = true(Y) and c ⇒ b = true(Y) holds c ⇒ a ∧ b = true(Y).

(19) For all elements a, b, c of BVF(Y) such that a ⇒ c = true(Y) and b ⇒ c = true(Y) holds a ∨ b ⇒ c = true(Y).

(20) For all elements a, b of BVF(Y) such that a ∨ b = true(Y) and ¬a = true(Y) holds b = true(Y).

(21) For all elements a, b, c, d of BVF(Y) such that a ⇒ b = true(Y) and c ⇒ d = true(Y) holds a ∧ c ⇒ b ∧ d = true(Y).

(22) For all elements a, b, c, d of BVF(Y) such that a ⇒ b = true(Y) and c ⇒ d = true(Y) holds a ∨ c ⇒ b ∨ d = true(Y).

(23) For all elements a, b of BVF(Y) such that a ∧ ¬b ⇒ ¬a = true(Y) holds a ⇒ b = true(Y).

(24) For all elements a, b of BVF(Y) such that ¬a ⇒ ¬b = true(Y) holds b ⇒ a = true(Y).

(25) For all elements a, b of BVF(Y) such that a ⇒ ¬b = true(Y) holds b ⇒ ¬a = true(Y).

(26) For all elements a, b of BVF(Y) such that ¬a ⇒ b = true(Y) holds ¬b ⇒ a = true(Y).

(27) For all elements a, b of BVF(Y) holds a ⇒ a ∨ b = true(Y).

(28) For all elements a, b of BVF(Y) holds a ∨ b ⇒ ¬a ⇒ b = true(Y).

(29) For all elements a, b of BVF(Y) holds ¬(a ∨ b) ⇒ ¬a ∧ ¬b = true(Y).

(30) For all elements a, b of BVF(Y) holds ¬a ∧ ¬b ⇒ ¬(a ∨ b) = true(Y).

(31) For all elements a, b of BVF(Y) holds ¬(a ∨ b) ⇒ ¬a = true(Y).

(32) For every element a of BVF(Y) holds a ∨ a ⇒ a = true(Y).

(33) For all elements a, b of BVF(Y) holds a ∧ ¬a ⇒ b = true(Y).

(34) For all elements a, b of BVF(Y) holds a ⇒ b ⇒ ¬a ∨ b = true(Y).

(35) For all elements a, b of BVF(Y) holds a ∧ b ⇒ ¬(a ⇒ ¬b) = true(Y).

(36) For all elements a, b of BVF(Y) holds ¬(a ⇒ ¬b) ⇒ a ∧ b = true(Y).

(37) For all elements a, b of BVF(Y) holds ¬(a ∧ b) ⇒ ¬a ∨ ¬b = true(Y).

(38) For all elements a, b of BVF(Y) holds ¬a ∨ ¬b ⇒ ¬(a ∧ b) = true(Y).

(39) For all elements a, b of BVF(Y) holds a ∧ b ⇒ a = true(Y).

(40) For all elements a, b of BVF(Y) holds a ∧ b ⇒ a ∨ b = true(Y).

(41) For all elements a, b of BVF(Y) holds a ∧ b ⇒ b = true(Y).

(42) For every element a of BVF(Y) holds a ⇒ a ∧ a = true(Y).

(43) For all elements a, b of BVF(Y) holds a ⇔ b ⇒ a ⇒ b = true(Y).

(44) For all elements a, b of BVF(Y) holds a ⇔ b ⇒ b ⇒ a = true(Y).

(45) For all elements a, b, c of BVF(Y) holds a∨b∨c ⇒ a∨(b∨c) = true(Y).

(46) For all elements a, b, c of BVF(Y) holds a∧b∧c ⇒ a∧(b∧c) = true(Y).

(47) For all elements a, b, c of BVF(Y) holds a∨(b∨c) ⇒ a∨b∨c = true(Y).

References

[1] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

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

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

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

118 shunichi kobayashi and yatsuka nakamura

Insert Sort on SCM _{FSA}^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. This article describes the insert sorting algorithm using macro instructions such as if-Macro (conditional branch macro instructions), for-loop macro instructions and While-Macro instructions etc. From the viewpoint of initialization, we generalize the halting and computing problem of the While-Macro. Generally speaking, it is difficult to judge whether the While-Macro is halting or not by way of loop inspection. For this reason, we introduce a practical and simple method, called body-inspection. That is, in many cases, we can prove the halting problem of the While-Macro by only verifying the nature of the body of the While-Macro, rather than the While-Macro itself. In fact, we have used this method in justifying the halting of the insert sorting algorithm. Finally, we prove that the insert sorting algorithm given in the article is autonomic and its computing result is correct.

MML Identifier: SCMISORT.

The articles [28], [39], [20], [8], [13], [40], [14], [38], [15], [16], [12], [7], [10], [9], [23], [30], [11], [26], [34], [31], [32], [33], [25], [5], [6], [3], [1], [17], [2], [35], [37], [18], [27], [29], [24], [4], [22], [19], [21], and [36] provide the terminology and notation for this paper.

1. Preliminaries

Let i be a good instruction of SCM _{FSA}. Observe that Macro(i) is good. Let a be a read-write integer location and let b be an integer location. Note

that AddTo(a, b) is good.

We now state several propositions:

1This research is supported by the National Natural Science Foundation of China Grant No. 69873033.

(1) For every function f and for all sets d, r such that d ∈ domf holds domf = dom(f+·(d⟼→ . r)).

(2) Let p be a programmed finite partial state of SCM _{FSA}, l be an instruction-location of SCM _{FSA}, and i_{1} be an instruction of SCM _{FSA}. Suppose l ∈ domp and there exists an instruction p_{1} of SCM _{FSA} such that p_{1} = p(l) and UsedIntLoc(p_{1}) = UsedIntLoc(i_{1}). Then UsedIntLoc(p) = UsedIntLoc(p+·(l⟼→ . i_{1})).

(3) For every integer location a and for every macro instruction I holds (if a > 0 then I; Goto(insloc(0)) else (Stop_{SCM_{FSA}} ))(insloc(cardI + 4)) = goto insloc(cardI + 4).

(4) Let p be a programmed finite partial state of SCM _{FSA}, l be an instruction-location of SCM _{FSA}, and i_{1} be an instruction of SCM _{FSA}. Suppose l ∈ domp and there exists an instruction p_{1} of SCM _{FSA} such that p_{1} = p(l) and UsedInt^{∗} Loc(p_{1}) = UsedInt^{∗} Loc(i_{1}). Then UsedInt^{∗} Loc(p) = UsedInt^{∗} Loc(p+·(l⟼→ . i_{1})).

(5) For every natural number k holds k + 1 > 0. For simplicity, we adopt the following convention: s is a state of SCM _{FSA}, I

is a macro instruction, a is a read-write integer location, and j, k, n are natural numbers.

Next we state a number of propositions:

(6) For every state s of SCM _{FSA} and for every macro instruction I such that s(intloc(0)) = 1 and IC_{s} = insloc(0) holds s+·I = s+·Initialized(I).

(7) Let I be a macro instruction and a, b be integer locations. If I does not destroy b, then while a > 0 do I does not destroy b.

(8) If n 11, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5 or n = 6 or n = 7 or n = 8 or n = 9 or n = 10 or n = 11.

(9) Let f, g be finite sequences of elements of ℤ and m, n be natural numbers. Suppose 1 n and n lenf and 1 m and m lenf and g = f +· (m, π_{n}f) +·(n, π_{m}f). Then

(i) f(m) = g(n), (ii) f(n) = g(m), (iii) for every set k such that k ≠ m and k ≠ n and k ∈ domf holds

f(k) = g(k), and (iv) f and g are fiberwise equipotent.

(10) Let s be a state of SCM _{FSA} and I be a macro instruction. Suppose I is halting on Initialize(s). Let a be an integer location. Then (IExec(I, s))(a) = (Computation(Initialize(s)+·(I+·Start-At(insloc(0))))) (LifeSpan(Initialize(s)+·(I+·Start-At(insloc(0)))))(a).

(11) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a InitHalting macro instruction. Suppose Initialized(I) ⊆ s_{1} and Initialized(I) ⊆ s_{2} and s_{1} and s_{2} are equal outside the instruction locations of SCM _{FSA}. Let k be a natural number. Then (Computation(s_{1}))(k) and (Computation(s_{2}))(k) are equal outside the instruction locations of SCM _{FSA} and CurInstr((Computation(s_{1}))(k)) = CurInstr((Computation (s_{2}))(k)).

(12) Let s_{1}, s_{2} be states of SCM _{FSA} and I be a InitHalting macro instruction. Suppose Initialized(I) ⊆ s_{1} and Initialized(I) ⊆ s_{2} and s_{1} and s_{2} are equal outside the instruction locations of SCM _{FSA}. Then LifeSpan(s_{1}) = LifeSpan(s_{2}) and Result(s_{1}) and Result(s_{2}) are equal outside the instruction locations of SCM _{FSA}.

(13) For every macro instruction I and for every finite sequence location f holds f ∉ domI.

(14) For every macro instruction I and for every integer location a holds a ∉ domI.

(15) Let N be a non empty set with non empty elements, S be a halting von Neumann definite AMI over N, and s be a state of S. If LifeSpan(s) j and s is halting, then (Computation(s))(j) = (Computation(s))(LifeSpan(s)).

2. Basic Property of while Macro

We now state several propositions:

(16) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. Suppose s(a) 0. Then while a > 0 do I is halting onInit s and while a > 0 do I is closed onInit s.

(17) Let a be an integer location, I be a macro instruction, s be a state of

SCM _{FSA}, and k be a natural number. Suppose that

(i) I is closed onInit s, (ii) I is halting onInit s, (iii) k < LifeSpan(s+·Initialized(I)), (iv) IC_{(Computation (s+·Initialized (while a>0 do I)))(1+k)} =

IC_{(Computation (s+·Initialized (I)))(k)} + 4, and

(v) (Computation(s+·Initialized(while a > 0 do I)))(1 + k)↾D = (Computation(s+·Initialized(I)))(k)↾D. Then IC_{(Computation (s+·Initialized (while a>0 do I)))(1+k+1)} = IC_{(Computation (s+·Initialized (I)))(k+1)} + 4 and (Computation(s+·Initialized (while a > 0 do I)))(1+k+1)↾D = (Computation(s+·Initialized(I)))(k+ 1)↾D, where D = Int-Locations∪ FinSeq-Locations.

(18) Let a be an integer location, I be a macro instruction, and s be a state of SCM _{FSA}. Suppose I is closed onInit s and I is halting onInit s and IC_{(Computation (s+·Initialized (while a>0 do I)))(1+LifeSpan (s+·Initialized (I)))} = IC_{(Computation (s+·Initialized (I)))(LifeSpan (s+·Initialized (I)))} + 4. Then CurInstr((Computation(s+·Initialized(while a > 0 do I)))(1 + LifeSpan(s+·Initialized(I)))) = goto insloc(cardI + 4).

(19) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. Suppose I is closed onInit s and I is halting onInit s and s(a) > 0. Then IC_{(Computation (s+·Initialized (while a>0 do I)))(LifeSpan (s+·Initialized (I))+3)} = insloc(0) and for every natural number k such that k LifeSpan(s+· Initialized(I)) + 3 holds IC_{(Computation (s+·Initialized (while a>0 do I)))(k)} ∈ dom(while a > 0 do I).

(20) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. Suppose I is closed onInit s and I is halting onInit s and s(a) > 0. Let k be a natural number. If k LifeSpan(s+·Initialized(I)) + 3, then IC_{(Computation (s+·Initialized (while a>0 do I)))(k)} ∈ dom(while a > 0 do I).

(21) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. Suppose I is closed onInit s and I is halting onInit s and s(a) > 0. Then IC_{(Computation (s+·Initialized (while a>0 do I)))(LifeSpan (s+·Initialized (I))+3)} = insloc(0) and (Computation(s+·Initialized(while a > 0 do I)))(LifeSpan (s+·Initialized(I)) + 3)↾D = (Computation(s+·Initialized(I)))(LifeSpan (s+·Initialized(I)))↾D, where D = Int-Locations∪ FinSeq-Locations.

(22) Let s be a state of SCM _{FSA}, I be a InitHalting macro instruction, and a be a read-write integer location. Suppose s(a) > 0. Then there exists a state s_{2} of SCM _{FSA} and there exists a natural number k such that

(i) s_{2} = s+·Initialized(while a > 0 do I), (ii) k = LifeSpan(s+·Initialized(I)) + 3, (iii) IC_{(Computation (s_{2}))(k)} = insloc(0), (iv) for every integer location b holds (Computation(s_{2}))(k)(b) =

(IExec(I, s))(b), and

(v) for every finite sequence location f holds (Computation(s_{2}))(k)(f) = (IExec(I, s))(f).

Let us consider s, I, a. The functor StepWhile>0(a, s, I) yields a function from ℕ into ∏ (the object kind of SCM _{FSA}) and is defined by the conditions (Def. 1).

(Def. 1)(i) (StepWhile>0(a, s, I))(0) = s qua element of ∏ (the object kind of SCM _{FSA}) qua non empty set, and (ii) for every natural number i and for every element x of

∏ (the object kind of SCM _{FSA}) qua non empty set such that x = (StepWhile>0(a, s, I))(i) holds (StepWhile>0(a, s, I))(i + 1) = (Computation(x+·Initialized(while a > 0 do I)))(LifeSpan(x+·Initialized (I)) + 3).

We now state several propositions:

(23) (StepWhile>0(a, s, I))(0) = s.

(24) (StepWhile>0(a, s, I))(k+1) = (Computation((StepWhile>0(a, s, I))(k) +·Initialized(while a > 0 do I)))(LifeSpan((StepWhile>0(a, s, I))(k)+· Initialized(I)) + 3).

(25) (StepWhile>0(a, s, I))(k+1) = (StepWhile>0(a, (StepWhile>0(a, s, I)) (k), I))(1).

(26) Let I be a macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Then (StepWhile>0(a, s, I))(0 + 1) = (Computation(s+·Initialized(while a > 0 do I)))(LifeSpan(s+·Initialized (I)) + 3).

(27) Let I be a macro instruction, a be a read-write integer location, s be a state of SCM _{FSA}, and k, n be natural numbers. Suppose IC_{(StepWhile>0 (a,s,I))(k)} = insloc(0) and (StepWhile>0(a, s, I))(k) = (Computation(s+·Initialized(while a > 0 do I)))(n) and (StepWhile>0 (a, s, I))(k)(intloc(0)) = 1. Then (StepWhile>0(a, s, I))(k) = (StepWhile>0(a, s, I))(k)+·Initialized (while a > 0 do I) and (StepWhile>0(a, s, I))(k+1) = (Computation(s+· Initialized(while a > 0 do I)))(n+(LifeSpan((StepWhile>0(a, s, I))(k)+· Initialized(I)) + 3)).

(28) Let I be a macro instruction, a be a read-write integer location, and s be a state of SCM _{FSA}. Given a function f from ∏ (the object kind of SCM _{FSA}) into ℕ such that let k be a natural number. Then

(i) if f((StepWhile>0(a, s, I))(k)) ≠ 0, then f((StepWhile>0(a, s, I))(k + 1)) < f((StepWhile>0(a, s, I))(k)) and I is closed onInit (StepWhile>0 (a, s, I))(k) and I is halting onInit (StepWhile>0(a, s, I))(k),

(ii) (StepWhile>0(a, s, I))(k + 1)(intloc(0)) = 1, and (iii) f((StepWhile>0(a, s, I))(k)) = 0 iff (StepWhile>0(a, s, I))(k)(a) 0.

Then while a > 0 do I is halting onInit s and while a > 0 do I is closed onInit s.

(29) Let I be a good InitHalting macro instruction and a be a read-write integer location. Suppose that for every state s of SCM _{FSA} such that s(a) > 0 holds (IExec(I, s))(a) < s(a). Then while a > 0 do I is InitHalting.

(30) Let I be a good InitHalting macro instruction and a be a readwrite integer location. Suppose that for every state s of SCM _{FSA} holds (IExec(I, s))(a) < s(a) or (IExec(I, s))(a) 0. Then while a > 0 do I is InitHalting.

Let D be a set, let f be a function from D into ℤ, and let d be an element of D. Then f(d) is an integer.

One can prove the following propositions:

(31) Let I be a good InitHalting macro instruction and a be a read-write integer location. Given a function f from ∏ (the object kind of SCM _{FSA}) into ℤ such that let s, t be states of SCM _{FSA}. Then

(i) if f(s) > 0, then f(IExec(I, s)) < f(s), (ii) if s↾D = t↾D, then f(s) = f(t), and (iii) f(s) 0 iff s(a) 0.

Then while a > 0 do I is InitHalting, where D = Int-Locations∪ FinSeq-Locations.

(32) Let s be a state of SCM _{FSA}, I be a macro instruction, and a be a read-write integer location. If s(a) 0, then IExec(while a > 0 do I, s)↾(Int-Locations∪ FinSeq-Locations) = Initialize(s)↾(Int-Locations∪ FinSeq-Locations).

(33) Let s be a state of SCM _{FSA}, I be a good InitHalting macro instruction, and a be a read-write integer location. If s(a) > 0 and while a > 0 do I is InitHalting, then IExec(while a > 0 do I, s)↾(Int-Locations∪ FinSeq-Locations) = IExec(while a > 0 do I, IExec(I, s))↾(Int-Locations∪ FinSeq-Locations).

(34) Let s be a state of SCM _{FSA}, I be a macro instruction, f be a finite sequence location, and a be a read-write integer location. If s(a) 0, then (IExec(while a > 0 do I, s))(f) = s(f).

(35) Let s be a state of SCM _{FSA}, I be a macro instruction, b be an integer location, and a be a read-write integer location. If s(a) 0, then (IExec(while a > 0 do I, s))(b) = (Initialize(s))(b).

(36) Let s be a state of SCM _{FSA}, I be a good InitHalting macro instruction, f be a finite sequence location, and a be a read-write integer location. If s(a) > 0 and while a > 0 do I is InitHalting, then (IExec(while a > 0 do I, s))(f) = (IExec(while a > 0 do I, IExec(I, s)))(f).

(37) Let s be a state of SCM _{FSA}, I be a good InitHalting macro instruction, b be an integer location, and a be a read-write integer location. If s(a) > 0 and while a > 0 do I is InitHalting, then (IExec(while a > 0 do I, s))(b) = (IExec(while a > 0 do I, IExec(I, s)))(b).

3. Insert Sort Algorithm

Let f be a finite sequence location. The functor insert − sort f yields a macro instruction and is defined as follows:

(Def. 2) insert − sort f = i_{2};(a_{1}:=lenf);SubFrom(a_{1}, a_{0}); Times(a_{1}, (a_{2}:=lenf); SubFrom(a_{2}, a_{1});(a_{3}:=a_{2});AddTo(a_{3}, a_{0});(a_{6}:=f_{a_{3}});SubFrom(a_{4}, a_{4}); (while a_{2} > 0 do ((a_{5}:=f_{a_{2}});SubFrom(a_{5}, a_{6});(if a_{5} > 0 then Macro (SubFrom(a_{2}, a_{2})) else (AddTo(a_{4}, a_{0});SubFrom(a_{2}, a_{0}))))); Times(a_{4}, (a_{2}:=a_{3});SubFrom(a_{3}, a_{0});(a_{5}:=f_{a_{2}});(a_{6}:=f_{a_{3}});(f_{a_{2}}:=a_{6});(f_{a_{3}}:=a_{5}))), where i_{2} = (a_{2}:=a_{0});(a_{3}:=a_{0});(a_{4}:=a_{0});(a_{5}:=a_{0});(a_{6}:=a_{0}), a_{2} = intloc(2), a_{0} = intloc(0), a_{3} = intloc(3), a_{4} = intloc(4), a_{5} = intloc(5), a_{6} = intloc(6), and a_{1} = intloc(1).

The macro instruction Insert − Sort − Algorithm is defined by: (Def. 3) Insert − Sort − Algorithm = insert − sort fsloc(0).

We now state a number of propositions:

(38) For every finite sequence location f holds UsedIntLoc(insert − sort f) = {a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}}, where a_{0} = intloc(0), a_{1} = intloc(1), a_{2} = intloc(2), a_{3} = intloc(3), a_{4} = intloc(4), a_{5} = intloc(5), and a_{6} = intloc(6).

(39) For every finite sequence location f holds UsedInt^{∗} Loc(insert − sort f) = {f}.

(40) For all instructions k_{1}, k_{2}, k_{3}, k_{4} of SCM _{FSA} holds card(k_{1};k_{2};k_{3};k_{4}) = 8.

(41) For all instructions k_{1}, k_{2}, k_{3}, k_{4}, k_{5} of SCM _{FSA} holds card(k_{1};k_{2};k_{3};k_{4};k_{5}) = 10.

(42) For every finite sequence location f holds cardinsert − sort f = 82.

(43) For every finite sequence location f and for every natural number k such that k < 82 holds insloc(k) ∈ dominsert − sort f.

(44) insert − sort fsloc(0) is keepInt0 1 and InitHalting.

(45) Let s be a state of SCM _{FSA}. Then (i) s(f_{0}) and (IExec(insert − sort f_{0}, s))(f_{0}) are fiberwise equipotent, and (ii) for all natural numbers i, j such that i 1 and j lens(f_{0}) and i < j

and for all integers x_{1}, x_{2} such that x_{1} = (IExec(insert − sort f_{0}, s))(f_{0})(i) and x_{2} = (IExec(insert − sort f_{0}, s))(f_{0})(j) holds x_{1} x_{2}, where f_{0} = fsloc(0).

(46) Let i be a natural number, s be a state of SCM _{FSA}, and w be a finite sequence of elements of ℤ. If Initialized(Insert − Sort − Algorithm )+·(fsloc(0) ⟼→ . w) ⊆ s, then IC_{(Computation (s))(i)} ∈ domInsert − Sort − Algorithm .

126 jing-chao chen

(47) Let s be a state of SCM _{FSA} and t be a finite sequence of elements of ℤ. Suppose Initialized(Insert − Sort − Algorithm )+·(fsloc(0)⟼→ . t) ⊆ s. Then there exists a finite sequence u of elements of ℝ such that (i) t and u are fiberwise equipotent, (ii) u is non-increasing and a finite sequence of elements of ℤ, and (iii) (Result(s))(fsloc(0)) = u.

(48) For every finite sequence w of elements of ℤ holds Initialized(Insert − Sort − Algorithm )+·(fsloc(0)⟼→ . w) is autonomic.

(49) Initialized(Insert − Sort − Algorithm ) computes Sorting-Function.

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Correctness of a Cyclic Redundancy Check Code Generator

Yuguang Yang Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Yatsuka Nakamura Shinshu University Nagano

Summary. We prove the correctness of the division circuit and the CRC (cyclic redundancy checks) circuit by verifying the contents of the register after one shift. Circuits with 12-bit register and 16-bit register are taken as examples. All the proofs are done formally.

MML Identifier: GATE 4.

The terminology and notation used here are introduced in the article [1].

1. Correctness of Division Circuits with 12-bit Register and 16-bit Register

One can prove the following propositions:

(1) Let g_{0}, g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}, g_{7}, g_{8}, g_{9}, g_{10}, g_{11}, g_{12}, a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}, a_{11}, b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}, b_{8}, b_{9}, b_{10}, b_{11}, p be sets such that NE g_{0} and NE g_{12} and NE b_{0} iff NE XOR2(p, AND2(g_{0}, a_{11})) and NE b_{1} iff NE XOR2(a_{0}, AND2(g_{1}, a_{11})) and NE b_{2} iff NE XOR2(a_{1}, AND2(g_{2}, a_{11})) and NE b_{3} iff NE XOR2(a_{2}, AND2(g_{3}, a_{11})) and NE b_{4} iff NE XOR2(a_{3}, AND2(g_{4}, a_{11})) and NE b_{5} iff NE XOR2(a_{4}, AND2(g_{5}, a_{11})) and NE b_{6} iff NE XOR2(a_{5}, AND2(g_{6}, a_{11})) and NE b_{7} iff NE XOR2(a_{6}, AND2(g_{7}, a_{11})) and NE b_{8} iff NE XOR2(a_{7}, AND2(g_{8}, a_{11})) and NE b_{9} iff NE XOR2(a_{8}, AND2(g_{9}, a_{11})) and NE b_{10} iff NE XOR2(a_{9}, AND2(g_{10}, a_{11})) and NE b_{11} iff NE XOR2(a_{10}, AND2(g_{11}, a_{11})). Then

(i) NE a_{11} iff NE AND2(g_{12}, a_{11}), (ii) NE a_{10} iff NE XOR2(b_{11}, AND2(g_{11}, a_{11})), (iii) NE a_{9} iff NE XOR2(b_{10}, AND2(g_{10}, a_{11})), (iv) NE a_{8} iff NE XOR2(b_{9}, AND2(g_{9}, a_{11})),

(v) NE a_{7} iff NE XOR2(b_{8}, AND2(g_{8}, a_{11})), (vi) NE a_{6} iff NE XOR2(b_{7}, AND2(g_{7}, a_{11})), (vii) NE a_{5} iff NE XOR2(b_{6}, AND2(g_{6}, a_{11})), (viii) NE a_{4} iff NE XOR2(b_{5}, AND2(g_{5}, a_{11})), (ix) NE a_{3} iff NE XOR2(b_{4}, AND2(g_{4}, a_{11})),

(x) NE a_{2} iff NE XOR2(b_{3}, AND2(g_{3}, a_{11})), (xi) NE a_{1} iff NE XOR2(b_{2}, AND2(g_{2}, a_{11})), (xii) NE a_{0} iff NE XOR2(b_{1}, AND2(g_{1}, a_{11})), and (xiii) NE p iff NE XOR2(b_{0}, AND2(g_{0}, a_{11})).

(2) Let g_{0}, g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}, g_{7}, g_{8}, g_{9}, g_{10}, g_{11}, g_{12}, g_{13}, g_{14}, g_{15}, g_{16}, a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{14}, a_{15}, b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}, b_{8}, b_{9}, b_{10}, b_{11}, b_{12}, b_{13}, b_{14}, b_{15}, p be sets such that NE g_{0} and NE g_{16} and NE b_{0} iff NE XOR2(p, AND2(g_{0}, a_{15})) and NE b_{1} iff NE XOR2(a_{0}, AND2(g_{1}, a_{15})) and NE b_{2} iff NE XOR2(a_{1}, AND2(g_{2}, a_{15})) and NE b_{3} iff NE XOR2(a_{2}, AND2(g_{3}, a_{15})) and NE b_{4} iff NE XOR2(a_{3}, AND2(g_{4}, a_{15})) and NE b_{5} iff NE XOR2(a_{4}, AND2(g_{5}, a_{15})) and NE b_{6} iff NE XOR2(a_{5}, AND2(g_{6}, a_{15})) and NE b_{7} iff NE XOR2(a_{6}, AND2(g_{7}, a_{15})) and NE b_{8} iff NE XOR2(a_{7}, AND2(g_{8}, a_{15})) and NE b_{9} iff NE XOR2(a_{8}, AND2(g_{9}, a_{15})) and NE b_{10} iff NE XOR2(a_{9}, AND2(g_{10}, a_{15})) and NE b_{11} iff NE XOR2(a_{10}, AND2(g_{11}, a_{15})) and NE b_{12} iff NE XOR2(a_{11}, AND2(g_{12}, a_{15})) and NE b_{13} iff NE XOR2(a_{12}, AND2(g_{13}, a_{15})) and NE b_{14} iff NE XOR2(a_{13}, AND2(g_{14}, a_{15})) and NE b_{15} iff NE XOR2(a_{14}, AND2(g_{15}, a_{15})). Then

(i) NE a_{15} iff NE AND2(g_{16}, a_{15}), (ii) NE a_{14} iff NE XOR2(b_{15}, AND2(g_{15}, a_{15})), (iii) NE a_{13} iff NE XOR2(b_{14}, AND2(g_{14}, a_{15})), (iv) NE a_{12} iff NE XOR2(b_{13}, AND2(g_{13}, a_{15})),

(v) NE a_{11} iff NE XOR2(b_{12}, AND2(g_{12}, a_{15})), (vi) NE a_{10} iff NE XOR2(b_{11}, AND2(g_{11}, a_{15})), (vii) NE a_{9} iff NE XOR2(b_{10}, AND2(g_{10}, a_{15})), (viii) NE a_{8} iff NE XOR2(b_{9}, AND2(g_{9}, a_{15})), (ix) NE a_{7} iff NE XOR2(b_{8}, AND2(g_{8}, a_{15})),

(x) NE a_{6} iff NE XOR2(b_{7}, AND2(g_{7}, a_{15})), (xi) NE a_{5} iff NE XOR2(b_{6}, AND2(g_{6}, a_{15})), (xii) NE a_{4} iff NE XOR2(b_{5}, AND2(g_{5}, a_{15})), (xiii) NE a_{3} iff NE XOR2(b_{4}, AND2(g_{4}, a_{15})), (xiv) NE a_{2} iff NE XOR2(b_{3}, AND2(g_{3}, a_{15})), (xv) NE a_{1} iff NE XOR2(b_{2}, AND2(g_{2}, a_{15})), (xvi) NE a_{0} iff NE XOR2(b_{1}, AND2(g_{1}, a_{15})), and (xvii) NE p iff NE XOR2(b_{0}, AND2(g_{0}, a_{15})).

2. Correctness of CRC Circuits with Generator Polynomial of Degree 12 and 16

Next we state two propositions:

(3) Let g_{0}, g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}, g_{7}, g_{8}, g_{9}, g_{10}, g_{11}, g_{12}, a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}, a_{11}, b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}, b_{8}, b_{9}, b_{10}, b_{11}, z, p be sets such that NE g_{0} and NE g_{12} and not NE z and NE b_{0} iff NE XOR2(p, a_{11}) and NE b_{1} iff NE XOR2(a_{0}, AND2(g_{1}, b_{0})) and NE b_{2} iff NE XOR2(a_{1}, AND2(g_{2}, b_{0})) and NE b_{3} iff NE XOR2(a_{2}, AND2(g_{3}, b_{0})) and NE b_{4} iff NE XOR2(a_{3}, AND2(g_{4}, b_{0})) and NE b_{5} iff NE XOR2(a_{4}, AND2(g_{5}, b_{0})) and NE b_{6} iff NE XOR2(a_{5}, AND2(g_{6}, b_{0})) and NE b_{7} iff NE XOR2(a_{6}, AND2(g_{7}, b_{0})) and NE b_{8} iff NE XOR2(a_{7}, AND2(g_{8}, b_{0})) and NE b_{9} iff NE XOR2(a_{8}, AND2(g_{9}, b_{0})) and NE b_{10} iff NE XOR2(a_{9}, AND2(g_{10}, b_{0})) and NE b_{11} iff NE XOR2(a_{10}, AND2(g_{11}, b_{0})). Then

(i) NE b_{11} iff NE XOR2(XOR2(a_{10}, AND2(g_{11}, a_{11})), XOR2(z, AND2(g_{11}, p))), (ii) NE b_{10} iff NE XOR2(XOR2(a_{9}, AND2(g_{10}, a_{11})), XOR2(z, AND2(g_{10}, p))), (iii) NE b_{9} iff NE XOR2(XOR2(a_{8}, AND2(g_{9}, a_{11})), XOR2(z, AND2(g_{9}, p))), (iv) NE b_{8} iff NE XOR2(XOR2(a_{7}, AND2(g_{8}, a_{11})), XOR2(z, AND2(g_{8}, p))),

(v) NE b_{7} iff NE XOR2(XOR2(a_{6}, AND2(g_{7}, a_{11})), XOR2(z, AND2(g_{7}, p))), (vi) NE b_{6} iff NE XOR2(XOR2(a_{5}, AND2(g_{6}, a_{11})), XOR2(z, AND2(g_{6}, p))), (vii) NE b_{5} iff NE XOR2(XOR2(a_{4}, AND2(g_{5}, a_{11})), XOR2(z, AND2(g_{5}, p))), (viii) NE b_{4} iff NE XOR2(XOR2(a_{3}, AND2(g_{4}, a_{11})), XOR2(z, AND2(g_{4}, p))), (ix) NE b_{3} iff NE XOR2(XOR2(a_{2}, AND2(g_{3}, a_{11})), XOR2(z, AND2(g_{3}, p))),

(x) NE b_{2} iff NE XOR2(XOR2(a_{1}, AND2(g_{2}, a_{11})), XOR2(z, AND2(g_{2}, p))), (xi) NE b_{1} iff NE XOR2(XOR2(a_{0}, AND2(g_{1}, a_{11})), XOR2(z, AND2(g_{1}, p))),

and (xii) NE b_{0} iff NE XOR2(XOR2(z, AND2(g_{0}, a_{11})), XOR2(z, AND2(g_{0}, p))).

(4) Let g_{0}, g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}, g_{7}, g_{8}, g_{9}, g_{10}, g_{11}, g_{12}, g_{13}, g_{14}, g_{15}, g_{16}, a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10}, a_{11}, a_{12}, a_{13}, a_{14}, a_{15}, b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}, b_{8}, b_{9}, b_{10}, b_{11}, b_{12}, b_{13}, b_{14}, b_{15}, z, p be sets such that NE g_{0} and NE g_{16} and not NE z and NE b_{0} iff NE XOR2(p, a_{15}) and NE b_{1} iff NE XOR2(a_{0}, AND2(g_{1}, b_{0})) and NE b_{2} iff NE XOR2(a_{1}, AND2(g_{2}, b_{0})) and NE b_{3} iff NE XOR2(a_{2}, AND2(g_{3}, b_{0})) and NE b_{4} iff NE XOR2(a_{3}, AND2(g_{4}, b_{0})) and NE b_{5} iff NE XOR2(a_{4}, AND2(g_{5}, b_{0})) and NE b_{6} iff NE XOR2(a_{5}, AND2(g_{6}, b_{0})) and NE b_{7} iff NE XOR2(a_{6}, AND2(g_{7}, b_{0})) and NE b_{8} iff NE XOR2(a_{7}, AND2(g_{8}, b_{0})) and NE b_{9} iff NE XOR2(a_{8}, AND2(g_{9}, b_{0})) and NE b_{10} iff NE XOR2(a_{9}, AND2(g_{10}, b_{0})) and NE b_{11} iff NE XOR2(a_{10}, AND2(g_{11}, b_{0})) and NE b_{12} iff NE XOR2(a_{11}, AND2(g_{12}, b_{0})) and NE b_{13} iff NE XOR2(a_{12}, AND2(g_{13}, b_{0})) and NE b_{14} iff NE XOR2(a_{13}, AND2(g_{14}, b_{0})) and NE b_{15} iff NE XOR2(a_{14}, AND2(g_{15}, b_{0})). Then

(i) NE b_{15} iff NE XOR2(XOR2(a_{14}, AND2(g_{15}, a_{15})), XOR2(z, AND2(g_{15}, p))), (ii) NE b_{14} iff NE XOR2(XOR2(a_{13}, AND2(g_{14}, a_{15})), XOR2(z, AND2(g_{14}, p))), (iii) NE b_{13} iff NE XOR2(XOR2(a_{12}, AND2(g_{13}, a_{15})), XOR2(z, AND2(g_{13}, p))), (iv) NE b_{12} iff NE XOR2(XOR2(a_{11}, AND2(g_{12}, a_{15})), XOR2(z, AND2(g_{12}, p))),

(v) NE b_{11} iff NE XOR2(XOR2(a_{10}, AND2(g_{11}, a_{15})), XOR2(z, AND2(g_{11}, p))), (vi) NE b_{10} iff NE XOR2(XOR2(a_{9}, AND2(g_{10}, a_{15})), XOR2(z, AND2(g_{10}, p))), (vii) NE b_{9} iff NE XOR2(XOR2(a_{8}, AND2(g_{9}, a_{15})), XOR2(z, AND2(g_{9}, p))), (viii) NE b_{8} iff NE XOR2(XOR2(a_{7}, AND2(g_{8}, a_{15})), XOR2(z, AND2(g_{8}, p))), (ix) NE b_{7} iff NE XOR2(XOR2(a_{6}, AND2(g_{7}, a_{15})), XOR2(z, AND2(g_{7}, p))),

(x) NE b_{6} iff NE XOR2(XOR2(a_{5}, AND2(g_{6}, a_{15})), XOR2(z, AND2(g_{6}, p))), (xi) NE b_{5} iff NE XOR2(XOR2(a_{4}, AND2(g_{5}, a_{15})), XOR2(z, AND2(g_{5}, p))), (xii) NE b_{4} iff NE XOR2(XOR2(a_{3}, AND2(g_{4}, a_{15})), XOR2(z, AND2(g_{4}, p))), (xiii) NE b_{3} iff NE XOR2(XOR2(a_{2}, AND2(g_{3}, a_{15})), XOR2(z, AND2(g_{3}, p))), (xiv) NE b_{2} iff NE XOR2(XOR2(a_{1}, AND2(g_{2}, a_{15})), XOR2(z, AND2(g_{2}, p))), (xv) NE b_{1} iff NE XOR2(XOR2(a_{0}, AND2(g_{1}, a_{15})), XOR2(z, AND2(g_{1}, p))),

and (xvi) NE b_{0} iff NE XOR2(XOR2(z, AND2(g_{0}, a_{15})), XOR2(z, AND2(g_{0}, p))).

References

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

Defining by Structural Induction in the Positive Propositional Language

Andrzej Trybulec University of Białystok

Summary. The main goal of the paper consists in proving schemes for defining by structural induction in the language defined by Adam Grabowski [13]. The article consists of four parts. Besides the preliminaries where we prove some simple facts still missing in the library, they are:

- “About the language” in which the consequences of the fact that the algebra of formulae is free are formulated,

- “Defining by structural induction” in which two schemes are proved,

- “The tree of the subformulae” in which a scheme proved in the previous section is used to define the tree of subformulae; also some simple facts about the tree are proved.

MML Identifier: HILBERT2.

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

1. Preliminaries

In this paper X, x denote sets. We now state four propositions:

(1) Let Z be a set and M be a many sorted set indexed by Z. Suppose that for every set x such that x ∈ Z holds M(x) is a many sorted set indexed by x. Let f be a function. If f = UnionM, then domf = ⋃ Z.

(2) For all sets x, y and for all finite sequences f, g such that ⟨x⟩^{⌢}f = ⟨y⟩^{⌢}g holds f = g.

(3) If ⟨x⟩ is a finite sequence of elements of X, then x ∈ X.

(4) Let given X and f be a finite sequence of elements of X. Suppose f ≠ ε. Then there exists a finite sequence g of elements of X and there exists an element d of X such that f = g^{⌢} ⟨d⟩.

We adopt the following rules: m, n are natural numbers, p, q, r, s are elements of HP-WFF, and T_{1}, T_{2} are trees.

Next we state the proposition

(5) ⟨x⟩ ∈ \overbrace{T_{1}, T_{2}} iff x = 0 or x = 1. Let us mention that ε is tree yielding. The scheme InTreeInd deals with a tree A and and states that:

For every element f of A holds P[f] provided the following conditions are satisfied:

• P[ε_{ℕ}], and

• For every element f of A such that P[f] and for every n such that f ^{⌢} ⟨n⟩ ∈ A holds P[f ^{⌢} ⟨n⟩].

In the sequel D is a non empty set and T_{1}, T_{2} are decorated trees. Next we state three propositions:

(6) For every set x and for all T_{1}, T_{2} holds (x-tree(T_{1}, T_{2}))(ε) = x.

(7) (x-tree(T_{1}, T_{2}))(⟨0⟩) = T_{1}(ε) and (x-tree(T_{1}, T_{2}))(⟨1⟩) = T_{2}(ε).

(8) (x-tree(T_{1}, T_{2}))↾⟨0⟩ = T_{1} and (x-tree(T_{1}, T_{2}))↾⟨1⟩ = T_{2}. Let us consider x and let p be a decorated tree yielding non empty finite

sequence. Observe that x-tree(p) is non root.

Let us consider x and let T_{1} be a decorated tree. Observe that x-tree(T_{1}) is non root. Let T_{2} be a decorated tree. Observe that x-tree(T_{1}, T_{2}) is non root.

2. About the Language

Let us consider n. The functor propn yielding an element of HP-WFF is defined as follows:

(Def. 1) propn = ⟨3 + n⟩.

Let D be a set. Let us observe that D has VERUM if and only if: (Def. 2) VERUM ∈ D.

Let us observe that D has propositional variables if and only if: (Def. 3) For every n holds propn ∈ D.

Let D be a subset of HP-WFF. Let us observe that D has implication if and only if:

(Def. 4) For all p, q such that p ∈ D and q ∈ D holds p ⇒ q ∈ D. Let us observe that D has conjunction if and only if:

(Def. 5) For all p, q such that p ∈ D and q ∈ D holds p ∧ q ∈ D. In the sequel t denotes a finite sequence. Let us consider p. We say that p is conjunctive if and only if:

(Def. 6) There exist r, s such that p = r ∧ s.

We say that p is conditional if and only if: (Def. 7) There exist r, s such that p = r ⇒ s.

We say that p is simple if and only if: (Def. 8) There exists n such that p = propn.

The scheme HP Ind concerns and states that:

For every r holds P[r] provided the following requirements are met:

• P[VERUM],

• For every n holds P[propn], and

• For all r, s such that P[r] and P[s] holds P[r ∧ s] and P[r ⇒ s]. Next we state a number of propositions:

(9) p is conjunctive, or conditional, or simple or p = VERUM.

(10) lenp 1.

(11) If p(1) = 1, then p is conditional.

(12) If p(1) = 2, then p is conjunctive.

(13) If p(1) = 3 + n, then p is simple.

(14) If p(1) = 0, then p = VERUM.

(15) lenp < len(p ∧ q) and lenq < len(p ∧ q).

(16) lenp < len(p ⇒ q) and lenq < len(p ⇒ q).

(17) If p = q^{⌢} t, then p = q.

(18) If p^{⌢} q = r^{⌢} s, then p = r and q = s.

(19) If p ∧ q = r ∧ s, then p = r and s = q.

(20) If p ⇒ q = r ⇒ s, then p = r and s = q.

(21) If propn = propm, then n = m.

(22) p ∧ q ≠ r ⇒ s.

(23) p ∧ q ≠ VERUM.

(24) p ∧ q ≠ propn.

(25) p ⇒ q ≠ VERUM.

(26) p ⇒ q ≠ propn.

(27) p ∧ q ≠ p and p ∧ q ≠ q.

(28) p ⇒ q ≠ p and p ⇒ q ≠ q.

(29) VERUM ≠ propn.

3. Defining by Structural Induction

Now we present two schemes. The scheme HP MSSExL deals with a set A, a unary functor F yielding a set, and a 5-ary predicate Q, and states that:

There exists a many sorted set M indexed by HP-WFF such that

(i) M(VERUM) = A, (ii) for every n holds M(propn) = F(n), and (iii) for all p, q and for all sets a, b, c, d such that a = M(p) and b = M(q) and c = M(p∧q) and d = M(p ⇒ q) holds P[p, q, a, b, c] and Q[p, q, a, b, d]

provided the following conditions are met:

• For all p, q and for all sets a, b there exists a set c such that P[p, q, a, b, c],

• For all p, q and for all sets a, b there exists a set d such that Q[p, q, a, b, d],

• For all p, q and for all sets a, b, c, d such that P[p, q, a, b, c] and P[p, q, a, b, d] holds c = d, and

• For all p, q and for all sets a, b, c, d such that Q[p, q, a, b, c] and Q[p, q, a, b, d] holds c = d.

The scheme HP MSSLambda deals with a set A, a unary functor F yielding a set, and two binary functors G and H yielding sets, and states that:

There exists a many sorted set M indexed by HP-WFF such that

(i) M(VERUM) = A, (ii) for every n holds M(propn) = F(n), and (iii) for all p, q and for all sets x, y such that x = M(p) and y = M(q) holds M(p ∧ q) = G(x, y) and M(p ⇒ q) = H(x, y)

for all values of the parameters.

4. The Tree of the Subformulae

The many sorted set HP-Subformulae indexed by HP-WFF is defined by the conditions (Def. 9).

(Def. 9)(i) (HP-Subformulae)(VERUM) = the root tree of VERUM,

(ii) for every n holds (HP-Subformulae)(propn) = the root tree of propn,

and (iii) for all p, q there exist trees p′, q′ decorated with elements of HP-WFF

such that p′ = (HP-Subformulae)(p) and q′ = (HP-Subformulae)(q) and (HP-Subformulae)(p ∧ q) = p ∧ q-tree(p′, q′) and (HP-Subformulae)(p ⇒ q) = (p ⇒ q)-tree(p′, q′).

defining by structural induction in the … 137

Let us consider p. The functor Subformulaep yielding a tree decorated with elements of HP-WFF is defined by:

(Def. 10) Subformulaep = (HP-Subformulae)(p). The following propositions are true:

(30) SubformulaeVERUM = the root tree of VERUM.

(31) Subformulaepropn = the root tree of propn.

(32) Subformulae(p ∧ q) = p ∧ q-tree(Subformulaep, Subformulaeq).

(33) Subformulae(p ⇒ q) = (p ⇒ q)-tree(Subformulaep, Subformulaeq).

(34) (Subformulaep)(ε) = p.

(35) For every element f of domSubformulaep holds Subformulaep↾f = Subformulae(Subformulaep)(f).

(36) If p ∈ Leaves(Subformulaeq), then p = VERUM or p is simple.

References

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

[2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[3] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547– 552, 1991.

[4] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

[5] Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195–204, 1992.

[6] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77–82, 1993.

[7] Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185–190, 1996.

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

[9] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[11] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

[14] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

[15] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

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

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

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

138 andrzej trybulec

Some Properties of Cells on Go-Board

Czesław Byliński University of Białystok

MML Identifier: GOBRD13.

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

We use the following convention: i, i_{1}, i_{2}, j, j_{1}, j_{2}, k, n are natural numbers, D is a non empty set, and f is a finite sequence of elements of D.

Let E be a non empty set, let S be a non empty set of finite sequences of the carrier of E_{T}^{2}, let F be a function from E into S, and let e be an element of E. Then F (e) is a finite sequence of elements of E_{T}^{2}.

Let F be a function. The functor ValuesF yielding a set is defined by: (Def. 1) ValuesF = Union(rng_{κ} F (κ)).

We now state three propositions:

(1) Let M be a finite sequence of elements of D^{∗}. If i ∈ domM, then M(i) is a finite sequence of elements of D.

(2) For every finite sequence M of elements of D^{∗} holds dom(rng_{κ} M(κ)) = domM.

(3) For every finite sequence M of elements of D^{∗} holds ValuesM = ⋃{rngf; f ranges over elements of D^{∗}: f ∈ rngM}.

Let D be a non empty set and let M be a finite sequence of elements of D^{∗}. Note that ValuesM is finite.

The following propositions are true:

(4) For every matrix M over D such that i ∈ domM and M(i) = f holds lenf = widthM.

(5) For every matrix M over D such that i ∈ domM and M(i) = f and j ∈ domf holds ⟨⟨i, j⟩⟩∈ the indices of M.

(6) For every matrix M over D such that ⟨⟨i, j⟩⟩ ∈ the indices of M and M(i) = f holds lenf = widthM and j ∈ domf.

(7) For every matrix M over D holds ValuesM = {M_{i,j} : ⟨⟨i, j⟩⟩∈ the indices of M}.

(8) For every non empty set D and for every matrix M over D holds cardValuesM lenM · widthM.

In the sequel f, f_{1}, f_{2} are finite sequences of elements of E_{T}^{2} and G is a Go-board.

Next we state a number of propositions:

(9) If f is a sequence which elements belong to G, then rngf ⊆ ValuesG.

(10) For all Go-boards G_{1}, G_{2} such that ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G_{1} and 1 j_{2} and j_{2} widthG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{1,j_{2}} holds i_{1} = 1.

(11) For all Go-boards G_{1}, G_{2} such that ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩ ∈ the indices of G_{1} and 1 j_{2} and j_{2} widthG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{len G_{2},j_{2}} holds i_{1} = lenG_{1}.

(12) For all Go-boards G_{1}, G_{2} such that ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G_{1} and 1 i_{2} and i_{2} lenG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},1} holds j_{1} = 1.

(13) For all Go-boards G_{1}, G_{2} such that ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G_{1} and 1 i_{2} and i_{2} lenG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},width G_{2}} holds j_{1} = widthG_{1}.

(14) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and 1 i_{1} and i_{1} < lenG_{1} and 1 j_{1} and j_{1} widthG_{1} and 1 i_{2} and i_{2} < lenG_{2} and 1 j_{2} and j_{2} widthG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then ((G_{2})_{i_{2}+1,j_{2}})_{1} ((G_{1})_{i_{1}+1,j_{1}})_{1}.

(15) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and 1 < i_{1} and i_{1} lenG_{1} and 1 j_{1} and j_{1} widthG_{1} and 1 < i_{2} and i_{2} lenG_{2} and 1 j_{2} and j_{2} widthG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then ((G_{1})_{i_{1}− ′1,j_{1}})_{1} ((G_{2})_{i_{2}− ′1,j_{2}})_{1}.

(16) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and 1 i_{1} and i_{1} lenG_{1} and 1 j_{1} and j_{1} < widthG_{1} and 1 i_{2} and i_{2} lenG_{2} and 1 j_{2} and j_{2} < widthG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then ((G_{2})_{i_{2},j_{2}+1})_{2} ((G_{1})_{i_{1},j_{1}+1})_{2}.

(17) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and 1 i_{1} and i_{1} lenG_{1} and 1 < j_{1} and j_{1} widthG_{1} and 1 i_{2} and i_{2} lenG_{2} and 1 < j_{2} and j_{2} widthG_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then ((G_{1})_{i_{1},j_{1}− ′1})_{2} ((G_{2})_{i_{2},j_{2}− ′1})_{2}.

(18) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G_{1} and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then cell(G_{2}, i_{2}, j_{2}) ⊆ cell(G_{1}, i_{1}, j_{1}).

(19) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G_{1} and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then cell(G_{2}, i_{2} − ′1, j_{2}) ⊆ cell(G_{1}, i_{1} − ′1, j_{1}).

(20) Let G_{1}, G_{2} be Go-boards. Suppose ValuesG_{1} ⊆ ValuesG_{2} and ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G_{1} and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G_{2} and (G_{1})_{i_{1},j_{1}} = (G_{2})_{i_{2},j_{2}}. Then cell(G_{2}, i_{2}, j_{2} − ′1) ⊆ cell(G_{1}, i_{1}, j_{1} − ′1).

(21) Let f be a standard special circular sequence. Suppose f is a sequence

which elements belong to G. Then Valuesthe Go-board of f ⊆ ValuesG. Let us consider f, G, k. Let us assume that 1 k and k + 1 lenf and f

is a sequence which elements belong to G. The functor right cell(f, k, G) yields a subset of E_{T}^{2} and is defined by the condition (Def. 2).

(Def. 2) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and right cell(f, k, G) = cell(G, i_{1}, j_{1}), or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and right cell(f, k, G) = cell(G, i_{1}, j_{1} − ′1), or (iii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and right cell(f, k, G) = cell(G, i_{2}, j_{2}), or (iv) i_{1} = i_{2} and j_{1} = j_{2} + 1 and right cell(f, k, G) = cell(G, i_{1} − ′1, j_{2}).

The functor left cell(f, k, G) yields a subset of E_{T}^{2} and is defined by the condition (Def. 3).

(Def. 3) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and left cell(f, k, G) = cell(G, i_{1} − ′1, j_{1}), or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and left cell(f, k, G) = cell(G, i_{1}, j_{1}), or (iii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and left cell(f, k, G) = cell(G, i_{2}, j_{2} − ′1), or (iv) i_{1} = i_{2} and j_{1} = j_{2} + 1 and left cell(f, k, G) = cell(G, i_{1}, j_{2}). We now state a number of propositions:

(22) Suppose that 1 k and k + 1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i, j + 1⟩⟩ ∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i,j+1}. Then left cell(f, k, G) = cell(G, i − ′1, j).

(23) Suppose that 1 k and k + 1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i, j + 1⟩⟩ ∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i,j+1}. Then right cell(f, k, G) = cell(G, i, j).

(24) Suppose that 1 k and k + 1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i + 1, j⟩⟩ ∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i+1,j}. Then left cell(f, k, G) = cell(G, i, j).

(25) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i + 1, j⟩⟩∈ the indices of G and π_{k}f = G_{i,j}

and π_{k+1}f = G_{i+1,j}. Then right cell(f, k, G) = cell(G, i, j − ′1).

(26) Suppose that 1 k and k + 1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i + 1, j⟩⟩ ∈ the indices of G and π_{k}f = G_{i+1,j} and π_{k+1}f = G_{i,j}. Then left cell(f, k, G) = cell(G, i, j − ′1).

(27) Suppose that 1 k and k + 1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i + 1, j⟩⟩ ∈ the indices of G and π_{k}f = G_{i+1,j} and π_{k+1}f = G_{i,j}. Then right cell(f, k, G) = cell(G, i, j).

(28) Suppose that 1 k and k + 1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j + 1⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G and π_{k}f = G_{i,j+1} and π_{k+1}f = G_{i,j}. Then left cell(f, k, G) = cell(G, i, j).

(29) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j +1⟩⟩∈ the indices of G and ⟨⟨i, j⟩⟩∈ the indices of G and π_{k}f = G_{i,j+1} and π_{k+1}f = G_{i,j}. Then right cell(f, k, G) = cell(G, i − ′1, j).

(30) If 1 k and k + 1 lenf and f is a sequence which elements belong to G, then left cell(f, k, G) ∩ right cell(f, k, G) = L(f, k).

(31) If 1 k and k + 1 lenf and f is a sequence which elements belong to G, then right cell(f, k, G) is closed.

(32) Suppose 1 k and k + 1 lenf and f is a sequence which elements belong to G and k + 1 n. Then left cell(f, k, G) = left cell(f↾n, k, G) and right cell(f, k, G) = right cell(f↾n, k, G).

(33) Suppose 1 k and k + 1 len(f_{⇂n}) and n lenf and f is a sequence which elements belong to G. Then left cell(f, k+n, G) = left cell(f_{⇂n}, k, G) and right cell(f, k + n, G) = right cell(f_{⇂n}, k, G).

(34) Let G be a Go-board and f be a standard special circular sequence. Suppose 1 n and n + 1 lenf and f is a sequence which elements belong to G. Then left cell(f, n, G) ⊆ leftcell(f, n) and right cell(f, n, G) ⊆ rightcell(f, n).

Let us consider f, G, k. Let us assume that 1 k and k + 1 lenf and f is a sequence which elements belong to G. The functor front right cell(f, k, G) yielding a subset of E_{T}^{2} is defined by the condition (Def. 4).

(Def. 4) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and front right cell(f, k, G) = cell(G, i_{2}, j_{2}), or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and front right cell(f, k, G) = cell(G, i_{2}, j_{2} − ′1),

or (iii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and front right cell(f, k, G) = cell(G, i_{2} − ′1, j_{2}),

or (iv) i_{1} = i_{2} and j_{1} = j_{2}+1 and front right cell(f, k, G) = cell(G, i_{2}− ′1, j_{2}− ′

1). The functor front left cell(f, k, G) yields a subset of E_{T}^{2} and is defined by the condition (Def. 5).

(Def. 5) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and front left cell(f, k, G) = cell(G, i_{2} − ′1, j_{2}),

or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and front left cell(f, k, G) = cell(G, i_{2}, j_{2}), or (iii) i_{1} = i_{2}+1 and j_{1} = j_{2} and front left cell(f, k, G) = cell(G, i_{2}− ′1, j_{2}− ′1),

or (iv) i_{1} = i_{2} and j_{1} = j_{2} + 1 and front left cell(f, k, G) = cell(G, i_{2}, j_{2} − ′1). Next we state several propositions:

(35) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i, j + 1⟩⟩∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i,j+1}. Then front left cell(f, k, G) = cell(G, i − ′1, j + 1).

(36) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i, j + 1⟩⟩∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i,j+1}. Then front right cell(f, k, G) = cell(G, i, j + 1).

(37) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i + 1, j⟩⟩∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i+1,j}. Then front left cell(f, k, G) = cell(G, i + 1, j).

(38) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i + 1, j⟩⟩∈ the indices of G and π_{k}f = G_{i,j} and π_{k+1}f = G_{i+1,j}. Then front right cell(f, k, G) = cell(G, i + 1, j − ′1).

(39) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i+1, j⟩⟩∈ the indices of G and π_{k}f = G_{i+1,j} and π_{k+1}f = G_{i,j}. Then front left cell(f, k, G) = cell(G, i − ′1, j − ′1).

(40) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j⟩⟩∈ the indices of G and ⟨⟨i+1, j⟩⟩∈ the indices of G and π_{k}f = G_{i+1,j} and π_{k+1}f = G_{i,j}. Then front right cell(f, k, G) = cell(G, i − ′1, j).

(41) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j +1⟩⟩∈ the indices of G and ⟨⟨i, j⟩⟩∈ the indices of G and π_{k}f = G_{i,j+1} and π_{k+1}f = G_{i,j}. Then front left cell(f, k, G) = cell(G, i, j − ′1).

(42) Suppose that 1 k and k+1 lenf and f is a sequence which elements belong to G and ⟨⟨i, j +1⟩⟩∈ the indices of G and ⟨⟨i, j⟩⟩∈ the indices of G and π_{k}f = G_{i,j+1} and π_{k+1}f = G_{i,j}. Then front right cell(f, k, G) = cell(G, i − ′1, j − ′1).

(43) Suppose 1 k and k + 1 lenf and f is a sequence which elements belong to G and k + 1 n. Then front left cell(f, k, G) = front left cell(f↾n, k, G) and front right cell(f, k, G) = front right cell(f↾n, k, G).

Let us consider f, G, k. We say that f turns right k, G if and only if the condition (Def. 6) is satisfied.

(Def. 6) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and ⟨⟨i_{2} + 1, j_{2}⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2}+1,j_{2}}, or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and ⟨⟨i_{2}, j_{2} − ′ 1⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2},j_{2}− ′1}, or (iii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and ⟨⟨i_{2}, j_{2} + 1⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2},j_{2}+1}, or (iv) i_{1} = i_{2} and j_{1} = j_{2} + 1 and ⟨⟨i_{2} − ′ 1, j_{2}⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2}− ′1,j_{2}}. We say that f turns left k, G if and only if the condition (Def. 7) is satisfied.

(Def. 7) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and ⟨⟨i_{2} − ′ 1, j_{2}⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2}− ′1,j_{2}}, or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and ⟨⟨i_{2}, j_{2} + 1⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2},j_{2}+1}, or (iii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and ⟨⟨i_{2}, j_{2} − ′ 1⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2},j_{2}− ′1}, or (iv) i_{1} = i_{2} and j_{1} = j_{2} + 1 and ⟨⟨i_{2} + 1, j_{2}⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2}+1,j_{2}}. We say that f goes straight k, G if and only if the condition (Def. 8) is satisfied.

(Def. 8) Let i_{1}, j_{1}, i_{2}, j_{2} be natural numbers. Suppose ⟨⟨i_{1}, j_{1}⟩⟩∈ the indices of G

and ⟨⟨i_{2}, j_{2}⟩⟩∈ the indices of G and π_{k}f = G_{i_{1},j_{1}} and π_{k+1}f = G_{i_{2},j_{2}}. Then (i) i_{1} = i_{2} and j_{1} + 1 = j_{2} and ⟨⟨i_{2}, j_{2} + 1⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2},j_{2}+1}, or (ii) i_{1} + 1 = i_{2} and j_{1} = j_{2} and ⟨⟨i_{2} + 1, j_{2}⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2}+1,j_{2}}, or (iii) i_{1} = i_{2} + 1 and j_{1} = j_{2} and ⟨⟨i_{2} − ′ 1, j_{2}⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2}− ′1,j_{2}}, or (iv) i_{1} = i_{2} and j_{1} = j_{2} + 1 and ⟨⟨i_{2}, j_{2} − ′ 1⟩⟩ ∈ the indices of G and

π_{k+2}f = G_{i_{2},j_{2}− ′1}. One can prove the following propositions:

(44) Suppose 1 k and k + 2 lenf and f is a sequence which elements belong to G and k + 2 n and f↾n turns right k, G. Then f turns right k, G.

(45) Suppose 1 k and k + 2 lenf and f is a sequence which elements belong to G and k + 2 n and f↾n turns left k, G. Then f turns left k, G.

(46) Suppose 1 k and k + 2 lenf and f is a sequence which elements belong to G and k+2 n and f↾n goes straight k, G. Then f goes straight k, G.

(47) Suppose that 1 < k and k + 1 lenf_{1} and k + 1 lenf_{2} and f_{1} is a sequence which elements belong to G and f_{2} is a sequence which elements belong to G and f_{1}↾k = f_{2}↾k and f_{1} turns right k − ′1, G and f_{2} turns right k − ′1, G. Then f_{1}↾(k + 1) = f_{2}↾(k + 1).

(48) Suppose that 1 < k and k + 1 lenf_{1} and k + 1 lenf_{2} and f_{1} is a sequence which elements belong to G and f_{2} is a sequence which elements belong to G and f_{1}↾k = f_{2}↾k and f_{1} turns left k − ′1, G and f_{2} turns left k − ′1, G. Then f_{1}↾(k + 1) = f_{2}↾(k + 1).

(49) Suppose that 1 < k and k + 1 lenf_{1} and k + 1 lenf_{2} and f_{1} is a sequence which elements belong to G and f_{2} is a sequence which elements belong to G and f_{1}↾k = f_{2}↾k and f_{1} goes straight k − ′1, G and f_{2} goes straight k − ′1, G. Then f_{1}↾(k + 1) = f_{2}↾(k + 1).

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Propositional Calculus for Boolean Valued Functions. Part III

Shunichi Kobayashi Shinshu University Nagano

Summary. In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC 7.

The articles [6], [8], [9], [2], [3], [5], [1], [7], and [4] provide the terminology and notation for this paper.

In this paper Y is a non empty set. Next we state a number of propositions:

(1) For all elements a, b of BVF(Y) holds (a ⇒ b) ∧ (¬a ⇒ b) = b.

(2) For all elements a, b of BVF(Y) holds (a ⇒ b) ∧ (a ⇒ ¬b) = ¬a.

(3) For all elements a, b, c of BVF(Y) holds a ⇒ b ∨ c = (a ⇒ b) ∨ (a ⇒ c).

(4) For all elements a, b, c of BVF(Y) holds a ⇒ b ∧ c = (a ⇒ b) ∧ (a ⇒ c).

(5) For all elements a, b, c of BVF(Y) holds a ∨ b ⇒ c = (a ⇒ c) ∧ (b ⇒ c).

(6) For all elements a, b, c of BVF(Y) holds a ∧ b ⇒ c = (a ⇒ c) ∨ (b ⇒ c).

(7) For all elements a, b, c of BVF(Y) holds a ∧ b ⇒ c = a ⇒ b ⇒ c.

(8) For all elements a, b, c of BVF(Y) holds a ∧ b ⇒ c = a ⇒ ¬b ∨ c.

(9) For all elements a, b, c of BVF(Y) holds a ⇒ b ∨ c = a ∧ ¬b ⇒ c.

(10) For all elements a, b of BVF(Y) holds a ∧ (a ⇒ b) = a ∧ b.

(11) For all elements a, b of BVF(Y) holds (a ⇒ b) ∧ ¬b = ¬a ∧ ¬b.

(12) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) = (a ⇒ b) ∧ (b ⇒ c) ∧ (a ⇒ c).

(13) For every element a of BVF(Y) holds true(Y) ⇒ a = a.

(14) For every element a of BVF(Y) holds a ⇒ false(Y) = ¬a.

148 shunichi kobayashi

(15) For every element a of BVF(Y) holds false(Y) ⇒ a = true(Y).

(16) For every element a of BVF(Y) holds a ⇒ true(Y) = true(Y).

(17) For every element a of BVF(Y) holds a ⇒ ¬a = ¬a.

(18) For all elements a, b, c of BVF(Y) holds a ⇒ b ⋐ c ⇒ a ⇒ c ⇒ b.

(19) For all elements a, b, c of BVF(Y) holds a ⇔ b ⋐ a ⇔ c ⇔ b ⇔ c.

(20) For all elements a, b, c of BVF(Y) holds a ⇔ b ⋐ a ⇒ c ⇔ b ⇒ c.

(21) For all elements a, b, c of BVF(Y) holds a ⇔ b ⋐ c ⇒ a ⇔ c ⇒ b.

(22) For all elements a, b, c of BVF(Y) holds a ⇔ b ⋐ a ∧ c ⇔ b ∧ c.

(23) For all elements a, b, c of BVF(Y) holds a ⇔ b ⋐ a ∨ c ⇔ b ∨ c.

(24) For all elements a, b of BVF(Y) holds a ⋐ a ⇔ b ⇔ b ⇔ a ⇔ a.

(25) For all elements a, b of BVF(Y) holds a ⋐ a ⇒ b ⇔ b.

(26) For all elements a, b of BVF(Y) holds a ⋐ b ⇒ a ⇔ a.

(27) For all elements a, b of BVF(Y) holds a ⋐ a ∧ b ⇔ b ∧ a ⇔ a.

References

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

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

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

[4] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

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

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

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

Propositional Calculus for Boolean Valued Functions. Part IV

Shunichi Kobayashi Shinshu University Nagano

Summary. In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC 8.

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

In this paper Y denotes a non empty set. One can prove the following propositions:

(1) For all elements a, b, c, d of BVF(Y) holds a ⇒ b∧c∧d = (a ⇒ b)∧(a ⇒ c) ∧ (a ⇒ d).

(2) For all elements a, b, c, d of BVF(Y) holds a ⇒ b∨c∨d = (a ⇒ b)∨(a ⇒ c) ∨ (a ⇒ d).

(3) For all elements a, b, c, d of BVF(Y) holds a∧b∧c ⇒ d = (a ⇒ d)∨(b ⇒ d) ∨ (c ⇒ d).

(4) For all elements a, b, c, d of BVF(Y) holds a∨b∨c ⇒ d = (a ⇒ d)∧(b ⇒ d) ∧ (c ⇒ d).

(5) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ a) = (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ a) ∧ (b ⇒ a) ∧ (a ⇒ c).

(6) For all elements a, b of BVF(Y) holds a = a ∧ b ∨ a ∧ ¬b.

(7) For all elements a, b of BVF(Y) holds a = (a ∨ b) ∧ (a ∨ ¬b).

(8) For all elements a, b, c of BVF(Y) holds a = a ∧ b ∧ c ∨ a ∧ b ∧ ¬c ∨ a ∧ ¬b ∧ c ∨ a ∧ ¬b ∧ ¬c.

(9) For all elements a, b, c of BVF(Y) holds a = (a ∨ b ∨ c) ∧ (a ∨ b ∨ ¬c) ∧ (a ∨ ¬b ∨ c) ∧ (a ∨ ¬b ∨ ¬c).

(10) For all elements a, b of BVF(Y) holds a ∧ b = a ∧ (¬a ∨ b).

(11) For all elements a, b of BVF(Y) holds a ∨ b = a ∨ ¬a ∧ b.

(12) For all elements a, b of BVF(Y) holds a ⊕ b = ¬(a ⇔ b).

(13) For all elements a, b of BVF(Y) holds a ⊕ b = (a ∨ b) ∧ (¬a ∨ ¬b).

(14) For every element a of BVF(Y) holds a ⊕ true(Y) = ¬a.

(15) For every element a of BVF(Y) holds a ⊕ false(Y) = a.

(16) For all elements a, b of BVF(Y) holds a ⊕ b = ¬a ⊕ ¬b.

(17) For all elements a, b of BVF(Y) holds ¬(a ⊕ b) = a ⊕ ¬b.

(18) For all elements a, b of BVF(Y) holds a ⇔ b = (a ∨ ¬b) ∧ (¬a ∨ b).

(19) For all elements a, b of BVF(Y) holds a ⇔ b = a ∧ b ∨ ¬a ∧ ¬b.

(20) For every element a of BVF(Y) holds a ⇔ true(Y) = a.

(21) For every element a of BVF(Y) holds a ⇔ false(Y) = ¬a.

(22) For all elements a, b of BVF(Y) holds ¬(a ⇔ b) = a ⇔ ¬b.

(23) For all elements a, b of BVF(Y) holds ¬a ⋐ a ⇒ b ⇔ ¬a.

(24) For all elements a, b of BVF(Y) holds ¬a ⋐ b ⇒ a ⇔ ¬b.

(25) For all elements a, b of BVF(Y) holds a ⋐ a ∨ b ⇔ b ∨ a ⇔ a.

(26) For every element a of BVF(Y) holds a ⇒ ¬a ⇔ ¬a = true(Y).

(27) For all elements a, b of BVF(Y) holds a ⇒ b ⇒ a ⇒ a = true(Y).

(28) For all elements a, b, c, d of BVF(Y) holds (a ⇒ c)∧(b ⇒ d)∧(¬c∨¬d) ⇒ ¬a ∨ ¬b = true(Y).

(29) For all elements a, b, c of BVF(Y) holds a ⇒ b ⇒ a ⇒ b ⇒ c ⇒ a ⇒ c = true(Y).

References

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

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

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

[4] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

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

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

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

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

Basic Properties of Genetic Algorithm

Akihiko Uchibori Yamaguchi University Ube

Noboru Endou Shinshu University Nagano

Summary. We defined the set of the gene, the space treated by the genetic algorithm and the individual of the space. Moreover, we defined some genetic operators such as one point crossover and two points crossover, and the validity of many characters were proven.

MML Identifier: GENEALG1.

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

1. Definitions of Gene-Set, GA-Space and Individual

We follow the rules: D is a non empty set, f_{1}, f_{2} are finite sequences of elements of D, and i, n, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6} are natural numbers.

We now state two propositions:

(1) If n lenf_{1}, then (f_{1} ^{⌢} f_{2})_{⇂n} = ((f_{1})_{⇂n})^{⌢} f_{2}.

(2) (f_{1} ^{⌢} f_{2})↾(lenf_{1} + i) = f_{1} ^{⌢} (f_{2}↾i). A Gene-Set is a non-empty non empty finite sequence. Let S be a Gene-Set. We introduce GA − Space S as a synonym of UnionS. Let f be a non-empty non empty function. Note that Unionf is non empty. Let S be a Gene-Set. A finite sequence of elements of GA − Space S is said

to be a Individual of S if: (Def. 1) lenit = lenS and for every i such that i ∈ domit holds it(i) ∈ S(i).

2. Definitions of Several Genetic Operators

Let S be a Gene-Set, let p_{1}, p_{2} be finite sequences of elements of GA − Space S, and let us consider n. The functor crossover(p_{1}, p_{2}, n) yields a finite sequence of elements of GA − Space S and is defined as follows:

(Def. 2) crossover(p_{1}, p_{2}, n) = (p_{1}↾n)^{⌢} ((p_{2})_{⇂n}).

Let S be a Gene-Set, let p_{1}, p_{2} be finite sequences of elements of GA − Space S, and let us consider n_{1}, n_{2}. The functor crossover(p_{1}, p_{2}, n_{1}, n_{2}) yields a finite sequence of elements of GA − Space S and is defined as follows:

(Def. 3) crossover(p_{1}, p_{2}, n_{1}, n_{2}) =

crossover(crossover(p_{1}, p_{2}, n_{1}), crossover(p_{2}, p_{1}, n_{1}), n_{2}). Let S be a Gene-Set, let p_{1}, p_{2} be finite sequences of ele-

ments of GA − Space S, and let us consider n_{1}, n_{2}, n_{3}. The functor crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) yields a finite sequence of elements of GA − Space S and is defined as follows:

(Def. 4) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) =

crossover(crossover(p_{1}, p_{2}, n_{1}, n_{2}), crossover(p_{2}, p_{1}, n_{1}, n_{2}), n_{3}). Let S be a Gene-Set, let p_{1}, p_{2} be finite sequences of ele-

ments of GA − Space S, and let us consider n_{1}, n_{2}, n_{3}, n_{4}. The functor crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) yields a finite sequence of elements of GA − Space S and is defined as follows:

(Def. 5) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}), crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}), n_{4}). Let S be a Gene-Set, let p_{1}, p_{2} be finite sequences of elements

of GA − Space S, and let us consider n_{1}, n_{2}, n_{3}, n_{4}, n_{5}. The functor crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) yielding a finite sequence of elements of GA − Space S is defined by:

(Def. 6) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}), crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{4}), n_{5}). Let S be a Gene-Set, let p_{1}, p_{2} be finite sequences of elements of

GA − Space S, and let us consider n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}. The functor crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) yielding a finite sequence of elements of GA − Space S is defined as follows:

(Def. 7) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}),

crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}), n_{6}).

3. Properties of 1-point Crossover

In the sequel S denotes a Gene-Set and p_{1}, p_{2} denote Individual of S. The following proposition is true

(3) crossover(p_{1}, p_{2}, n) is a Individual of S. Let S be a Gene-Set, let p_{1}, p_{2} be Individual of S, and let us consider n.

Then crossover(p_{1}, p_{2}, n) is a Individual of S.

One can prove the following propositions:

(4) crossover(p_{1}, p_{2}, 0) = p_{2}.

(5) If n lenp_{1}, then crossover(p_{1}, p_{2}, n) = p_{1}.

4. Properties of 2-points Crossover

We now state the proposition

(6) crossover(p_{1}, p_{2}, n_{1}, n_{2}) is a Individual of S. Let S be a Gene-Set, let p_{1}, p_{2} be Individual of S, and let us consider n_{1},

n_{2}. Then crossover(p_{1}, p_{2}, n_{1}, n_{2}) is a Individual of S. We now state several propositions:

(7) crossover(p_{1}, p_{2}, 0, n) = crossover(p_{2}, p_{1}, n).

(8) crossover(p_{1}, p_{2}, n, 0) = crossover(p_{2}, p_{1}, n).

(9) If n_{1} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}) = crossover(p_{1}, p_{2}, n_{2}).

(10) If n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}) = crossover(p_{1}, p_{2}, n_{1}).

(11) If n_{1} lenp_{1} and n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}) = p_{1}.

(12) crossover(p_{1}, p_{2}, n_{1}, n_{1}) = p_{1}.

(13) crossover(p_{1}, p_{2}, n_{1}, n_{2}) = crossover(p_{1}, p_{2}, n_{2}, n_{1}).

5. Properties of 3-points Crossover

Next we state the proposition

(14) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) is a Individual of S.

Let S be a Gene-Set, let p_{1}, p_{2} be Individual of S, and let us consider n_{1}, n_{2}, n_{3}. Then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) is a Individual of S.

We now state a number of propositions:

(15) crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}) = crossover(p_{2}, p_{1}, n_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}) = crossover(p_{2}, p_{1}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{2}).

(16) crossover(p_{1}, p_{2}, 0, 0, n_{3}) = crossover(p_{1}, p_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, 0, 0) = crossover(p_{1}, p_{2}, n_{1}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0) = crossover(p_{1}, p_{2}, n_{2}).

(17) crossover(p_{1}, p_{2}, 0, 0, 0) = p_{2}.

(18) If n_{1} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}).

(19) If n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}).

(20) If n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}).

(21) If n_{1} lenp_{1} and n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{3}).

(22) If n_{1} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{2}).

(23) If n_{2} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{1}).

(24) If n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = p_{1}.

(25) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{2}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{2}).

(26) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}) = crossover(p_{1}, p_{2}, n_{3}, n_{1}, n_{2}).

(27) crossover(p_{1}, p_{2}, n_{1}, n_{1}, n_{3}) = crossover(p_{1}, p_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{1}) = crossover(p_{1}, p_{2}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{2}) = crossover(p_{1}, p_{2}, n_{1}).

6. Properties of 4-points Crossover

Next we state the proposition

(28) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) is a Individual of S.

Let S be a Gene-Set, let p_{1}, p_{2} be Individual of S, and let us consider n_{1}, n_{2}, n_{3}, n_{4}. Then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) is a Individual of S.

The following propositions are true:

(29) crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, n_{4}) = crossover(p_{2}, p_{1}, n_{2}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, n_{4}) = crossover(p_{2}, p_{1}, n_{1}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, n_{4}) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}).

(30) crossover(p_{1}, p_{2}, 0, 0, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, 0) = crossover(p_{1}, p_{2}, n_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, 0) = crossover(p_{1}, p_{2}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, 0, 0, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}) and

crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, 0) = crossover(p_{1}, p_{2}, n_{1}, n_{2}).

(31) crossover(p_{1}, p_{2}, n_{1}, 0, 0, 0) = crossover(p_{2}, p_{1}, n_{1}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0, 0) = crossover(p_{2}, p_{1}, n_{2}) and crossover(p_{1}, p_{2}, 0, 0, n_{3}, 0) = crossover(p_{2}, p_{1}, n_{3}) and crossover(p_{1}, p_{2}, 0, 0, 0, n_{4}) = crossover(p_{2}, p_{1}, n_{4}).

(32) crossover(p_{1}, p_{2}, 0, 0, 0, 0) = p_{1}. (33)(i) If n_{1} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}), (ii) if n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{4}), (iii) if n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{4}), and (iv) if n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}). (34)(i) If n_{1} lenp_{1} and n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{3}, n_{4}), (ii) if n_{1} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{2}, n_{4}), (iii) if n_{1} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{2}, n_{3}), (iv) if n_{2} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{1}, n_{4}),

(v) if n_{2} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}), and

(vi) if n_{3} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}). (35)(i) If n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}), (ii) if n_{1} lenp_{1} and n_{2} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}), (iii) if n_{1} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}), and (iv) if n_{2} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}).

(36) If n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = p_{1}.

(37) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{4}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{4}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}, n_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}, n_{3}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{1}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{1}, n_{4}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{1}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}, n_{1}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}, n_{3}, n_{1}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{1}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{1}, n_{4}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{2}, n_{1}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{2}, n_{4}, n_{1}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}, n_{1}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}, n_{2}, n_{1}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}, n_{1}, n_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}, n_{1}, n_{3}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}, n_{2}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}, n_{2}, n_{3}, n_{1}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}, n_{3}, n_{1}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{4}, n_{3}, n_{2}, n_{1}).

(38) crossover(p_{1}, p_{2}, n_{1}, n_{1}, n_{3}, n_{4}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{1}, n_{4}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{1}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{2}, n_{4}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{2}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{3}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}).

(39) crossover(p_{1}, p_{2}, n_{1}, n_{1}, n_{3}, n_{3}) = p_{1} and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{1}, n_{2}) = p_{1} and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{2}, n_{1}) = p_{1}.

7. Properties of 5-points Crossover

Next we state the proposition

(40) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) is a Individual of S.

Let S be a Gene-Set, let p_{1}, p_{2} be Individual of S, and let us consider n_{1}, n_{2}, n_{3}, n_{4}, n_{5}. Then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) is a Individual of S.

Next we state a number of propositions:

(41) crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{2}, p_{1}, n_{2}, n_{3}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, n_{4}, n_{5}) = crossover(p_{2}, p_{1}, n_{1}, n_{3}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, n_{4}, n_{5}) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, 0, n_{5}) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{4}).

(42) crossover(p_{1}, p_{2}, 0, 0, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, 0, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{5}) and crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, n_{4}, 0) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, 0, 0, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, 0, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, n_{4}, 0) = crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, 0, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, n_{4}, 0) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, 0, 0) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}).

(43) crossover(p_{1}, p_{2}, 0, 0, 0, n_{4}, n_{5}) = crossover(p_{2}, p_{1}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, 0, 0, n_{3}, 0, n_{5}) = crossover(p_{2}, p_{1}, n_{3}, n_{5}) and crossover(p_{1}, p_{2}, 0, 0, n_{3}, n_{4}, 0) = crossover(p_{2}, p_{1}, n_{3}, n_{4}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0, 0, n_{5}) = crossover(p_{2}, p_{1}, n_{2}, n_{5}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0, n_{4}, 0) = crossover(p_{2}, p_{1}, n_{2}, n_{4}) and crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, 0, 0) = crossover(p_{2}, p_{1}, n_{2}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, 0, 0, 0, n_{5}) = crossover(p_{2}, p_{1}, n_{1}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, 0, 0, n_{4}, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{4}) and crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, 0, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{3}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, 0, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{2}).

(44) crossover(p_{1}, p_{2}, 0, 0, 0, 0, n_{5}) = crossover(p_{1}, p_{2}, n_{5}) and crossover(p_{1}, p_{2}, 0, 0, 0, n_{4}, 0) = crossover(p_{1}, p_{2}, n_{4}) and crossover(p_{1}, p_{2}, 0, 0, n_{3}, 0, 0) = crossover(p_{1}, p_{2}, n_{3}) and crossover(p_{1}, p_{2}, 0, n_{2}, 0, 0, 0) = crossover(p_{1}, p_{2}, n_{2}) and crossover(p_{1}, p_{2}, n_{1}, 0, 0, 0, 0) = crossover(p_{1}, p_{2}, n_{1}).

(45) crossover(p_{1}, p_{2}, 0, 0, 0, 0, 0) = p_{2}. (46)(i) If n_{1} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}, n_{5}), (ii) if n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{4}, n_{5}), (iii) if n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{4}, n_{5}), (iv) if n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{5}), and

(v) if n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}).

(47)(i) If n_{1} lenp_{1} and n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{3}, n_{4}, n_{5}), (ii) if n_{1} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{2}, n_{4}, n_{5}), 158 akihiko uchibori and noboru endou

(iii) if n_{1} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{5}), (iv) if n_{1} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}),

(v) if n_{2} lenp_{1} and n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}, n_{5}),

(vi) if n_{2} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{5}), (vii) if n_{2} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{4}), (viii) if n_{3} lenp_{1} and n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{5}), (ix) if n_{3} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{4}), and

(x) if n_{4} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}).

(48)(i) If n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{4}, n_{5}), (ii) if n_{1} lenp_{1} and n_{2} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{3}, n_{5}), (iii) if n_{1} lenp_{1} and n_{2} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}), (iv) if n_{1} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{5}),

(v) if n_{1} lenp_{1} and n_{3} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}),

(vi) if n_{1} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}), (vii) if n_{2} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{5}), (viii) if n_{2} lenp_{1} and n_{3} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{4}), (ix) if n_{2} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{3}), and

(x) if n_{3} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}).

(49)(i) If n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{5}), (ii) if n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{4}), (iii) if n_{1} lenp_{1} and n_{2} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{3}), (iv) if n_{1} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}), and

(v) if n_{2} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{1}).

(50) If n_{1} lenp_{1} and n_{2} lenp_{1} and n_{3} lenp_{1} and n_{4} lenp_{1} and n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = p_{1}.

(51) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{1}, n_{3}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{3}, n_{2}, n_{1}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{4}, n_{2}, n_{3}, n_{1}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{5}, n_{2}, n_{3}, n_{4}, n_{1}).

(52) crossover(p_{1}, p_{2}, n_{1}, n_{1}, n_{3}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{1}, n_{4}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{1}, n_{5}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{5}) and crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{1}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}).

8. Properties of 6-points Crossover

Next we state the proposition

(53) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) is a Individual of S.

Let S be a Gene-Set, let p_{1}, p_{2} be Individual of S, and let us consider n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}. Then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) is a Individual of S.

We now state four propositions: (54)(i) crossover(p_{1}, p_{2}, 0, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{2}, p_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}), (ii) crossover(p_{1}, p_{2}, n_{1}, 0, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{2}, p_{1}, n_{1}, n_{3}, n_{4}, n_{5}, n_{6}), (iii) crossover(p_{1}, p_{2}, n_{1}, n_{2}, 0, n_{4}, n_{5}, n_{6}) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{4}, n_{5}, n_{6}), (iv) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, 0, n_{5}, n_{6}) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{5}, n_{6}),

(v) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, 0, n_{6}) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{4}, n_{6}), and

(vi) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, 0) = crossover(p_{2}, p_{1}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}). (55)(i) If n_{1} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) =

crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}), (ii) if n_{2} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) =

crossover(p_{1}, p_{2}, n_{1}, n_{3}, n_{4}, n_{5}, n_{6}), (iii) if n_{3} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{4}, n_{5}, n_{6}), (iv) if n_{4} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) =

crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{5}, n_{6}), 160 akihiko uchibori and noboru endou

(v) if n_{5} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{6}), and

(vi) if n_{6} lenp_{1}, then crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}).

(56)(i) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{2}, n_{1}, n_{3}, n_{4}, n_{5}, n_{6}), (ii) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{3}, n_{2}, n_{1}, n_{4}, n_{5}, n_{6}), (iii) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{4}, n_{2}, n_{3}, n_{1}, n_{5}, n_{6}), (iv) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{5}, n_{2}, n_{3}, n_{4}, n_{1}, n_{6}),

and

(v) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{6}, n_{2}, n_{3}, n_{4}, n_{5}, n_{1}). (57)(i) crossover(p_{1}, p_{2}, n_{1}, n_{1}, n_{3}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{3}, n_{4}, n_{5}, n_{6}), (ii) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{1}, n_{4}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{2}, n_{4}, n_{5}, n_{6}), (iii) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{1}, n_{5}, n_{6}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{5}, n_{6}), (iv) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{1}, n_{6}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}, n_{6}),

and

(v) crossover(p_{1}, p_{2}, n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{1}) = crossover(p_{1}, p_{2}, n_{2}, n_{3}, n_{4}, n_{5}).

References

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

[2] Grzegorz Bancerek. The reflection theorem. Formalized Mathematics, 1(5):973–977, 1990.

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

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

[5] 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.

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

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

[8] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

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

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

Propositional Calculus for Boolean Valued Functions. Part V

Shunichi Kobayashi Shinshu University Nagano

Summary. In this paper, we have proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC 9.

The terminology and notation used here have been introduced in the following articles: [3], [4], [5], [2], and [1].

In this paper Y denotes a non empty set. We now state a number of propositions:

(1) For all elements a, b, c of BVF(Y) holds (a ∨ b) ∧ (b ⇒ c) ⋐ a ∨ c.

(2) For all elements a, b of BVF(Y) holds a ∧ (a ⇒ b) ⋐ b.

(3) For all elements a, b of BVF(Y) holds (a ⇒ b) ∧ ¬b ⋐ ¬a.

(4) For all elements a, b of BVF(Y) holds (a ∨ b) ∧ ¬a ⋐ b.

(5) For all elements a, b of BVF(Y) holds (a ⇒ b) ∧ (¬a ⇒ b) ⋐ b.

(6) For all elements a, b of BVF(Y) holds (a ⇒ b) ∧ (a ⇒ ¬b) ⋐ ¬a.

(7) For all elements a, b, c of BVF(Y) holds a ⇒ b ∧ c ⋐ a ⇒ b.

(8) For all elements a, b, c of BVF(Y) holds a ∨ b ⇒ c ⋐ a ⇒ c.

(9) For all elements a, b, c of BVF(Y) holds a ⇒ b ⋐ a ∧ c ⇒ b.

(10) For all elements a, b, c of BVF(Y) holds a ⇒ b ⋐ a ∧ c ⇒ b ∧ c.

(11) For all elements a, b, c of BVF(Y) holds a ⇒ b ⋐ a ⇒ b ∨ c.

(12) For all elements a, b, c of BVF(Y) holds a ⇒ b ⋐ a ∨ c ⇒ b ∨ c.

(13) For all elements a, b, c of BVF(Y) holds a ∧ b ∨ c ⋐ a ∨ c.

(14) For all elements a, b, c, d of BVF(Y) holds a ∧ b ∨ c ∧ d ⋐ a ∨ c.

(15) For all elements a, b, c of BVF(Y) holds (a ∨ b) ∧ (b ⇒ c) ⋐ a ∨ c.

(16) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (¬a ⇒ c) ⋐ b ∨ c.

(17) For all elements a, b, c of BVF(Y) holds (a ⇒ c) ∧ (b ⇒ ¬c) ⋐ ¬a ∨ ¬b.

(18) For all elements a, b, c of BVF(Y) holds (a ∨ b) ∧ (¬a ∨ c) ⋐ b ∨ c.

(19) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (a ⇒ c) ⋐ a ⇒ b ∧ c.

(20) For all elements a, b, c, d of BVF(Y) holds (a ⇒ b) ∧ (c ⇒ d) ⋐ a ∧ c ⇒ b ∧ d.

(21) For all elements a, b, c of BVF(Y) holds (a ⇒ c) ∧ (b ⇒ c) ⋐ a ∨ b ⇒ c.

(22) For all elements a, b, c, d of BVF(Y) holds (a ⇒ b) ∧ (c ⇒ d) ⋐ a ∨ c ⇒ b ∨ d.

(23) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (a ⇒ c) ⋐ a ⇒ b ∨ c.

(24) For all elements a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} of BVF(Y) holds (b_{1} ⇒ b_{2}) ∧ (c_{1} ⇒ c_{2}) ∧ (a_{1} ∨ b_{1} ∨ c_{1}) ∧ ¬(a_{2} ∧ b_{2}) ∧ ¬(a_{2} ∧ c_{2}) ⋐ a_{2} ⇒ a_{1}.

(25) For all elements a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} of BVF(Y) holds (a_{1} ⇒ a_{2}) ∧ (b_{1} ⇒ b_{2}) ∧ (c_{1} ⇒ c_{2}) ∧ (a_{1} ∨ b_{1} ∨ c_{1}) ∧ ¬(a_{2} ∧ b_{2}) ∧ ¬(a_{2} ∧ c_{2}) ∧ ¬(b_{2} ∧ c_{2}) ⋐ (a_{2} ⇒ a_{1}) ∧ (b_{2} ⇒ b_{1}) ∧ (c_{2} ⇒ c_{1}).

(26) For all elements a_{1}, b_{1}, a_{2}, b_{2} of BVF(Y) holds (a_{1} ⇒ a_{2}) ∧ (b_{1} ⇒ b_{2}) ∧ ¬(a_{2} ∧ b_{2}) ⇒ ¬(a_{1} ∧ b_{1}) = true(Y).

(27) For all elements a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} of BVF(Y) holds (a_{1} ⇒ a_{2}) ∧ (b_{1} ⇒ b_{2}) ∧ (c_{1} ⇒ c_{2}) ∧ ¬(a_{2} ∧ b_{2}) ∧ ¬(a_{2} ∧ c_{2}) ∧ ¬(b_{2} ∧ c_{2}) ⋐ ¬(a_{1} ∧ b_{1}) ∧ ¬(a_{1} ∧ c_{1}) ∧ ¬(b_{1} ∧ c_{1}).

References

[1] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

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

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

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

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

Properties of Left and Right Components

Artur Korniłowicz^{1} University of Białystok

MML Identifier: GOBRD14.

The notation and terminology used here have been introduced in the following papers: [33], [42], [43], [6], [7], [41], [5], [16], [35], [1], [30], [38], [31], [17], [27], [8], [19], [39], [18], [20], [15], [4], [2], [3], [40], [32], [29], [44], [12], [28], [11], [13], [14], [21], [22], [25], [34], [10], [24], [23], [37], [36], [26], and [9].

1. Components

For simplicity, we adopt the following rules: r denotes a real number, i, j, n denote natural numbers, f denotes a non constant standard special circular sequence, g denotes a clockwise oriented non constant standard special circular sequence, p, q denote points of E_{T}^{2}, P, Q, R denote subsets of E_{T}^{2}, C denotes a compact non vertical non horizontal subset of E_{T}^{2}, and G denotes a Go-board.

Next we state several propositions:

(1) Let T be a topological space, A be a subset of the carrier of T, and B be a subset of T. If B is a component of A, then B is connected.

(2) Let A be a subset of the carrier of E_{T}^{n} and B be a subset of E_{T}^{n}. If B is inside component of A, then B is connected.

(3) Let A be a subset of the carrier of E_{T}^{n} and B be a subset of E_{T}^{n}. If B is outside component of A, then B is connected.

(4) For every subset A of the carrier of E_{T}^{n} and for every subset B of E_{T}^{n} such that B is a component of A^{c} holds A ∩ B = ∅.

(5) If P is outside component of Q and R is inside component of Q, then P ∩ R = ∅.

1This paper was written while the author visited Shinshu University, winter 1999.

(6) Let A, B be subsets of E_{T}^{2}. Suppose A is outside component of L̃(f) and B is outside component of L̃(f). Then A = B.

(7) Let p be a point of E^{2}. Suppose p = 0_{E_{T}^{2}} and P is outside component of L̃(f). Then there exists a real number r such that r > 0 and Ball(p, r)^{c} ⊆ P.

Let C be a closed subset of E_{T}^{2}. Observe that BDDC is open and UBDC is open.

Let C be a compact subset of E_{T}^{2}. Observe that UBDC is connected.

2. Go-Boards

One can prove the following proposition

(8) For every finite sequence f of elements of E_{T}^{n} such that L̃(f) ≠ ∅ holds 2 lenf.

Let n be a natural number and let a, b be points of E_{T}^{n}. The functor ρ(a, b) yields a real number and is defined by:

(Def. 1) There exist points p, q of E^{n} such that p = a and q = b and ρ(a, b) =

ρ(p, q). Let us notice that the functor ρ(a, b) is commutative.

The following propositions are true:

(9) ρ(p, q) = \sqrt{(p_{1} − q_{1})^{2} + (p_{2} − q_{2})^{2}} .

(10) For every point p of E_{T}^{n} holds ρ(p, p) = 0.

(11) For all points p, q, r of E_{T}^{n} holds ρ(p, r) ρ(p, q) + ρ(q, r).

(12) Let x_{1}, x_{2}, y_{1}, y_{2} be real numbers and a, b be points of E_{T}^{2}. Suppose x_{1} a_{1} and a_{1} x_{2} and y_{1} a_{2} and a_{2} y_{2} and x_{1} b_{1} and b_{1} x_{2} and y_{1} b_{2} and b_{2} y_{2}. Then ρ(a, b) |x_{2} − x_{1}| + |y_{2} − y_{1}|.

(13) If 1 i and i < lenG and 1 j and j < widthG, then cell(G, i, j) = ∏[1 ⟼→ [(G_{i,1})_{1}, (G_{i+1,1})_{1}], 2 ⟼→ [(G_{1,j})_{2}, (G_{1,j+1})_{2}]].

(14) If 1 i and i < lenG and 1 j and j < widthG, then cell(G, i, j) is compact.

(15) If ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i + n, j⟩⟩ ∈ the indices of G, then ρ(G_{i,j}, G_{i+n,j}) = (G_{i+n,j})_{1} − (G_{i,j})_{1}.

(16) If ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨i, j + n⟩⟩ ∈ the indices of G, then ρ(G_{i,j}, G_{i,j+n}) = (G_{i,j+n})_{2} − (G_{i,j})_{2}.

(17) 3 lenGauge(C, n) − ′1.

(18) Suppose i j. Let a, b be natural numbers. Suppose 2 a and a lenGauge(C, i) − 1 and 2 b and b lenGauge(C, i) − 1. Then there exist natural numbers c, d such that 2 c and c lenGauge(C, j) − 1 and 2 d and d lenGauge(C, j) − 1 and ⟨⟨c, d⟩⟩ ∈ the indices of Gauge(C, j) and (Gauge(C, i))_{a,b} = (Gauge(C, j))_{c,d} and c = 2 + 2^{j− ′i} · (a − ′2) and d = 2 + 2^{j− ′i} · (b − ′2).

(19) If ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, n) and ⟨⟨i, j + 1⟩⟩ ∈ the indices of Gauge(C, n), then ρ((Gauge(C, n))_{i,j}, (Gauge(C, n))_{i,j+1}) = \frac{N-bound C−S-bound C}{2^{n}} .

(20) If ⟨⟨i, j⟩⟩ ∈ the indices of Gauge(C, n) and ⟨⟨i + 1, j⟩⟩ ∈ the indices of Gauge(C, n), then ρ((Gauge(C, n))_{i,j}, (Gauge(C, n))_{i+1,j}) = \frac{E-bound C−W-bound C}{2^{n}} .

(21) If r > 0, then there exists a natural number n such that ρ((Gauge(C, n))_{1,1}, (Gauge(C, n))_{1,2}) < r and ρ((Gauge(C, n))_{1,1}, (Gauge(C, n))_{2,1}) < r.

3. LeftComp and RightComp

One can prove the following propositions:

(22) For every subset P of (E_{T}^{2})↾(L̃(f))^{c} such that P is a component of (E_{T}^{2})↾(L̃(f))^{c} holds P = RightComp(f) or P = LeftComp(f).

(23) Let A_{1}, A_{2} be subsets of E_{T}^{2}. Suppose that (i) (L̃(f))^{c} = A_{1} ∪ A_{2}, (ii) A_{1} ∩ A_{2} = ∅, and (iii) for all subsets C_{1}, C_{2} of (E_{T}^{2})↾(L̃(f))^{c} such that C_{1} = A_{1} and C_{2} =

A_{2} holds C_{1} is a component of (E_{T}^{2})↾(L̃(f))^{c} and C_{2} is a component of (E_{T}^{2})↾(L̃(f))^{c}. Then A_{1} = RightComp(f) and A_{2} = LeftComp(f) or A_{1} = LeftComp(f) and A_{2} = RightComp(f).

(24) LeftComp(f) ∩ RightComp(f) = ∅.

(25) L̃(f) ∪ RightComp(f) ∪ LeftComp(f) = the carrier of E_{T}^{2}.

(26) p ∈ L̃(f) iff p ∉ LeftComp(f) and p ∉ RightComp(f).

(27) p ∈ LeftComp(f) iff p ∉ L̃(f) and p ∉ RightComp(f).

(28) p ∈ RightComp(f) iff p ∉ L̃(f) and p ∉ LeftComp(f).

(29) L̃(f) = \overline{RightComp(f)} \ RightComp(f).

(30) L̃(f) = \overline{LeftComp(f)} \ LeftComp(f).

(31) \overline{RightComp(f)} = RightComp(f) ∪ L̃(f).

(32) \overline{LeftComp(f)} = LeftComp(f) ∪ L̃(f).

Let f be a non constant standard special circular sequence. One can verify that L̃(f) is Jordan.

The following propositions are true:

(33) If π_{1}g = N-min L̃(g) and p ∈ RightComp(g), then W-bound L̃(g) < p_{1}.

(34) If π_{1}g = N-min L̃(g) and p ∈ RightComp(g), then E-bound L̃(g) > p_{1}.

(35) If π_{1}g = N-min L̃(g) and p ∈ RightComp(g), then N-bound L̃(g) > p_{2}.

(36) If π_{1}g = N-min L̃(g) and p ∈ RightComp(g), then S-bound L̃(g) < p_{2}.

(37) If p ∈ RightComp(f) and q ∈ LeftComp(f), then L(p, q) ∩ L̃(f) ≠ ∅.

(38) \overline{RightComp(SpStSeqC)} = ∏[1 ⟼→ [W-bound L̃(SpStSeqC), E-bound L̃(SpStSeqC)], 2 ⟼→ [S-bound L̃(SpStSeqC), N-bound L̃(SpStSeqC)]].

(39) (proj1)^{∘}L̃(f) ⊆ (proj1)^{∘}\overline{RightComp(f)} and if π_{1}f = N-min L̃(f) and f is clockwise oriented, then (proj1)^{∘}\overline{RightComp(f)} = (proj1)^{∘}L̃(f).

(40) (proj2)^{∘}L̃(f) ⊆ (proj2)^{∘}\overline{RightComp(f)} and if π_{1}f = N-min L̃(f) and f is clockwise oriented, then (proj2)^{∘}\overline{RightComp(f)} = (proj2)^{∘}L̃(f).

(41) If π_{1}g = N-min L̃(g), then RightComp(g) ⊆ RightComp(SpStSeq L̃(g)).

(42) If π_{1}g = N-min L̃(g), then \overline{RightComp(g)} is compact.

(43) If π_{1}g = N-min L̃(g), then LeftComp(g) is non Bounded.

(44) If π_{1}g = N-min L̃(g), then LeftComp(g) is outside component of L̃(g).

(45) If π_{1}g = N-min L̃(g), then RightComp(g) is inside component of L̃(g).

(46) If π_{1}g = N-min L̃(g), then UBD L̃(g) = LeftComp(g).

(47) If π_{1}g = N-min L̃(g), then BDD L̃(g) = RightComp(g).

(48) If π_{1}g = N-min L̃(g) and P is outside component of L̃(g), then P = LeftComp(g).

(49) If π_{1}g = N-min L̃(g) and P is inside component of L̃(g), then P ∩ RightComp(g) ≠ ∅.

(50) If π_{1}g = N-min L̃(g) and P is inside component of L̃(g), then P = BDD L̃(g).

(51) If π_{1}g = N-min L̃(g), then W-bound L̃(g) = W-boundRightComp(g).

(52) If π_{1}g = N-min L̃(g), then E-bound L̃(g) = E-boundRightComp(g).

(53) If π_{1}g = N-min L̃(g), then N-bound L̃(g) = N-boundRightComp(g).

(54) If π_{1}g = N-min L̃(g), then S-bound L̃(g) = S-boundRightComp(g).

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in the preparation of the article.

References

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

properties of left and right components 167

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

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Received May 5, 1999

Noetherian Lattices

Christoph Schwarzweller University of Tuebingen

Summary. In this article we define noetherian and co-noetherian lattices and show how some properties concerning upper and lower neighbours, irreducibility and density can be improved when restricted to these kinds of lattices. In addition we define atomic lattices.

MML Identifier: LATTICE6.

The notation and terminology used here are introduced in the following papers: [18], [13], [17], [14], [19], [7], [1], [8], [6], [20], [3], [9], [2], [10], [15], [16], [5], [11], [4], and [12].

Let us observe that there exists a lattice which is finite. Let us mention that every lattice which is finite is also complete. Let L be a lattice and let D be a subset of the carrier of L. The functor D^{·}

yields a subset of Poset(L) and is defined by: (Def. 1) D^{·}= {d^{·}; d ranges over elements of the carrier of L: d ∈ D}.

Let L be a lattice and let D be a subset of the carrier of Poset(L). The functor ^{·}D yielding a subset of the carrier of L is defined by:

(Def. 2) ^{·}D = {^{·}d; d ranges over elements of Poset(L): d ∈ D}. Let L be a finite lattice. Note that Poset(L) is well founded. Let L be a lattice. We say that L is noetherian if and only if:

(Def. 3) Poset(L) is well founded.

We say that L is co-noetherian if and only if: (Def. 4) Poset(L)^{⌣} is well founded.

One can verify the following observations:

∗ there exists a lattice which is noetherian and upper-bounded,

∗ there exists a lattice which is noetherian and lower-bounded, and

∗ there exists a lattice which is noetherian and complete.

One can verify the following observations:

∗ there exists a lattice which is co-noetherian and upper-bounded,

∗ there exists a lattice which is co-noetherian and lower-bounded, and ∗ there exists a lattice which is co-noetherian and complete. Next we state the proposition

(1) For every lattice L holds L is noetherian iff L^{∘} is co-noetherian.

One can check that every lattice which is finite is also noetherian and every lattice which is finite is also co-noetherian.

Let L be a lattice and let a, b be elements of the carrier of L. We say that a is-upper-neighbour-of b if and only if:

(Def. 5) a ≠ b and b ⊑ a and for every element c of the carrier of L such that

b ⊑ c and c ⊑ a holds c = a or c = b. We introduce b is-lower-neighbour-of a as a synonym of a is-upper-neighbour-of b.

We now state several propositions:

(2) Let L be a lattice, a be an element of the carrier of L, and b, c be elements of the carrier of L such that b ≠ c. Then

(i) if b is-upper-neighbour-of a and c is-upper-neighbour-of a, then a = c⊓b, and

(ii) if b is-lower-neighbour-of a and c is-lower-neighbour-of a, then a = c⊔b.

(3) Let L be a noetherian lattice, a be an element of the carrier of L, and d be an element of the carrier of L. Suppose a ⊑ d and a ≠ d. Then there exists an element c of the carrier of L such that c ⊑ d and c is-upper-neighbour-of a.

(4) Let L be a co-noetherian lattice, a be an element of the carrier of L, and d be an element of the carrier of L. Suppose d ⊑ a and a ≠ d. Then there exists an element c of the carrier of L such that d ⊑ c and c is-lower-neighbour-of a.

(5) Let L be an upper-bounded lattice. Then it is not true that there exists an element b of the carrier of L such that b is-upper-neighbour-of ⊤_{L}.

(6) Let L be a noetherian upper-bounded lattice and a be an element of the carrier of L. Then a = ⊤_{L} if and only if it is not true that there exists an element b of the carrier of L such that b is-upper-neighbour-of a.

(7) Let L be a lower-bounded lattice. Then it is not true that there exists an element b of the carrier of L such that b is-lower-neighbour-of ⊥_{L}.

(8) Let L be a co-noetherian lower-bounded lattice and a be an element of the carrier of L. Then a = ⊥_{L} if and only if it is not true that there exists an element b of the carrier of L such that b is-lower-neighbour-of a.

Let L be a complete lattice and let a be an element of the carrier of L. The functor a^{∗} yielding an element of the carrier of L is defined by:

(Def. 6) a^{∗} = ⌈−⌉_{L}{d; d ranges over elements of the carrier of L: a ⊑ d ∧ d ≠ a}.

The functor ^{∗}a yields an element of the carrier of L and is defined as follows: (Def. 7) ^{∗}a = ⨆_{L}{d; d ranges over elements of the carrier of L: d ⊑ a ∧ d ≠ a}.

Let L be a complete lattice and let a be an element of the carrier of L. We say that a is completely-meet-irreducible if and only if:

(Def. 8) a^{∗} ≠ a.

We say that a is completely-join-irreducible if and only if: (Def. 9) ^{∗}a ≠ a.

The following propositions are true:

(9) For every complete lattice L and for every element a of the carrier of L holds a ⊑ a^{∗} and ^{∗}a ⊑ a.

(10) For every complete lattice L holds (⊤_{L})^{∗} = ⊤_{L} and (⊤_{L})^{·} is meetirreducible.

(11) For every complete lattice L holds ^{∗}(⊥_{L}) = ⊥_{L} and (⊥_{L})^{·} is joinirreducible.

(12) Let L be a complete lattice and a be an element of the carrier of L.

Suppose a is completely-meet-irreducible. Then

(i) a^{∗} is-upper-neighbour-of a, and (ii) for every element c of the carrier of L such that c is-upper-neighbour-of

a holds c = a^{∗}.

(13) Let L be a complete lattice and a be an element of the carrier of L.

Suppose a is completely-join-irreducible. Then

(i) ^{∗}a is-lower-neighbour-of a, and (ii) for every element c of the carrier of L such that c is-lower-neighbour-of

a holds c = ^{∗}a.

(14) Let L be a noetherian complete lattice and a be an element of the carrier of L. Suppose a ≠ ⊤_{L}. Then a is completely-meet-irreducible if and only if there exists an element b of the carrier of L such that b is-upper-neighbour-of a and for every element c of the carrier of L such that c is-upper-neighbour-of a holds c = b.

(15) Let L be a co-noetherian complete lattice and a be an element of the carrier of L. Suppose a ≠ ⊥_{L}. Then a is completely-join-irreducible if and only if there exists an element b of the carrier of L such that b is-lower-neighbour-of a and for every element c of the carrier of L such that c is-lower-neighbour-of a holds c = b.

(16) Let L be a complete lattice and a be an element of the carrier of L. If a is completely-meet-irreducible, then a^{·} is meet-irreducible.

(17) Let L be a complete noetherian lattice and a be an element of the carrier of L. Suppose a ≠ ⊤_{L}. Then a is completely-meet-irreducible if and only if a^{·} is meet-irreducible.

(18) Let L be a complete lattice and a be an element of the carrier of L. If a is completely-join-irreducible, then a^{·} is join-irreducible.

(19) Let L be a complete co-noetherian lattice and a be an element of the carrier of L. Suppose a ≠ ⊥_{L}. Then a is completely-join-irreducible if and only if a^{·} is join-irreducible.

(20) Let L be a finite lattice and a be an element of the carrier of L such that

a ≠ ⊥_{L} and a ≠ ⊤_{L}. Then

(i) a is completely-meet-irreducible iff a^{·} is meet-irreducible, and (ii) a is completely-join-irreducible iff a^{·} is join-irreducible. Let L be a lattice and let a be an element of the carrier of L. We say that

a is atomic if and only if: (Def. 10) a is-upper-neighbour-of ⊥_{L}.

We say that a is co-atomic if and only if: (Def. 11) a is-lower-neighbour-of ⊤_{L}.

One can prove the following propositions:

(21) Let L be a complete lattice and a be an element of the carrier of L. If a is atomic, then a is completely-join-irreducible.

(22) Let L be a complete lattice and a be an element of the carrier of L. If a

is co-atomic, then a is completely-meet-irreducible. Let L be a lattice. We say that L is atomic if and only if the condition

(Def. 12) is satisfied. (Def. 12) Let a be an element of the carrier of L. Then there exists a subset X of

the carrier of L such that for every element x of the carrier of L such that x ∈ X holds x is atomic and a = ⨆_{L} X.

One can verify that there exists a lattice which is atomic and complete. Let L be a complete lattice and let D be a subset of L. We say that D is

supremum-dense if and only if: (Def. 13) For every element a of the carrier of L there exists a subset D′of D such

that a = ⨆_{L} D′. We say that D is infimum-dense if and only if:

(Def. 14) For every element a of the carrier of L there exists a subset D′of D such

that a = ⌈−⌉_{L}D′. One can prove the following propositions:

(23) Let L be a complete lattice and D be a subset of L. Then D is supremumdense if and only if for every element a of the carrier of L holds a = ⨆_{L}{d; d ranges over elements of the carrier of L: d ∈ D ∧ d ⊑ a}.

noetherian lattices 173

(24) Let L be a complete lattice and D be a subset of L. Then D is infimumdense if and only if for every element a of the carrier of L holds a = ⌈−⌉_{L}{d; d ranges over elements of the carrier of L: d ∈ D ∧ a ⊑ d}.

(25) Let L be a complete lattice and D be a subset of L. Then D is infimum-

dense if and only if D^{·} is order-generating. Let L be a complete lattice. The functor MIRRSL yields a subset of L and

is defined by: (Def. 15) MIRRSL = {a; a ranges over elements of the carrier of L: a is

completely-meet-irreducible}. The functor JIRRSL yielding a subset of L is defined by:

(Def. 16) JIRRSL = {a; a ranges over elements of the carrier of L: a is completely-

join-irreducible}. One can prove the following two propositions:

(26) For every complete lattice L and for every subset D of L such that D is supremum-dense holds JIRRSL ⊆ D.

(27) For every complete lattice L and for every subset D of L such that D is

infimum-dense holds MIRRSL ⊆ D. Let L be a co-noetherian complete lattice. Note that MIRRSL is infimum-

dense.

Let L be a noetherian complete lattice. One can check that JIRRSL is supremum-dense.

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174 christoph schwarzweller

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A Small Computer Model with Push-Down Stack^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. The SCMFSA computer can prove the correctness of many algorithms. Unfortunately, it cannot prove the correctness of recursive algorithms. For this reason, this article improves the SCMFSA computer and presents a Small Computer Model with Push-Down Stack (called SCMPDS for short). In addition to conventional arithmetic and ”goto” instructions, we increase two new instructions such as ”return” and ”save instruction-counter” in order to be able to design recursive programs.

MML Identifier: SCMPDS 1.

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

1. Preliminaries

For simplicity, we follow the rules: x_{1}, x_{2}, x_{3}, x_{4}, x_{5} are sets, i, j, k are natural numbers, I, I_{2}, I_{3}, I_{4} are elements of ℤ_{14}, i_{1} is an element of Instr-Loc_{SCM}, d_{1}, d_{2}, d_{3}, d_{4}, d_{5} are elements of Data-Loc_{SCM}, and k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6} are integers.

Let x_{1}, x_{2}, x_{3}, x_{4} be sets. The functor < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ > yields a set and is defined as follows:

(Def. 1) < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ >= ⟨x_{1}, x_{2}, x_{3}⟩^{⌢} ⟨x_{4}⟩.

Let x_{5} be a set. The functor < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ > yielding a set is defined by: (Def. 2) < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ >= ⟨x_{1}, x_{2}, x_{3}⟩^{⌢} ⟨x_{4}, x_{5}⟩.

1This work was done while the author visited Shinshu University March–April 1999.

Let x_{1}, x_{2}, x_{3}, x_{4} be sets. One can verify that < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ > is functionlike and relation-like. Let x_{5} be a set. One can verify that < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ > is function-like and relation-like.

Let x_{1}, x_{2}, x_{3}, x_{4} be sets. One can verify that < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ > is finite sequence-like. Let x_{5} be a set. One can check that < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ > is finite sequence-like.

Let D be a non empty set and let x_{1}, x_{2}, x_{3}, x_{4} be elements of D. Then < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ > is a finite sequence of elements of D.

Let D be a non empty set and let x_{1}, x_{2}, x_{3}, x_{4}, x_{5} be elements of D. Then < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ > is a finite sequence of elements of D.

One can prove the following propositions:

(1) < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ >= ⟨x_{1}, x_{2}, x_{3}⟩ ^{⌢} ⟨x_{4}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ >= ⟨x_{1}, x_{2}⟩ ^{⌢} ⟨x_{3}, x_{4}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ >= ⟨x_{1}⟩ ^{⌢} ⟨x_{2}, x_{3}, x_{4}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ >= ⟨x_{1}⟩^{⌢} ⟨x_{2}⟩^{⌢} ⟨x_{3}⟩^{⌢} ⟨x_{4}⟩.

(2) < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ >= ⟨x_{1}, x_{2}, x_{3}⟩^{⌢}⟨x_{4}, x_{5}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ > =< ∗x_{1}, x_{2}, x_{3}, x_{4}∗ > ^{⌢}⟨x_{5}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ >= ⟨x_{1}⟩ ^{⌢} ⟨x_{2}⟩ ^{⌢} ⟨x_{3}⟩^{⌢} ⟨x_{4}⟩^{⌢} ⟨x_{5}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ >= ⟨x_{1}, x_{2}⟩^{⌢} ⟨x_{3}, x_{4}, x_{5}⟩ and < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ >= ⟨x_{1}⟩^{⌢} < ∗x_{2}, x_{3}, x_{4}, x_{5}∗ > .

We adopt the following rules: N_{1} is a non empty set, y_{1}, y_{2}, y_{3}, y_{4}, y_{5} are elements of N_{1}, and p is a finite sequence.

We now state several propositions:

(3) p =< ∗x_{1}, x_{2}, x_{3}, x_{4}∗ > iff lenp = 4 and p(1) = x_{1} and p(2) = x_{2} and p(3) = x_{3} and p(4) = x_{4}.

(4) dom < ∗x_{1}, x_{2}, x_{3}, x_{4}∗ >= Seg4.

(5) p =< ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ > iff lenp = 5 and p(1) = x_{1} and p(2) = x_{2} and p(3) = x_{3} and p(4) = x_{4} and p(5) = x_{5}.

(6) dom < ∗x_{1}, x_{2}, x_{3}, x_{4}, x_{5}∗ >= Seg5.

(7) π_{1} < ∗y_{1}, y_{2}, y_{3}, y_{4}∗ >= y_{1} and π_{2} < ∗y_{1}, y_{2}, y_{3}, y_{4}∗ >= y_{2} and π_{3} < ∗y_{1}, y_{2}, y_{3}, y_{4}∗ >= y_{3} and π_{4} < ∗y_{1}, y_{2}, y_{3}, y_{4}∗ >= y_{4}.

(8) π_{1} < ∗y_{1}, y_{2}, y_{3}, y_{4}, y_{5}∗ >= y_{1} and π_{2} < ∗y_{1}, y_{2}, y_{3}, y_{4}, y_{5}∗ >= y_{2} and π_{3} < ∗y_{1}, y_{2}, y_{3}, y_{4}, y_{5}∗ >= y_{3} and π_{4} < ∗y_{1}, y_{2}, y_{3}, y_{4}, y_{5}∗ >= y_{4} and π_{5} < ∗y_{1}, y_{2}, y_{3}, y_{4}, y_{5}∗ >= y_{5}.

(9) For every integer k holds k ∈ ⋃{ℤ} ∪ ℕ.

(10) For every integer k holds k ∈ Data-Loc_{SCM} ∪ ℤ.

(11) For every element d of Data-Loc_{SCM} holds d ∈ Data-Loc_{SCM} ∪ ℤ. m

2. The Construction of SCM with Push-Down Stack

The subset SCMPDS − Instr of [: ℤ_{14}, (⋃{ℤ} ∪ ℕ)^{∗} :] is defined by the condition (Def. 3).

(Def. 3) SCMPDS − Instr = {⟨⟨0, ⟨l⟩⟩⟩ : l ranges over integers} ∪ {⟨⟨1, ⟨s_{1}⟩⟩⟩ : s_{1} ranges over elements of Data-Loc_{SCM}} ∪ {⟨⟨I, ⟨v, c⟩⟩⟩; I ranges over elements of ℤ_{14}, v ranges over elements of Data-Loc_{SCM}, c ranges over integers: I ∈ {2, 3}} ∪ {⟨⟨I, ⟨v, c_{1}, c_{2}⟩⟩⟩; I ranges over elements of ℤ_{14}, v ranges over elements of Data-Loc_{SCM}, c_{1} ranges over integers, c_{2} ranges over integers: I ∈ {4, 5, 6, 7, 8}} ∪ {⟨⟨I, < ∗v_{1}, v_{2}, c_{1}, c_{2}∗ > ⟩⟩; I ranges over elements of ℤ_{14}, v_{1} ranges over elements of Data-Loc_{SCM}, v_{2} ranges over elements of Data-Loc_{SCM}, c_{1} ranges over integers, c_{2} ranges over integers: I ∈ {9, 10, 11, 12, 13}}.

We now state two propositions:

(12) SCMPDS − Instr = {⟨⟨0, ⟨k_{1}⟩⟩⟩} ∪ {⟨⟨1, ⟨d_{1}⟩⟩⟩} ∪ {⟨⟨I_{2}, ⟨d_{2}, k_{2}⟩⟩⟩ : I_{2} ∈ {2, 3}} ∪ {⟨⟨I_{3}, ⟨d_{3}, k_{3}, k_{4}⟩⟩⟩: I_{3} ∈ {4, 5, 6, 7, 8}} ∪ {⟨⟨I_{4}, < ∗d_{4}, d_{5}, k_{5}, k_{6}∗ > ⟩⟩: I_{4} ∈ {9, 10, 11, 12, 13}}.

(13) ⟨⟨0, ⟨0⟩⟩⟩∈ SCMPDS − Instr .

One can verify that SCMPDS − Instr is non empty. We now state three propositions:

(14) k = 0 or there exists j such that k = 2 · j + 1 or there exists j such that k = 2 · j + 2.

(15) If k = 0, then it is not true that there exists j such that k = 2 · j + 1 and it is not true that there exists j such that k = 2 · j + 2.

(16)(i) If there exists j such that k = 2 · j + 1, then k ≠ 0 and it is not true

that there exists j such that k = 2 · j + 2, and (ii) if there exists j such that k = 2 · j + 2, then k ≠ 0 and it is not true

that there exists j such that k = 2 · j + 1. The function SCMPDS − OK from ℕ into {ℤ}∪{SCMPDS − Instr , Instr-Loc_{SCM}}

is defined as follows: (Def. 4) (SCMPDS − OK )(0) = Instr-Loc_{SCM} and for every natural number k

holds (SCMPDS − OK )(2 · k + 1) = ℤ and (SCMPDS − OK )(2 · k + 2) = SCMPDS − Instr .

A SCMPDS-State is an element of ∏ SCMPDS − OK . Next we state several propositions:

(17) Instr-Loc_{SCM} ≠ SCMPDS − Instr and SCMPDS − Instr ≠ ℤ.

(18) (SCMPDS − OK )(i) = Instr-Loc_{SCM} iff i = 0.

(19) (SCMPDS − OK )(i) = ℤ iff there exists k such that i = 2 · k + 1.

(20) (SCMPDS − OK )(i) = SCMPDS − Instr iff there exists k such that i = 2 · k + 2.

(21) (SCMPDS − OK )(d_{1}) = ℤ.

(22) (SCMPDS − OK )(i_{1}) = SCMPDS − Instr .

(23) π_{0} ∏ SCMPDS − OK = Instr-Loc_{SCM}.

(24) π_{d_{1}} ∏ SCMPDS − OK = ℤ.

(25) π_{i_{1}} ∏ SCMPDS − OK = SCMPDS − Instr .

Let s be a SCMPDS-State. The functor IC_{s} yielding an element of Instr-Loc_{SCM} is defined as follows:

(Def. 5) IC_{s} = s(0).

Let s be a SCMPDS-State and let u be an element of Instr-Loc_{SCM}. The functor Chg_{SCM}(s, u) yielding a SCMPDS-State is defined as follows:

(Def. 6) Chg_{SCM}(s, u) = s+·(0⟼→ . u).

We now state three propositions:

(26) For every SCMPDS-State s and for every element u of Instr-Loc_{SCM} holds (Chg_{SCM}(s, u))(0) = u.

(27) For every SCMPDS-State s and for every element u of Instr-Loc_{SCM} and for every element m_{1} of Data-Loc_{SCM} holds (Chg_{SCM}(s, u))(m_{1}) = s(m_{1}).

(28) For every SCMPDS-State s and for all elements u, v of Instr-Loc_{SCM}

holds (Chg_{SCM}(s, u))(v) = s(v). Let s be a SCMPDS-State, let t be an element of Data-Loc_{SCM}, and let u be

an integer. The functor Chg_{SCM}(s, t, u) yields a SCMPDS-State and is defined as follows:

(Def. 7) Chg_{SCM}(s, t, u) = s+·(t⟼→ . u).

The following propositions are true:

(29) For every SCMPDS-State s and for every element t of Data-Loc_{SCM} and for every integer u holds (Chg_{SCM}(s, t, u))(0) = s(0).

(30) For every SCMPDS-State s and for every element t of Data-Loc_{SCM} and for every integer u holds (Chg_{SCM}(s, t, u))(t) = u.

(31) Let s be a SCMPDS-State, t be an element of Data-Loc_{SCM}, u be an integer, and m_{1} be an element of Data-Loc_{SCM}. If m_{1} ≠ t, then (Chg_{SCM}(s, t, u))(m_{1}) = s(m_{1}).

(32) Let s be a SCMPDS-State, t be an element of Data-Loc_{SCM}, u be an integer, and v be an element of Instr-Loc_{SCM}. Then (Chg_{SCM}(s, t, u))(v) = s(v).

Let s be a SCMPDS-State and let a be an element of Data-Loc_{SCM}. Then s(a) is an integer.

Let s be a SCMPDS-State, let a be an element of Data-Loc_{SCM}, and let n be an integer. The functor Address Add(s, a, n) yields an element of Data-Loc_{SCM} and is defined by:

(Def. 8) Address Add(s, a, n) = 2 · |s(a) + n| + 1.

Let s be a SCMPDS-State and let n be an integer. The functor jump address(s, n) yielding an element of Instr-Loc_{SCM} is defined as follows:

(Def. 9) jump address(s, n) = |((IC_{s} qua natural number)−2) + 2 · n| + 2.

Let d be an element of Data-Loc_{SCM} and let s be an integer. Then ⟨d, s⟩ is a finite sequence of elements of Data-Loc_{SCM} ∪ ℤ.

Let x be an element of SCMPDS − Instr . Let us assume that there exist an element m_{1} of Data-Loc_{SCM} and I such that x = ⟨⟨I, ⟨m_{1}⟩⟩⟩. The functor x address_{1} yielding an element of Data-Loc_{SCM} is defined as follows:

(Def. 10) There exists a finite sequence f of elements of Data-Loc_{SCM} such that

f = x_{2} and x address_{1} = π_{1}f. The following proposition is true

(33) For every element x of SCMPDS − Instr and for every element m_{1} of

Data-Loc_{SCM} such that x = ⟨⟨I, ⟨m_{1}⟩⟩⟩ holds x address_{1} = m_{1}. Let x be an element of SCMPDS − Instr . Let us assume that there exist

an integer r and I such that x = ⟨⟨I, ⟨r⟩⟩⟩. The functor x const INT yielding an integer is defined by:

(Def. 11) There exists a finite sequence f of elements of ℤ such that f = x_{2} and

x const INT = π_{1}f. The following proposition is true

(34) For every element x of SCMPDS − Instr and for every integer k such

that x = ⟨⟨I, ⟨k⟩⟩⟩ holds x const INT = k. Let x be an element of SCMPDS − Instr . Let us assume that there exist an

element m_{1} of Data-Loc_{SCM}, an integer r, and I such that x = ⟨⟨I, ⟨m_{1}, r⟩⟩⟩. The functor x P21address yielding an element of Data-Loc_{SCM} is defined as follows:

(Def. 12) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P21address = π_{1}f. The functor x P22const yielding an integer is defined as follows:

(Def. 13) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P22const = π_{2}f. The following proposition is true

(35) Let x be an element of SCMPDS − Instr , m_{1} be an element of Data-Loc_{SCM}, and r be an integer. If x = ⟨⟨I, ⟨m_{1}, r⟩⟩⟩, then x P21address = m_{1} and x P22const = r.

Let x be an element of SCMPDS − Instr . Let us assume that there exist an element m_{2} of Data-Loc_{SCM}, integers k_{1}, k_{2}, and I such that x = ⟨⟨I, ⟨m_{2}, k_{1}, k_{2}⟩⟩⟩. The functor x P31address yielding an element of Data-Loc_{SCM} is defined as follows:

(Def. 14) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P31address = π_{1}f. The functor x P32const yielding an integer is defined as follows:

(Def. 15) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such that f = x_{2} and x P32const = π_{2}f.

The functor x P33const yields an integer and is defined by: (Def. 16) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P33const = π_{3}f. We now state the proposition

(36) Let x be an element of SCMPDS − Instr , d_{1} be an element of Data-Loc_{SCM}, and k_{1}, k_{2} be integers. If x = ⟨⟨I, ⟨d_{1}, k_{1}, k_{2}⟩⟩⟩, then x P31address = d_{1} and x P32const = k_{1} and x P33const = k_{2}.

Let x be an element of SCMPDS − Instr . Let us assume that there exist elements m_{2}, m_{3} of Data-Loc_{SCM}, integers k_{1}, k_{2}, and I such that x = ⟨⟨I, < ∗m_{2}, m_{3}, k_{1}, k_{2}∗ > ⟩⟩. The functor x P41address yields an element of Data-Loc_{SCM} and is defined by:

(Def. 17) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P41address = π_{1}f. The functor x P42address yields an element of Data-Loc_{SCM} and is defined as follows:

(Def. 18) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P42address = π_{2}f. The functor x P43const yielding an integer is defined as follows:

(Def. 19) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P43const = π_{3}f. The functor x P44const yielding an integer is defined as follows:

(Def. 20) There exists a finite sequence f of elements of Data-Loc_{SCM} ∪ ℤ such

that f = x_{2} and x P44const = π_{4}f. We now state the proposition

(37) Let x be an element of SCMPDS − Instr , d_{1}, d_{2} be elements of Data-Loc_{SCM}, and k_{1}, k_{2} be integers. If x = ⟨⟨I, < ∗d_{1}, d_{2}, k_{1}, k_{2}∗ > ⟩⟩, then x P41address = d_{1} and x P42address = d_{2} and x P43const = k_{1} and x P44const = k_{2}.

Let s be a SCMPDS-State and let a be an element of Data-Loc_{SCM}. The functor PopInstrLoc(s, a) yielding an element of Instr-Loc_{SCM} is defined as follows:

(Def. 21) PopInstrLoc(s, a) = 2 · (|s(a)| ÷ 2) + 4.

The natural number RetSP is defined as follows: (Def. 22) RetSP = 0.

The natural number RetIC is defined as follows: (Def. 23) RetIC = 1.

Let x be an element of SCMPDS − Instr and let s be a SCMPDS-State. The functor Exec-Res_{SCM}(x, s) yielding a SCMPDS-State is defined as follows:

(Def. 24) Exec-Res_{SCM}(x, s) = Chg_{SCM}(s, jump address(s, x const INT)), if there exists k_{1} such that x = ⟨⟨0, ⟨k_{1}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, x P21address, x P22const), Next(IC_{s})), if there exist d_{1}, k_{1} such that x = ⟨⟨2, ⟨d_{1}, k_{1}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P21address, x P22const), (IC_{s} qua natural

number)), Next(IC_{s})), if there exist d_{1}, k_{1} such that x = ⟨⟨3, ⟨d_{1}, k_{1}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, x address_{1}, s(Address Add(s, x address_{1}, RetSP))), PopInstrLoc (s, Address Add(s, x address_{1}, RetIC))), if there exists d_{1} such that x = ⟨⟨1, ⟨d_{1}⟩⟩⟩, Chg_{SCM}(s, (s(Address Add(s, x P31address, x P32const)) = 0 → Next(IC_{s}), jump address(s, x P33const))), if there exist d_{1}, k_{1}, k_{2} such that x = ⟨⟨4, ⟨d_{1}, k_{1}, k_{2}⟩⟩⟩, Chg_{SCM}(s, (s(Address Add(s, x P31address, x P32const)) > 0 → Next(IC_{s}), jump address(s, x P33const))), if there exist d_{1}, k_{1}, k_{2} such that x = ⟨⟨5, ⟨d_{1}, k_{1}, k_{2}⟩⟩⟩, Chg_{SCM}(s, (0 > s(Address Add(s, x P31address, x P32const)) → Next(IC_{s}), jump address(s, x P33const))), if there exist d_{1}, k_{1}, k_{2} such that x = ⟨⟨6, ⟨d_{1}, k_{1}, k_{2}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P31address, x P32const), x P33const), Next(IC_{s})), if there exist d_{1}, k_{1}, k_{2} such that x = ⟨⟨7, ⟨d_{1}, k_{1}, k_{2}⟩⟩⟩, Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P31address, x P32const), s(Address Add(s, x P31address, x P32const)) + x P33const), Next(IC_{s})), if there exist d_{1}, k_{1}, k_{2} such that x = ⟨⟨8, ⟨d_{1}, k_{1}, k_{2}⟩⟩⟩,

Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P41address, x P43const), s(Address Add (s, x P41address, x P43const)) + s(Address Add(s, x P42address, x P44const))), Next(IC_{s})), if there exist d_{1}, d_{2}, k_{1}, k_{2} such that x = ⟨⟨9, < ∗d_{1}, d_{2}, k_{1}, k_{2}∗ > ⟩⟩, Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P41address, x P43const), s(Address Add (s, x P41address, x P43const)) − s(Address Add(s, x P42address, x P44const))), Next(IC_{s})), if there exist d_{1}, d_{2}, k_{1}, k_{2} such that x = ⟨⟨10, < ∗d_{1}, d_{2}, k_{1}, k_{2}∗ > ⟩⟩, Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P41address, x P43const), s(Address Add (s, x P41address, x P43const)) · s(Address Add(s, x P42address, x P44const))), Next(IC_{s})), if there exist d_{1}, d_{2}, k_{1}, k_{2} such that x = ⟨⟨11, < ∗d_{1}, d_{2}, k_{1}, k_{2}∗ > ⟩⟩, Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P41address, x P43const), s(Address Add(s, x P42address, x P44const))), Next(IC_{s})), if there exist d_{1}, d_{2}, k_{1}, k_{2} such that x = ⟨⟨13, < ∗d_{1}, d_{2}, k_{1}, k_{2}∗ > ⟩⟩, Chg_{SCM}(Chg_{SCM}(Chg_{SCM}(s, Address Add(s, x P41address, x P43const), s(Address Add(s, x P41address, x P43const)) ÷ s(Address Add(s, x P42address, x P44const))), Address Add(s, x P42address, x P44const), s(Address Add(s, x P41address, x P43const)) mod s(Address Add(s, x P42address, x P44const))), Next(IC_{s})), if there exist d_{1}, d_{2}, k_{1}, k_{2} such that x = ⟨⟨12, < ∗d_{1}, d_{2}, k_{1}, k_{2}∗ > ⟩⟩, s, otherwise. Let f be a function from SCMPDS − Instr into (∏ SCMPDS − OK )^{∏ SCMPDS−OK} and let x be an element of SCMPDS − Instr .

Note that f(x) is function-like and relation-like. The function SCMPDS − Exec from SCMPDS − Instr into

(∏ SCMPDS − OK )^{∏ SCMPDS−OK} is defined by: (Def. 25) For every element x of SCMPDS − Instr and for every SCMPDS-State

y holds (SCMPDS − Exec )(x)(y) = Exec-Res_{SCM}(x, y).

Acknowledgments

We wish to thank Prof. Y. Nakamura for many helpful suggestions.

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The SCMPDS Computer and the Basic Semantics of its Instructions^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. The article defines the SCMPDS computer and its instructions. The SCMPDS computer consists of such instructions as conventional arithmetic, ”goto”, ”return” and ”save instruction-counter” (”saveIC” for short). The address used in the ”goto” instruction is an offset value rather than a pointer in the standard sense. Thus, we don’t define halting instruction directly but define it by ”goto 0” instruction. The ”saveIC” and ”return” equal almost call and return statements in the usual high programming language. Theoretically, the SCMPDS computer can implement all algorithms described by the usual high programming language including recursive routine. In addition, we describe the execution semantics and halting properties of each instruction.

MML Identifier: SCMPDS 2.

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

1. The SCMPDS Computer

In this paper x denotes a set and i, k denote natural numbers. The strict AMI SCMPDS over {ℤ} is defined as follows:

(Def. 1) SCMPDS = ⟨ℕ, 0, Instr-Loc_{SCM}, ℤ_{14}, SCMPDS − Instr , SCMPDS − OK , SCMPDS − Exec ⟩.

Next we state three propositions:

1This work was done while the author visited Shinshu University March–April 1999.

(1) There exists k such that x = 2 · k + 2 iff x ∈ Instr-Loc_{SCM}.

(2) SCMPDS is data-oriented.

(3) SCMPDS is definite. Let us note that SCMPDS is von Neumann data-oriented and definite. The following two propositions are true: (4)(i) The instruction locations of SCMPDS ≠ ℤ, (ii) the instructions of SCMPDS ≠ ℤ, and (iii) the instruction locations of SCMPDS ≠ the instructions of SCMPDS.

(5) ℕ = {0} ∪ Data-Loc_{SCM} ∪ Instr-Loc_{SCM}. In the sequel s is a state of SCMPDS. One can prove the following propositions:

(6) IC_{SCMPDS} = 0.

(7) For every SCMPDS-State S such that S = s holds IC_{s} = IC_{S}.

2. The Memory Structure

An object of SCMPDS is called a Int position if: (Def. 2) It ∈ Data-Loc_{SCM}.

In the sequel d_{1} denotes a Int position. The following propositions are true:

(8) d_{1} ∈ Data-Loc_{SCM}.

(9) If x ∈ Data-Loc_{SCM}, then x is a Int position.

(10) Data-Loc_{SCM} misses the instruction locations of SCMPDS.

(11) The instruction locations of SCMPDS are infinite.

(12) Every Int position is a data-location.

(13) For every Int position l holds ObjectKind(l) = ℤ.

(14) For every set x such that x ∈ Instr-Loc_{SCM} holds x is an instructionlocation of SCMPDS.

3. The Instruction Structure

We use the following convention: d_{2}, d_{3}, d_{4}, d_{5}, d_{6} are elements of Data-Loc_{SCM} and k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6} are integers.

Let I be an instruction of SCMPDS. The functor InsCode(I) yields a natural number and is defined by:

(Def. 3) InsCode(I) = I_{1}.

In the sequel I is an instruction of SCMPDS.

Next we state the proposition

(15) For every instruction I of SCMPDS holds InsCode(I) 13.

Let s be a state of SCMPDS and let d be a Int position. Then s(d) is an integer.

Let m, n be integers. The functor DataLoc(m, n) yields a Int position and is defined as follows:

(Def. 4) DataLoc(m, n) = 2 · |m + n| + 1.

One can prove the following propositions:

(16) ⟨⟨0, ⟨k_{1}⟩⟩⟩∈ SCMPDS − Instr .

(17) ⟨⟨1, ⟨d_{2}⟩⟩⟩∈ SCMPDS − Instr .

(18) If x ∈ {2, 3}, then ⟨⟨x, ⟨d_{3}, k_{2}⟩⟩⟩∈ SCMPDS − Instr .

(19) If x ∈ {4, 5, 6, 7, 8}, then ⟨⟨x, ⟨d_{4}, k_{3}, k_{4}⟩⟩⟩∈ SCMPDS − Instr .

(20) If x ∈ {9, 10, 11, 12, 13}, then ⟨⟨x, < ∗d_{5}, d_{6}, k_{5}, k_{6}∗ > ⟩⟩ ∈

SCMPDS − Instr . In the sequel a, b, c are Int position. Let us consider k_{1}. The functor goto k_{1} yielding an instruction of SCMPDS

is defined as follows: (Def. 5) goto k_{1} = ⟨⟨0, ⟨k_{1}⟩⟩⟩.

Let us consider a. The functor returna yields an instruction of SCMPDS and is defined by:

(Def. 6) returna = ⟨⟨1, ⟨a⟩⟩⟩.

Let us consider a, k_{1}. The functor a:=k_{1} yields an instruction of SCMPDS and is defined as follows:

(Def. 7) a:=k_{1} = ⟨⟨2, ⟨a, k_{1}⟩⟩⟩.

The functor saveIC(a, k_{1}) yields an instruction of SCMPDS and is defined as follows:

(Def. 8) saveIC(a, k_{1}) = ⟨⟨3, ⟨a, k_{1}⟩⟩⟩.

Let us consider a, k_{1}, k_{2}. The functor (a, k_{1}) <> 0 gotok_{2} yields an instruction of SCMPDS and is defined as follows:

(Def. 9) (a, k_{1}) <> 0 gotok_{2} = ⟨⟨4, ⟨a, k_{1}, k_{2}⟩⟩⟩. The functor (a, k_{1}) <= 0 gotok_{2} yielding an instruction of SCMPDS is defined as follows:

(Def. 10) (a, k_{1}) <= 0 gotok_{2} = ⟨⟨5, ⟨a, k_{1}, k_{2}⟩⟩⟩. The functor (a, k_{1}) >= 0 gotok_{2} yielding an instruction of SCMPDS is defined by:

(Def. 11) (a, k_{1}) >= 0 gotok_{2} = ⟨⟨6, ⟨a, k_{1}, k_{2}⟩⟩⟩. The functor a_{k_{1}}:=k_{2} yielding an instruction of SCMPDS is defined as follows:

(Def. 12) a_{k_{1}}:=k_{2} = ⟨⟨7, ⟨a, k_{1}, k_{2}⟩⟩⟩.

The functor AddTo(a, k_{1}, k_{2}) yielding an instruction of SCMPDS is defined by: (Def. 13) AddTo(a, k_{1}, k_{2}) = ⟨⟨8, ⟨a, k_{1}, k_{2}⟩⟩⟩.

Let us consider a, b, k_{1}, k_{2}. The functor AddTo(a, k_{1}, b, k_{2}) yields an instruction of SCMPDS and is defined by:

(Def. 14) AddTo(a, k_{1}, b, k_{2}) = ⟨⟨9, < ∗a, b, k_{1}, k_{2}∗ > ⟩⟩. The functor SubFrom(a, k_{1}, b, k_{2}) yielding an instruction of SCMPDS is defined by:

(Def. 15) SubFrom(a, k_{1}, b, k_{2}) = ⟨⟨10, < ∗a, b, k_{1}, k_{2}∗ > ⟩⟩. The functor MultBy(a, k_{1}, b, k_{2}) yielding an instruction of SCMPDS is defined as follows:

(Def. 16) MultBy(a, k_{1}, b, k_{2}) = ⟨⟨11, < ∗a, b, k_{1}, k_{2}∗ > ⟩⟩. The functor Divide(a, k_{1}, b, k_{2}) yielding an instruction of SCMPDS is defined by:

(Def. 17) Divide(a, k_{1}, b, k_{2}) = ⟨⟨12, < ∗a, b, k_{1}, k_{2}∗ > ⟩⟩.

The functor (a, k_{1}) := (b, k_{2}) yielding an instruction of SCMPDS is defined by: (Def. 18) (a, k_{1}) := (b, k_{2}) = ⟨⟨13, < ∗a, b, k_{1}, k_{2}∗ > ⟩⟩.

One can prove the following propositions:

(21) InsCode(goto k_{1}) = 0.

(22) InsCode(returna) = 1.

(23) InsCode(a:=k_{1}) = 2.

(24) InsCode(saveIC(a, k_{1})) = 3.

(25) InsCode((a, k_{1}) <> 0 gotok_{2}) = 4.

(26) InsCode((a, k_{1}) <= 0 gotok_{2}) = 5.

(27) InsCode((a, k_{1}) >= 0 gotok_{2}) = 6.

(28) InsCode(a_{k_{1}}:=k_{2}) = 7.

(29) InsCode(AddTo(a, k_{1}, k_{2})) = 8.

(30) InsCode(AddTo(a, k_{1}, b, k_{2})) = 9.

(31) InsCode(SubFrom(a, k_{1}, b, k_{2})) = 10.

(32) InsCode(MultBy(a, k_{1}, b, k_{2})) = 11.

(33) InsCode(Divide(a, k_{1}, b, k_{2})) = 12.

(34) InsCode((a, k_{1}) := (b, k_{2})) = 13.

(35) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 0 there exists k_{1} such that i_{1} = goto k_{1}.

(36) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 1 there exists a such that i_{1} = returna.

(37) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 2 there exist a, k_{1} such that i_{1} = a:=k_{1}.

(38) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 3 there exist a, k_{1} such that i_{1} = saveIC(a, k_{1}).

(39) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 4 there exist a, k_{1}, k_{2} such that i_{1} = (a, k_{1}) <> 0 gotok_{2}.

(40) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 5 there exist a, k_{1}, k_{2} such that i_{1} = (a, k_{1}) <= 0 gotok_{2}.

(41) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 6 there exist a, k_{1}, k_{2} such that i_{1} = (a, k_{1}) >= 0 gotok_{2}.

(42) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 7 there exist a, k_{1}, k_{2} such that i_{1} = a_{k_{1}}:=k_{2}.

(43) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 8 there exist a, k_{1}, k_{2} such that i_{1} = AddTo(a, k_{1}, k_{2}).

(44) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 9 there exist a, b, k_{1}, k_{2} such that i_{1} = AddTo(a, k_{1}, b, k_{2}).

(45) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 10 there exist a, b, k_{1}, k_{2} such that i_{1} = SubFrom(a, k_{1}, b, k_{2}).

(46) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 11 there exist a, b, k_{1}, k_{2} such that i_{1} = MultBy(a, k_{1}, b, k_{2}).

(47) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 12 there exist a, b, k_{1}, k_{2} such that i_{1} = Divide(a, k_{1}, b, k_{2}).

(48) For every instruction i_{1} of SCMPDS such that InsCode(i_{1}) = 13 there exist a, b, k_{1}, k_{2} such that i_{1} = (a, k_{1}) := (b, k_{2}).

(49) For every state s of SCMPDS and for every Int position d holds d ∈ doms.

(50) For every state s of SCMPDS holds Data-Loc_{SCM} ⊆ doms.

(51) For every state s of SCMPDS holds dom(s↾Data-Loc_{SCM}) = Data-Loc_{SCM}.

(52) For every Int position d_{7} holds d_{7} ≠ IC_{SCMPDS} .

(53) For every instruction-location i_{2} of SCMPDS and for every Int position d_{7} holds i_{2} ≠ d_{7}.

(54) Let s_{1}, s_{2} be states of SCMPDS. Suppose IC_{(s_{1})} = IC_{(s_{2})} and for every Int position a holds s_{1}(a) = s_{2}(a) and for every instruction-location i of SCMPDS holds s_{1}(i) = s_{2}(i). Then s_{1} = s_{2}.

Let l_{1} be an instruction-location of SCMPDS. The functor Next(l_{1}) yields an instruction-location of SCMPDS and is defined by:

(Def. 19) There exists an element m_{1} of Instr-Loc_{SCM} such that m_{1} = l_{1} and Next(l_{1}) = Next(m_{1}).

One can prove the following propositions:

(55) For every instruction-location l_{1} of SCMPDS and for every element m_{1} of Instr-Loc_{SCM} such that m_{1} = l_{1} holds Next(m_{1}) = Next(l_{1}).

(56) For every element i of SCMPDS − Instr such that i = I and for every SCMPDS-State S such that S = s holds Exec(I, s) = Exec-Res_{SCM}(i, S).

4. Execution Semantics of the SCMPDS instructions

The following propositions are true:

(57) (Exec(a:=k_{1}, s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec(a:=k_{1}, s))(a) = k_{1} and for every b such that b ≠ a holds (Exec(a:=k_{1}, s))(b) = s(b).

(58) (Exec(a_{k_{1}}:=k_{2}, s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec(a_{k_{1}}:=k_{2}, s)) (DataLoc(s(a), k_{1})) = k_{2} and for every b such that b ≠ DataLoc(s(a), k_{1}) holds (Exec(a_{k_{1}}:=k_{2}, s))(b) = s(b).

(59) (Exec((a, k_{1}) := (b, k_{2}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec((a, k_{1}) := (b, k_{2}), s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(b), k_{2})) and for every c such that c ≠ DataLoc(s(a), k_{1}) holds (Exec((a, k_{1}) := (b, k_{2}), s))(c) = s(c).

(60) (Exec(AddTo(a, k_{1}, k_{2}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec(AddTo (a, k_{1}, k_{2}), s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(a), k_{1})) + k_{2} and for every b such that b ≠ DataLoc(s(a), k_{1}) holds (Exec(AddTo(a, k_{1}, k_{2}), s))(b) = s(b).

(61) (Exec(AddTo(a, k_{1}, b, k_{2}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec(AddTo (a, k_{1}, b, k_{2}), s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(a), k_{1}))+s(DataLoc(s(b), k_{2})) and for every c such that c ≠ DataLoc(s(a), k_{1}) holds (Exec(AddTo(a, k_{1}, b, k_{2}), s))(c) = s(c).

(62) (Exec(SubFrom(a, k_{1}, b, k_{2}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec (SubFrom(a, k_{1}, b, k_{2}), s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(a), k_{1})) − s(DataLoc(s(b), k_{2})) and for every c such that c ≠ DataLoc(s(a), k_{1}) holds (Exec(SubFrom(a, k_{1}, b, k_{2}), s))(c) = s(c).

(63) (Exec(MultBy(a, k_{1}, b, k_{2}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec (MultBy(a, k_{1}, b, k_{2}), s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(a), k_{1})) · s(DataLoc(s(b), k_{2})) and for every c such that c ≠ DataLoc(s(a), k_{1}) holds (Exec(MultBy(a, k_{1}, b, k_{2}), s))(c) = s(c).

(64)(i) (Exec(Divide(a, k_{1}, b, k_{2}), s))(IC_{SCMPDS} ) = Next(IC_{s}), (ii) if DataLoc(s(a), k_{1}) ≠ DataLoc(s(b), k_{2}), then (Exec(Divide(a, k_{1}, b, k_{2}),

s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(a), k_{1})) ÷ s(DataLoc(s(b), k_{2})), (iii) (Exec(Divide(a, k_{1}, b, k_{2}), s))(DataLoc(s(b), k_{2})) = s(DataLoc(s(a), k_{1}))

mod s(DataLoc(s(b), k_{2})), and (iv) for every c such that c ≠ DataLoc(s(a), k_{1}) and c ≠ DataLoc(s(b), k_{2})

holds (Exec(Divide(a, k_{1}, b, k_{2}), s))(c) = s(c).

(65) (Exec(Divide(a, k_{1}, a, k_{1}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec(Divide (a, k_{1}, a, k_{1}), s))(DataLoc(s(a), k_{1})) = s(DataLoc(s(a), k_{1}))mods(DataLoc (s(a), k_{1})) and for every c such that c ≠ DataLoc(s(a), k_{1}) holds (Exec(Divide(a, k_{1}, a, k_{1}), s))(c) = s(c).

Let s be a state of SCMPDS and let c be an integer. The functor ICplusConst(s, c) yields an instruction-location of SCMPDS and is defined by:

(Def. 20) There exists a natural number m such that m = IC_{s} and

ICplusConst(s, c) = |(m − 2) + 2 · c| + 2. The following propositions are true:

(66) (Exec(goto k_{1}, s))(IC_{SCMPDS} ) = ICplusConst(s, k_{1}) and for every a holds (Exec(goto k_{1}, s))(a) = s(a).

(67) If s(DataLoc(s(a), k_{1})) ≠ 0, then (Exec((a, k_{1}) <> 0 gotok_{2}, s))(IC_{SCMPDS} ) = ICplusConst(s, k_{2}) and if s(DataLoc(s(a), k_{1})) = 0, then (Exec((a, k_{1}) <> 0 gotok_{2}, s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec((a, k_{1}) <> 0 gotok_{2}, s))(b) = s(b).

(68) If s(DataLoc(s(a), k_{1})) 0, then (Exec((a, k_{1}) <= 0 gotok_{2}, s))(IC_{SCMPDS} ) = ICplusConst(s, k_{2}) and if s(DataLoc(s(a), k_{1})) > 0, then (Exec((a, k_{1}) <= 0 gotok_{2}, s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec((a, k_{1}) <= 0 gotok_{2}, s))(b) = s(b).

(69) If s(DataLoc(s(a), k_{1})) 0, then (Exec((a, k_{1}) >= 0 gotok_{2}, s))(IC_{SCMPDS} ) = ICplusConst(s, k_{2}) and if s(DataLoc(s(a), k_{1})) < 0, then (Exec((a, k_{1}) >= 0 gotok_{2}, s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec((a, k_{1}) >= 0 gotok_{2}, s))(b) = s(b).

(70) (Exec(returna, s))(IC_{SCMPDS} ) = 2 · (|s(DataLoc(s(a), RetIC))| ÷ 2) + 4 and (Exec(returna, s))(a) = s(DataLoc(s(a), RetSP)) and for every b such that a ≠ b holds (Exec(returna, s))(b) = s(b).

(71) (Exec(saveIC(a, k_{1}), s))(IC_{SCMPDS} ) = Next(IC_{s}) and (Exec(saveIC(a, k_{1}), s))(DataLoc(s(a), k_{1})) = IC_{s} and for every b such that DataLoc(s(a), k_{1}) ≠ b holds (Exec(saveIC(a, k_{1}), s))(b) = s(b).

(72) For every integer k there exists a function f from Data-Loc_{SCM} into ℤ such that for every element x of Data-Loc_{SCM} holds f(x) = k.

(73) For every integer k there exists a state s of SCMPDS such that for every Int position d holds s(d) = k.

(74) Let k be an integer and l_{1} be an instruction-location of SCMPDS. Then there exists a state s of SCMPDS such that s(0) = l_{1} and for every Int position d holds s(d) = k.

(75) goto 0 is halting.

(76) For every instruction I of SCMPDS such that there exists s such that (Exec(I, s))(IC_{SCMPDS} ) = Next(IC_{s}) holds I is non halting.

(77) a:=k_{1} is non halting.

(78) a_{k_{1}}:=k_{2} is non halting.

(79) (a, k_{1}) := (b, k_{2}) is non halting.

(80) AddTo(a, k_{1}, k_{2}) is non halting.

(81) AddTo(a, k_{1}, b, k_{2}) is non halting.

(82) SubFrom(a, k_{1}, b, k_{2}) is non halting.

(83) MultBy(a, k_{1}, b, k_{2}) is non halting.

(84) Divide(a, k_{1}, b, k_{2}) is non halting.

(85) If k_{1} ≠ 0, then goto k_{1} is non halting.

(86) (a, k_{1}) <> 0 gotok_{2} is non halting.

(87) (a, k_{1}) <= 0 gotok_{2} is non halting.

(88) (a, k_{1}) >= 0 gotok_{2} is non halting.

(89) returna is non halting.

(90) saveIC(a, k_{1}) is non halting.

(91) Let I be a set. Then I is an instruction of SCMPDS if and only if one of the following conditions is satisfied: there exists k_{1} such that I = goto k_{1} or there exists a such that I = returna or there exist a, k_{1} such that I = saveIC(a, k_{1}) or there exist a, k_{1} such that I = a:=k_{1} or there exist a, k_{1}, k_{2} such that I = a_{k_{1}}:=k_{2} or there exist a, k_{1}, k_{2} such that I = (a, k_{1}) <> 0 gotok_{2} or there exist a, k_{1}, k_{2} such that I = (a, k_{1}) <= 0 gotok_{2} or there exist a, k_{1}, k_{2} such that I = (a, k_{1}) >= 0 gotok_{2} or there exist a, b, k_{1}, k_{2} such that I = AddTo(a, k_{1}, k_{2}) or there exist a, b, k_{1}, k_{2} such that I = AddTo(a, k_{1}, b, k_{2}) or there exist a, b, k_{1}, k_{2} such that I = SubFrom(a, k_{1}, b, k_{2}) or there exist a, b, k_{1}, k_{2} such that I = MultBy(a, k_{1}, b, k_{2}) or there exist a, b, k_{1}, k_{2} such that I = Divide(a, k_{1}, b, k_{2}) or there exist a, b, k_{1}, k_{2} such that I = (a, k_{1}) := (b, k_{2}).

Let us observe that SCMPDS is halting.

We now state several propositions:

(92) For every instruction I of SCMPDS such that I is halting holds I = halt_{SCMPDS} .

(93) halt_{SCMPDS} = goto 0.

(94) Exec(halt_{SCMPDS} , s) = s.

(95) For every state s of SCMPDS and for every instruction-location i of SCMPDS holds s(i) is an instruction of SCMPDS.

(96) For every state s of SCMPDS and for every instruction i of SCMPDS and for every instruction-location l of SCMPDS holds (Exec(i, s))(l) = s(l).

the scmpds computer and the basic … 191

(97) SCMPDS is realistic.

Let us observe that SCMPDS is steady-programmed and realistic. One can prove the following propositions:

(98) IC_{SCMPDS} ≠ d_{i} and IC_{SCMPDS} ≠ i_{i}.

(99) For every instruction I of SCMPDS such that I = goto 0 holds I is halting.

Acknowledgments

We wish to thank Prof. Y. Nakamura for many helpful suggestions.

References

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192 jing-chao chen

Computation and Program Shift in the SCMPDS Computer^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. A finite partial state is said to be autonomic if the computation results in any two states containing it are same on its domain. On the basis of this definition, this article presents some computation results about autonomic finite partial states of the SCMPDS computer. Because the instructions of the SCMPDS computer are more complicated than those of the SCMFSA computer, the results given by this article are weaker than those reported previously by the article on the SCMFSA computer. The second task of this article is to define the notion of program shift. The importance of this notion is that the computation of some program blocks can be simplified by shifting a program block to the initial position.

MML Identifier: SCMPDS 3.

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

1. Preliminaries

In this paper k, m, n denote natural numbers. Next we state several propositions:

(1) Suppose n 13. Then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5 or n = 6 or n = 7 or n = 8 or n = 9 or n = 10 or n = 11 or n = 12 or n = 13.

1This work was done while the author visited Shinshu University March–April 1999.

(2) For every integer k_{1} and for all states s_{1}, s_{2} of SCMPDS such that IC_{(s_{1})} = IC_{(s_{2})} holds ICplusConst(s_{1}, k_{1}) = ICplusConst(s_{2}, k_{1}).

(3) Let k_{1} be an integer, a be a Int position, and s_{1}, s_{2} be states of SCMPDS. If s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}, then s_{1}(DataLoc(s_{1}(a), k_{1})) = s_{2}(DataLoc(s_{2}(a), k_{1})).

(4) For every Int position a and for all states s_{1}, s_{2} of SCMPDS such that s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM} holds s_{1}(a) = s_{2}(a).

(5) The objects of SCMPDS = {IC_{SCMPDS} }∪Data-Loc_{SCM} ∪the instruction locations of SCMPDS.

(6) IC_{SCMPDS} ∉ Data-Loc_{SCM}.

(7) For all states s_{1}, s_{2} of SCMPDS such that s_{1}↾(Data-Loc_{SCM} ∪ {IC_{SCMPDS} }) = s_{2}↾(Data-Loc_{SCM} ∪ {IC_{SCMPDS} }) and for every instruction l of SCMPDS holds Exec(l, s_{1})↾(Data-Loc_{SCM} ∪ {IC_{SCMPDS} }) = Exec(l, s_{2})↾(Data-Loc_{SCM} ∪ {IC_{SCMPDS} }).

(8) For every instruction i of SCMPDS and for every state s of SCMPDS holds Exec(i, s)↾Instr-Loc_{SCM} = s↾Instr-Loc_{SCM}.

2. Finite Partial States of SCMPDS

Next we state two propositions:

(9) For every finite partial state p of SCMPDS holds DataPart(p) = p↾Data-Loc_{SCM}.

(10) For every finite partial state p of SCMPDS holds p is data-only iff

domp ⊆ Data-Loc_{SCM}. Let us mention that there exists a finite partial state of SCMPDS which is

data-only.

Next we state two propositions:

(11) For every finite partial state p of SCMPDS holds domDataPart(p) ⊆ Data-Loc_{SCM}.

(12) For every finite partial state p of SCMPDS holds domProgramPart(p) ⊆

the instruction locations of SCMPDS. Let I_{1} be a partial function from FinPartSt(SCMPDS) to FinPartSt(SCMPDS).

We say that I_{1} is data-only if and only if the condition (Def. 1) is satisfied. (Def. 1) Let p be a finite partial state of SCMPDS. Suppose p ∈ domI_{1}. Then

p is data-only and for every finite partial state q of SCMPDS such that q = I_{1}(p) holds q is data-only.

Let us observe that there exists a partial function from FinPartSt(SCMPDS) to FinPartSt(SCMPDS) which is data-only.

Next we state three propositions:

(13) Let i be an instruction of SCMPDS, s be a state of SCMPDS, and p be a programmed finite partial state of SCMPDS. Then Exec(i, s+·p) = Exec(i, s)+·p.

(14) For every state s of SCMPDS and for every instruction-location i_{1} of SCMPDS and for every Int position a holds s(a) = (s+·Start-At(i_{1}))(a).

(15) For all states s, t of SCMPDS holds s+·t↾Data-Loc_{SCM} is a state of SCMPDS.

3. Autonomic Finite Partial States of SCMPDS and its Computation

Let l_{1} be a Int position and let a be an integer. Then l_{1}⟼→ . a is a finite partial state of SCMPDS.

Next we state the proposition

(16) For every autonomic finite partial state p of SCMPDS such that

DataPart(p) ≠ ∅ holds IC_{SCMPDS} ∈ domp. Let us observe that there exists a finite partial state of SCMPDS which is

autonomic and non programmed.

One can prove the following propositions:

(17) For every autonomic non programmed finite partial state p of SCMPDS holds IC_{SCMPDS} ∈ domp.

(18) Let s_{1}, s_{2} be states of SCMPDS and k_{1}, k_{2}, m be integers. If IC_{(s_{1})} = IC_{(s_{2})} and k_{1} ≠ k_{2} and m = IC_{(s_{1})} and (m − 2) + 2 · k_{1} 0 and (m − 2) + 2 · k_{2} 0, then ICplusConst(s_{1}, k_{1}) ≠ ICplusConst(s_{2}, k_{2}).

(19) For all states s_{1}, s_{2} of SCMPDS and for all natural numbers k_{1}, k_{2} such that IC_{(s_{1})} = IC_{(s_{2})} and k_{1} ≠ k_{2} holds ICplusConst(s_{1}, k_{1}) ≠ ICplusConst(s_{2}, k_{2}).

(20) For every state s of SCMPDS holds Next(IC_{s}) = ICplusConst(s, 1).

(21) For every autonomic finite partial state p of SCMPDS such that IC_{SCMPDS} ∈ domp holds IC_{p} ∈ domp.

(22) Let p be an autonomic non programmed finite partial state of SCMPDS and s be a state of SCMPDS. If p ⊆ s, then for every natural number i holds IC_{(Computation (s))(i)} ∈ domProgramPart(p).

(23) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)).

(24) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i be a natural number, k_{1}, k_{2} be integers, and a, b be Int position. Suppose CurInstr((Computation(s_{1}))(i)) = (a, k_{1}) := (b, k_{2}) and a ∈ domp and DataLoc((Computation(s_{1}))(i)(a), k_{1}) ∈ domp. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(b), k_{2})) = (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(b), k_{2})).

(25) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i be a natural number, k_{1}, k_{2} be integers, and a, b be Int position. Suppose CurInstr((Computation(s_{1}))(i)) = AddTo(a, k_{1}, b, k_{2}) and a ∈ domp and DataLoc((Computation(s_{1}))(i)(a), k_{1}) ∈ domp. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(b), k_{2})) = (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(b), k_{2})).

(26) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i be a natural number, k_{1}, k_{2} be integers, and a, b be Int position. Suppose CurInstr((Computation(s_{1}))(i)) = SubFrom(a, k_{1}, b, k_{2}) and a ∈ domp and DataLoc((Computation(s_{1}))(i)(a), k_{1}) ∈ domp. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(b), k_{2})) = (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(b), k_{2})).

(27) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i be a natural number, k_{1}, k_{2} be integers, and a, b be Int position. Suppose CurInstr((Computation(s_{1}))(i)) = MultBy(a, k_{1}, b, k_{2}) and a ∈ domp and DataLoc((Computation(s_{1}))(i)(a), k_{1}) ∈ domp. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(a), k_{1}))· (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(b), k_{2})) = (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(a), k_{1}))· (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(b), k_{2})).

(28) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i, m be natural numbers, a be a Int position, and k_{1}, k_{2} be integers. Suppose CurInstr((Computation(s_{1}))(i)) = (a, k_{1}) <> 0 gotok_{2} and m = IC_{(Computation (s_{1}))(i)} and (m − 2) + 2 · k_{2} 0 and k_{2} ≠ 1. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(a), k_{1})) = 0 if and only if (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(a), k_{1})) = 0.

(29) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i, m be natural numbers, a be a Int position, and k_{1}, k_{2} be integers. Suppose CurInstr((Computation(s_{1}))(i)) = (a, k_{1}) <= 0 gotok_{2} and m = IC_{(Computation (s_{1}))(i)} and (m − 2) + 2 · k_{2} 0 and k_{2} ≠ 1. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(a), k_{1})) > 0 if and only if (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(a), k_{1})) > 0.

(30) Let p be an autonomic non programmed finite partial state of SCMPDS and s_{1}, s_{2} be states of SCMPDS. Suppose p ⊆ s_{1} and p ⊆ s_{2}. Let i, m be natural numbers, a be a Int position, and k_{1}, k_{2} be integers. Suppose CurInstr((Computation(s_{1}))(i)) = (a, k_{1}) >= 0 gotok_{2} and m = IC_{(Computation (s_{1}))(i)} and (m − 2) + 2 · k_{2} 0 and k_{2} ≠ 1. Then (Computation(s_{1}))(i)(DataLoc((Computation(s_{1}))(i)(a), k_{1})) < 0 if and only if (Computation(s_{2}))(i)(DataLoc((Computation(s_{2}))(i)(a), k_{1})) < 0.

4. Program Shift in the SCMPDS Computer

Let us consider k. The functor insposk yielding an instruction-location of SCMPDS is defined by:

(Def. 2) insposk = i_{k}.

One can prove the following two propositions:

(31) For all natural numbers k_{1}, k_{2} such that k_{1} ≠ k_{2} holds insposk_{1} ≠ insposk_{2}.

(32) For every instruction-location i_{2} of SCMPDS there exists a natural num-

ber i such that i_{2} = insposi. Let l_{2} be an instruction-location of SCMPDS and let k be a natural number.

The functor l_{2} + k yields an instruction-location of SCMPDS and is defined as follows:

(Def. 3) There exists a natural number m such that l_{2} = insposm and l_{2} + k =

insposm + k. The functor l_{2} − ′k yielding an instruction-location of SCMPDS is defined as follows:

(Def. 4) There exists a natural number m such that l_{2} = insposm and l_{2} − ′k =

insposm − ′k. Next we state four propositions:

(33) For every instruction-location l of SCMPDS and for all m, n holds (l + m) + n = l + (m + n).

(34) For every instruction-location l_{2} of SCMPDS and for every natural number k holds (l_{2} + k) − ′k = l_{2}.

(35) For all instructions-locations l_{3}, l_{4} of SCMPDS and for every natural number k holds Start-At(l_{3} + k) = Start-At(l_{4} + k) iff Start-At(l_{3}) = Start-At(l_{4}).

(36) For all instructions-locations l_{3}, l_{4} of SCMPDS and for every natural number k such that Start-At(l_{3}) = Start-At(l_{4}) holds Start-At(l_{3} − ′k) = Start-At(l_{4} − ′k).

Let I_{1} be a finite partial state of SCMPDS. We say that I_{1} is initial if and only if:

(Def. 5) For all m, n such that insposn ∈ domI_{1} and m < n holds insposm ∈

domI_{1}. The finite partial state SCMPDS − Stop of SCMPDS is defined as follows:

(Def. 6) SCMPDS − Stop = inspos0⟼→ . halt_{SCMPDS} .

Let us observe that SCMPDS − Stop is non empty initial and programmed. Let us observe that there exists a finite partial state of SCMPDS which is

initial, programmed, and non empty.

Let p be a programmed finite partial state of SCMPDS and let k be a natural number. The functor Shift(p, k) yielding a programmed finite partial state of SCMPDS is defined as follows:

(Def. 7) domShift(p, k) = {insposm+k : insposm ∈ domp} and for every m such

that insposm ∈ domp holds (Shift(p, k))(insposm + k) = p(insposm). We now state several propositions:

(37) Let l be an instruction-location of SCMPDS, k be a natural number, and p be a programmed finite partial state of SCMPDS. If l ∈ domp, then (Shift(p, k))(l + k) = p(l).

(38) Let p be a programmed finite partial state of SCMPDS and k be a natural number. Then domShift(p, k) = {i_{2}+k; i_{2} ranges over instructionslocations of SCMPDS: i_{2} ∈ domp}.

(39) For every programmed finite partial state I of SCMPDS holds Shift(Shift(I, m), n) = Shift(I, m + n).

(40) Let s be a programmed finite partial state of SCMPDS, f be a function from the instructions of SCMPDS into the instructions of SCMPDS, and given n. Then Shift(f · s, n) = f · Shift(s, n).

(41) For all programmed finite partial states I, J of SCMPDS holds Shift(I+·J, n) = Shift(I, n)+·Shift(J, n).

Acknowledgments

We wish to thank Prof. Y. Nakamura for many helpful suggestions.

References

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

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

[3] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676,

computation and program shift in the … 199

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[8] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

[9] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

[10] Jing-Chao Chen. A small computer model with push-down stack. Formalized Mathematics, 8(1):175–182, 1999.

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

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

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

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200 jing-chao chen

The Construction and Shiftability of Program Blocks for SCMPDS^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. In this article,a program block is defined as a finite sequence of instructions stored consecutively on initial positions. Based on this definition,any program block with more than two instructions can be viewed as the combination of two smaller program blocks. To describe the computation of a program block by the result of its two sub-blocks, we introduce the notions of paraclosed, parahalting, valid, and shiftable, the meaning of which may be stated as follows:

- a program is paraclosed if and only if any state containing it is closed,

- a program is parahalting if and only if any state containing it is halting,

- in a program block, a jumping instruction is valid if its jumping offset is valid,

- a program block is shiftable if it does not contain any return and saveIC instructions,and each instruction in it is valid.

When a program block is shiftable, its computing result does not depend on its storage position.

MML Identifier: SCMPDS 4.

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

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

1. Definition of a Program Block and its Basic Properties

A Program-block is an initial programmed finite partial state of SCMPDS. We adopt the following convention: m, n are natural numbers, i, j, k are

instructions of SCMPDS, and I, J, K are Program-block.

Let us consider i. The functor Load(i) yielding a Program-block is defined as follows:

(Def. 1) Load(i) = inspos0⟼→ . i.

Let us consider i. Note that Load(i) is non empty. Next we state the proposition

(1) For every Program-block P and for every n holds n < cardP iff insposn ∈ domP.

Let I be an initial finite partial state of SCMPDS. Note that ProgramPart(I) is initial.

Next we state four propositions:

(2) domI misses domShift(J, cardI).

(3) For every programmed finite partial state I of SCMPDS holds cardShift(I, m) = cardI.

(4) For all finite partial states I, J of SCMPDS holds ProgramPart(I+·J) = ProgramPart(I)+·ProgramPart(J).

(5) For all finite partial states I, J of SCMPDS holds Shift(ProgramPart (I+·J), n) = Shift(ProgramPart(I), n)+·Shift(ProgramPart(J), n).

We use the following convention: a, b are Int position, s, s_{1}, s_{2} are states of SCMPDS, and k_{1}, k_{2} are integers.

Let us consider I. The functor Initialized(I) yields a finite partial state of SCMPDS and is defined as follows:

(Def. 2) Initialized(I) = I+·Start-At(inspos0). We now state a number of propositions:

(6) InsCode(i) ∈ {0, 1, 4, 5, 6} or (Exec(i, s))(IC_{SCMPDS} ) = Next(IC_{s}).

(7) IC_{SCMPDS} ∈ domInitialized(I).

(8) IC_{Initialized (I)} = inspos0.

(9) I ⊆ Initialized(I).

(10) s and s+·I are equal outside the instruction locations of SCMPDS.

(11) Let s_{1}, s_{2} be states of SCMPDS. Suppose IC_{(s_{1})} = IC_{(s_{2})} and for every Int position a holds s_{1}(a) = s_{2}(a). Then s_{1} and s_{2} are equal outside the instruction locations of SCMPDS.

(13)^{2} Suppose s_{1} and s_{2} are equal outside the instruction locations of

SCMPDS. Let a be a Int position. Then s_{1}(a) = s_{2}(a).

(14) If s_{1} and s_{2} are equal outside the instruction locations of SCMPDS, then s_{1}(DataLoc(s_{1}(a), k_{1})) = s_{2}(DataLoc(s_{2}(a), k_{1})).

(15) Suppose s_{1} and s_{2} are equal outside the instruction locations of SCMPDS. Then Exec(i, s_{1}) and Exec(i, s_{2}) are equal outside the instruction locations of SCMPDS.

(16) Initialized(I)↾the instruction locations of SCMPDS = I.

(17) For all natural numbers k_{1}, k_{2} such that k_{1} ≠ k_{2} holds DataLoc(k_{1}, 0) ≠ DataLoc(k_{2}, 0).

(18) For every Int position d_{1} there exists a natural number i such that d_{1} =

DataLoc(i, 0). The scheme SCMPDSEx deals with a unary functor F yielding an instruction

of SCMPDS, a unary functor G yielding an integer, and an instruction-location A of SCMPDS, and states that:

There exists a state S of SCMPDS such that IC_{S} = A and for every natural number i holds S(insposi) = F(i) and S(DataLoc(i, 0)) = G(i)

for all values of the parameters.

Next we state a number of propositions:

(19) For every state s of SCMPDS holds doms = {IC_{SCMPDS} } ∪ Data-Loc_{SCM} ∪ the instruction locations of SCMPDS.

(20) Let s be a state of SCMPDS and x be a set. Suppose x ∈ doms. Then x is a Int position or x = IC_{SCMPDS} or x is an instruction-location of SCMPDS.

(21) Let s_{1}, s_{2} be states of SCMPDS. Then for every instruction-location l of SCMPDS holds s_{1}(l) = s_{2}(l) if and only if s_{1}↾the instruction locations of SCMPDS = s_{2}↾the instruction locations of SCMPDS.

(22) For every instruction-location i of SCMPDS holds i ∉ Data-Loc_{SCM}.

(23) For all states s_{1}, s_{2} of SCMPDS holds for every Int position a holds s_{1}(a) = s_{2}(a) iff s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}.

(24) Let s_{1}, s_{2} be states of SCMPDS. Suppose s_{1} and s_{2} are equal outside the instruction locations of SCMPDS. Then s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}.

(25) For all states s, s_{3} of SCMPDS and for every set A holds (s_{3}+·s↾A)↾A = s↾A.

(26) For all Program-block I, J holds I and J are equal outside the instruction locations of SCMPDS.

2The proposition (12) has been removed.

(27) For every Program-block I holds domInitialized(I) = domI ∪ {IC_{SCMPDS} }.

(28) For every Program-block I and for every set x such that x ∈ domInitialized(I) holds x ∈ domI or x = IC_{SCMPDS} .

(29) For every Program-block I holds (Initialized(I))(IC_{SCMPDS} ) = inspos0.

(30) For every Program-block I holds IC_{SCMPDS} ∉ domI.

(31) For every Program-block I and for every Int position a holds a ∉

domInitialized(I). In the sequel x denotes a set. The following propositions are true:

(32) If x ∈ domI, then I(x) = (I+·Start-At(insposn))(x).

(33) For every Program-block I and for every set x such that x ∈ domI holds I(x) = (Initialized(I))(x).

(34) For all Program-block I, J and for every state s of SCMPDS such that Initialized(J) ⊆ s holds s+·Initialized(I) = s+·I.

(35) For all Program-block I, J and for every state s of SCMPDS such that Initialized(J) ⊆ s holds Initialized(I) ⊆ s+·I.

(36) Let I, J be Program-block and s be a state of SCMPDS. Then s+·Initialized(I) and s+·Initialized(J) are equal outside the instruction locations of SCMPDS.

2. Combining two Consecutive Blocks into One Program Block

Let I, J be Program-block. The functor I;J yields a Program-block and is defined by:

(Def. 3) I;J = I+·Shift(J, cardI).

One can prove the following propositions:

(37) For all Program-block I, J and for every instruction-location l of SCMPDS such that l ∈ domI holds (I;J)(l) = I(l).

(38) For all Program-block I, J and for every instruction-location l of SCMPDS such that l ∈ domJ holds (I;J)(l + cardI) = J(l).

(39) For all Program-block I, J holds domI ⊆ dom(I;J).

(40) For all Program-block I, J holds I ⊆ I;J.

(41) For all Program-block I, J holds I+·(I;J) = I;J.

(42) For all Program-block I, J holds Initialized(I)+·(I;J) = Initialized(I;J).

3. Combining a Block and a Instruction into One Program Block

Let us consider i, J. The functor i;J yielding a Program-block is defined by: (Def. 4) i;J = Load(i);J.

Let us consider I, j. The functor I;j yields a Program-block and is defined by:

(Def. 5) I;j = I; Load(j).

Let us consider i, j. The functor i;j yielding a Program-block is defined as follows:

(Def. 6) i;j = Load(i); Load(j).

The following propositions are true:

(43) i;j = Load(i);j.

(44) i;j = i; Load(j).

(45) card(I;J) = cardI + cardJ.

(46) (I;J);K = I;(J;K).

(47) (I;J);k = I;(J;k).

(48) (I;j);K = I;(j;K).

(49) (I;j);k = I;(j;k).

(50) (i;J);K = i;(J;K).

(51) (i;J);k = i;(J;k).

(52) (i;j);K = i;(j;K).

(53) (i;j);k = i;(j;k).

(54) domI ∩ domStart-At(insposn) = ∅.

(55) Start-At(inspos0) ⊆ Initialized(I).

(56) If I+·Start-At(insposn) ⊆ s, then I ⊆ s.

(57) If Initialized(I) ⊆ s, then I ⊆ s.

(58) (I+·Start-At(insposn))↾the instruction locations of SCMPDS = I. In the sequel l, l_{1} denote instructions-locations of SCMPDS. Next we state four propositions:

(59) a ∉ domStart-At(l).

(60) l_{1} ∉ domStart-At(l).

(61) a ∉ dom(I+·Start-At(l)).

(62) s+·I+·Start-At(inspos0) = s+·Start-At(inspos0)+·I.

Let s be a state of SCMPDS, let l_{2} be a Int position, and let k be an integer. Then s +·(l_{2}, k) is a state of SCMPDS.

4. The Notions of Paraclosed, Parahalting and their Basic Properties

Let I be a Program-block. The functor stopI yielding a Program-block is defined as follows:

(Def. 7) stopI = I; SCMPDS − Stop .

Let I be a Program-block and let s be a state of SCMPDS. The functor IExec(I, s) yielding a state of SCMPDS is defined as follows:

(Def. 8) IExec(I, s) = Result(s+·Initialized(stopI))+·s↾the instruction loca-

tions of SCMPDS. Let I be a Program-block. We say that I is paraclosed if and only if:

(Def. 9) For every state s of SCMPDS and for every natural number n such that

Initialized(stopI) ⊆ s holds IC_{(Computation (s))(n)} ∈ domstopI. We say that I is parahalting if and only if:

(Def. 10) Initialized(stopI) is halting. Let us note that there exists a Program-block which is parahalting. One can prove the following proposition

(63) For every parahalting Program-block I such that Initialized(stopI) ⊆ s

holds s is halting. Let I be a parahalting Program-block. Note that Initialized(stopI) is hal-

ting.

Let l_{3}, l_{4} be instructions-locations of SCMPDS and let a, b be instructions of SCMPDS. Then [l_{3} ⟼→ a, l_{4} ⟼→ b] is a finite partial state of SCMPDS.

One can prove the following propositions:

(64) For every integer k such that k ≠ 0 holds goto k ≠ halt_{SCMPDS} .

(65) IC_{s} ≠ Next(IC_{s}).

(66) s_{2}+·[IC_{(s_{2})} ⟼→ goto 1, Next(IC_{(s_{2})}) ⟼→ goto (− 1)] is not halting.

(67) Suppose that (i) s_{1} and s_{2} are equal outside the instruction locations of SCMPDS, (ii) I ⊆ s_{1}, (iii) I ⊆ s_{2}, and (iv) for every m such that m < n holds IC_{(Computation (s_{2}))(m)} ∈ domI.

Let given m. Suppose m n. Then (Computation(s_{1}))(m) and (Computation(s_{2}))(m) are equal outside the instruction locations of SCMPDS.

(68) For every state s of SCMPDS and for every instruction-location l of

SCMPDS holds l ∈ doms. In the sequel l_{1}, l_{5} are instructions-locations of SCMPDS and i_{1}, i_{2} are

instructions of SCMPDS.

The following propositions are true:

(69) s+·[l_{1} ⟼→ i_{1}, l_{5} ⟼→ i_{2}] = s +·(l_{1}, i_{1}) +·(l_{5}, i_{2}).

(70) Next(insposn) = insposn + 1.

(71) If IC_{s} ∉ domI, then Next(IC_{s}) ∉ domI.

Let us mention that every Program-block which is parahalting is also paraclosed.

We now state several propositions:

(72) domSCMPDS − Stop = {inspos0}.

(73) inspos0 ∈ domSCMPDS − Stop and (SCMPDS − Stop )(inspos0) = halt_{SCMPDS} .

(74) cardSCMPDS − Stop = 1.

(75) inspos0 ∈ domstopI.

(76) Let p be a programmed finite partial state of SCMPDS, k be a natural number, and i_{3} be an instruction-location of SCMPDS. If i_{3} ∈ domp, then i_{3} + k ∈ domShift(p, k).

5. Shiftability of Program Blocks and Instructions

Let i be an instruction of SCMPDS and let n be a natural number. We say that i valid at n if and only if the conditions (Def. 11) are satisfied.

(Def. 11)(i) If InsCode(i) = 0, then there exists k_{1} such that i = goto k_{1} and

n + k_{1} 0, (ii) if InsCode(i) = 4, then there exist a, k_{1}, k_{2} such that i = (a, k_{1}) <>

0 gotok_{2} and n + k_{2} 0, (iii) if InsCode(i) = 5, then there exist a, k_{1}, k_{2} such that i = (a, k_{1}) <=

0 gotok_{2} and n + k_{2} 0, and (iv) if InsCode(i) = 6, then there exist a, k_{1}, k_{2} such that i = (a, k_{1}) >=

0 gotok_{2} and n + k_{2} 0. One can prove the following proposition

(77) Let i be an instruction of SCMPDS and m, n be natural numbers. If i

valid at m and m n, then i valid at n. Let I_{1} be a finite partial state of SCMPDS. We say that I_{1} is shiftable if and

only if: (Def. 12) For all n, i such that insposn ∈ domI_{1} and i = I_{1}(insposn) holds

InsCode(i) ≠ 1 and InsCode(i) ≠ 3 and i valid at n. Let us mention that there exists a Program-block which is parahalting and

shiftable.

Let i be an instruction of SCMPDS. We say that i is shiftable if and only if:

(Def. 13) InsCode(i) = 2 or InsCode(i) > 6. One can check that there exists an instruction of SCMPDS which is shiftable. Let us consider a, k_{1}. Observe that a:=k_{1} is shiftable. Let us consider a, k_{1}, k_{2}. One can check that a_{k_{1}}:=k_{2} is shiftable. Let us consider a, k_{1}, k_{2}. Observe that AddTo(a, k_{1}, k_{2}) is shiftable. Let us consider a, b, k_{1}, k_{2}. One can check the following observations:

∗ AddTo(a, k_{1}, b, k_{2}) is shiftable,

∗ SubFrom(a, k_{1}, b, k_{2}) is shiftable,

∗ MultBy(a, k_{1}, b, k_{2}) is shiftable,

∗ Divide(a, k_{1}, b, k_{2}) is shiftable, and

∗ (a, k_{1}) := (b, k_{2}) is shiftable. Let I, J be shiftable Program-block. Observe that I;J is shiftable. Let i be a shiftable instruction of SCMPDS. Observe that Load(i) is shifta-

ble.

Let i be a shiftable instruction of SCMPDS and let J be a shiftable Programblock. Observe that i;J is shiftable.

Let I be a shiftable Program-block and let j be a shiftable instruction of SCMPDS. Observe that I;j is shiftable.

Let i, j be shiftable instructions of SCMPDS. Note that i;j is shiftable. Let us note that SCMPDS − Stop is parahalting and shiftable. Let I be a shiftable Program-block. One can verify that stopI is shiftable. Next we state the proposition

(78) For every shiftable Program-block I and for every integer k_{1} such that

cardI + k_{1} 0 holds I;goto k_{1} is shiftable. Let n be a natural number. Note that Load(goto n) is shiftable. One can prove the following proposition

(79) Let I be a shiftable Program-block, k_{1}, k_{2} be integers, and a be a Int

position. If cardI + k_{2} 0, then I;((a, k_{1}) <> 0 gotok_{2}) is shiftable. Let k_{1} be an integer, let a be a Int position, and let n be a natural number.

Note that Load((a, k_{1}) <> 0 goton) is shiftable.

Next we state the proposition

(80) Let I be a shiftable Program-block, k_{1}, k_{2} be integers, and a be a Int

position. If cardI + k_{2} 0, then I;((a, k_{1}) <= 0 gotok_{2}) is shiftable. Let k_{1} be an integer, let a be a Int position, and let n be a natural number.

Observe that Load((a, k_{1}) <= 0 goton) is shiftable.

One can prove the following proposition

(81) Let I be a shiftable Program-block, k_{1}, k_{2} be integers, and a be a Int

position. If cardI + k_{2} 0, then I;((a, k_{1}) >= 0 gotok_{2}) is shiftable. Let k_{1} be an integer, let a be a Int position, and let n be a natural number.

Observe that Load((a, k_{1}) >= 0 goton) is shiftable.

the construction and shiftability of program … 209

We now state three propositions:

(82) Let s_{1}, s_{2} be states of SCMPDS, n, m be natural numbers, and k_{1} be an integer. If IC_{(s_{1})} = insposm and m + k_{1} 0 and IC_{(s_{1})} + n = IC_{(s_{2})}, then ICplusConst(s_{1}, k_{1}) + n = ICplusConst(s_{2}, k_{1}).

(83) Let s_{1}, s_{2} be states of SCMPDS, n, m be natural numbers, and i be an instruction of SCMPDS. Suppose IC_{(s_{1})} = insposm and i valid at m and InsCode(i) ≠ 1 and InsCode(i) ≠ 3 and IC_{(s_{1})} + n = IC_{(s_{2})} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then IC_{Exec(i,s_{1})} + n = IC_{Exec(i,s_{2})} and Exec(i, s_{1})↾Data-Loc_{SCM} = Exec(i, s_{2})↾Data-Loc_{SCM}.

(84) Let J be a parahalting shiftable Program-block. Suppose Initialized(stop J) ⊆ s_{1}. Let n be a natural number. Suppose Shift(stopJ, n) ⊆ s_{2} and IC_{(s_{2})} = insposn and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} + n = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

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

[4] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[6] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[7] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

[8] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Formalized Mathematics, 8(1):193–199, 1999.

[9] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

[10] Jing-Chao Chen. A small computer model with push-down stack. Formalized Mathematics, 8(1):175–182, 1999.

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

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

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

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

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

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

[19] Andrzej Trybulec and Yatsuka Nakamura. Modifying addresses of instructions of SCM _{FSA} . Formalized Mathematics, 5(4):571–576, 1996.

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

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

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

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

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

Computation of Two Consecutive Program Blocks for SCMPDS^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. In this article, a program block without halting instructions is called No-StopCode program block. If a program consists of two blocks, where the first block is parahalting (i.e. halt for all states) and No-StopCode, and the second block is parahalting and shiftable, it can be computed by combining the computation results of the two blocks. For a program which consists of a instruction and a block, we obtain a similar conclusion. For a large amount of programs, the computation method given in the article is useful, but it is not suitable to recursive programs.

MML Identifier: SCMPDS 5.

The terminology and notation used here have been introduced in the following articles: [16], [20], [11], [21], [5], [6], [18], [2], [12], [13], [17], [14], [4], [10], [9], [19], [7], [1], [15], [8], and [3].

1. Preliminaries

For simplicity, we use the following convention: x denotes a set, m, n denote natural numbers, a, b denote Int position, i denotes an instruction of SCMPDS, s, s_{1}, s_{2} denote states of SCMPDS, k_{1}, k_{2} denote integers, l_{1} denotes an instruction-location of SCMPDS, I, J denote Program-block, and N denotes a set with non empty elements.

One can prove the following propositions:

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

(1) Let S be a halting von Neumann definite AMI over N and s be a state of S. If s = Following(s), then for every n holds (Computation(s))(n) = s.

(2) x ∈ domLoad(i) iff x = inspos0.

(3) If l_{1} ∈ domstopI and (stopI)(l_{1}) ≠ halt_{SCMPDS} , then l_{1} ∈ domI.

(4) domLoad(i) = {inspos0} and (Load(i))(inspos0) = i.

(5) inspos0 ∈ domLoad(i).

(6) cardLoad(i) = 1.

(7) cardstopI = cardI + 1.

(8) cardstopLoad(i) = 2.

(9) inspos0 ∈ domstopLoad(i) and inspos1 ∈ domstopLoad(i).

(10) (stopLoad(i))(inspos0) = i and (stopLoad(i))(inspos1) = halt_{SCMPDS} .

(11) x ∈ domstopLoad(i) iff x = inspos0 or x = inspos1.

(12) domstopLoad(i) = {inspos0, inspos1}.

(13) inspos0 ∈ domInitialized(stopLoad(i)) and inspos1 ∈ domInitialized (stopLoad(i)) and (Initialized(stopLoad(i)))(inspos0) = i and (Initialized (stopLoad(i)))(inspos1) = halt_{SCMPDS} .

(14) For all Program-block I, J holds Initialized(stopI;J) = (I;(J; SCMPDS − Stop ))+·Start-At(inspos0).

(15) For all Program-block I, J holds Initialized(I) ⊆ Initialized(stopI;J).

(16) domstopI ⊆ domstopI;J.

(17) For all Program-block I, J holds Initialized(stopI)+·Initialized(stopI;J) = Initialized(stopI;J).

(18) If Initialized(I) ⊆ s, then IC_{s} = inspos0.

(19) (s+·Initialized(I))(a) = s(a).

(20) Let I be a parahalting Program-block. Suppose Initialized(stopI) ⊆ s_{1} and Initialized(stopI) ⊆ s_{2} and s_{1} and s_{2} are equal outside the instruction locations of SCMPDS. Let k be a natural number. Then (Computation(s_{1}))(k) and (Computation(s_{2}))(k) are equal outside the instruction locations of SCMPDS and CurInstr((Computation(s_{1}))(k)) = CurInstr((Computation(s_{2}))(k)).

(21) Let I be a parahalting Program-block. Suppose Initialized(stopI) ⊆ s_{1} and Initialized(stopI) ⊆ s_{2} and s_{1} and s_{2} are equal outside the instruction locations of SCMPDS. Then LifeSpan(s_{1}) = LifeSpan(s_{2}) and Result(s_{1}) and Result(s_{2}) are equal outside the instruction locations of SCMPDS.

(22) For every Program-block I holds IC_{IExec(I,s)} = IC_{Result(s+·Initialized (stop I))}.

(23) Let I be a parahalting Program-block and J be a Program-block. Suppose Initialized(stopI) ⊆ s. Let given m. Suppose m LifeSpan(s). Then (Computation(s))(m) and (Computation(s+·(I;J)))(m) are equal outside the instruction locations of SCMPDS.

(24) Let I be a parahalting Program-block and J be a Program-block. Suppose Initialized(stopI) ⊆ s. Let given m. Suppose m LifeSpan(s). Then (Computation(s))(m) and (Computation(s+·Initialized(stopI;J)))(m) are equal outside the instruction locations of SCMPDS.

2. Non Halting Instructions and Parahalting Instructions

Let i be an instruction of SCMPDS. We say that i is No-StopCode if and only if:

(Def. 1) i ≠ halt_{SCMPDS} .

Let i be an instruction of SCMPDS. We say that i is parahalting if and only if:

(Def. 2) Load(i) is parahalting.

One can verify that there exists an instruction of SCMPDS which is No-StopCode, shiftable, and parahalting.

One can prove the following proposition

(25) If k_{1} ≠ 0, then goto k_{1} is No-StopCode.

Let us consider a. Observe that returna is No-StopCode. Let us consider a, k_{1}. Note that a:=k_{1} is No-StopCode and saveIC(a, k_{1}) is

No-StopCode.

Let us consider a, k_{1}, k_{2}. One can check the following observations:

∗ (a, k_{1}) <> 0 gotok_{2} is No-StopCode,

∗ (a, k_{1}) <= 0 gotok_{2} is No-StopCode,

∗ (a, k_{1}) >= 0 gotok_{2} is No-StopCode, and

∗ a_{k_{1}}:=k_{2} is No-StopCode. Let us consider a, k_{1}, k_{2}. Note that AddTo(a, k_{1}, k_{2}) is No-StopCode. Let us consider a, b, k_{1}, k_{2}. One can verify the following observations:

∗ AddTo(a, k_{1}, b, k_{2}) is No-StopCode,

∗ SubFrom(a, k_{1}, b, k_{2}) is No-StopCode,

∗ MultBy(a, k_{1}, b, k_{2}) is No-StopCode,

∗ Divide(a, k_{1}, b, k_{2}) is No-StopCode, and

∗ (a, k_{1}) := (b, k_{2}) is No-StopCode. Let us note that halt_{SCMPDS} is parahalting. Let i be a parahalting instruction of SCMPDS. Observe that Load(i) is

parahalting.

Let us consider a, k_{1}. Observe that a:=k_{1} is parahalting. Let us consider a, k_{1}, k_{2}. Note that a_{k_{1}}:=k_{2} is parahalting and

AddTo(a, k_{1}, k_{2}) is parahalting.

Let us consider a, b, k_{1}, k_{2}. One can check the following observations:

∗ AddTo(a, k_{1}, b, k_{2}) is parahalting,

∗ SubFrom(a, k_{1}, b, k_{2}) is parahalting,

∗ MultBy(a, k_{1}, b, k_{2}) is parahalting,

∗ Divide(a, k_{1}, b, k_{2}) is parahalting, and

∗ (a, k_{1}) := (b, k_{2}) is parahalting. Next we state the proposition

(26) If InsCode(i) = 1, then i is not parahalting.

Let I_{1} be a finite partial state of SCMPDS. We say that I_{1} is No-StopCode if and only if:

(Def. 3) For every instruction-location x of SCMPDS such that x ∈ domI_{1} holds

I_{1}(x) ≠ halt_{SCMPDS} . Let us observe that there exists a Program-block which is parahalting, shi-

ftable, and No-StopCode.

Let I, J be No-StopCode Program-block. Observe that I;J is No-StopCode. Let i be a No-StopCode instruction of SCMPDS. Observe that Load(i) is

No-StopCode.

Let i be a No-StopCode instruction of SCMPDS and let J be a No-StopCode Program-block. Note that i;J is No-StopCode.

Let I be a No-StopCode Program-block and let j be a No-StopCode instruction of SCMPDS. Observe that I;j is No-StopCode.

Let i, j be No-StopCode instructions of SCMPDS. Observe that i;j is No-StopCode.

Next we state several propositions:

(27) For every parahalting No-StopCode Program-block I such that Initialized(stopI) ⊆ s holds IC_{(Computation (s))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(28) For every parahalting Program-block I and for every natural number k such that k < LifeSpan(s+·Initialized(stopI)) holds IC_{(Computation (s+·Initialized (stop I)))(k)} ∈ domI.

(29) Let I be a parahalting Program-block and k be a natural number. Suppose Initialized(I) ⊆ s and k LifeSpan(s+·Initialized(stopI)). Then (Computation(s))(k) and (Computation(s+·Initialized(stopI)))(k) are equal outside the instruction locations of SCMPDS.

(30) For every parahalting No-StopCode Program-block I such that Initialized(I) ⊆ s holds IC_{(Computation (s))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(31) For every parahalting Program-block I such that Initialized(I) ⊆ s holds CurInstr((Computation(s))(LifeSpan(s+·Initialized(stopI)))) = halt_{SCMPDS} or IC_{(Computation (s))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(32) Let I be a parahalting No-StopCode Program-block and k be a natural number. If Initialized(I) ⊆ s and k < LifeSpan(s+·Initialized(stopI)), then CurInstr((Computation(s))(k)) ≠ halt_{SCMPDS} .

(33) Let I be a parahalting Program-block, J be a Program-block, and k be a natural number. Suppose k LifeSpan(s+·Initialized(stopI)). Then (Computation(s+·Initialized(stopI)))(k) and (Computation(s+·((I;J)+· Start-At(inspos0))))(k) are equal outside the instruction locations of SCMPDS.

(34) Let I be a parahalting Program-block, J be a Program-block, and k be a natural number. Suppose k LifeSpan(s+·Initialized(stopI)). Then (Computation(s+·Initialized(stopI)))(k) and (Computation(s+·Initialized (stopI;J)))(k) are equal outside the instruction locations of SCMPDS.

Let I be a parahalting Program-block and let J be a parahalting shiftable Program-block. One can verify that I;J is parahalting.

Let i be a parahalting instruction of SCMPDS and let J be a parahalting shiftable Program-block. Note that i;J is parahalting.

Let I be a parahalting Program-block and let j be a parahalting shiftable instruction of SCMPDS. Observe that I;j is parahalting.

Let i be a parahalting instruction of SCMPDS and let j be a parahalting shiftable instruction of SCMPDS. One can check that i;j is parahalting.

Next we state the proposition

(35) Let s, s_{1} be states of SCMPDS and J be a parahalting shiftable Program-block. If s = (Computation(s_{1}+·Initialized(stopJ)))(m), then Exec(CurInstr(s), s+·Start-At(IC_{s} + n)) = Following(s)+·Start-At(IC_{Following (s)} + n).

3. Computation of two Consecutive Program Blocks

The following propositions are true:

(36) Let I be a parahalting No-StopCode Program-block, J be a parahalting shiftable Program-block, and k be a natural number. Suppose Initialized(stopI;J) ⊆ s. Then (Computation(Result(s+·Initialized (stopI))+·Initialized(stopJ)))(k)+·Start-At (IC_{(Computation (Result(s+·Initialized (stop I))+·Initialized (stop J)))(k)} + cardI) and (Computation(s+·Initialized(stopI;J)))(LifeSpan(s+·Initialized(stopI))+ k) are equal outside the instruction locations of SCMPDS.

(37) Let I be a parahalting No-StopCode Program-block and J be a parahalting shiftable Program-block. Then LifeSpan(s+·Initialized(stopI;J)) = LifeSpan(s+·Initialized(stopI))+LifeSpan(Result(s+·Initialized(stopI))+· Initialized(stopJ)).

(38) Let I be a parahalting No-StopCode Program-block and J be a parahalting shiftable Program-block. Then IExec(I;J, s) = IExec(J, IExec(I, s))+·Start-At(IC_{IExec(J,IExec(I,s))} + cardI).

(39) Let I be a parahalting No-StopCode Program-block and J be a parahalting shiftable Program-block. Then (IExec(I;J, s))(a) = (IExec(J, IExec(I, s)))(a).

4. Computation of the Program Consisting of a Instruction and a Block

Let s be a state of SCMPDS. The functor Initialized(s) yields a state of SCMPDS and is defined by:

(Def. 4) Initialized(s) = s+·Start-At(inspos0).

Next we state several propositions:

(40) IC_{Initialized (s)} = inspos0 and (Initialized(s))(a) = s(a) and (Initialized(s))(l_{1}) = s(l_{1}).

(41) s_{1} and s_{2} are equal outside the instruction locations of SCMPDS iff s_{1}↾(Data-Loc_{SCM} ∪ {IC_{SCMPDS} }) = s_{2}↾(Data-Loc_{SCM} ∪ {IC_{SCMPDS} }).

(42) If s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}, then s_{1}(DataLoc(s_{1}(a), k_{1})) = s_{2}(DataLoc(s_{2}(a), k_{1})).

(43) If s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM} and InsCode(i) ≠ 3, then Exec(i, s_{1})↾Data-Loc_{SCM} = Exec(i, s_{2})↾Data-Loc_{SCM}.

(44) For every shiftable instruction i of SCMPDS such that s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM} holds Exec(i, s_{1})↾Data-Loc_{SCM} = Exec(i, s_{2})↾Data-Loc_{SCM}.

(45) For every parahalting instruction i of SCMPDS holds Exec(i, Initialized(s)) = IExec(Load(i), s).

(46) Let I be a parahalting No-StopCode Program-block and j be a parahalting shiftable instruction of SCMPDS. Then (IExec(I;j, s))(a) = (Exec(j, IExec(I, s)))(a).

(47) Let i be a No-StopCode parahalting instruction of SCMPDS and j be a shiftable parahalting instruction of SCMPDS. Then (IExec(i;j, s))(a) = (Exec(j, Exec(i, Initialized(s))))(a).

References

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

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

[3] Grzegorz Bancerek and Piotr Rudnicki. Development of terminology for scm. Formalized Mathematics, 4(1):61–67, 1993.

computation of two consecutive program … 217

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

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

[6] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[7] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Formalized Mathematics, 8(1):193–199, 1999.

[8] Jing-Chao Chen. The construction and shiftability of program blocks for SCMPDS. Formalized Mathematics, 8(1):201–210, 1999.

[9] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

[10] Jing-Chao Chen. A small computer model with push-down stack. Formalized Mathematics, 8(1):175–182, 1999.

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

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

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

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

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

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

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

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

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

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

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

218 jing-chao chen

The Construction and Computation of Conditional Statements for SCMPDS^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. We construct conditional statements like the usual high level program language by program blocks of SCMPDS. Roughly speaking, the article justifies such a fact that when the condition of a conditional statement is true (false), and the true (false) branch is shiftable, parahalting and does not contain any halting instruction, and the false branch is shiftable, then it is halting and its computation result equals that of the true (false) branch. The parahalting means some program halts for all states, this is strong condition. For this reason, we introduce the notions of ”is closed on” and ”is halting on”. The predicate ”A is closed on B” denotes program A is closed on state B, and ”A is halting on B” denotes program A is halting on state B. We obtain a similar theorem to the above fact by replacing parahalting by ”is closed on” and ”is halting on”.

MML Identifier: SCMPDS 6.

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

1. Preliminaries

For simplicity, we follow the rules: a denotes a Int position, i denotes an instruction of SCMPDS, s, s_{1}, s_{2} denote states of SCMPDS, k_{1} denotes an integer, l_{1} denotes an instruction-location of SCMPDS, and I, J denote Program-block.

One can prove the following propositions:

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

(1) For every state s of SCMPDS holds dom(s↾the instruction locations of SCMPDS) = the instruction locations of SCMPDS.

(2) For every state s of SCMPDS such that s is halting and for every natural number k such that LifeSpan(s) k holds CurInstr((Computation(s))(k)) = halt_{SCMPDS} .

(3) For every state s of SCMPDS such that s is halting and for every natural number k such that LifeSpan(s) k holds IC_{(Computation (s))(k)} = IC_{(Computation (s))(LifeSpan (s))}.

(4) Let s_{1}, s_{2} be states of SCMPDS. Then s_{1} and s_{2} are equal outside the instruction locations of SCMPDS if and only if IC_{(s_{1})} = IC_{(s_{2})} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}.

(5) For every state s of SCMPDS and for every Program-block I holds Initialized(s)+·Initialized(I) = s+·Initialized(I).

(6) For every Program-block I and for every instruction-location l of SCMPDS holds I ⊆ I+·Start-At(l).

(7) For every state s of SCMPDS and for every instruction-location l of SCMPDS holds s↾Data-Loc_{SCM} = (s+·Start-At(l))↾Data-Loc_{SCM}.

(8) For every state s of SCMPDS and for every Program-block I and for every instruction-location l of SCMPDS holds s↾Data-Loc_{SCM} = (s+·(I+·Start-At(l)))↾Data-Loc_{SCM}.

(9) For every state s of SCMPDS and for every Program-block I holds s↾Data-Loc_{SCM} = (s+·Initialized(I))↾Data-Loc_{SCM}.

(10) Let s be a state of SCMPDS and l be an instruction-location of SCMPDS. Then dom(s↾the instruction locations of SCMPDS) misses domStart-At(l).

(11) Let s be a state of SCMPDS, I, J be Program-block, and l be an instruction-location of SCMPDS. Then s+·(I+·Start-At(l)) and s+·(J+·Start-At(l)) are equal outside the instruction locations of SCMPDS.

(12) Let s_{1}, s_{2} be states of SCMPDS and I, J be Program-block. Suppose s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then s_{1}+·Initialized(I) and s_{2}+·Initialized(J) are equal outside the instruction locations of SCMPDS.

(13) Let I be a programmed finite partial state of SCMPDS and x be a set. If x ∈ domI, then I(x) is an instruction of SCMPDS.

(14) For every state s of SCMPDS and for all instructions-locations l_{2}, l_{3} of SCMPDS holds s+·Start-At(l_{2})+·Start-At(l_{3}) = s+·Start-At(l_{3}).

(15) card(i;I) = cardI + 1.

(16) (i;I)(inspos0) = i.

(17) I ⊆ Initialized(stopI).

(18) If l_{1} ∈ domI, then l_{1} ∈ domstopI.

(19) If l_{1} ∈ domI, then (stopI)(l_{1}) = I(l_{1}).

(20) If l_{1} ∈ domI, then (Initialized(stopI))(l_{1}) = I(l_{1}).

(21) IC_{s+·Initialized (I)} = inspos0.

(22) CurInstr(s+·Initialized(stopi;I)) = i.

(23) For every state s of SCMPDS and for all natural numbers m_{1}, m_{2} such that IC_{s} = insposm_{1} holds ICplusConst(s, m_{2}) = insposm_{1} + m_{2}.

(24) For all Program-block I, J holds Shift(stopJ, cardI) ⊆ stopI;J.

(25) insposcardI ∈ domstopI and (stopI)(insposcardI) = halt_{SCMPDS} .

(26) For all instructions-locations x, l of SCMPDS holds (IExec(J, s))(x) = (IExec(I, s)+·Start-At(l))(x).

(27) For all instructions-locations x, l of SCMPDS holds (IExec(I, s))(x) = (s+·Start-At(l))(x).

(28) Let s be a state of SCMPDS, i be a No-StopCode parahalting instruction of SCMPDS, J be a parahalting shiftable Program-block, and a be a Int position. Then (IExec(i;J, s))(a) = (IExec(J, Exec(i, Initialized(s))))(a).

(29) For every Int position a and for all integers k_{1}, k_{2} holds (a, k_{1}) <> 0 gotok_{2} ≠ halt_{SCMPDS} .

(30) For every Int position a and for all integers k_{1}, k_{2} holds (a, k_{1}) <= 0 gotok_{2} ≠ halt_{SCMPDS} .

(31) For every Int position a and for all integers k_{1}, k_{2} holds (a, k_{1}) >=

0 gotok_{2} ≠ halt_{SCMPDS} . Let us consider k_{1}. The functor Goto(k_{1}) yielding a Program-block is defined

as follows: (Def. 1) Goto(k_{1}) = Load(goto k_{1}).

Let n be a natural number. One can verify that goto (n+1) is No-StopCode and goto (− (n + 1)) is No-StopCode.

Let n be a natural number. Observe that Goto(n + 1) is No-StopCode and Goto(− (n + 1)) is No-StopCode.

The following two propositions are true:

(32) cardGoto(k_{1}) = 1.

(33) inspos0 ∈ domGoto(k_{1}) and (Goto(k_{1}))(inspos0) = goto k_{1}.

2. The Predicates of is closed on and is halting on

Let I be a Program-block and let s be a state of SCMPDS. We say that I is closed on s if and only if:

(Def. 2) For every natural number k holds IC_{(Computation (s+·Initialized (stop I)))(k)} ∈

domstopI. We say that I is halting on s if and only if:

(Def. 3) s+·Initialized(stopI) is halting.

We now state a number of propositions:

(34) For every Program-block I holds I is paraclosed iff for every state s of SCMPDS holds I is closed on s.

(35) For every Program-block I holds I is parahalting iff for every state s of SCMPDS holds I is halting on s.

(36) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. If s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}, then if I is closed on s_{1}, then I is closed on s_{2}.

(37) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Suppose I is closed on s_{1} and halting on s_{1}. Then I is closed on s_{2} and halting on s_{2}.

(38) For every state s of SCMPDS and for all Program-block I, J holds I is closed on s iff I is closed on s+·Initialized(J).

(39) Let I, J be Program-block and s be a state of SCMPDS. Suppose I is

closed on s and halting on s. Then

(i) for every natural number k such that k LifeSpan(s+·Initialized(stopI)) holds IC_{(Computation (s+·Initialized (stop I)))(k)} = IC_{(Computation (s+·Initialized (stop I;J)))(k)}, and

(ii) (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI))) ↾Data-Loc_{SCM} = (Computation(s+·Initialized(stopI;J)))(LifeSpan(s+· Initialized(stopI)))↾Data-Loc_{SCM}.

(40) Let I be a Program-block and k be a natural number. If I is closed on s and halting on s and k < LifeSpan(s+·Initialized(stopI)), then IC_{(Computation (s+·Initialized (stop I)))(k)} ∈ domI.

(41) Let I, J be Program-block, s be a state of SCMPDS, and k be a natural number. Suppose I is closed on s and halting on s and k < LifeSpan(s+·Initialized(stopI)). Then CurInstr((Computation(s+· Initialized(stopI)))(k)) = CurInstr((Computation(s+·Initialized(stopI;J)))(k)).

(42) Let I be a No-StopCode Program-block, s be a state of SCMPDS, and k be a natural number. If I is closed on s and halting on s and k < LifeSpan(s+·Initialized(stopI)), then CurInstr((Computation(s+·Initialized(stopI)))(k)) ≠ halt_{SCMPDS} .

(43) Let I be a No-StopCode Program-block and s be a state of SCMPDS. If I is closed on s and halting on s, then IC_{(Computation (s+·Initialized (stop I)))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(44) Let I, J be Program-block and s be a state of SCMPDS. Suppose I is closed on s and halting on s. Then I; Goto(cardJ + 1);J is halting on s and I; Goto(cardJ + 1);J is closed on s.

(45) Let I be a shiftable Program-block. Suppose Initialized(stopI) ⊆ s_{1} and I is closed on s_{1}. Let n be a natural number. Suppose Shift(stopI, n) ⊆ s_{2} and IC_{(s_{2})} = insposn and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} + n = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

(46) Let s be a state of SCMPDS, I be a No-StopCode Program-block, and J be a Program-block. If I is closed on s and halting on s, then IC_{IExec(I; Goto(card J+1);J,s)} = insposcardI + cardJ + 1.

(47) Let s be a state of SCMPDS, I be a No-StopCode Program-block, and J be a Program-block. If I is closed on s and halting on s, then IExec(I; Goto(cardJ + 1);J, s) = IExec(I, s)+·Start-At(insposcardI + cardJ + 1).

(48) Let s be a state of SCMPDS and I be a No-StopCode Program-block. If I is closed on s and halting on s, then IC_{IExec(I,s)} = insposcardI.

3. The Construction of Conditional Statements

Let a be a Int position, let k be an integer, and let I, J be Program-block. The functor if a = k then I else J yielding a Program-block is defined by:

(Def. 4) if a = k then I else J = ((a, k) <> 0 goto cardI + 2);I; Goto(cardJ +

1);J. The functor if a > k then I else J yielding a Program-block is defined by:

(Def. 5) if a > k then I else J = ((a, k) <= 0 goto cardI + 2);I; Goto(cardJ +

1);J. The functor if a < k then I else J yielding a Program-block is defined by:

(Def. 6) if a < k then I else J = ((a, k) >= 0 goto cardI + 2);I; Goto(cardJ +

1);J. Let a be a Int position, let k be an integer, and let I be a Program-block.

The functor if a = 0 then k else I yields a Program-block and is defined as follows:

(Def. 7) if a = 0 then k else I = ((a, k) <> 0 goto cardI + 1);I.

The functor if a ≠ 0 then k else I yielding a Program-block is defined by: (Def. 8) if a ≠ 0 then k else I = ((a, k) <> 0 goto2);goto (cardI + 1);I.

The functor if a > 0 then k else I yielding a Program-block is defined as follows:

(Def. 9) if a > 0 then k else I = ((a, k) <= 0 goto cardI + 1);I. The functor if a 0 then k else I yields a Program-block and is defined as follows:

(Def. 10) if a 0 then k else I = ((a, k) <= 0 goto2);goto (cardI + 1);I. The functor if a < 0 then k else I yields a Program-block and is defined as follows:

(Def. 11) if a < 0 then k else I = ((a, k) >= 0 goto cardI + 1);I. The functor if a 0 then k else I yields a Program-block and is defined as follows:

(Def. 12) if a 0 then k else I = ((a, k) >= 0 goto2);goto (cardI + 1);I.

4. The Computation of ”if var=0 then block1 else block2”

One can prove the following propositions:

(49) card(if a = k_{1} then I else J) = cardI + cardJ + 2.

(50) inspos0 ∈ dom(if a = k_{1} then I else J) and inspos1 ∈ dom(if a = k_{1} then I else J).

(51) (if a = k_{1} then I else J)(inspos0) = (a, k_{1}) <> 0 goto cardI + 2.

(52) Let s be a state of SCMPDS, I, J be shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) = 0 and I is closed on s and halting on s. Then if a = k_{1} then I else J is closed on s and if a = k_{1} then I else J is halting on s.

(53) Let s be a state of SCMPDS, I be a Program-block, J be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) ≠ 0 and J is closed on s and halting on s. Then if a = k_{1} then I else J is closed on s and if a = k_{1} then I else J is halting on s.

(54) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, J be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) = 0 and I is closed on s and halting on s. Then IExec(if a = k_{1} then I else J, s) = IExec(I, s)+·Start-At(insposcardI + cardJ + 2).

(55) Let s be a state of SCMPDS, I be a Program-block, J be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) ≠ 0 and J is closed on s and halting on s. Then IExec(if a = k_{1} then I else J, s) = IExec(J, s)+·Start-At(insposcardI + cardJ + 2).

Let I, J be shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a = k_{1} then I else J is shiftable and parahalting.

Let I, J be No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a = k_{1} then I else J is No-StopCode.

We now state three propositions:

(56) Let s be a state of SCMPDS, I, J be No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a=k_{1} then I else J,s)} = insposcardI + cardJ + 2.

(57) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, J be a shiftable Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) = 0, then (IExec(if a = k_{1} then I else J, s))(b) = (IExec(I, s))(b).

(58) Let s be a state of SCMPDS, I be a Program-block, J be a No-StopCode parahalting shiftable Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) ≠ 0, then (IExec(if a = k_{1} then I else J, s))(b) = (IExec(J, s))(b).

5. The Computation of ”if var=0 then block”

One can prove the following propositions:

(59) card(if a = 0 then k_{1} else I) = cardI + 1.

(60) inspos0 ∈ dom(if a = 0 then k_{1} else I).

(61) (if a = 0 then k_{1} else I)(inspos0) = (a, k_{1}) <> 0 goto cardI + 1.

(62) Let s be a state of SCMPDS, I be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) = 0 and I is closed on s and halting on s. Then if a = 0 then k_{1} else I is closed on s and if a = 0 then k_{1} else I is halting on s.

(63) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) ≠ 0, then if a = 0 then k_{1} else I is closed on s and if a = 0 then k_{1} else I is halting on s.

(64) Let s be a state of SCMPDS, I be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) = 0 and I is closed on s and halting on s. Then IExec(if a = 0 then k_{1} else I, s) = IExec(I, s)+·Start-At(insposcardI+ 1).

(65) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) ≠ 0, then IExec(if a =

0 then k_{1} else I, s) = s+·Start-At(insposcardI + 1). Let I be a shiftable parahalting Program-block, let a be a Int position, and

let k_{1} be an integer. One can verify that if a = 0 then k_{1} else I is shiftable and parahalting.

Let I be a No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a = 0 then k_{1} else I is No-StopCode.

Next we state three propositions:

(66) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a=0 then k_{1} else I,s)} = insposcardI + 1.

(67) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) = 0, then (IExec(if a = 0 then k_{1} else I, s))(b) = (IExec(I, s))(b).

(68) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) ≠ 0, then (IExec(if a = 0 then k_{1} else I, s))(b) = s(b).

6. The Computation of ”if var¡¿0 then block”

One can prove the following propositions:

(69) card(if a ≠ 0 then k_{1} else I) = cardI + 2.

(70) inspos0 ∈ dom(if a ≠ 0 then k_{1} else I) and inspos1 ∈ dom(if a ≠ 0 then k_{1} else I).

(71) (if a ≠ 0 then k_{1} else I)(inspos0) = (a, k_{1}) <> 0 goto2 and (if a ≠ 0 then k_{1} else I)(inspos1) = goto (cardI + 1).

(72) Let s be a state of SCMPDS, I be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) ≠ 0 and I is closed on s and halting on s. Then if a ≠ 0 then k_{1} else I is closed on s and if a ≠ 0 then k_{1} else I is halting on s.

(73) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) = 0, then if a ≠ 0 then k_{1} else I is closed on s and if a ≠ 0 then k_{1} else I is halting on s.

(74) Let s be a state of SCMPDS, I be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) ≠ 0 and I is closed on s and halting on s. Then IExec(if a ≠ 0 then k_{1} else I, s) = IExec(I, s)+·Start-At(insposcardI+ 2).

(75) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) = 0, then IExec(if a ≠ 0 then k_{1} else I, s) = s+·Start-At(insposcardI + 2).

Let I be a shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a ≠ 0 then k_{1} else I is shiftable and parahalting.

Let I be a No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. One can verify that if a ≠ 0 then k_{1} else I is No-StopCode.

One can prove the following three propositions:

(76) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a≠0 then k_{1} else I,s)} = insposcardI + 2.

(77) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) ≠ 0, then (IExec(if a ≠ 0 then k_{1} else I, s))(b) = (IExec(I, s))(b).

(78) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) = 0, then (IExec(if a ≠ 0 then k_{1} else I, s))(b) = s(b).

7. The Computation of ”if var¿0 then block1 else block2”

We now state several propositions:

(79) card(if a > k_{1} then I else J) = cardI + cardJ + 2.

(80) inspos0 ∈ dom(if a > k_{1} then I else J) and inspos1 ∈ dom(if a > k_{1} then I else J).

(81) (if a > k_{1} then I else J)(inspos0) = (a, k_{1}) <= 0 goto cardI + 2.

(82) Let s be a state of SCMPDS, I, J be shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) > 0 and I is closed on s and halting on s. Then if a > k_{1} then I else J is closed on s and if a > k_{1} then I else J is halting on s.

(83) Let s be a state of SCMPDS, I be a Program-block, J be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and J is closed on s and halting on s. Then if a > k_{1} then I else J is closed on s and if a > k_{1} then I else J is halting on s.

(84) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, J be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) > 0 and I is closed on s and halting on s. Then IExec(if a > k_{1} then I else J, s) = IExec(I, s)+·Start-At(insposcardI + cardJ + 2).

(85) Let s be a state of SCMPDS, I be a Program-block, J be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and J is closed on s and halting on s. Then IExec(if a > k_{1} then I else J, s) = IExec(J, s)+·Start-At(insposcardI + cardJ + 2).

Let I, J be shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a > k_{1} then I else J is shiftable and parahalting.

Let I, J be No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a > k_{1} then I else J is No-StopCode.

Next we state three propositions:

(86) Let s be a state of SCMPDS, I, J be No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a>k_{1} then I else J,s)} = insposcardI + cardJ + 2.

(87) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, J be a shiftable Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) > 0, then (IExec(if a > k_{1} then I else J, s))(b) = (IExec(I, s))(b).

(88) Let s be a state of SCMPDS, I be a Program-block, J be a No-StopCode parahalting shiftable Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then (IExec(if a > k_{1} then I else J, s))(b) = (IExec(J, s))(b).

8. The Computation of ”if var¿0 then block”

The following propositions are true:

(89) card(if a > 0 then k_{1} else I) = cardI + 1.

(90) inspos0 ∈ dom(if a > 0 then k_{1} else I).

(91) (if a > 0 then k_{1} else I)(inspos0) = (a, k_{1}) <= 0 goto cardI + 1.

(92) Let s be a state of SCMPDS, I be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) > 0 and I is closed on s and halting on s. Then if a > 0 then k_{1} else I is closed on s and if a > 0 then k_{1} else I is halting on s.

(93) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then if a > 0 then k_{1} else I is closed on s and if a > 0 then k_{1} else I is halting on s.

(94) Let s be a state of SCMPDS, I be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) > 0 and I is closed on s and halting on s. Then IExec(if a > 0 then k_{1} else I, s) = IExec(I, s)+·Start-At(insposcardI+ 1).

(95) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then IExec(if a > 0 then k_{1} else I, s) = s+·Start-At(insposcardI + 1).

Let I be a shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a > 0 then k_{1} else I is shiftable and parahalting.

Let I be a No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a > 0 then k_{1} else I is No-StopCode.

The following propositions are true:

(96) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a>0 then k_{1} else I,s)} = insposcardI + 1.

(97) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) > 0, then (IExec(if a > 0 then k_{1} else I, s))(b) = (IExec(I, s))(b).

(98) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then (IExec(if a > 0 then k_{1} else I, s))(b) = s(b).

9. The Computation of ”if var¡=0 then block”

We now state several propositions:

(99) card(if a 0 then k_{1} else I) = cardI + 2.

(100) inspos0 ∈ dom(if a 0 then k_{1} else I) and inspos1 ∈ dom(if a 0 then k_{1} else I).

(101) (if a 0 then k_{1} else I)(inspos0) = (a, k_{1}) <= 0 goto2 and (if a 0 then k_{1} else I)(inspos1) = goto (cardI + 1).

(102) Let s be a state of SCMPDS, I be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and I is closed on s and halting on s. Then if a 0 then k_{1} else I is closed on s and if a 0 then k_{1} else I is halting on s.

(103) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) > 0, then if a 0 then k_{1} else I is closed on s and if a 0 then k_{1} else I is halting on s.

(104) Let s be a state of SCMPDS, I be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and I is closed on s and halting on s. Then IExec(if a 0 then k_{1} else I, s) = IExec(I, s)+·Start-At(insposcardI+ 2).

(105) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) > 0, then IExec(if a 0 then k_{1} else I, s) = s+·Start-At(insposcardI + 2).

Let I be a shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a 0 then k_{1} else I is shiftable and parahalting.

Let I be a No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a 0 then k_{1} else I is No-StopCode.

We now state three propositions:

(106) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a 0 then k_{1} else I,s)} = insposcardI + 2.

(107) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then (IExec(if a 0 then k_{1} else I, s))(b) = (IExec(I, s))(b).

(108) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) > 0, then (IExec(if a 0 then k_{1} else I, s))(b) = s(b).

10. The Computation of ”if var¡0 then block1 else block2”

One can prove the following propositions:

(109) card(if a < k_{1} then I else J) = cardI + cardJ + 2.

(110) inspos0 ∈ dom(if a < k_{1} then I else J) and inspos1 ∈ dom(if a < k_{1} then I else J).

(111) (if a < k_{1} then I else J)(inspos0) = (a, k_{1}) >= 0 goto cardI + 2.

(112) Let s be a state of SCMPDS, I, J be shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) < 0 and I is closed on s and halting on s. Then if a < k_{1} then I else J is closed on s and if a < k_{1} then I else J is halting on s.

(113) Let s be a state of SCMPDS, I be a Program-block, J be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and J is closed on s and halting on s. Then if a < k_{1} then I else J is closed on s and if a < k_{1} then I else J is halting on s.

(114) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, J be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) < 0 and I is closed on s and halting on s. Then IExec(if a < k_{1} then I else J, s) = IExec(I, s)+·Start-At(insposcardI + cardJ + 2).

(115) Let s be a state of SCMPDS, I be a Program-block, J be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and J is closed on s and halting on s. Then IExec(if a < k_{1} then I else J, s) = IExec(J, s)+·Start-At(insposcardI + cardJ + 2).

Let I, J be shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a < k_{1} then I else J is shiftable and parahalting.

Let I, J be No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a < k_{1} then I else J is No-StopCode.

Next we state three propositions:

(116) Let s be a state of SCMPDS, I, J be No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a<k_{1} then I else J,s)} = insposcardI + cardJ + 2.

(117) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, J be a shiftable Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) < 0, then (IExec(if a < k_{1} then I else J, s))(b) = (IExec(I, s))(b).

(118) Let s be a state of SCMPDS, I be a Program-block, J be a No-StopCode parahalting shiftable Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then (IExec(if a < k_{1} then I else J, s))(b) = (IExec(J, s))(b).

11. The Computation of ”if var¡0 then block”

One can prove the following propositions:

(119) card(if a < 0 then k_{1} else I) = cardI + 1.

(120) inspos0 ∈ dom(if a < 0 then k_{1} else I).

(121) (if a < 0 then k_{1} else I)(inspos0) = (a, k_{1}) >= 0 goto cardI + 1.

(122) Let s be a state of SCMPDS, I be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) < 0 and I is closed on s and halting on s. Then if a < 0 then k_{1} else I is closed on s and if a < 0 then k_{1} else I is halting on s.

(123) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then if a < 0 then k_{1} else I is closed on s and if a < 0 then k_{1} else I is halting on s.

(124) Let s be a state of SCMPDS, I be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) < 0 and I is closed on s and halting on s. Then IExec(if a < 0 then k_{1} else I, s) = IExec(I, s)+·Start-At(insposcardI+ 1).

(125) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then IExec(if a < 0 then k_{1} else I, s) = s+·Start-At(insposcardI + 1).

Let I be a shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a < 0 then k_{1} else I is shiftable and parahalting.

Let I be a No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. One can check that if a < 0 then k_{1} else I is No-StopCode.

Next we state three propositions:

(126) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a<0 then k_{1} else I,s)} = insposcardI + 1.

(127) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) < 0, then (IExec(if a < 0 then k_{1} else I, s))(b) = (IExec(I, s))(b).

(128) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then (IExec(if a < 0 then k_{1} else I, s))(b) = s(b).

12. The Computation of ”if var¿=0 then block”

The following propositions are true:

(129) card(if a 0 then k_{1} else I) = cardI + 2.

(130) inspos0 ∈ dom(if a 0 then k_{1} else I) and inspos1 ∈ dom(if a 0 then k_{1} else I).

(131) (if a 0 then k_{1} else I)(inspos0) = (a, k_{1}) >= 0 goto2 and (if a 0 then k_{1} else I)(inspos1) = goto (cardI + 1).

(132) Let s be a state of SCMPDS, I be a shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and I is closed on s and halting on s. Then if a 0 then k_{1} else I is closed on s and if a 0 then k_{1} else I is halting on s.

(133) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) < 0, then if a 0 then k_{1} else I is closed on s and if a 0 then k_{1} else I is halting on s.

(134) Let s be a state of SCMPDS, I be a No-StopCode shiftable Program-block, a be a Int position, and k_{1} be an integer. Suppose s(DataLoc(s(a), k_{1})) 0 and I is closed on s and halting on s. Then IExec(if a 0 then k_{1} else I, s) = IExec(I, s)+·Start-At(insposcardI+ 2).

(135) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) < 0, then IExec(if a 0 then k_{1} else I, s) = s+·Start-At(insposcardI + 2).

Let I be a shiftable parahalting Program-block, let a be a Int position, and let k_{1} be an integer. Note that if a 0 then k_{1} else I is shiftable and parahalting.

Let I be a No-StopCode Program-block, let a be a Int position, and let k_{1} be an integer. Observe that if a 0 then k_{1} else I is No-StopCode.

We now state three propositions:

(136) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a be a Int position, and k_{1} be an integer. Then IC_{IExec(if a 0 then k_{1} else I,s)} = insposcardI + 2.

(137) Let s be a state of SCMPDS, I be a No-StopCode shiftable parahalting Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) 0, then (IExec(if a 0 then k_{1} else I, s))(b) = (IExec(I, s))(b).

(138) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, and k_{1} be an integer. If s(DataLoc(s(a), k_{1})) < 0, then (IExec(if a 0 then k_{1} else I, s))(b) = s(b).

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 Piotr Rudnicki. Development of terminology for scm. Formalized Mathematics, 4(1):61–67, 1993.

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

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

[6] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[7] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Formalized Mathematics, 8(1):193–199, 1999.

[8] Jing-Chao Chen. Computation of two consecutive program blocks for SCMPDS. Formalized Mathematics, 8(1):211–217, 1999.

[9] Jing-Chao Chen. The construction and shiftability of program blocks for SCMPDS. Formalized Mathematics, 8(1):201–210, 1999.

[10] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

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

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

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

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

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

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

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

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

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

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

Recursive Euclide Algorithm ^{1}

Jing-Chao Chen Shanghai Jiaotong University

Summary. The earlier SCM computer is not of any recursive function, so Trybulec and Nakamura proved the correctness of the Euclid’s algorithm only by way of an iterative program. However, the recursive method is a very important programming method, furthermore, for some algorithms, for example Quicksort, only by employing a recursive method (note push-down stack is essentially also a recursive method) can they be implemented. The main goal of the article is to test the recursive function of the SCMPDS computer by proving the correctness of the Euclid’s algorithm by way of a recursive program. In this article, we observed that the memory required by the recursive Euclide algorithm is variable but it is still autonomic. Although the algorithm here is more complicated than the non-recursive algorithm, its interest is that the SCMPDS computer will be able to implement many algorithms like Quicksort which the SCM computer cannot do.

MML Identifier: SCMP GCD.

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

1. Preliminaries

For simplicity, we adopt the following rules: m, n denote natural numbers, i, j denote instructions of SCMPDS, s denotes a state of SCMPDS, and I, J denote Program-block.

The following propositions are true:

(1) If m > 0, then gcd(n, m) = gcd(m, n mod m).

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

(2) For all integers i, j such that i 0 and j > 0 holds i gcdj = j gcdimodj.

(3) For every natural number m and for every integer j such that insposm = j holds insposm + 2 = 2 · (|j| ÷ 2) + 4.

Let k be a natural number. The functor intposk yielding a Int position is defined by:

(Def. 1) intposk = d_{k}.

The following propositions are true:

(4) For all natural numbers n_{1}, n_{2} such that n_{1} ≠ n_{2} holds intposn_{1} ≠ intposn_{2}.

(5) For all natural numbers n_{1}, n_{2} holds DataLoc(n_{1}, n_{2}) = intposn_{1} + n_{2}.

(6) For every state s of SCMPDS and for all natural numbers m_{1}, m_{2} such that IC_{s} = insposm_{1} + m_{2} holds ICplusConst(s, − m_{2}) = insposm_{1}.

The Int position GBP is defined by: (Def. 2) GBP = intpos0.

The Int position SBP is defined by: (Def. 3) SBP = intpos1.

One can prove the following propositions:

(7) GBP ≠ SBP.

(8) card(I;i) = cardI + 1.

(9) card(i;j) = 2.

(10) (I;i)(insposcardI) = i and insposcardI ∈ dom(I;i).

(11) (I;i;J)(insposcardI) = i.

2. The Construction of Recursive Euclide Algorithm

The Program-block GCD − Algorithm is defined by: (Def. 4) GCD − Algorithm = (GBP:=0);(SBP:=7); saveIC(SBP, RetIC);goto 2;halt_{SCMPDS} ;((SBP, 3) <=

0 goto9);((SBP, 6) := (SBP, 3)); Divide(SBP, 2, SBP, 3);((SBP, 7) :=

(SBP, 3));((SBP, 4+RetSP) := (GBP, 1)); AddTo(GBP, 1, 4); saveIC(SBP, RetIC);goto (− 7);((SBP, 2) := (SBP, 6)); returnSBP.

3. The Computation of Recursive Euclide Algorithm

The following propositions are true:

(12) cardGCD − Algorithm = 15.

(13) n < 15 iff insposn ∈ domGCD − Algorithm .

(14) (GCD − Algorithm )(inspos0) = GBP:=0 and (GCD − Algorithm )(inspos1) = SBP:=7 and (GCD − Algorithm )(inspos2) = saveIC(SBP, RetIC) and (GCD − Algorithm )(inspos3) = goto 2 and (GCD − Algorithm )(inspos4) = halt_{SCMPDS} and (GCD − Algorithm )(inspos5) = (SBP, 3) <= 0 goto9 and (GCD − Algorithm )(inspos6) = (SBP, 6) := (SBP, 3) and (GCD − Algorithm )(inspos7) = Divide(SBP, 2, SBP, 3) and (GCD − Algorithm )(inspos8) = (SBP, 7) := (SBP, 3) and (GCD − Algorithm )(inspos9) = (SBP, 4 + RetSP) := (GBP, 1) and (GCD − Algorithm )(inspos10) = AddTo(GBP, 1, 4) and (GCD − Algorithm )(inspos11) = saveIC(SBP, RetIC) and (GCD − Algorithm )(inspos12) = goto (− 7) and (GCD − Algorithm )(inspos13) = (SBP, 2) := (SBP, 6) and (GCD − Algorithm )(inspos14) = returnSBP.

(15) Let s be a state of SCMPDS. Suppose Initialized(GCD − Algorithm ) ⊆ s. Then IC_{(Computation (s))(4)} = inspos5 and (Computation(s))(4)(GBP) = 0 and (Computation(s))(4)(SBP) = 7 and (Computation(s))(4)(intpos7+ RetIC) = inspos2 and (Computation(s))(4)(intpos9) = s(intpos9) and (Computation(s))(4)(intpos10) = s(intpos10).

(16) Let s be a state of SCMPDS. Suppose GCD − Algorithm ⊆ s and IC_{s} = inspos5 and s(SBP) > 0 and s(GBP) = 0 and s(DataLoc(s(SBP), 3)) 0 and s(DataLoc(s(SBP), 2)) s(DataLoc(s(SBP), 3)). Then there exists n such that

(i) CurInstr((Computation(s))(n)) = returnSBP, (ii) s(SBP) = (Computation(s))(n)(SBP), (iii) (Computation(s))(n)(DataLoc(s(SBP), 2)) = s(DataLoc(s(SBP), 2)) gcds(DataLoc(s(SBP), 3)),

and (iv) for every natural number j such that 1 < j and j s(SBP) + 1 holds

s(intposj) = (Computation(s))(n)(intposj).

(17) Let s be a state of SCMPDS. Suppose GCD − Algorithm ⊆ s and IC_{s} = inspos5 and s(SBP) > 0 and s(GBP) = 0 and s(DataLoc(s(SBP), 3)) 0 and s(DataLoc(s(SBP), 2)) 0. Then there exists n such that

(i) CurInstr((Computation(s))(n)) = returnSBP, (ii) s(SBP) = (Computation(s))(n)(SBP), (iii) (Computation(s))(n)(DataLoc(s(SBP), 2)) = s(DataLoc(s(SBP), 2)) gcds(DataLoc(s(SBP), 3)),

and (iv) for every natural number j such that 1 < j and j s(SBP) + 1 holds

s(intposj) = (Computation(s))(n)(intposj).

4. The Correctness of Recursive Euclide Algorithm (18) Let s be a state of SCMPDS. Suppose Initialized(GCD − Algorithm ) ⊆

s. Let x, y be integers. If s(intpos9) = x and s(intpos10) = y and x 0 and y 0, then (Result(s))(intpos9) = x gcdy.

5. The Autonomy of Recursive Euclide Algorithm

The following proposition is true

(19) Let p be a finite partial state of SCMPDS and x, y be integers. If y 0 and x y and p = [intpos9 ⟼→ x, intpos10 ⟼→ y], then Initialized(GCD − Algorithm )+·p is autonomic.

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

[4] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

[5] Jing-Chao Chen. Computation and program shift in the SCMPDS computer. Formalized Mathematics, 8(1):193–199, 1999.

[6] Jing-Chao Chen. The construction and shiftability of program blocks for SCMPDS. Formalized Mathematics, 8(1):201–210, 1999.

[7] Jing-Chao Chen. The SCMPDS computer and the basic semantics of its instructions. Formalized Mathematics, 8(1):183–191, 1999.

[8] Jing-Chao Chen. A small computer model with push-down stack. Formalized Mathematics, 8(1):175–182, 1999.

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

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

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

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

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

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

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

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

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

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

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

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

Scott-Continuous Functions. Part II^{1}

Adam Grabowski University of Białystok

MML Identifier: WAYBEL24.

The terminology and notation used here are introduced in the following articles: [20], [26], [25], [27], [9], [10], [1], [17], [4], [2], [19], [21], [11], [18], [24], [3], [22], [8], [5], [12], [6], [14], [28], [7], [16], [23], [13], and [15].

1. Preliminaries

The following proposition is true

(1) Let S, T be up-complete Scott top-lattices and M be a subset of SCMaps(S, T). Then ⨆_{SCMaps (S,T)} M is a continuous map from S into T.

Let S be a non empty relational structure and let T be a non empty reflexive relational structure. Observe that every map from S into T which is constant is also monotone.

Let S be a non empty relational structure, let T be a reflexive non empty relational structure, and let a be an element of the carrier of T. One can verify that S ⟼→ a is monotone.

Next we state several propositions:

(2) Let S be a non empty relational structure and T be a lower-bounded antisymmetric reflexive non empty relational structure. Then ⊥_{MonMaps (S,T)} = S ⟼→ ⊥_{T} .

(3) Let S be a non empty relational structure and T be an upperbounded antisymmetric reflexive non empty relational structure. Then ⊤_{MonMaps (S,T)} = S ⟼→ ⊤_{T} .

1This work has been supported by KBN Grant 8 T11C 018 12.

(4) Let S, T be complete lattices, f be a monotone map from S into T, and x be an element of S. Then f(x) = sup(f^{∘}↓x).

(5) Let S, T be complete lower-bounded lattices, f be a monotone map from S into T, and x be an element of S. Then f(x) = ⨆_{T} {f(w); w ranges over elements of S: w x}.

(6) Let S be a relational structure, T be a non empty relational structure, and F be a subset of T^{the carrier of S}. Then supF is a map from S into T.

2. On the Scott Continuity of Maps

Let X_{1}, X_{2}, Y be non empty relational structures, let f be a map from [: X_{1}, X_{2} :] into Y, and let x be an element of the carrier of X_{1}. The functor Proj(f, x) yielding a map from X_{2} into Y is defined by:

(Def. 1) Proj(f, x) = (curryf)(x).

For simplicity, we use the following convention: X_{1}, X_{2}, Y are non empty relational structures, f is a map from [: X_{1}, X_{2} :] into Y, x is an element of the carrier of X_{1}, and y is an element of the carrier of X_{2}.

One can prove the following proposition

(7) For every element y of the carrier of X_{2} holds (Proj(f, x))(y) = f(⟨⟨x, y⟩⟩).

Let X_{1}, X_{2}, Y be non empty relational structures, let f be a map from [: X_{1}, X_{2} :] into Y, and let y be an element of the carrier of X_{2}. The functor Proj(f, y) yielding a map from X_{1} into Y is defined as follows:

(Def. 2) Proj(f, y) = (curry′f)(y). One can prove the following propositions:

(8) For every element x of the carrier of X_{1} holds (Proj(f, y))(x) = f(⟨⟨x, y⟩⟩).

(9) Let R, S, T be non empty relational structures, f be a map from [: R, S:] into T, a be an element of R, and b be an element of S. Then (Proj(f, a))(b) = (Proj(f, b))(a).

Let S be a non empty relational structure and let T be a non empty reflexive relational structure. Observe that there exists a map from S into T which is antitone.

The following two propositions are true:

(10) Let R, S, T be non empty reflexive relational structures, f be a map from [: R, S:] into T, a be an element of the carrier of R, and b be an element of the carrier of S. If f is monotone, then Proj(f, a) is monotone and Proj(f, b) is monotone.

(11) Let R, S, T be non empty reflexive relational structures, f be a map from [: R, S:] into T, a be an element of the carrier of R, and b be an element of the carrier of S. If f is antitone, then Proj(f, a) is antitone and Proj(f, b) is antitone.

Let R, S, T be non empty reflexive relational structures, let f be a monotone map from [: R, S:] into T, and let a be an element of the carrier of R. Observe that Proj(f, a) is monotone.

Let R, S, T be non empty reflexive relational structures, let f be a monotone map from [: R, S:] into T, and let b be an element of the carrier of S. Observe that Proj(f, b) is monotone.

Let R, S, T be non empty reflexive relational structures, let f be an antitone map from [: R, S:] into T, and let a be an element of the carrier of R. Observe that Proj(f, a) is antitone.

Let R, S, T be non empty reflexive relational structures, let f be an antitone map from [: R, S:] into T, and let b be an element of the carrier of S. One can verify that Proj(f, b) is antitone.

We now state several propositions:

(12) Let R, S, T be lattices and f be a map from [: R, S:] into T. Suppose that for every element a of R and for every element b of S holds Proj(f, a) is monotone and Proj(f, b) is monotone. Then f is monotone.

(13) Let R, S, T be lattices and f be a map from [: R, S:] into T. Suppose that for every element a of R and for every element b of S holds Proj(f, a) is antitone and Proj(f, b) is antitone. Then f is antitone.

(14) Let R, S, T be lattices, f be a map from [: R, S:] into T, b be an element of S, and X be a subset of R. Then (Proj(f, b))^{∘}X = f^{∘}[: X, {b}:].

(15) Let R, S, T be lattices, f be a map from [: R, S:] into T, b be an element of R, and X be a subset of S. Then (Proj(f, b))^{∘}X = f^{∘}[: {b}, X:].

(16) Let R, S, T be lattices, f be a map from [: R, S:] into T, a be an element of R, and b be an element of S. Suppose f is directed-sups-preserving. Then Proj(f, a) is directed-sups-preserving and Proj(f, b) is directed-sups-preserving.

(17) Let R, S, T be lattices, f be a monotone map from [: R, S:] into T, a be an element of R, b be an element of S, and X be a directed subset of [: R, S:]. If sup f^{∘}X exists in T and a ∈ π_{1}(X) and b ∈ π_{2}(X), then f(⟨⟨a, b⟩⟩) sup(f^{∘}X).

(18) Let R, S, T be complete lattices and f be a map from [: R, S:] into T. Suppose that for every element a of R and for every element b of S holds Proj(f, a) is directed-sups-preserving and Proj(f, b) is directed-sups-preserving. Then f is directed-sups-preserving.

(19) Let S be a non empty 1-sorted structure, T be a non empty relational structure, and f be a set. Then f is an element of T^{the carrier of S} if and only if f is a map from S into T.

3. The Poset of Continuous Maps

Let S be a topological structure and let T be a non empty FR-structure. The functor [S → T] yields a strict relational structure and is defined by the conditions (Def. 3).

(Def. 3)(i) [S → T] is a full relational substructure of T^{the carrier of S}, and

(ii) for every set x holds x ∈ the carrier of ([S → T]) iff there exists a map

f from S into T such that x = f and f is continuous. Let S be a non empty topological space and let T be a non empty topological

space-like FR-structure. One can verify that [S → T] is non empty.

Let S be a non empty topological space and let T be a non empty topological space-like FR-structure. Note that [S → T] is constituted functions.

One can prove the following two propositions:

(20) Let S be a non empty topological space, T be a non empty reflexive topological space-like FR-structure, and x, y be elements of [S → T]. Then x y if and only if for every element i of S holds ⟨⟨x(i), y(i)⟩⟩∈ the internal relation of T.

(21) Let S be a non empty topological space, T be a non empty reflexive topological space-like FR-structure, and x be a set. Then x is a continuous map from S into T if and only if x is an element of [S → T].

Let S be a non empty topological space and let T be a non empty reflexive topological space-like FR-structure. Note that [S → T] is reflexive.

Let S be a non empty topological space and let T be a non empty transitive topological space-like FR-structure. One can check that [S → T] is transitive.

Let S be a non empty topological space and let T be a non empty antisymmetric topological space-like FR-structure. Note that [S → T] is antisymmetric.

Let S be a non empty 1-sorted structure and let T be a non empty topological space-like FR-structure. Observe that T^{the carrier of S} is constituted functions.

Next we state three propositions:

(22) Let S be a non empty 1-sorted structure, T be a complete lattice, f, g, h be maps from S into T, and i be an element of S. If h = ⨆_{(T^{the carrier of S})}{f, g}, then h(i) = sup{f(i), g(i)}.

(23) Let I be a non empty set and J be a relational structure yielding nonempty reflexive-yielding many sorted set indexed by I. Suppose that for every element i of I holds J(i) is a complete lattice. Let X be a subset of ∏ J and i be an element of I. Then (inf X)(i) = inf π_{i}X.

(24) Let S be a non empty 1-sorted structure, T be a complete lattice, f, g, h be maps from S into T, and i be an element of S. If h = ⌈−⌉_{(T^{the carrier of S})}{f, g}, then h(i) = inf{f(i), g(i)}.

Let S be a non empty 1-sorted structure and let T be a lattice. Note that every element of T^{the carrier of S} is function-like and relation-like.

Let S, T be top-lattices. One can verify that every element of [S → T] is function-like and relation-like.

One can prove the following two propositions:

(25) Let S be a non empty relational structure, T be a complete lattice, F be a non empty subset of T^{the carrier of S}, and i be an element of the carrier of S. Then (supF)(i) = ⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }.

(26) Let S, T be complete top-lattices, F be a non empty subset of [S → T], and i be an element of the carrier of S. Then (⨆_{(T^{the carrier of S})} F)(i) = ⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }.

In the sequel S denotes a non empty relational structure, T denotes a complete lattice, and i denotes an element of S.

We now state two propositions:

(27) Let F be a non empty subset of T^{the carrier of S} and D be a non empty subset of S. Then (supF)^{∘}D = {⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }; i ranges over elements of S: i ∈ D}.

(28) Let S, T be complete Scott top-lattices, F be a non empty subset of [S → T], and D be a non empty subset of S. Then (⨆_{(T^{the carrier of S})} F)^{∘}D = {⨆_{T} {f(i); f ranges over elements of T^{the carrier of S}: f ∈ F }; i ranges over elements of S: i ∈ D}.

The scheme FraenkelF’RSS deals with a non empty relational structure A, a unary functor F yielding a set, a unary functor G yielding a set, and and states that:

{F(v_{1}); v_{1} ranges over elements of A : P[v_{1}]} = {G(v_{2}); v_{2} ranges over elements of A : P[v_{2}]}

provided the following condition is met:

• For every element v of A such that P[v] holds F(v) = G(v). One can prove the following propositions:

(29) Let S, T be complete Scott top-lattices and F be a non empty subset of [S → T]. Then ⨆_{(T^{the carrier of S})} F is a monotone map from S into T.

(30) Let S, T be complete Scott top-lattices, F be a non empty subset of [S → T], and D be a directed non empty subset of S. Then ⨆_{T} {⨆_{T} {g(i); i ranges over elements of S: i ∈ D}; g ranges over elements of T^{the carrier of S}: g ∈ F } = ⨆_{T} {⨆_{T} {g′(i′); g′ ranges over elements of T^{the carrier of S}: g′ ∈ F }; i′ ranges over elements of S: i′∈ D}.

(31) Let S, T be complete Scott top-lattices, F be a non empty subset of [S → T], and D be a directed non empty subset of S. Then ⨆_{T} ((⨆_{(T^{the carrier of S})} F)^{∘}D) = (⨆_{(T^{the carrier of S})} F)(supD).

(32) Let S, T be complete Scott top-lattices and F be a non empty subset of [S → T]. Then ⨆_{(T^{the carrier of S})} F ∈ the carrier of ([S → T]).

(33) Let S be a non empty relational structure and T be a lower-bounded antisymmetric non empty relational structure. Then ⊥_{T^{the carrier of S}} = S ⟼→ ⊥_{T} .

(34) Let S be a non empty relational structure and T be an upper-bounded antisymmetric non empty relational structure. Then ⊤_{T^{the carrier of S}} = S ⟼→ ⊤_{T} .

Let S be a non empty reflexive relational structure, let T be a complete lattice, and let a be an element of T. One can verify that S ⟼→ a is directedsups-preserving.

The following proposition is true

(35) Let S, T be complete Scott top-lattices. Then [S → T] is a sups-

inheriting relational substructure of T^{the carrier of S}. Let S, T be complete Scott top-lattices. Note that [S → T] is complete. Next we state three propositions:

(36) For all non empty Scott complete top-lattices S, T holds ⊥_{[S→T]} = S ⟼→ ⊥_{T} .

(37) For all non empty Scott complete top-lattices S, T holds ⊤_{[S→T]} = S ⟼→ ⊤_{T} .

(38) For all Scott complete top-lattices S, T holds SCMaps(S, T) = [S → T].

References

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

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

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

[4] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.

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

[6] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.

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

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

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

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

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

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

scott-continuous functions. part ii 245

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

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

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

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

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

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

[19] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

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

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

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

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

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

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

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

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

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

246 adam grabowski

Some Properties of Isomorphism between Relational Structures.

On the Product of Topological Spaces^{1}

Jarosław Gryko University of Bialystok

Artur Korniłowicz University of Bialystok

MML Identifier: YELLOW14.

The terminology and notation used in this paper are introduced in the following papers: [30], [37], [36], [38], [39], [16], [10], [11], [12], [9], [7], [20], [29], [1], [3], [31], [24], [26], [27], [25], [40], [15], [14], [42], [22], [21], [8], [28], [35], [2], [19], [4], [5], [13], [18], [41], [6], [33], [32], [23], [34], and [17].

1. Preliminaries

One can prove the following three propositions:

(1) 2^{1} = {0, 1}.

(2) For every set X and for every subset Y of X holds rng(id_{X}↾Y) = Y.

(3) For every function f and for all sets a, b holds (f+·(a⟼→ . b))(a) = b. Let us mention that there exists a relational structure which is strict and

empty.

Next we state four propositions:

(4) Let S be an empty 1-sorted structure, T be a 1-sorted structure, and f be a map from S into T. If rngf = Ω_{T} , then T is empty.

(5) Let S be a 1-sorted structure, T be an empty 1-sorted structure, and f be a map from S into T. If domf = Ω_{S}, then S is empty.

(6) Let S be a non empty 1-sorted structure, T be a 1-sorted structure, and f be a map from S into T. If domf = Ω_{S}, then T is non empty.

1This work has been supported by KBN Grant 8 T11C 018 12.

(7) Let S be a 1-sorted structure, T be a non empty 1-sorted structure, and f be a map from S into T. If rngf = Ω_{T} , then S is non empty.

Let S be a non empty reflexive relational structure, let T be a non empty relational structure, and let f be a map from S into T. Let us observe that f is directed-sups-preserving if and only if:

(Def. 1) For every non empty directed subset X of S holds f preserves sup of X.

Let R be a 1-sorted structure and let N be a net structure over R. We say that N is function yielding if and only if:

(Def. 2) The mapping of N is function yielding.

Let us observe that there exists a 1-sorted structure which is strict, non empty, and constituted functions.

One can verify that there exists a relational structure which is strict, non empty, and constituted functions.

Let R be a constituted functions 1-sorted structure. Note that every net structure over R is function yielding.

Let R be a constituted functions 1-sorted structure. Note that there exists a net structure over R which is strict and function yielding.

Let R be a non empty constituted functions 1-sorted structure. Note that there exists a net structure over R which is strict, non empty, and function yielding.

Let R be a constituted functions 1-sorted structure and let N be a function yielding net structure over R. One can check that the mapping of N is function yielding.

Let R be a non empty constituted functions 1-sorted structure. One can check that there exists a net in R which is strict and function yielding.

Let S be a non empty 1-sorted structure and let N be a non empty net structure over S. One can check that rng(the mapping of N) is non empty.

Let S be a non empty 1-sorted structure and let N be a non empty net structure over S. Observe that rngnetmap(N, S) is non empty.

One can prove the following two propositions:

(8) Let A, B, C be non empty relational structures, f be a map from B into C, and g, h be maps from A into B. If g h and f is monotone, then f · g f · h.

(9) Let S be a non empty topological space, T be a non empty topological space-like FR-structure, f, g be maps from S into T, and x, y be elements of [S → T]. If x = f and y = g, then x y iff f g.

Let I be a set and let R be a non empty relational structure. One can verify that every element of the carrier of R^{I} is function-like and relation-like.

Let I be a non empty set, let R be a non empty relational structure, let f be an element of the carrier of R^{I}, and let i be an element of I. Then f(i) is an element of R.

2. Some Properties of Isomorphism between Relational Structures

The following proposition is true

(10) For all relational structures S, T and for every map f from S into T

such that f is isomorphic holds f is onto. Let S, T be relational structures. One can verify that every map from S into

T which is isomorphic is also onto.

Next we state four propositions:

(11) Let S, T be non empty relational structures and f be a map from S into T. If f is isomorphic, then f^{−1} is isomorphic.

(12) For all non empty relational structures S, T such that S and T are isomorphic and S has g.l.b.’s holds T has g.l.b.’s.

(13) For all non empty relational structures S, T such that S and T are isomorphic and S has l.u.b.’s holds T has l.u.b.’s.

(14) For every relational structure L such that L is empty holds L is bounded. Let us note that every relational structure which is empty is also bounded. One can prove the following propositions:

(15) Let S, T be relational structures. Suppose S and T are isomorphic and S is lower-bounded. Then T is lower-bounded.

(16) Let S, T be relational structures. Suppose S and T are isomorphic and S is upper-bounded. Then T is upper-bounded.

(17) Let S, T be non empty relational structures, A be a subset of S, and f be a map from S into T. Suppose f is isomorphic and sup A exists in S. Then sup f^{∘}A exists in T.

(18) Let S, T be non empty relational structures, A be a subset of S, and f be a map from S into T. Suppose f is isomorphic and inf A exists in S. Then inf f^{∘}A exists in T.

3. On the Product of Topological Spaces

Next we state two propositions:

(19) Let S, T be topological structures. Suppose S and T are homeomorphic or there exists a map f from S into T such that domf = Ω_{S} and rngf = Ω_{T} . Then S is empty if and only if T is empty.

(20) For every non empty topological space T holds T and the topological structure of T are homeomorphic.

Let T be a Scott reflexive non empty FR-structure. Note that every subset of T which is open is also inaccessible and upper and every subset of T which is inaccessible and upper is also open.

One can prove the following propositions:

(21) Let T be a topological structure, x, y be points of T, and X, Y be subsets of T. If X = \{x\} and \overline{X} ⊆ \overline{Y} , then x ∈ \overline{Y} .

(22) Let T be a topological structure, x, y be points of T, and Y, V be subsets of T. If Y = \{y\} and x ∈ \overline{Y} and V is open and x ∈ V, then y ∈ V.

(23) Let T be a topological structure, x, y be points of T, and X, Y be subsets of T. Suppose X = {x} and Y = {y}. Suppose that for every subset V of T such that V is open holds if x ∈ V, then y ∈ V. Then \overline{X} ⊆ \overline{Y} .

(24) Let S, T be non empty topological spaces, A be an irreducible subset of S, and B be a subset of T. Suppose A = B and the topological structure of S = the topological structure of T. Then B is irreducible.

(25) Let S, T be non empty topological spaces, a be a point of S, b be a point of T, A be a subset of the carrier of S, and B be a subset of the carrier of T. Suppose a = b and A = B and the topological structure of S = the topological structure of T and a is dense point of A. Then b is dense point of B.

(26) Let S, T be topological structures, A be a subset of S, and B be a subset of T. Suppose A = B and the topological structure of S = the topological structure of T and A is compact. Then B is compact.

(27) Let S, T be non empty topological spaces. Suppose the topological structure of S = the topological structure of T and S is sober. Then T is sober.

(28) Let S, T be non empty topological spaces. Suppose the topological structure of S = the topological structure of T and S is locally-compact. Then T is locally-compact.

(29) Let S, T be topological structures. Suppose the topological structure of

S = the topological structure of T and S is compact. Then T is compact. Let I be a non empty set, let T be a non empty topological space, let x be

a point of ∏(I ⟼→ T), and let i be an element of I. Then x(i) is an element of T.

Next we state several propositions:

(30) Let M be a non empty set, J be a topological space yielding nonempty many sorted set indexed by M, and x, y be points of ∏ J. Then x ∈ \overline{\{y\}} if and only if for every element i of M holds x(i) ∈ \overline{\{y(i)\}}.

(31) Let M be a non empty set, T be a non empty topological space, and x, y be points of ∏(M ⟼→ T). Then x ∈ \overline{\{y\}} if and only if for every element i of M holds x(i) ∈ \overline{\{y(i)\}}.

(32) Let M be a non empty set, i be an element of M, J be a topological space yielding nonempty many sorted set indexed by M, and x be a point of ∏ J. Then π_{i}\overline{\{x\}} = \overline{\{x(i)\}}.

(33) Let M be a non empty set, i be an element of M, T be a non empty topological space, and x be a point of ∏(M ⟼→ T). Then π_{i}\overline{\{x\}} = \overline{\{x(i)\}}.

(34) Let X, Y be non empty topological structures, f be a map from X into Y, and g be a map from Y into X. Suppose f = id_{X} and g = id_{X} and f is continuous and g is continuous. Then the topological structure of X = the topological structure of Y.

(35) Let X, Y be non empty topological spaces and f be a map from X into

Y. If f^{∘} is continuous, then f is continuous. Let X, Y be non empty topological spaces. One can verify that every con-

tinuous map from X into Y is continuous.

Let X be a non empty topological space and let Y be a non empty subspace of X. Note that {Y \atop ↪} is continuous.

We now state three propositions:

(36) For every non empty topological space T and for every map f from T into T such that f · f = f holds f^{∘} · ({Im f \atop ↪}) = id_{Im f}.

(37) For every non empty topological space Y and for every non empty subspace W of Y holds ({W \atop ↪})^{∘} is a homeomorphism.

(38) Let M be a non empty set and J be a topological space yielding nonempty many sorted set indexed by M. Suppose that for every element i of M holds J(i) is a T_{0} topological space. Then ∏ J is T_{0}.

Let I be a non empty set and let T be a non empty T_{0} topological space. One can check that ∏(I ⟼→ T) is T_{0}.

We now state the proposition

(39) Let M be a non empty set and J be a topological space yielding nonempty many sorted set indexed by M. Suppose that for every element i of M holds J(i) is T_{1} and topological space-like. Then ∏ J is a T_{1} space.

Let I be a non empty set and let T be a non empty T_{1} topological space. Observe that ∏(I ⟼→ T) is T_{1}.

References

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

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

[3] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.

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

[5] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.

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

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

252 jarosław gryko and artur korniłowicz

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

[9] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

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

[12] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

[17] Adam Grabowski. Scott-continuous functions. Part II. Formalized Mathematics, 8(1):239– 245, 1999.

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

[19] Jarosław Gryko. Injective spaces. Formalized Mathematics, 7(1):57–62, 1998.

[20] Krzysztof Hryniewiecki. Relations of tolerance. Formalized Mathematics, 2(1):105–109, 1991.

[21] Zbigniew Karno. Continuity of mappings over the union of subspaces. Formalized Mathematics, 3(1):1–16, 1992.

[22] Zbigniew Karno. On Kolmogorov topological spaces. Formalized Mathematics, 5(1):119– 124, 1996.

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

[24] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

some properties of isomorphism between … 253

[42] Mariusz Żynel and Adam Guzowski. T_{0} topological spaces. Formalized Mathematics,

5(1):75–77, 1996.

Received June 22, 1999 254 jarosław gryko and artur korniłowicz

Cages - the External Aproximation of Jordan’s Curve

Czesław Byliński^{1} University of Białystok

Mariusz Żynel^{2} University of Białystok

Summary. On the Euclidean plane Jordan’s curve may be approximated with a polygonal path of sides parallel to coordinate axes, either externally, or internally. The paper deals with external approximation, and existence of a Cage

– an external polygonal path – is proved.

MML Identifier: JORDAN9.

The papers [27], [22], [32], [31], [16], [2], [23], [26], [33], [7], [8], [1], [3], [4], [6], [30], [9], [19], [17], [18], [25], [34], [13], [24], [5], [14], [15], [20], [21], [12], [29], [28], [10], and [11] provide the terminology and notation for this paper.

1. Generalities

We adopt the following rules: k, n denote natural numbers, D denotes a non empty set, and f, g denote finite sequences of elements of D.

The following propositions are true:

(1) For all sets A, B such that A meets B holds A ∩ B meets B.

(2) For every non empty set A and for all sets B_{1}, B_{2} such that A ⊆ B_{1} and A ⊆ B_{2} holds B_{1} meets B_{2}.

1The paper was started during the author’s visit in the Shinshu University, Nagano, Japan, summer 1998.

2The paper was finished during the author’s visit in the Shinshu University, Nagano, Japan, summer 1999.

(3) Let T be a non empty topological space and B, C_{1}, C_{2}, D be subsets of T. Suppose B is connected and C_{1} is a component of D and C_{2} is a component of D and B meets C_{1} and B meets C_{2} and B ⊆ D. Then C_{1} = C_{2}.

(4) If for every n holds f↾n = g↾n, then f = g.

(5) If n ∈ domf, then there exists k such that k ∈ domRev(f) and n + k = lenf + 1 and π_{n}f = π_{k} Rev(f).

(6) If n ∈ domRev(f), then there exists k such that k ∈ domf and n + k = lenf + 1 and π_{n} Rev(f) = π_{k}f.

2. Go-Board Preliminaries

For simplicity, we adopt the following convention: G denotes a Go-board, f, g denote finite sequences of elements of E_{T}^{2}, p denotes a point of E_{T}^{2}, r, s denote real numbers, i, j, k denote natural numbers, and x denotes a set.

One can prove the following propositions:

(7) f is a sequence which elements belong to G iff Rev(f) is a sequence which elements belong to G.

(8) If f is a sequence which elements belong to G and 1 k and k lenf, then π_{k}f ∈ ValuesG.

(9) If n lenf and x ∈ L̃(f_{⇂n}), then there exists a natural number i such that n + 1 i and i + 1 lenf and x ∈ L(f, i).

(10) If f is a sequence which elements belong to G and 1 k and k+1 lenf, then π_{k}f ∈ left cell(f, k, G) and π_{k}f ∈ right cell(f, k, G).

(11) If f is a sequence which elements belong to G and 1 k and k+1 lenf, then Intleft cell(f, k, G) ≠ ∅ and Intright cell(f, k, G) ≠ ∅.

(12) Suppose f is a sequence which elements belong to G and 1 k and k + 1 lenf. Then Intleft cell(f, k, G) is connected and Intright cell(f, k, G) is connected.

(13) If f is a sequence which elements belong to G and 1 k and k+1 lenf, then \overline{Intleft cell(f, k, G)} = left cell(f, k, G) and \overline{Intright cell(f, k, G)} = right cell(f, k, G).

(14) Suppose f is a sequence which elements belong to G and L(f, k) is horizontal. Then there exists j such that 1 j and j widthG and for every p such that p ∈ L(f, k) holds p_{2} = (G_{1,j})_{2}.

(15) Suppose f is a sequence which elements belong to G and L(f, k) is vertical. Then there exists i such that 1 i and i lenG and for every p such that p ∈ L(f, k) holds p_{1} = (G_{i,1})_{1}.

(16) If f is a sequence which elements belong to G and special and i lenG and j widthG, then Intcell(G, i, j) misses L̃(f).

(17) Suppose f is a sequence which elements belong to G and special and 1 k and k + 1 lenf. Then Intleft cell(f, k, G) misses L̃(f) and Intright cell(f, k, G) misses L̃(f).

(18) Suppose 1 i and i + 1 lenG and 1 j and j + 1 widthG. Then (G_{i,j})_{1} = (G_{i,j+1})_{1} and (G_{i,j})_{2} = (G_{i+1,j})_{2} and (G_{i+1,j+1})_{1} = (G_{i+1,j})_{1} and (G_{i+1,j+1})_{2} = (G_{i,j+1})_{2}.

(19) Let i, j be natural numbers. Suppose 1 i and i + 1 lenG and 1 j and j + 1 widthG. Then p ∈ cell(G, i, j) if and only if the following conditions are satisfied:

(i) (G_{i,j})_{1} p_{1}, (ii) p_{1} (G_{i+1,j})_{1}, (iii) (G_{i,j})_{2} p_{2}, and (iv) p_{2} (G_{i,j+1})_{2}.

(20) If 1 i and i + 1 lenG and 1 j and j + 1 widthG, then cell(G, i, j) = {[r, s] : (G_{i,j})_{1} r ∧ r (G_{i+1,j})_{1} ∧ (G_{i,j})_{2} s ∧ s (G_{i,j+1})_{2}}.

(21) Suppose 1 i and i + 1 lenG and 1 j and j + 1 widthG and p ∈ ValuesG and p ∈ cell(G, i, j). Then p = G_{i,j} or p = G_{i,j+1} or p = G_{i+1,j+1} or p = G_{i+1,j}.

(22) If 1 i and i + 1 lenG and 1 j and j + 1 widthG, then G_{i,j} ∈ cell(G, i, j) and G_{i,j+1} ∈ cell(G, i, j) and G_{i+1,j+1} ∈ cell(G, i, j) and G_{i+1,j} ∈ cell(G, i, j).

(23) If 1 i and i+1 lenG and 1 j and j+1 widthG and p ∈ ValuesG and p ∈ cell(G, i, j), then p is extremal in cell(G, i, j).

(24) Suppose 2 lenG and 2 widthG and f is a sequence which elements belong to G and 1 k and k + 1 lenf. Then there exist i, j such that 1 i and i + 1 lenG and 1 j and j + 1 widthG and L(f, k) ⊆ cell(G, i, j).

(25) Suppose 2 lenG and 2 widthG and f is a sequence which elements belong to G and 1 k and k +1 lenf and p ∈ ValuesG and p ∈ L(f, k). Then p = π_{k}f or p = π_{k+1}f.

(26) If ⟨⟨i, j⟩⟩ ∈ the indices of G and 1 k and k widthG, then (G_{i,j})_{1} (G_{len G,k})_{1}.

(27) If ⟨⟨i, j⟩⟩ ∈ the indices of G and 1 k and k lenG, then (G_{i,j})_{2} (G_{k,width G})_{2}.

(28) Suppose f is a sequence which elements belong to G and special and L̃(g) ⊆ L̃(f) and 1 k and k + 1 lenf. Let A be a subset of E_{T}^{2}. If A = right cell(f, k, G) \ L̃(g) or A = left cell(f, k, G) \ L̃(g), then A is

connected.

(29) Let f be a non constant standard special circular sequence. Suppose f is a sequence which elements belong to G. Let given k. If 1 k and k + 1 lenf, then right cell(f, k, G)\L̃(f) ⊆ RightComp(f) and left cell(f, k, G)\ L̃(f) ⊆ LeftComp(f).

3. Cages

We use the following convention: C is a compact non vertical non horizontal non empty subset of E_{T}^{2} and i, k, n, i_{1}, i_{2} are natural numbers.

We now state three propositions:

(30) There exists i such that 1 i and i + 1 lenGauge(C, n) and N-minC ∈ cell(Gauge(C, n), i, widthGauge(C, n) − ′ 1) and N-minC ≠ (Gauge(C, n))_{i,width Gauge(C,n)− ′1}.

(31) Suppose that 1 i_{1} and i_{1} + 1 lenGauge(C, n) and N-minC ∈ cell(Gauge(C, n), i_{1}, widthGauge(C, n)− ′1) and N-minC ≠ (Gauge(C, n))_{i_{1},width Gauge(C,n)− ′1} and 1 i_{2} and i_{2} + 1 lenGauge(C, n) and N-minC ∈ cell(Gauge(C, n), i_{2}, widthGauge(C, n)− ′1) and N-minC ≠ (Gauge(C, n))_{i_{2},width Gauge(C,n)− ′1}. Then i_{1} = i_{2}.

(32) Let f be a standard non constant special circular sequence. Suppose that (i) f is a sequence which elements belong to Gauge(C, n), (ii) for every k such that 1 k and k + 1 lenf holds

left cell(f, k, Gauge(C, n)) ∩ C = ∅ and right cell(f, k, Gauge(C, n)) ∩ C ≠ ∅, and

(iii) there exists i such that 1 i and i + 1 lenGauge(C, n) and π_{1}f = (Gauge(C, n))_{i,width Gauge(C,n)} and π_{2}f = (Gauge(C, n))_{i+1,width Gauge(C,n)} and N-minC ∈ cell(Gauge(C, n), i, widthGauge(C, n)− ′1) and N-minC ≠ (Gauge(C, n))_{i,width Gauge(C,n)− ′1}. Then N-min L̃(f) = π_{1}f.

Let C be a compact non vertical non horizontal non empty subset of E_{T}^{2} and let n be a natural number. Let us assume that C is connected. The functor Cage(C, n) yields a clockwise oriented standard non constant special circular sequence and is defined by the conditions (Def. 1).

(Def. 1)(i) Cage(C, n) is a sequence which elements belong to Gauge(C, n),

(ii) there exists i such that 1 i and i + 1 lenGauge(C, n) and π_{1} Cage(C, n) = (Gauge(C, n))_{i,width Gauge(C,n)} and π_{2} Cage(C, n) = (Gauge(C, n))_{i+1,width Gauge(C,n)} and N-minC ∈ cell(Gauge(C, n), i, widthGauge(C, n)− ′

1) and N-minC ≠ (Gauge(C, n))_{i,width Gauge(C,n)− ′1}, and

cages - the external aproximation of … 259

(iii) for every k such that 1  k and k + 2  lenCage(C, n)

holds if front left cell(Cage(C, n), k, Gauge(C, n)) ∩ C = ∅ and front right cell(Cage(C, n), k, Gauge(C, n))∩C = ∅, then Cage(C, n) turns right k, Gauge(C, n) and if front left cell(Cage(C, n), k, Gauge(C, n)) ∩ C = ∅ and front right cell(Cage(C, n), k, Gauge(C, n)) ∩ C ≠ ∅, then Cage(C, n) goes straight k, Gauge(C, n) and if front left cell(Cage(C, n), k, Gauge(C, n))∩ C ≠ ∅, then Cage(C, n) turns left k, Gauge(C, n).

We now state two propositions:

(33) If C is connected and 1 k and k + 1 lenCage(C, n), then left cell(Cage(C, n), k, Gauge(C, n))∩C = ∅ and right cell(Cage(C, n), k, Gauge(C, n))∩ C ≠ ∅.

(34) If C is connected, then N-min L̃(Cage(C, n)) = π_{1} Cage(C, n).

Acknowledgments

Thanks are due to Professor Yatsuka Nakamura and people in the Kiso Laboratory for their gracious support during the work on this paper. Authors would like also to express their gratitude for Andrzej Trybulec for his help in preparing the paper.

References

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Components and Basis of Topological Spaces^{1}

Robert Milewski University in Białystok

Summary. This article contains many facts about components and basis of topological spaces.

MML Identifier: YELLOW15.

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

1. Preliminaries

The scheme SeqLambda1C deals with a natural number A, a non empty set B, a unary functor F yielding a set, a unary functor G yielding a set, and and states that:

There exists a finite sequence p of elements of B such that lenp = A and for every natural number i such that i ∈ SegA holds if P[i], then p(i) = F(i) and if not P[i], then p(i) = G(i)

provided the parameters satisfy the following condition:

• For every natural number i such that i ∈ SegA holds if P[i], then F(i) ∈ B and if not P[i], then G(i) ∈ B.

Let X be a set and let p be a finite sequence of elements of 2^{X}. Then rngp is a family of subsets of X.

Let us observe that Boolean is finite.

Next we state two propositions:

1This work has been supported by KBN Grant 8 T11C 018 12.

(2)^{2} For every natural number i and for every finite set D holds D^{i} is finite.

(3) For every finite set T holds every family of subsets of T is finite. Let T be a finite set. Note that every family of subsets of T is finite. Let T be a finite 1-sorted structure. Observe that every family of subsets of

T is finite.

The following proposition is true

(4) For every infinite set X there exist sets x, y such that x ∈ X and y ∈ X and x ≠ y.

2. Components

Let X be a set, let p be a finite sequence of elements of 2^{X}, and let q be a finite sequence of elements of Boolean. The functor MergeSequence(p, q) yielding a finite sequence of elements of 2^{X} is defined by:

(Def. 1) lenMergeSequence(p, q) = lenp and for every natural number i such that

i ∈ domp holds (MergeSequence(p, q))(i) = (q(i) = true → p(i), X \ p(i)). The following propositions are true:

(5) Let X be a set, p be a finite sequence of elements of 2^{X}, and q be a finite sequence of elements of Boolean. Then domMergeSequence(p, q) = domp.

(6) Let X be a set, p be a finite sequence of elements of 2^{X}, q be a finite sequence of elements of Boolean, and i be a natural number. If q(i) = true, then (MergeSequence(p, q))(i) = p(i).

(7) Let X be a set, p be a finite sequence of elements of 2^{X}, q be a finite sequence of elements of Boolean, and i be a natural number. If i ∈ domp and q(i) = false, then (MergeSequence(p, q))(i) = X \ p(i).

(8) For every set X and for every finite sequence q of elements of Boolean holds lenMergeSequence(ε_{2^{X}} , q) = 0.

(9) For every set X and for every finite sequence q of elements of Boolean holds MergeSequence(ε_{2^{X}} , q) = ε_{2^{X}} .

(10) For every set X and for every element x of 2^{X} and for every finite sequence q of elements of Boolean holds lenMergeSequence(⟨x⟩, q) = 1.

(11) Let X be a set, x be an element of 2^{X}, and q be a finite sequence of

elements of Boolean. Then

(i) if q(1) = true, then (MergeSequence(⟨x⟩, q))(1) = x, and (ii) if q(1) = false, then (MergeSequence(⟨x⟩, q))(1) = X \ x.

(12) For every set X and for all elements x, y of 2^{X} and for every finite sequence q of elements of Boolean holds lenMergeSequence(⟨x, y⟩, q) = 2.

2The proposition (1) has been removed.

(13) Let X be a set, x, y be elements of 2^{X}, and q be a finite sequence of

elements of Boolean. Then

(i) if q(1) = true, then (MergeSequence(⟨x, y⟩, q))(1) = x, (ii) if q(1) = false, then (MergeSequence(⟨x, y⟩, q))(1) = X \ x, (iii) if q(2) = true, then (MergeSequence(⟨x, y⟩, q))(2) = y, and (iv) if q(2) = false, then (MergeSequence(⟨x, y⟩, q))(2) = X \ y.

(14) Let X be a set, x, y, z be elements of 2^{X}, and q be a finite sequence of elements of Boolean. Then lenMergeSequence(⟨x, y, z⟩, q) = 3.

(15) Let X be a set, x, y, z be elements of 2^{X}, and q be a finite sequence of

elements of Boolean. Then

(i) if q(1) = true, then (MergeSequence(⟨x, y, z⟩, q))(1) = x, (ii) if q(1) = false, then (MergeSequence(⟨x, y, z⟩, q))(1) = X \ x, (iii) if q(2) = true, then (MergeSequence(⟨x, y, z⟩, q))(2) = y, (iv) if q(2) = false, then (MergeSequence(⟨x, y, z⟩, q))(2) = X \ y,

(v) if q(3) = true, then (MergeSequence(⟨x, y, z⟩, q))(3) = z, and (vi) if q(3) = false, then (MergeSequence(⟨x, y, z⟩, q))(3) = X \ z.

(16) Let X be a set and p be a finite sequence of elements of 2^{X}. Then {Intersect(rngMergeSequence(p, q)); q ranges over finite sequences of elements of Boolean: lenq = lenp} is a family of subsets of X.

Let X be a set and let Y be a finite family of subsets of X. The functor ComponentsY yields a family of subsets of X and is defined by the condition (Def. 2).

(Def. 2) There exists a finite sequence p of elements of 2^{X} such that lenp = cardY and rngp = Y and ComponentsY = {Intersect(rngMergeSequence(p, q)); q ranges over finite sequences of elements of Boolean: lenq = lenp}.

Let X be a set and let Y be a finite family of subsets of X. Observe that ComponentsY is finite.

The following four propositions are true:

(17) For every set X and for every empty family Y of subsets of X holds ComponentsY = {X}.

(18) For every set X and for all finite families Y, Z of subsets of X such that Z ⊆ Y holds ComponentsY is finer than ComponentsZ.

(19) For every set X and for every finite family Y of subsets of X holds ⋃ ComponentsY = X.

(20) Let X be a set, Y be a finite family of subsets of X, and A, B be sets. If

A ∈ ComponentsY and B ∈ ComponentsY and A ≠ B, then A ∩ B = ∅. Let X be a set and let Y be a finite family of subsets of X. We say that Y

is in general position if and only if: (Def. 3) ∅ ∉ ComponentsY.

Next we state three propositions:

(21) Let X be a set and Y, Z be finite families of subsets of X. If Z is in general position and Y ⊆ Z, then Y is in general position.

(22) For every non empty set X holds every empty family of subsets of X is in general position.

(23) Let X be a non empty set and Y be a finite family of subsets of X. If Y is in general position, then ComponentsY is a partition of X.

3. About Basis of Topological Spaces

The following propositions are true:

(24) For every non empty relational structure L holds Ω_{L} is infs-closed and sups-closed.

(25) For every non empty relational structure L holds Ω_{L} has bottom and

top. Let L be a non empty relational structure. Observe that Ω_{L} is infs-closed

and sups-closed and has bottom and top.

One can prove the following propositions:

(26) For every continuous sup-semilattice L holds Ω_{L} is a CLbasis of L.

(27) For every up-complete non empty poset L such that L is finite holds the carrier of L = the carrier of CompactSublatt(L).

(28) For every lower-bounded sup-semilattice L and for every subset B of L such that B is infinite holds \overline{\overline{B}} = \overline{\overline{finsups(B)}}.

(29) For every T_{0} non empty topological space T holds \overline{\overline{the carrier of T}} ⊆ \overline{\overline{the topology of T}} .

(30) Let T be a topological structure and X be a subset of T. Suppose X is open. Let B be a finite family of subsets of T. Suppose B is a basis of T. Let Y be a set. If Y ∈ ComponentsB, then X ∩ Y = ∅ or Y ⊆ X.

(31) For every T_{0} topological space T such that T is infinite holds every basis of T is infinite.

(32) Let T be a non empty topological space. Suppose T is finite. Let B be a basis of T and x be an element of T. Then ⋂{A; A ranges over elements of the topology of T: x ∈ A} ∈ B.

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

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

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266 robert milewski

Properties of the External Approximation of Jordan’s Curve

Artur Korniłowicz^{1} University of Bialystok

MML Identifier: JORDAN10.

The articles [28], [32], [31], [13], [1], [27], [24], [15], [12], [2], [18], [30], [3], [26], [33], [25], [21], [9], [4], [14], [10], [11], [17], [22], [20], [7], [29], [19], [23], [5], [6], [16], and [8] provide the terminology and notation for this paper.

Let us observe that there exists a subset of E_{T}^{2} which is connected, compact, non vertical, and non horizontal.

We use the following convention: i, j, k, n denote natural numbers, P denotes a subset of E_{T}^{2}, and C denotes a connected compact non vertical non horizontal subset of E_{T}^{2}.

The following propositions are true:

(1) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i, j + 1⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i,j}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i,j+1}.

Then i < lenGauge(C, n).

(2) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i, j + 1⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i,j+1}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i,j}.

1This paper was written while the author visited Shinshu University, winter 1999.

Then i > 1.

(3) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i + 1, j⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i,j}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i+1,j}.

Then j > 1.

(4) Suppose that

(i) 1 k, (ii) k + 1 lenCage(C, n), (iii) ⟨⟨i, j⟩⟩∈ the indices of Gauge(C, n), (iv) ⟨⟨i + 1, j⟩⟩∈ the indices of Gauge(C, n),

(v) π_{k} Cage(C, n) = (Gauge(C, n))_{i+1,j}, and (vi) π_{k+1} Cage(C, n) = (Gauge(C, n))_{i,j}.

Then j < widthGauge(C, n).

(5) C ∩ L̃(Cage(C, n)) = ∅.

(6) N-bound L̃(Cage(C, n)) = N-boundC + \frac{N-bound C−S-bound C}{2^{n}} .

(7) If i < j, then N-bound L̃(Cage(C, j)) < N-bound L̃(Cage(C, i)). Let C be a connected compact non vertical non horizontal subset of E_{T}^{2} and

let n be a natural number. Note that \overline{RightComp(Cage(C, n))} is compact. One can prove the following propositions:

(8) N-minC ∈ RightComp(Cage(C, n)).

(9) C ∩ RightComp(Cage(C, n)) ≠ ∅.

(10) C ∩ LeftComp(Cage(C, n)) = ∅.

(11) C ⊆ RightComp(Cage(C, n)).

(12) C ⊆ BDD L̃(Cage(C, n)).

(13) UBD L̃(Cage(C, n)) ⊆ UBDC. Let C be a compact non vertical non horizontal subset of E_{T}^{2}. The functor

UBD-FamilyC is defined as follows: (Def. 1) UBD-FamilyC = {UBD L̃(Cage(C, n)) : n ranges over natural

numbers}. The functor BDD-FamilyC is defined as follows:

(Def. 2) BDD-FamilyC = {BDD L̃(Cage(C, n)) : n ranges over natural numbers}.

properties of the external approximation of … 269

Let C be a compact non vertical non horizontal subset of E_{T}^{2}. Then UBD-FamilyC is a family of subsets of E_{T}^{2}. Then BDD-FamilyC is a family of subsets of E_{T}^{2}.

Let C be a compact non vertical non horizontal subset of E_{T}^{2}. Observe that UBD-FamilyC is non empty and BDD-FamilyC is non empty.

One can prove the following propositions:

(14) ⋃ UBD-FamilyC = UBDC.

(15) C ⊆ ⋂ BDD-FamilyC.

(16) BDDC ∩ LeftComp(Cage(C, n)) = ∅.

(17) BDDC ⊆ RightComp(Cage(C, n)).

(18) If P is inside component of C, then P ∩ L̃(Cage(C, n)) = ∅.

(19) BDDC ∩ L̃(Cage(C, n)) = ∅.

(20) ⋂ BDD-FamilyC = C ∪ BDDC.

Acknowledgments

I would like to thank Professor Yatsuka Nakamura for his help in preparation of the article.

References

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

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

[3] Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353–359, 1996.

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

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

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

[12] Alicia de la Cruz. Totally bounded metric spaces. Formalized Mathematics, 2(4):559–562, 1991.

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

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

[16] Artur Korniłowicz. Properties of left and right components. Formalized Mathematics, 8(1):163–168, 1999.

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

[19] Yatsuka Nakamura and Czesław Byliński. Extremal properties of vertices on special polygons. Part I. Formalized Mathematics, 5(1):97–102, 1996.

[20] Yatsuka Nakamura and Jarosław Kotowicz. Connectedness conditions using polygonal arcs. Formalized Mathematics, 3(1):101–106, 1992.

[21] Yatsuka Nakamura and Andrzej Trybulec. Components and unions of components. Formalized Mathematics, 5(4):513–517, 1996.

[22] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-board into cells. Formalized Mathematics, 5(3):323–328, 1996.

[23] Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1–13, 1999.

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

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

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

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Irrationality of e

Freek Wiedijk^{1} University of Nijmegen

Summary. We prove the irrationality of square roots of prime numbers and of the number e. In order to be able to prove the last, a proof is given that number e = exp(1) as defined in the Mizar library, that is that

lim_{n→∞} (1 + \frac{1}{n})^{n} = ∑_{k=0}^{∞} \frac{1}{k!}

MML Identifier: IRRAT 1.

The notation and terminology used in this paper are introduced in the following papers: [23], [21], [1], [14], [7], [11], [9], [17], [18], [16], [10], [2], [4], [19], [3], [13], [22], [24], [8], [20], [12], [5], [15], and [6].

1. Square Roots of Primes are Irrational

For simplicity, we adopt the following convention: k, n, p, K, N are natural numbers, x, y, e_{1} are real numbers, s_{1}, s_{2}, s_{3} are sequences of real numbers, and s_{4} is a finite sequence of elements of ℝ.

Let us consider x. We introduce x is irrational as an antonym of x is rational.

Let us consider x, y. We introduce x^{y} as a synonym of x^{y}.

Next we state two propositions:

(1) If p is prime, then \sqrt{p} is irrational.

(2) There exist x, y such that x is irrational and y is irrational and x^{y} is rational.

1Written while a guest of the Mathematical Institute of the University of Bialystok.

2. A proof that e = e

The scheme LambdaRealSeq deals with a unary functor F yielding a real number, and states that:

There exists s_{1} such that for every n holds s_{1}(n) = F(n) and for all s_{2}, s_{3} such that for every n holds s_{2}(n) = F(n) and for every n holds s_{3}(n) = F(n) holds s_{2} = s_{3}

for all values of the parameter.

Let us consider k. The functor a_{k} is a sequence of real numbers and is defined by:

(Def. 1) For every n holds a_{k}(n) = \frac{n−k}{n} .

Let us consider k. The functor b_{k} is a sequence of real numbers and is defined as follows:

(Def. 2) For every n holds b_{k}(n) = \binom{n}{k} · n^{− k}.

Let us consider n. The functor c_{n} is a sequence of real numbers and is defined by:

(Def. 3) For every k holds c_{n}(k) = \binom{n}{k} · n^{− k}. The following proposition is true

(3) c_{n}(k) = b_{k}(n). The sequence d of real numbers is defined as follows:

(Def. 4) For every n holds d(n) = (1 + \frac{1}{n})^{n}.

The sequence e of real numbers is defined as follows: (Def. 5) For every k holds e(k) = \frac{1}{k!}.

The following propositions are true:

(4) If n > 0, then n^{− (k+1)} = \frac{n^{− k}}{n} .

(5) For all real numbers x, y, z, v, w such that y ≠ 0 and z ≠ 0 and v ≠ 0 and w ≠ 0 holds \frac{x}{y·z· \frac{v}{w}} = \frac{w}{z} · \frac{x}{y·v}.

(6) \binom{n}{k+1} = \frac{n−k}{k+1} · \binom{n}{k}.

(7) If n > 0, then b_{k+1}(n) = \frac{1}{k+1} · b_{k}(n) · a_{k}(n).

(8) If n > 0, then a_{k}(n) = 1 − \frac{k}{n}.

(9) a_{k} is convergent and lim(a_{k}) = 1.

(10) For every s_{1} such that for every n holds s_{1}(n) = x holds s_{1} is convergent and lims_{1} = x.

(11) For every n such that n > 0 holds b_{0}(n) = 1.

(12) \frac{1}{k+1} · \frac{1}{k!} = \frac{1}{(k+1)!}.

(13) b_{k} is convergent and lim(b_{k}) = \frac{1}{k!} and lim(b_{k}) = e(k).

(14) If k < n, then 0 < a_{k}(n) and a_{k}(n) 1.

irrationality of e 273

(15) If n > 0, then 0 b_{k}(n) and b_{k}(n) \frac{1}{k!} and b_{k}(n) e(k) and 0 c_{n}(k) and c_{n}(k) \frac{1}{k!} and c_{n}(k) e(k).

(16) For every s_{1} such that s_{1} ↑ 1 is summable holds s_{1} is summable and ∑ s_{1} = s_{1}(0) + ∑(s_{1} ↑ 1).

(17) For every s_{4} such that lens_{4} = n and 1 k and k < n holds (s_{4})_{⇂1}(k) = s_{4}(k + 1).

(18) For every s_{4} such that lens_{4} > 0 holds ∑ s_{4} = s_{4}(1) + ∑((s_{4})_{⇂1}).

(19) Let given n and given s_{1}, s_{4}. Suppose lens_{4} = n and for every k such that k < n holds s_{1}(k) = s_{4}(k + 1) and for every k such that k n holds s_{1}(k) = 0. Then s_{1} is summable and ∑ s_{1} = ∑ s_{4}.

(20) If x ≠ 0 and y ≠ 0 and k n, then ⟨\binom{n}{0}x^{0}y^{n}, …, \binom{n}{n}x^{n}y^{0}⟩(k + 1) = \binom{n}{k} · x^{n−k} · y^{k}.

(21) If n > 0 and k n, then c_{n}(k) = ⟨\binom{n}{0}1^{0}(\frac{1}{n})^{n}, …, \binom{n}{n}1^{n}(\frac{1}{n})^{0}⟩(k + 1).

(22) If n > 0, then c_{n} is summable and ∑(c_{n}) = (1 + \frac{1}{n})^{n} and ∑(c_{n}) = d(n).

(23) d is convergent and limd = e.

(24) e is summable and ∑ e = exp1.

(25) Let given K and d_{1} be a sequence of real numbers. If for every n holds d_{1}(n) = (∑_{α=0}^{κ}(c_{n})(α))_{κ∈ℕ}(K), then d_{1} is convergent and limd_{1} = (∑_{α=0}^{κ} e(α))_{κ∈ℕ}(K).

(26) If s_{1} is convergent and lims_{1} = x, then for every e_{1} such that e_{1} > 0 there exists N such that for every n such that n N holds s_{1}(n) > x−e_{1}.

(27) Suppose that

(i) for every e_{1} such that e_{1} > 0 there exists N such that for every n such that n N holds s_{1}(n) > x − e_{1}, and

(ii) there exists N such that for every n such that n N holds s_{1}(n) x.

Then s_{1} is convergent and lims_{1} = x.

(28) If s_{1} is summable, then for every e_{1} such that e_{1} > 0 there exists K such that (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(K) > ∑ s_{1} − e_{1}.

(29) If n 1, then d(n) ∑ e.

(30) If s_{1} is summable and for every k holds s_{1}(k) 0, then ∑ s_{1} (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(K).

(31) d is convergent and limd = ∑ e.

e can be characterized by the condition: (Def. 6) e = ∑ e.

e can be characterized by the condition: (Def. 7) e = exp1.

274 freek wiedijk

3. The Number e is Irrational

One can prove the following propositions:

(32) If x is rational, then there exists n such that n 2 and n! · x is integer.

(33) n! · e(k) = \frac{n!}{k!}.

(34) \frac{n!}{k!} > 0.

(35) If s_{1} is summable and for every n holds s_{1}(n) > 0, then ∑ s_{1} > 0.

(36) n! · ∑(e↑(n + 1)) > 0.

(37) If k n, then \frac{n!}{k!} is integer.

(38) n! · (∑_{α=0}^{κ} e(α))_{κ∈ℕ}(n) is integer.

(39) If x = \frac{1}{n+1}, then \frac{n!}{(n+k+1)!} x^{k+1}.

(40) If n > 0 and x = \frac{1}{n+1}, then n! · ∑(e↑(n + 1)) \frac{x}{1−x} .

(41) If n 2 and x = \frac{1}{n+1}, then \frac{x}{1−x} < 1.

(42) e is irrational.

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

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[6] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin’s test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317–321, 1998.

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

[8] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.

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

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

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

[12] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

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

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

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

irrationality of e 275

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

[17] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

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

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

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

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

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

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

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

Received July 2, 1999 276 freek wiedijk

Injective Spaces. Part II^{1}

Artur Korniłowicz University of Bialystok

Jarosław Gryko University of Bialystok

MML Identifier: WAYBEL25.

The terminology and notation used in this paper have been introduced in the following articles: [31], [39], [30], [38], [40], [41], [10], [11], [8], [33], [21], [3], [1], [25], [26], [28], [19], [29], [27], [42], [14], [13], [32], [22], [44], [9], [16], [37], [2], [4], [5], [43], [12], [23], [35], [6], [34], [24], [36], [7], [15], [18], [17], and [20].

1. Injective Spaces

One can prove the following two propositions:

(1) For every point p of the Sierpiński space such that p = 0 holds {p} is closed.

(2) For every point p of the Sierpiński space such that p = 1 holds {p} is non closed.

Let us observe that the Sierpiński space is non T_{1}.

Let us note that every top-lattice which is complete and Scott is also discernible.

Let us note that there exists a T_{0}-space which is injective and strict.

One can check that there exists a top-lattice which is complete, Scott, and strict.

The following propositions are true:

(3) Let I be a non empty set and T be a Scott topological augmentation of ∏(I ⟼→ 2_{⊆}^{1} ). Then the carrier of T = the carrier of ∏(I ⟼→ the Sierpiński space).

1This work has been supported by KBN Grant 8 T11C 018 12.

(4) Let L_{1}, L_{2} be complete lattices, T_{1} be a Scott topological augmentation of L_{1}, T_{2} be a Scott topological augmentation of L_{2}, h be a map from L_{1} into L_{2}, and H be a map from T_{1} into T_{2}. If h = H and h is isomorphic, then H is a homeomorphism.

(5) Let L_{1}, L_{2} be complete lattices, T_{1} be a Scott topological augmentation of L_{1}, and T_{2} be a Scott topological augmentation of L_{2}. If L_{1} and L_{2} are isomorphic, then T_{1} and T_{2} are homeomorphic.

(6) Let S, T be non empty topological spaces. If S is injective and S and T are homeomorphic, then T is injective.

(7) Let L_{1}, L_{2} be complete lattices, T_{1} be a Scott topological augmentation of L_{1}, and T_{2} be a Scott topological augmentation of L_{2}. If L_{1} and L_{2} are isomorphic and T_{1} is injective, then T_{2} is injective.

Let X, Y be non empty topological spaces. Let us observe that X is a topological retract of Y if and only if:

(Def. 1) There exists a continuous map c from X into Y and there exists a con-

tinuous map r from Y into X such that r · c = id_{X}. We now state several propositions:

(8) Let S, T, X, Y be non empty topological spaces. Suppose that

(i) the topological structure of S = the topological structure of T, (ii) the topological structure of X = the topological structure of Y, and (iii) S is a topological retract of X.

Then T is a topological retract of Y.

(9) Let T, S_{1}, S_{2} be non empty topological structures. Suppose S_{1} and S_{2} are homeomorphic and S_{1} is a topological retract of T. Then S_{2} is a topological retract of T.

(10) Let S, T be non empty topological spaces. Suppose T is injective and S is a topological retract of T. Then S is injective.

(11) Let J be an injective non empty topological space and Y be a non empty topological space. If J is a subspace of Y, then J is a topological retract of Y.

(12) For every complete continuous lattice L holds every Scott topological

augmentation of L is injective. Let L be a complete continuous lattice. Observe that every topological au-

gmentation of L which is Scott is also injective.

Let T be an injective non empty topological space. Observe that the topological structure of T is injective.

2. Specialization Order

Let T be a topological structure. The functor ΩT yielding a strict FRstructure is defined by the conditions (Def. 2).

(Def. 2)(i) The topological structure of ΩT = the topological structure of T, and

(ii) for all elements x, y of ΩT holds x y iff there exists a subset Y of T

such that Y = \{y\} and x ∈ \overline{Y} . Let T be an empty topological structure. One can verify that ΩT is empty. Let T be a non empty topological structure. Note that ΩT is non empty. Let T be a topological space. Note that ΩT is topological space-like. Let T be a topological structure. Observe that ΩT is reflexive. Let T be a topological structure. Observe that ΩT is transitive. Let T be a T_{0}-space. Note that ΩT is antisymmetric. Next we state four propositions:

(13) Let S, T be topological spaces. Suppose the topological structure of S = the topological structure of T. Then ΩS = ΩT.

(14) Let M be a non empty set and T be a non empty topological space. Then the relational structure of Ω ∏(M ⟼→ T) = the relational structure of ∏(M ⟼→ ΩT).

(15) For every Scott complete top-lattice S holds ΩS = the FR-structure of S.

(16) Let L be a complete lattice and S be a Scott topological augmentation

of L. Then the relational structure of ΩS = the relational structure of L. Let S be a Scott complete top-lattice. One can verify that ΩS is complete. One can prove the following four propositions:

(17) Let T be a non empty topological space and S be a non empty subspace of T. Then ΩS is a full relational substructure of ΩT.

(18) Let S, T be topological spaces, h be a map from S into T, and g be a map from ΩS into ΩT. If h = g and h is a homeomorphism, then g is isomorphic.

(19) For all topological spaces S, T such that S and T are homeomorphic holds ΩS and ΩT are isomorphic.

(20) For every injective T_{0}-space T holds ΩT is a complete continuous lattice. Let T be an injective T_{0}-space. Observe that ΩT is complete and continuous. The following proposition is true

(21) For all non empty topological spaces X, Y holds every continuous map

from ΩX into ΩY is monotone. Let X, Y be non empty topological spaces. Observe that every map from

ΩX into ΩY which is continuous is also monotone.

Next we state the proposition

(22) For every non empty topological space T and for every element x of the carrier of ΩT holds \overline{\{x\}} = ↓x.

Let T be a non empty topological space and let x be an element of the carrier of ΩT. Observe that \overline{\{x\}} is non empty lower and directed and ↓x is closed.

One can prove the following proposition

(23) For every topological space X holds every open subset of ΩX is upper. Let T be a topological space. One can check that every subset of ΩT which

is open is also upper.

Let I be a non empty set, let S, T be non empty relational structures, let N be a net in T, and let i be an element of I. Let us assume that the carrier of T ⊆ the carrier of S^{I}. The functor commute(N, i, S) yielding a strict net in S is defined by the conditions (Def. 3).

(Def. 3)(i) The relational structure of commute(N, i, S) = the relational struc-

ture of N, and (ii) the mapping of commute(N, i, S) = (commute(the mapping of N))(i). The following propositions are true:

(24) Let X, Y be non empty topological spaces, N be a net in [X → ΩY], i be an element of the carrier of N, and x be a point of X. Then (the mapping of commute(N, x, ΩY))(i) = (the mapping of N)(i)(x).

(25) Let X, Y be non empty topological spaces, N be an eventually-directed net in [X → ΩY], and x be a point of X. Then commute(N, x, ΩY) is eventually-directed.

Let X, Y be non empty topological spaces, let N be an eventually-directed net in [X → ΩY], and let x be a point of X. Note that commute(N, x, ΩY) is eventually-directed.

Let X, Y be non empty topological spaces. One can check that every net in [X → ΩY] is function yielding.

We now state the proposition

(26) Let X be a non empty topological space, Y be a T_{0}-space, and N be a net in [X → ΩY]. Suppose that for every point x of X holds sup commute(N, x, ΩY) exists. Then sup rng(the mapping of N) exists in (ΩY)^{the carrier of X}.

3. Monotone Convergence Topological Spaces

Let T be a non empty topological space. We say that T is monotone convergence if and only if the condition (Def. 4) is satisfied.

(Def. 4) Let D be a non empty directed subset of ΩT. Then sup D exists in ΩT

and for every open subset V of T such that supD ∈ V holds D meets V. Next we state the proposition (27) Let S, T be non empty topological spaces. Suppose the topological struc-

ture of S = the topological structure of T and S is monotone convergence. Then T is monotone convergence.

Let us observe that every T_{0}-space which is trivial is also monotone convergence.

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

One can prove the following two propositions:

(28) Let S be a monotone convergence T_{0}-space and T be a T_{0}-space. If S and T are homeomorphic, then T is monotone convergence.

(29) Every Scott complete top-lattice is monotone convergence.

Let L be a complete lattice. One can check that every Scott topological augmentation of L is monotone convergence.

Let L be a complete lattice and let S be a Scott topological augmentation of L. One can check that the topological structure of S is monotone convergence.

One can prove the following proposition

(30) For every monotone convergence T_{0}-space X holds ΩX is up-complete. Let X be a monotone convergence T_{0}-space. Note that ΩX is up-complete. One can prove the following propositions:

(31) Let X be a monotone convergence non empty topological space and N be an eventually-directed prenet over ΩX. Then sup N exists.

(32) Let X be a monotone convergence non empty topological space and N be an eventually-directed net in ΩX. Then supN ∈ LimN.

(33) For every monotone convergence non empty topological space X holds

every eventually-directed net in ΩX is convergent. Let X be a monotone convergence non empty topological space. One can

verify that every eventually-directed net in ΩX is convergent.

We now state two propositions:

(34) Let X be a non empty topological space. Suppose that for every eventually-directed net N in ΩX holds sup N exists and supN ∈ LimN. Then X is monotone convergence.

(35) Let X be a monotone convergence non empty topological space and Y be a T_{0}-space. Then every continuous map from ΩX into ΩY is directedsups-preserving.

Let X be a monotone convergence non empty topological space and let Y be a T_{0}-space. Observe that every map from ΩX into ΩY which is continuous is also directed-sups-preserving.

Next we state four propositions:

(36) Let T be a monotone convergence T_{0}-space and R be a T_{0}-space. If R is a topological retract of T, then R is monotone convergence.

(37) Let T be an injective T_{0}-space and S be a Scott topological augmentation of ΩT. Then the topological structure of S = the topological structure of T.

(38) Every injective T_{0}-space is compact, locally-compact, and sober.

(39) Every injective T_{0}-space is monotone convergence.

One can verify that every T_{0}-space which is injective is also monotone convergence.

We now state four propositions:

(40) Let X be a non empty topological space, Y be a monotone convergence T_{0}-space, N be a net in [X → ΩY], and f, g be maps from X into ΩY. Suppose that

(i) f = ⨆_{((ΩY)^{the carrier of X})} rng(the mapping of N), (ii) sup rng(the mapping of N) exists in (ΩY)^{the carrier of X}, and (iii) g ∈ rng(the mapping of N).

Then g f.

(41) Let X be a non empty topological space, Y be a monotone convergence T_{0}-space, N be a net in [X → ΩY], x be a point of X, and f be a map from X into ΩY. Suppose for every point a of X holds commute(N, a, ΩY) is eventually-directed and f = ⨆_{((ΩY)^{the carrier of X})} rng(the mapping of N). Then f(x) = supcommute(N, x, ΩY).

(42) Let X be a non empty topological space, Y be a monotone convergence T_{0}-space, and N be a net in [X → ΩY]. Suppose that for every point x of X holds commute(N, x, ΩY) is eventually-directed. Then ⨆_{((ΩY)^{the carrier of X})} rng(the mapping of N) is a continuous map from X into Y.

(43) Let X be a non empty topological space and Y be a monotone convergence T_{0}-space. Then [X → ΩY] is a directed-sups-inheriting relational substructure of (ΩY)^{the carrier of X}.

References

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

[3] Grzegorz Bancerek. Quantales. Formalized Mathematics, 5(1):85–91, 1996.

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

[5] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.

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

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

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

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

injective spaces. part ii 283

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

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

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[22] Zbigniew Karno. Continuity of mappings over the union of subspaces. Formalized Mathematics, 3(1):1–16, 1992.

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

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

[25] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

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

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

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

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

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

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

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

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

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284 artur korniłowicz and jarosław gryko

[44] Mariusz Żynel and Adam Guzowski. T_{0} topological spaces. Formalized Mathematics,

5(1):75–77, 1996.

Received July 3, 1999

Propositional Calculus for Boolean Valued Functions. Part VI

Shunichi Kobayashi Shinshu University Nagano

Summary. In this paper, we proved some elementary propositional calculus formulae for Boolean valued functions.

MML Identifier: BVFUNC10.

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

In this paper Y is a non empty set. Next we state a number of propositions:

(1) For all elements a, b, c of BVF(Y) holds a ∧ b ∨ b ∧ c ∨ c ∧ a = (a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a).

(2) For all elements a, b, c of BVF(Y) holds a ∧ ¬b ∨ b ∧ ¬c ∨ c ∧ ¬a = b ∧ ¬a ∨ c ∧ ¬b ∨ a ∧ ¬c.

(3) For all elements a, b, c of BVF(Y) holds (a ∨ ¬b) ∧ (b ∨ ¬c) ∧ (c ∨ ¬a) = (b ∨ ¬a) ∧ (c ∨ ¬b) ∧ (a ∨ ¬c).

(4) For all elements a, b, c of BVF(Y) such that c ⇒ a = true(Y) and c ⇒ b = true(Y) holds c ⇒ a ∨ b = true(Y).

(5) For all elements a, b, c of BVF(Y) such that a ⇒ c = true(Y) and b ⇒ c = true(Y) holds a ∧ b ⇒ c = true(Y).

(6) For all elements a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2} of BVF(Y) holds (a_{1} ⇒ a_{2}) ∧ (b_{1} ⇒ b_{2}) ∧ (c_{1} ⇒ c_{2}) ∧ (a_{1} ∨ b_{1} ∨ c_{1}) ⋐ a_{2} ∨ b_{2} ∨ c_{2}.

(7) For all elements a_{1}, a_{2}, b_{1}, b_{2} of BVF(Y) holds (a_{1} ⇒ b_{1}) ∧ (a_{2} ⇒ b_{2}) ∧ (a_{1} ∨ a_{2}) ∧ ¬(b_{1} ∧ b_{2}) = (b_{1} ⇒ a_{1}) ∧ (b_{2} ⇒ a_{2}) ∧ (b_{1} ∨ b_{2}) ∧ ¬(a_{1} ∧ a_{2}).

(8) For all elements a, b, c, d of BVF(Y) holds (a ∨ b) ∧ (c ∨ d) = a ∧ c ∨ a ∧ d ∨ b ∧ c ∨ b ∧ d.

(9) For all elements a_{1}, a_{2}, b_{1}, b_{2}, b_{3} of BVF(Y) holds a_{1} ∧ a_{2} ∨ b_{1} ∧ b_{2} ∧ b_{3} = (a_{1} ∨ b_{1}) ∧ (a_{1} ∨ b_{2}) ∧ (a_{1} ∨ b_{3}) ∧ (a_{2} ∨ b_{1}) ∧ (a_{2} ∨ b_{2}) ∧ (a_{2} ∨ b_{3}).

(10) For all elements a, b, c, d of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ d) = (a ⇒ b ∧ c ∧ d) ∧ (b ⇒ c ∧ d) ∧ (c ⇒ d).

(11) For all elements a, b, c, d of BVF(Y) holds (a ⇒ c) ∧ (b ⇒ d) ∧ (a ∨ b) ⋐ c ∨ d.

(12) For all elements a, b, c of BVF(Y) holds (a ∧ b ⇒ ¬c) ∧ a ∧ c ⋐ ¬b.

(13) For all elements a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} of BVF(Y) holds a_{1} ∧ a_{2} ∧ a_{3} ⇒ b_{1} ∨ b_{2} ∨ b_{3} = ¬b_{1} ∧ ¬b_{2} ∧ a_{3} ⇒ ¬a_{1} ∨ ¬a_{2} ∨ b_{3}.

(14) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ a) = a ∧ b ∧ c ∨ ¬a ∧ ¬b ∧ ¬c.

(15) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ∧ (c ⇒ a) ∧ (a ∨ b ∨ c) = a ∧ b ∧ c.

(16) For all elements a, b, c of BVF(Y) holds (a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a) ∧ ¬(a ∧ b ∧ c) = ¬a ∧ b ∧ c ∨ a ∧ ¬b ∧ c ∨ a ∧ b ∧ ¬c.

(17) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ⇒ b ∧ c.

(18) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ∨ b ⇒ c.

(19) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ⇒ b ∨ c.

(20) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ a ⇒ b ∨ ¬c.

(21) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ b ⇒ c ∨ a.

(22) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ b ⇒ c ∨ ¬a.

(23) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b) ∧ (b ⇒ c ∨ a).

(24) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ ¬c) ∧ (b ⇒ c).

(25) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ c) ∧ (b ⇒ c ∨ a).

(26) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ ¬c) ∧ (b ⇒ c ∨ a).

(27) For all elements a, b, c of BVF(Y) holds (a ⇒ b) ∧ (b ⇒ c) ⋐ (a ⇒ b ∨ ¬c) ∧ (b ⇒ c ∨ ¬a).

References

[1] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

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

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

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

propositional calculus for boolean valued … 287

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

1990.

Received July 14, 1999 288 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part III

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC11.

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

1. Preliminaries

In this paper Y is a non empty set. Next we state several propositions:

(1) For every element z of Y and for all partitions P_{1}, P_{2} of Y such that P_{1} ⋐ P_{2} holds EqClass(z, P_{1}) ⊆ EqClass(z, P_{2}).

(2) For every element z of Y and for all partitions P_{1}, P_{2} of Y holds EqClass(z, P_{1}) ⊆ EqClass(z, P_{1} ∨ P_{2}).

(3) For every element z of Y and for all partitions P_{1}, P_{2} of Y holds EqClass(z, P_{1} ∧ P_{2}) ⊆ EqClass(z, P_{1}).

(4) For every element z of Y and for every partition P_{1} of Y holds EqClass(z, P_{1}) ⊆ EqClass(z, O(Y)) and EqClass(z, I(Y)) ⊆ EqClass(z, P_{1}).

(5) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. Suppose G is an independent family of partitions and G = {A, B} and A ≠ B. Let a, b be sets. If a ∈ A and b ∈ B, then a ∩ b ≠ ∅.

(6) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ⋀ G = A ∧ B.

(7) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then CompF(A, G) = B and CompF(B, G) = A.

2. Predicate Calculus

We now state a number of propositions:

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B}, then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B}, then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(14) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(17) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(19) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,A} G,B}G.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

[4] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

[5] Shunichi Kobayashi and Kui Jia. A theory of partitions. Part I. Formalized Mathematics, 7(2):243–247, 1998.

[6] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

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

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

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

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

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

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

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

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

Received July 14, 1999 292 shunichi kobayashi and yatsuka nakamura

A Characterization of Concept Lattices. Dual Concept Lattices

Christoph Schwarzweller University of Tuebingen

Summary. In this article we continue the formalization of concept lattices following [5]. We prove a theorem stating necessary and sufficient conditions for a complete lattice to be isomorphic to a given formal context. As a by-product we get that a lattice is complete if and only if it is isomorphic to a concept lattice. In addition we introduce dual formal concepts and dual concept lattices.

MML Identifier: CONLAT 2.

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

1. Preliminaries

Let C be a FormalContext and let C_{1} be a strict FormalConcept of C. The functor ^{@}C_{1} yields an element of ConceptLatticeC and is defined by:

(Def. 1) ^{@}C_{1} = C_{1}. One can prove the following propositions:

(1) For every FormalContext C holds ⊥_{ConceptLattice C} = Concept − with − all − Attributes C and ⊤_{ConceptLattice C} = Concept − with − all − Objects C.

(2) Let C be a FormalContext and D be a non empty subset of 2^{the objects of C}. Then (ObjectDerivationC)(⋃ D) = ⋂{(ObjectDerivationC)(O); O ranges over subsets of the objects of C: O ∈ D}.

(3) Let C be a FormalContext and D be a non empty subset of 2^{the Attributes of C}. Then (AttributeDerivationC)(⋃ D) = ⋂{(AttributeDerivationC)(A); A ranges over subsets of the Attributes of C: A ∈ D}.

(4) Let C be a FormalContext and D be a subset of the carrier of ConceptLatticeC. Then ⌈−⌉_{ConceptLattice C}D is a FormalConcept of C and ⨆_{ConceptLattice C} D is a FormalConcept of C.

Let C be a FormalContext and let D be a subset of the carrier of ConceptLatticeC. The functor ⌈−⌉_{C}D yields a FormalConcept of C and is defined by:

(Def. 2) ⌈−⌉_{C}D = ⌈−⌉_{ConceptLattice C}D.

The functor ⨆_{C} D yields a FormalConcept of C and is defined as follows: (Def. 3) ⨆_{C} D = ⨆_{ConceptLattice C} D.

One can prove the following propositions:

(5) For every FormalContext C holds ⨆_{C}(∅_{ConceptLattice C}) = Concept − with − all − Attributes C and ⌈−⌉_{C}(∅_{ConceptLattice C}) = Concept − with − all − Objects C.

(6) For every FormalContext C holds ⨆_{C}(Ω_{the carrier of ConceptLattice C}) = Concept − with − all − Objects C and ⌈−⌉_{C}(Ω_{the carrier of ConceptLattice C}) = Concept − with − all − Attributes C.

(7) Let C be a FormalContext and D be a non empty subset of ConceptLatticeC. Then

(i) the Extent of ⨆_{C} D = (AttributeDerivationC)((ObjectDerivationC)(⋃{the Extent of ⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D})), and

(ii) the Intent of ⨆_{C} D = ⋂{the Intent of ⟨E, I⟩; E ranges over subsets of

the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D}.

(8) Let C be a FormalContext and D be a non empty subset of ConceptLatticeC. Then

(i) the Extent of ⌈−⌉_{C}D = ⋂{the Extent of ⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D}, and

(ii) the Intent of ⌈−⌉_{C}D = (ObjectDerivationC)((AttributeDerivationC)(⋃{the Intent of ⟨E, I⟩; E ranges over subsets of the objects of C, I ranges over subsets of the Attributes of C: ⟨E, I⟩ ∈ D})).

(9) Let C be a FormalContext and C_{1} be a strict FormalConcept of C.

Then ⨆_{ConceptLattice C}{⟨O, A⟩; O ranges over subsets of the objects of C, A ranges over subsets of the Attributes of C: ⋁_{o: object of C} (o ∈ the Extent of C_{1} ∧ O = (AttributeDerivationC)((ObjectDerivationC)({o})) ∧ A = (ObjectDerivationC)({o}))} = C_{1}.

(10) Let C be a FormalContext and C_{1} be a strict FormalConcept of C. Then ⌈−⌉_{ConceptLattice C}{⟨O, A⟩; O ranges over subsets of the objects of C, A ranges over subsets of the Attributes of C: ⋁_{a: Attribute of C} (a ∈ the Intent of C_{1} ∧ O = (AttributeDerivationC)({a}) ∧ A = (ObjectDerivationC)((AttributeDerivationC)({a})))} = C_{1}.

Let C be a FormalContext. The functor gammaC yields a function from the objects of C into the carrier of ConceptLatticeC and is defined by the condition (Def. 4).

(Def. 4) Let o be an element of the objects of C. Then there exists a subset O of the objects of C and there exists a subset A of the Attributes of C such that (gammaC)(o) = ⟨O, A⟩ and O = (AttributeDerivationC)((ObjectDerivationC)({o})) and A = (ObjectDerivationC)({o}).

Let C be a FormalContext. The functor δ_{C} yields a function from the Attributes of C into the carrier of ConceptLatticeC and is defined by the condition (Def. 5).

(Def. 5) Let a be an element of the Attributes of C. Then there exists a subset O of the objects of C and there exists a subset A of the Attributes of C such that δ_{C}(a) = ⟨O, A⟩ and O = (AttributeDerivationC)({a}) and A = (ObjectDerivationC)((AttributeDerivationC)({a})).

We now state three propositions:

(11) Let C be a FormalContext, o be an object of C, and a be a Attribute of C. Then (gammaC)(o) is a FormalConcept of C and δ_{C}(a) is a Formal-Concept of C.

(12) For every FormalContext C holds rnggammaC is supremum-dense and rng(δ_{C}) is infimum-dense.

(13) Let C be a FormalContext, o be an object of C, and a be a Attribute of C. Then o is connected with a if and only if (gammaC)(o) ⊑ δ_{C}(a).

2. The Characterization

One can prove the following proposition

(14) Let L be a complete lattice and C be a FormalContext. Then ConceptLatticeC and L are isomorphic if and only if there exists a function g from the objects of C into the carrier of L and there exists a function d from the Attributes of C into the carrier of L such that rngg is supremum-dense and rngd is infimum-dense and for every object o of C and for every Attribute a of C holds o is connected with a iff g(o) ⊑ d(a).

Let L be a lattice. The functor ContextL yielding a strict non quasi-empty ContextStr is defined by:

(Def. 6) ContextL = ⟨the carrier of L, the carrier of L, LattRel(L)⟩.

The following proposition is true

(15) For every complete lattice L holds ConceptLatticeContextL and L are isomorphic.

Let L_{1}, L_{2} be lattices. Let us note that the predicate L_{1} and L_{2} are isomorphic is symmetric.

The following proposition is true

(16) For every lattice L holds L is complete iff there exists a FormalContext C such that ConceptLatticeC and L are isomorphic.

3. Dual Concept Lattices

Let L be a complete lattice. Note that L^{∘} is complete. Let C be a FormalContext. The functor C^{∘} yields a strict non quasi-empty

ContextStr and is defined as follows: (Def. 7) C^{∘} = ⟨the Attributes of C, the objects of C, (the Information of C)^{⌣}⟩.

The following propositions are true:

(17) For every strict FormalContext C holds (C^{∘})^{∘} = C.

(18) For every FormalContext C and for every subset O of the objects of C holds (ObjectDerivationC)(O) = (AttributeDerivationC^{∘})(O).

(19) For every FormalContext C and for every subset A of the Attributes of

C holds (AttributeDerivationC)(A) = (ObjectDerivationC^{∘})(A). Let C be a FormalContext and let C_{1} be a ConceptStr over C. The functor

C_{1}^{∘} yielding a strict ConceptStr over C^{∘} is defined by: (Def. 8) The Extent of C_{1}^{∘} = the Intent of C_{1} and the Intent of C_{1}^{∘} = the Extent

of C_{1}. Let C be a FormalContext and let C_{1} be a FormalConcept of C. Then C_{1}^{∘}

is a strict FormalConcept of C^{∘}.

The following proposition is true

(20) For every FormalContext C and for every strict FormalConcept C_{1} of C

holds (C_{1}^{∘})^{∘} = C_{1}. Let C be a FormalContext. The functor DualHomomorphismC yielding a

homomorphism from (ConceptLatticeC)^{∘} to ConceptLatticeC^{∘} is defined by: (Def. 9) For every strict FormalConcept C_{1} of C holds (DualHomomorphismC)(C_{1}) =

C_{1}^{∘}. Next we state two propositions:

(21) For every FormalContext C holds DualHomomorphismC is isomorphism.

(22) For every FormalContext C holds ConceptLatticeC^{∘} and (ConceptLatticeC)^{∘} are isomorphic. a characterization of concept lattices. dual … 297

References

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

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

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

[5] Bernhard Ganter and Rudolf Wille. Formal Concept Analysis. Springer Verlag, Berlin, Heidelberg, Ney York, 1996. (written in German).

[6] Jolanta Kamieńska and Jarosław Stanisław Walijewski. Homomorphisms of lattices, finite join and finite meet. Formalized Mathematics, 4(1):35–40, 1993.

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

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

[9] Christoph Schwarzweller. Introduction to concept lattices. Formalized Mathematics, 7(2):233–241, 1998.

[10] Christoph Schwarzweller. Noetherian lattices. Formalized Mathematics, 8(1):169–174, 1999.

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

[12] Andrzej Trybulec. Finite join and finite meet and dual lattices. Formalized Mathematics, 1(5):983–988, 1990.

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

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

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

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

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

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

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Received August 17, 1999 298 christoph schwarzweller

Predicate Calculus for Boolean Valued Functions. Part IV

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC12.

The terminology and notation used in this paper have been introduced in the following articles: [4], [6], [5], [3], [1], and [2].

In this paper Y is a non empty set. Next we state a number of propositions:

(1) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∀_{∀_{a,A}G,B}G = ∃_{¬∀_{a,A}G,B} G.

(2) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∀_{a,A}G,B}G = ∀_{¬∀_{a,A}G,B} G.

(3) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{¬∀_{a,A}G,B} G = ∀_{∃_{¬a,A} G,B}G.

(4) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{¬∃_{a,A}G,B} G = ∀_{∀_{¬a,A} G,B}G.

(5) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∀_{∃_{a,A}G,B}G = ∃_{∀_{¬a,A} G,B}G.

(6) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∀_{a,A}G,B}G = ∀_{∃_{¬a,A} G,B}G.

(7) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∀_{∀_{a,A}G,B}G = ∃_{∃_{¬a,A} G,B}G.

(8) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∃_{¬∀_{a,A}G,B} G = ∃_{∃_{¬a,A} G,B}G.

(9) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∃_{¬∃_{a,A}G,B} G = ∃_{∀_{¬a,A} G,B}G.

(10) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∃_{a,A}G,B}G = ∀_{¬∃_{a,A}G,B} G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(12) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,A}G,B}G.

(13) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,A}G,B}G.

(14) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,A}G,B}G.

(15) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,A}G,B}G.

(16) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,A}G,B}G.

References

[1] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

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

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

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

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

Received August 17, 1999 302 shunichi kobayashi and yatsuka nakamura

Predicate Calculus for Boolean Valued Functions. Part V

Shunichi Kobayashi Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC13.

The notation and terminology used in this paper have been introduced in the following articles: [4], [6], [7], [5], [3], [1], and [2].

In this paper Y denotes a non empty set. The following propositions are true:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(2) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(3) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(11) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(17) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(19) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(21) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(22) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(23) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(24) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(25) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(26) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(27) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(28) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(29) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

(30) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(31) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(32) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B,

then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∃_{a,B}G,A}G.

(33) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∃_{a,B}G,A}G.

(34) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(35) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(36) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(37) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(38) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(39) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(40) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∃_{a,B}G,A} G.

(41) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∃_{a,B}G,A} G.

(42) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(43) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(44) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(45) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(46) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∀_{¬a,B} G,A}G.

(47) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∀_{¬a,B} G,A}G.

(48) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(49) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(50) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(51) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(52) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(53) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(54) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(55) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(56) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(57) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(58) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(59) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and 308 shunichi kobayashi and yatsuka nakamura

A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(61)^{1} Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(62) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

References

[1] Shunichi Kobayashi and Kui Jia. A theory of Boolean valued functions and partitions. Formalized Mathematics, 7(2):249–254, 1998.

[2] Shunichi Kobayashi and Yatsuka Nakamura. A theory of Boolean valued functions and quantifiers with respect to partitions. Formalized Mathematics, 7(2):307–312, 1998.

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

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

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

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

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

Received August 17, 1999

1The proposition (60) has been removed.

Definitions of Radix-2^{k} Signed-Digit Number and its Adder Algorithm

Yoshinori Fujisawa Shinshu University Nagano

Yasushi Fuwa Shinshu University Nagano

Summary. In this article, a radix-2^{k} signed-digit number (Radix-2^{k} SD number) is defined and a high-speed adder algorithm is discussed based on it. The processes of coding and encoding for public-key cryptograms require a great deal of addition operations of natural number of many figures. This results in a long time for the encoding and decoding processes. It is possible to reduce the processing time using the high-speed adder algorithm in this article. In the first section, we prepared some useful theorems for natural numbers and Integers.In the second section, we defined the concept of radix-2^{k}, a set named k-SD and proved some properties about them. In the third section, we provide some important functions for generating Radix-2^{k} SD numbers from natural numbers and natural numbers from Radix-2^{k} SD numbers. In the fourth section, we defined the carry and data components of addition with Radix-2^{k} SD numbers and some properties about them. In the fifth section, we defined a theorem for checking whether or not a natural number can be expressed as n digits Radix-2^{k} SD number. In the last section, a high-speed adder algorithm on Radix-2^{k} SD numbers is proposed and we provided some properties. In this algorithm, the carry of each digit has an effect on only the next digit. Properties of the relationships of the results of this algorithm to the operations of natural numbers are also given.

MML Identifier: RADIX 1.

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

1. Some Useful Theorems

We adopt the following rules: i, k, m, n, x, y denote natural numbers, i_{1}, i_{2}, i_{3} denote integers, and e denotes a set.

The following propositions are true:

(1) If n ≠ 0, then m ÷ n = (m qua integer) ÷n qua integer and m modn = (m qua integer) mod n qua integer.

(2) If k ≠ 0 and n mod k = k − 1, then (n + 1) mod k = 0.

(3) If k ≠ 0 and n mod k < k − 1, then (n + 1) mod k = (n mod k) + 1.

(4) If m ≠ 0 and n ≠ 0, then k mod m · n mod n = k mod n.

(5) If k ≠ 0, then (n + 1) mod k = 0 or (n + 1) mod k = (n mod k) + 1.

(6) If i ≠ 0 and k ≠ 0, then (n mod i_{ℕ}^{k}) ÷ i_{ℕ}^{k− ′1} < i.

(7) If k n, then m_{ℕ}^{k} | m_{ℕ}^{n}.

(8) If i_{3} > 0, then i_{1} mod i_{2} · i_{3} mod i_{3} = i_{1} mod i_{3}.

2. Definition for Radix-2^{k}, k-SD

Let us consider n. The functor Radixn yields a natural number and is defined as follows:

(Def. 1) Radixn = 2^{n}.

Let us consider k. The functor k −SD yielding a set is defined as follows: (Def. 2) k −SD = {e; e ranges over integers: e Radixk−1 ∧ e − Radixk+1}.

One can prove the following propositions:

(9) Radixn ≠ 0.

(10) For every e holds e ∈ 0 −SD iff e = 0.

(11) 0 −SD = {0}.

(12) k −SD ⊆ k + 1 −SD .

(13) If e ∈ k −SD , then e is an integer.

(14) k −SD ⊆ ℤ.

(15) If i_{1} ∈ k −SD , then i_{1} Radixk − 1 and i_{1} − Radixk + 1.

(16) 0 ∈ k −SD .

Let us consider k. Observe that k −SD is non empty. Let us consider k. Then k −SD is a non empty subset of ℤ.

3. Functions for Generating Radix-2^{k} SD Numbers from Natural Numbers and Natural Numbers from Radix-2^{k} SD Numbers

In the sequel a denotes a tuple of n and k −SD .

definitions of radix-2^{k} signed-digit … 311

We now state the proposition (18)^{1} If i ∈ Segn, then a(i) is an element of k −SD .

Let i, k, n be natural numbers and let x be a tuple of n and k −SD . The functor DigA(x, i) yields an integer and is defined as follows:

(Def. 3)(i) DigA(x, i) = x(i) if i ∈ Segn, (ii) DigA(x, i) = 0 if i = 0. Let i, k, n be natural numbers and let x be a tuple of n and k −SD . The

functor DigB(x, i) yielding an element of ℤ is defined by: (Def. 4) DigB(x, i) = DigA(x, i).

We now state two propositions:

(19) If i ∈ Segn, then DigA(a, i) is an element of k −SD .

(20) For every tuple x of 1 and ℤ such that π_{1}x = m holds x = ⟨m⟩. Let i, k, n be natural numbers and let x be a tuple of n and k −SD . The

functor SubDigit(x, i, k) yielding an element of ℤ is defined as follows: (Def. 5) SubDigit(x, i, k) = ((Radixk)_{ℕ}^{i− ′1}) · DigB(x, i).

Let n, k be natural numbers and let x be a tuple of n and k −SD . The functor DigitSDx yields a tuple of n and ℤ and is defined by:

(Def. 6) For every natural number i such that i ∈ Segn holds π_{i} DigitSDx =

SubDigit(x, i, k). Let n, k be natural numbers and let x be a tuple of n and k −SD . The

functor SDDecx yielding an integer is defined by: (Def. 7) SDDecx = ∑ DigitSDx.

Let i, k, x be natural numbers. The functor DigitDC(x, i, k) yields an element of k −SD and is defined as follows:

(Def. 8) DigitDC(x, i, k) = (x mod (Radixk)_{ℕ}^{i}) ÷ (Radixk)_{ℕ}^{i− ′1}.

Let k, n, x be natural numbers. The functor DecSD(x, n, k) yields a tuple of n and k −SD and is defined by:

(Def. 9) For every natural number i such that i ∈ Segn holds DigA(DecSD(x, n, k), i) = DigitDC(x, i, k).

4. Definition for Carry and Data Components of Addition

Let x be an integer. The functor SD Add Carryx yields an integer and is defined by:

(Def. 10) SD Add Carryx = { (i) 1, if x > 2, (ii) − 1, if x < − 2, 0, otherwise.

1The proposition (17) has been removed.

We now state the proposition

(21) SD Add Carry0 = 0.

Let x be an integer and let k be a natural number. The functor SD Add Data(x, k) yields an integer and is defined by:

(Def. 11) SD Add Data(x, k) = x − SD Add Carryx · Radixk.

Next we state two propositions:

(22) SD Add Data(0, k) = 0.

(23) If k 2 and i_{1} ∈ k −SD and i_{2} ∈ k −SD , then − Radixk + 2 SD Add Data(i_{1} + i_{2}, k) and SD Add Data(i_{1} + i_{2}, k) Radixk − 2.

5. Definition for Checking whether or not a Natural Number can be Expressed as n Digits Radix-2^{k} SD Number

Let n, x, k be natural numbers. We say that x is represented by n, k if and only if:

(Def. 12) x < (Radixk)_{ℕ}^{n}.

We now state four propositions:

(24) If m is represented by 1, k, then DigA(DecSD(m, 1, k), 1) = m.

(25) For every n such that n 1 and for every m such that m is represented by n, k holds m = SDDecDecSD(m, n, k).

(26) If k 2 and m is represented by 1, k and n is represented by 1, k, then SD Add CarryDigA(DecSD(m, 1, k), 1) + DigA(DecSD(n, 1, k), 1) = SD Add Carrym + n.

(27) If m is represented by n + 1, k, then DigA(DecSD(m, n + 1, k), n + 1) = m ÷ (Radixk)_{ℕ}^{n}.

6. Definition for Addition Operation for a High-Speed Adder Algorithm on Radix-2^{k} SD Number

Let k, i, n be natural numbers and let x, y be tuples of n and k −SD . Let us assume that i ∈ Segn and k 2. The functor Add(x, y, i, k) yields an element of k −SD and is defined as follows:

(Def. 13) Add(x, y, i, k) = SD Add Data(DigA(x, i)+DigA(y, i), k)+SD Add CarryDigA(x, i− ′

1) + DigA(y, i − ′1). Let n, k be natural numbers and let x, y be tuples of n and k −SD . The

functor x′+′y yields a tuple of n and k −SD and is defined as follows: (Def. 14) For every i such that i ∈ Segn holds DigA(x′+′y, i) = Add(x, y, i, k).

definitions of radix-2^{k} signed-digit … 313

We now state two propositions:

(28) If k 2 and m is represented by 1, k and n is represented by 1, k, then SDDecDecSD(m, 1, k)′+′DecSD(n, 1, k) = SD Add Data(m + n, k).

(29) Let given n. Suppose n 1. Let given k, x, y. Suppose k 2 and x is represented by n, k and y is represented by n, k. Then x + y = SDDecDecSD(x, n, k)′+′DecSD(y, n, k) + ((Radixk)_{ℕ}^{n}) · SD Add CarryDigA(DecSD(x, n, k), n) + DigA(DecSD(y, n, k), n).

References

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

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

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

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

[5] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

[6] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Euler’s Theorem and small Fermat’s Theorem. Formalized Mathematics, 7(1):123–126, 1998.

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

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

[9] Michał Muzalewski. Reper algebras. Formalized Mathematics, 3(1):23–28, 1992.

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

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

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

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

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

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

314 yoshinori fujisawa and yasushi fuwa

Retracts and Inheritance^{1}

Grzegorz Bancerek University of Białystok

MML Identifier: YELLOW16.

The papers [26], [31], [30], [13], [25], [32], [11], [12], [10], [9], [1], [20], [28], [21], [23], [14], [2], [29], [3], [17], [22], [24], [16], [34], [27], [4], [5], [6], [18], [33], [15], [8], [7], and [19] provide the notation and terminology for this paper.

1. Poset Retracts

The following three propositions are true:

(1) For all binary relations a, b holds a · b = a b.

(2) Let X be a set, L be a non empty relational structure, S be a non empty relational substructure of L, f, g be functions from X into the carrier of S, and f′, g′be functions from X into the carrier of L. If f′= f and g′= g and f g, then f′ g′.

(3) Let X be a set, L be a non empty relational structure, S be a full non empty relational substructure of L, f, g be functions from X into the carrier of S, and f′, g′be functions from X into the carrier of L. If f′= f and g′= g and f′ g′, then f g.

Let S be a non empty relational structure and let T be a non empty reflexive antisymmetric relational structure. One can check that there exists a map from S into T which is directed-sups-preserving and monotone.

The following proposition is true

(4) For all functions f, g such that f is idempotent and rngg ⊆ rngf and rngg ⊆ domf holds f · g = g.

1This work has been supported by KBN Grant 8 T11C 018 12.

Let S be a 1-sorted structure. Observe that there exists a map from S into S which is idempotent.

Next we state several propositions:

(5) For every up-complete non empty poset L holds every directed-sups-inheriting full non empty relational substructure of L is up-complete.

(6) Let L be an up-complete non empty poset and f be a map from L into L. Suppose f is idempotent and directed-sups-preserving. Then Imf is directed-sups-inheriting.

(7) Let T be an up-complete non empty poset and S be a directed-sups-inheriting full non empty relational substructure of T. Then incl(S, T) is directed-sups-preserving.

(8) Let S, T be non empty relational structures, f be a map from T into S, and g be a map from S into T. If f · g = id_{S}, then rngf = the carrier of S.

(9) Let T be a non empty relational structure, S be a non empty relational substructure of T, and f be a map from T into S. If f · incl(S, T) = id_{S}, then f is an idempotent map from T into T.

Let S, T be non empty posets and let f be a function. We say that f is a retraction of T into S if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) f is a directed-sups-preserving map from T into S, (ii) f↾the carrier of S = id_{S}, and (iii) S is a directed-sups-inheriting full relational substructure of T.

We say that f is a UPS retraction of T into S if and only if the conditions (Def. 2) are satisfied.

(Def. 2)(i) f is a directed-sups-preserving map from T into S, and

(ii) there exists a directed-sups-preserving map g from S into T such that

f · g = id_{S}. Let S, T be non empty posets. We say that S is a retract of T if and only if:

(Def. 3) There exists a map f from T into S such that f is a retraction of T into

S. We say that S is a UPS retract of T if and only if:

(Def. 4) There exists a map f from T into S such that f is a UPS retraction of

T into S. The following propositions are true:

(10) For all non empty posets S, T and for every function f such that f is a retraction of T into S holds f · incl(S, T) = id_{S}.

(11) Let S be a non empty poset, T be an up-complete non empty poset, and f be a function. Suppose f is a retraction of T into S. Then f is a UPS retraction of T into S.

(12) Let S, T be non empty posets and f be a function. If f is a retraction of T into S, then rngf = the carrier of S.

(13) Let S, T be non empty posets and f be a function. If f is a UPS retraction of T into S, then rngf = the carrier of S.

(14) Let S, T be non empty posets and f be a function. Suppose f is a retraction of T into S. Then f is an idempotent map from T into T.

(15) Let T, S be non empty posets and f be a map from T into T. Suppose f is a retraction of T into S. Then Imf = the relational structure of S.

(16) Let T be an up-complete non empty poset, S be a non empty poset, and f be a map from T into T. Suppose f is a retraction of T into S. Then f is directed-sups-preserving and projection.

(17) Let S, T be non empty reflexive transitive relational structures and f be a map from S into T. Then f is isomorphic if and only if the following conditions are satisfied:

(i) f is monotone, and (ii) there exists a monotone map g from T into S such that f · g = id_{T} and

g · f = id_{S}.

(18) Let S, T be non empty posets. Then S and T are isomorphic if and only if there exists a monotone map f from S into T and there exists a monotone map g from T into S such that f · g = id_{T} and g · f = id_{S}.

(19) Let S, T be up-complete non empty posets. Suppose S and T are isomorphic. Then S is a UPS retract of T and T is a UPS retract of S.

(20) Let T, S be non empty posets, f be a monotone map from T into S, and g be a monotone map from S into T. Suppose f · g = id_{S}. Then there exists a projection map h from T into T such that h = g · f and h↾the carrier of Imh = id_{Im h} and S and Imh are isomorphic.

(21) Let T be an up-complete non empty poset, S be a non empty poset, and f be a function. Suppose f is a UPS retraction of T into S. Then there exists a directed-sups-preserving projection map h from T into T such that h is a retraction of T into Imh and S and Imh are isomorphic.

(22) For every up-complete non empty poset L and for every non empty poset S such that S is a retract of L holds S is up-complete.

(23) For every complete non empty poset L and for every non empty poset S such that S is a retract of L holds S is complete.

(24) Let L be a continuous complete lattice and S be a non empty poset. If S is a retract of L, then S is continuous.

(25) Let L be an up-complete non empty poset and S be a non empty poset. If S is a UPS retract of L, then S is up-complete.

(26) Let L be a complete non empty poset and S be a non empty poset. If S is a UPS retract of L, then S is complete.

(27) Let L be a continuous complete lattice and S be a non empty poset. If S is a UPS retract of L, then S is continuous.

(28) Let L be a relational structure, S be a full relational substructure of L, and R be a relational substructure of S. Then R is full if and only if R is a full relational substructure of L.

(29) Let L be a non empty transitive relational structure and S be a directed-sups-inheriting non empty full relational substructure of L. Then every directed-sups-inheriting non empty relational substructure of S is a directed-sups-inheriting relational substructure of L.

(30) Let L be an up-complete non empty poset and S_{1}, S_{2} be directed-sups-inheriting full non empty relational substructures of L. Suppose S_{1} is a relational substructure of S_{2}. Then S_{1} is a directed-sups-inheriting full relational substructure of S_{2}.

Let X, Y be non empty topological spaces. Observe that every continuous map from X into Y is continuous.

2. Products

Let R be a binary relation. We say that R is poset-yielding if and only if: (Def. 5) For every set x such that x ∈ rngR holds x is a poset.

One can verify that every binary relation which is poset-yielding is also relational structure yielding and reflexive-yielding.

Let X be a non empty set, let L be a non empty relational structure, let i be an element of X, and let Y be a subset of L^{X}. Then π_{i}Y is a subset of L.

Let X be a set and let S be a poset. Observe that X ⟼→ S is poset-yielding. Let I be a set. One can verify that there exists a many sorted set indexed

by I which is poset-yielding and nonempty.

Let I be a non empty set and let J be a poset-yielding nonempty many sorted set indexed by I. One can verify that ∏ J is transitive and antisymmetric.

Let I be a non empty set, let J be a poset-yielding nonempty many sorted set indexed by I, and let i be an element of I. Then J(i) is a non empty poset.

Next we state a number of propositions:

(31) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, f be an element of ∏ J, and X be a subset of ∏ J. Then f X if and only if for every element i of I holds f(i) π_{i}X.

(32) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, f be an element of ∏ J, and X be a subset of ∏ J. Then f X if and only if for every element i of I holds f(i) π_{i}X.

(33) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. Then sup X exists in ∏ J if and only if for every element i of I holds sup π_{i}X exists in J(i).

(34) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. Then inf X exists in ∏ J if and only if for every element i of I holds inf π_{i}X exists in J(i).

(35) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. If sup X exists in ∏ J, then for every element i of I holds (supX)(i) = supπ_{i}X.

(36) Let I be a non empty set, J be a poset-yielding nonempty many sorted set indexed by I, and X be a subset of ∏ J. If inf X exists in ∏ J, then for every element i of I holds (inf X)(i) = inf π_{i}X.

(37) Let I be a non empty set, J be a relational structure yielding nonempty reflexive-yielding many sorted set indexed by I, X be a directed subset of ∏ J, and i be an element of I. Then π_{i}X is directed.

(38) Let I be a non empty set and J, K be relational structure yielding nonempty many sorted sets indexed by I. Suppose that for every element i of I holds K(i) is a relational substructure of J(i). Then ∏ K is a relational substructure of ∏ J.

(39) Let I be a non empty set and J, K be relational structure yielding nonempty many sorted sets indexed by I. Suppose that for every element i of I holds K(i) is a full relational substructure of J(i). Then ∏ K is a full relational substructure of ∏ J.

(40) Let L be a non empty relational structure, S be a non empty relational substructure of L, and X be a set. Then S^{X} is a relational substructure of L^{X}.

(41) Let L be a non empty relational structure, S be a full non empty relational substructure of L, and X be a set. Then S^{X} is a full relational substructure of L^{X}.

3. Inheritance

Let S, T be non empty relational structures and let X be a set. We say that S inherits sup of X from T if and only if:

(Def. 6) If sup X exists in T, then ⨆_{T} X ∈ the carrier of S.

We say that S inherits inf of X from T if and only if: (Def. 7) If inf X exists in T, then ⌈−⌉_{T} X ∈ the carrier of S.

One can prove the following propositions:

(42) Let T be a non empty transitive relational structure, S be a full non empty relational substructure of T, and X be a subset of S. Then S inherits sup of X from T if and only if if sup X exists in T, then sup X exists in S and supX = ⨆_{T} X.

(43) Let T be a non empty transitive relational structure, S be a full non empty relational substructure of T, and X be a subset of S. Then S inherits inf of X from T if and only if if inf X exists in T, then inf X exists in S and inf X = ⌈−⌉_{T} X.

In this article we present several logical schemes. The scheme ProductSupsInher deals with a non empty set A, poset-yielding nonempty many sorted sets B, C indexed by A, and and states that:

For every subset X of ∏ C such that P[X, ∏ C] holds ∏ C inherits sup of X from ∏ B

provided the following requirements are met:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L],

• For every subset X of ∏ C such that P[X, ∏ C] and for every element i of A holds P[π_{i}X, C(i)],

• For every element i of A holds C(i) is a full relational substructure of B(i), and

• For every element i of A and for every subset X of C(i) such that P[X, C(i)] holds C(i) inherits sup of X from B(i).

The scheme PowerSupsInherit deals with a non empty set A, a non empty poset B, a non empty full relational substructure C of B, and and states that:

For every subset X of C^{A} such that P[X, C^{A}] holds C^{A} inherits sup of X from B^{A}

provided the parameters satisfy the following conditions:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L],

• For every subset X of C^{A} such that P[X, C^{A}] and for every element i of A holds P[π_{i}X, C], and

• For every subset X of C such that P[X, C] holds C inherits sup of X from B.

The scheme ProductInfsInher deals with a non empty set A, poset-yielding nonempty many sorted sets B, C indexed by A, and and states that:

For every subset X of ∏ C such that P[X, ∏ C] holds ∏ C inherits inf of X from ∏ B

provided the parameters meet the following requirements:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L], • For every subset X of ∏ C such that P[X, ∏ C] and for every

element i of A holds P[π_{i}X, C(i)],

• For every element i of A holds C(i) is a full relational substructure of B(i), and

• For every element i of A and for every subset X of C(i) such that P[X, C(i)] holds C(i) inherits inf of X from B(i).

The scheme PowerInfsInherit deals with a non empty set A, a non empty poset B, a non empty full relational substructure C of B, and and states that:

For every subset X of C^{A} such that P[X, C^{A}] holds C^{A} inherits inf of X from B^{A}

provided the following conditions are satisfied:

• Let L be a non empty poset, S be a non empty full relational substructure of L, and X be a subset of S. If P[X, S], then P[X, L],

• For every subset X of C^{A} such that P[X, C^{A}] and for every element i of A holds P[π_{i}X, C], and

• For every subset X of C such that P[X, C] holds C inherits inf of X from B.

Let I be a set, let L be a non empty relational structure, let X be a non empty subset of L^{I}, and let i be a set. One can verify that π_{i}X is non empty.

One can prove the following proposition

(44) Let L be a non empty poset, S be a directed-sups-inheriting non empty full relational substructure of L, and X be a non empty set. Then S^{X} is a directed-sups-inheriting relational substructure of L^{X}.

Let I be a non empty set, let J be a relational structure yielding nonempty many sorted set indexed by I, let X be a non empty subset of ∏ J, and let i be a set. Note that π_{i}X is non empty.

One can prove the following proposition

(45) For every non empty set X and for every up-complete non empty poset

L holds L^{X} is up-complete. Let L be an up-complete non empty poset and let X be a non empty set.

One can check that L^{X} is up-complete.

4. Topological Retracts

Let T be a topological space. Observe that the topology of T is non empty. We now state a number of propositions:

(46) Let T be a non empty topological space, S be a non empty subspace of T, and f be a continuous map from T into S. If f is a retraction, then rngf = the carrier of S.

(47) Let T be a non empty topological space, S be a non empty subspace of T, and f be a continuous map from T into S. If f is a retraction, then f is idempotent.

(48) Let T be a non empty topological space and V be an open subset of T. Then χ_{V,the carrier of T} is a continuous map from T into the Sierpiński space.

(49) Let T be a non empty topological space and x, y be elements of T. Suppose that for every open subset W of T such that y ∈ W holds x ∈ W. Then [0 ⟼→ y, 1 ⟼→ x] is a continuous map from the Sierpiński space into T.

(50) Let T be a non empty topological space, x, y be elements of T, and V be an open subset of T. Suppose x ∈ V and y ∉ V. Then χ_{V,the carrier of T} · [0 ⟼→ y, 1 ⟼→ x] = id_{the Sierpiński space}.

(51) Let T be a non empty 1-sorted structure, V, W be subsets of T, S be a topological augmentation of 2_{⊆}^{1} , and f, g be maps from T into S. Suppose f = χ_{V,the carrier of T} and g = χ_{W,the carrier of T} . Then V ⊆ W if and only if f g.

(52) Let L be a non empty relational structure, X be a non empty set, and R be a full non empty relational substructure of L^{X}. Suppose that for every set a holds a is an element of R iff there exists an element x of L such that a = X ⟼→ x. Then L and R are isomorphic.

(53) Let S, T be non empty topological spaces. Then S and T are homeomorphic if and only if there exists a continuous map f from S into T and there exists a continuous map g from T into S such that f · g = id_{T} and g · f = id_{S}.

(54) Let T, S, R be non empty topological spaces, f be a map from T into S, g be a map from S into T, and h be a map from S into R. If f · g = id_{S} and h is a homeomorphism, then h · f · (g · h^{−1} ) = id_{R}.

(55) Let T, S, R be non empty topological spaces. Suppose S is a topological retract of T and S and R are homeomorphic. Then R is a topological retract of T.

(56) For every non empty topological space T and for every non empty subspace S of T holds incl(S, T) is continuous.

(57) Let T be a non empty topological space, S be a non empty subspace of T, and f be a continuous map from T into S. If f is a retraction, then f · incl(S, T) = id_{S}.

(58) Let T be a non empty topological space and S be a non empty subspace of T. If S is a retract of T, then S is a topological retract of T.

(59) Let R, T be non empty topological spaces. Then R is a topological retract of T if and only if there exists a non empty subspace S of T such that S

retracts and inheritance 323

is a retract of T and S and R are homeomorphic.

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324 grzegorz bancerek

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Technical Preliminaries to Algebraic Specifications

Grzegorz Bancerek University of Białystok

MML Identifier: ALGSPEC1.

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

One can prove the following propositions:

(1) For all functions f, g, h such that domf ∩ domg ⊆ domh holds f+·g+·h = g+·f+·h.

(2) For all functions f, g, h such that f ⊆ g and rngh ∩ domg ⊆ domf holds g · h = f · h.

(3) For all functions f, g, h such that domf ⊆ rngg and domf misses rngh and g^{∘} domh misses domf holds f · (g+·h) = f · g.

(4) For all functions f_{1}, f_{2}, g_{1}, g_{2} such that f_{1} ≈ f_{2} and g_{1} ≈ g_{2} holds f_{1} · g_{1} ≈ f_{2} · g_{2}.

(5) Let X_{1}, Y_{1}, X_{2}, Y_{2} be non empty sets, f be a function from X_{1} into X_{2}, and g be a function from Y_{1} into Y_{2}. If f ⊆ g, then f^{∗} ⊆ g^{∗}.

(6) Let X_{1}, Y_{1}, X_{2}, Y_{2} be non empty sets, f be a function from X_{1} into X_{2}, and g be a function from Y_{1} into Y_{2}. If f ≈ g, then f^{∗} ≈ g^{∗}.

Let X be a set and let f be a function. The functor X -indexingf yielding a many sorted set indexed by X is defined by:

(Def. 1) X -indexingf = id_{X}+·f↾X. Next we state a number of propositions:

(7) For every set X and for every function f holds rng(X -indexingf) = (X \ domf) ∪ f^{∘}X.

(8) For every non empty set X and for every function f and for every element x of X holds (X -indexingf)(x) = (id_{X}+·f)(x).

(9) For all sets X, x and for every function f such that x ∈ X holds if x ∈ domf, then (X -indexingf)(x) = f(x) and if x ∉ domf, then (X -indexingf)(x) = x.

(10) For every set X and for every function f such that domf = X holds X -indexingf = f.

(11) For every set X and for every function f holds X -indexing(X -indexingf) = X -indexingf.

(12) For every set X and for every function f holds X -indexing(id_{X}+·f) = X -indexingf.

(13) For every set X and for every function f such that f ⊆ id_{X} holds X -indexingf = id_{X}.

(14) For every set X holds X -indexing∅ = id_{X}.

(15) For every set X and for every function f holds X -indexingf↾X = X -indexingf.

(16) For every set X and for every function f such that X ⊆ domf holds X -indexingf = f↾X.

(17) For every function f holds ∅ -indexingf = ∅.

(18) For all sets X, Y and for every function f such that X ⊆ Y holds (Y -indexingf)↾X = X -indexingf.

(19) For all sets X, Y and for every function f holds (X ∪ Y) -indexingf = (X -indexingf)+·(Y -indexingf).

(20) For all sets X, Y and for every function f holds X -indexingf ≈ Y -indexingf.

(21) For all sets X, Y and for every function f holds (X ∪ Y) -indexingf = (X -indexingf) ∪ (Y -indexingf).

(22) For every non empty set X and for all functions f, g such that rngg ⊆ X holds (X -indexingf) · g = (id_{X}+·f) · g.

(23) For all functions f, g such that domf misses domg and rngg misses

domf and for every set X holds f · (X -indexingg) = f↾X. Let f be a function. A function is called a rng-retraction of f if:

(Def. 2) domit = rngf and f · it = id_{rng f}.

One can prove the following propositions:

(24) For every function f and for every rng-retraction g of f holds rngg ⊆ domf.

(25) Let f be a function, g be a rng-retraction of f, and x be a set. If x ∈ rngf, then g(x) ∈ domf and f(g(x)) = x.

(26) For every function f such that f is one-to-one holds f^{−1} is a rngretraction of f.

(27) For every function f such that f is one-to-one and for every rngretraction g of f holds g = f^{−1} .

(28) Let f_{1}, f_{2} be functions. Suppose f_{1} ≈ f_{2}. Let g_{1} be a rng-retraction of f_{1} and g_{2} be a rng-retraction of f_{2}. Then g_{1}+·g_{2} is a rng-retraction of f_{1}+·f_{2}.

(29) Let f_{1}, f_{2} be functions. Suppose f_{1} ⊆ f_{2}. Let g_{1} be a rng-retraction of f_{1}. Then there exists a rng-retraction g_{2} of f_{2} such that g_{1} ⊆ g_{2}.

2. ?????????????????????

Let S be a non empty non void many sorted signature and let f, g be functions. We say that f and g form a replacement in S if and only if the condition (Def. 3) is satisfied.

(Def. 3) Let o_{1}, o_{2} be operation symbols of S. Suppose (id_{the operation symbols of S}+·g)(o_{1}) =

(id_{the operation symbols of S}+·g)(o_{2}). Then

(i) (id_{the carrier of S}+·f) · Arity(o_{1}) = (id_{the carrier of S}+·f) · Arity(o_{2}), and (ii) (id_{the carrier of S}+·f)(the result sort of o_{1}) = (id_{the carrier of S}+·f)(the

result sort of o_{2}). Next we state two propositions:

(30) Let S be a non empty non void many sorted signature and f, g be functions. Then f and g form a replacement in S if and only if for all operation symbols o_{1}, o_{2} of S such that ((the operation symbols of S) -indexingg)(o_{1}) = ((the operation symbols of S) -indexingg)(o_{2}) holds ((the carrier of S) -indexingf)·Arity(o_{1}) = ((the carrier of S) -indexingf)· Arity(o_{2}) and ((the carrier of S) -indexingf)(the result sort of o_{1}) = ((the carrier of S) -indexingf)(the result sort of o_{2}).

(31) Let S be a non empty non void many sorted signature and f, g be functions. Then f and g form a replacement in S if and only if (the carrier of S) -indexingf and (the operation symbols of S) -indexingg form a replacement in S.

In the sequel S, S′ denote non void signatures and f, g denote functions. Next we state four propositions:

(32) If f and g form morphism between S and S′, then f and g form a replacement in S.

(33) f and ∅ form a replacement in S.

(34) If g is one-to-one and (the operation symbols of S)∩rngg ⊆ domg, then f and g form a replacement in S.

(35) If g is one-to-one and rngg misses the operation symbols of S, then f and g form a replacement in S.

Let X be a set, let Y be a non empty set, let a be a function from Y into X^{∗}, and let r be a function from Y into X. One can verify that ⟨X, Y, a, r⟩ is non void.

Let S be a non empty non void many sorted signature and let f, g be functions. Let us assume that f and g form a replacement in S. The functor S with-replacement(f, g) yields a strict non empty non void many sorted signature and is defined by the conditions (Def. 4).

(Def. 4)(i) (The carrier of S) -indexingf and (the operation symbols of

S) -indexingg form morphism between S and S with-replacement(f, g), (ii) the carrier of S with-replacement(f, g) = rng((the carrier of

S) -indexingf), and (iii) the operation symbols of S with-replacement(f, g) = rng((the operation

symbols of S) -indexingg). Next we state a number of propositions:

(36) Let S_{1}, S_{2} be non void signatures, f be a function from the carrier of S_{1} into the carrier of S_{2}, and g be a function. Suppose f and g form morphism between S_{1} and S_{2}. Then f^{∗} · the arity of S_{1} = (the arity of S_{2}) · g.

(37) Suppose f and g form a replacement in S. Then (the carrier of S) -indexingf is a function from the carrier of S into the carrier of S with-replacement(f, g).

(38) Suppose f and g form a replacement in S. Let f′ be a function from the carrier of S into the carrier of S with-replacement(f, g). Suppose f′= (the carrier of S) -indexingf. Let g′ be a rng-retraction of (the operation symbols of S) -indexingg. Then the arity of S with-replacement(f, g) = f^{′∗}· the arity of S · g′.

(39) Suppose f and g form a replacement in S. Let g′ be a rng-retraction of (the operation symbols of S) -indexingg. Then the result sort of S with-replacement(f, g) = ((the carrier of S) -indexingf) · the result sort of S · g′.

(40) If f and g form morphism between S and S′, then S with-replacement(f, g) is a subsignature of S′.

(41) f and g form a replacement in S if and only if (the carrier of S) -indexingf and (the operation symbols of S) -indexingg form morphism between S and S with-replacement(f, g).

(42) Suppose domf ⊆ the carrier of S and domg ⊆ the operation symbols of S and f and g form a replacement in S. Then id_{the carrier of S}+·f and id_{the operation symbols of S}+·g form morphism between S and S with-replacement(f, g).

(43) Suppose domf = the carrier of S and domg = the operation symbols of S and f and g form a replacement in S. Then f and g form morphism

between S and S with-replacement(f, g).

(44) If f and g form a replacement in S, then S with-replacement((the carrier of S) -indexingf, g) = S with-replacement(f, g).

(45) If f and g form a replacement in S, then S with-replacement(f, (the operation symbols of S) -indexingg) = S with-replacement(f, g).

3. ?????????????????????

Let S be a signature. A signature is called an extension of S if: (Def. 5) S is a subsignature of it.

One can prove the following propositions:

(46) For all signatures S, E holds S is a subsignature of E iff E is an extension of S.

(47) Every signature S is an extension of S.

(48) For every signature S_{1} and for every extension S_{2} of S_{1} holds every extension of S_{2} is an extension of S_{1}.

(49) For all non empty signatures S_{1}, S_{2} such that S_{1} ≈ S_{2} holds S_{1}+·S_{2} is an extension of S_{1}.

(50) For all non empty signatures S_{1}, S_{2} holds S_{1}+·S_{2} is an extension of S_{2}.

(51) Let S_{1}, S_{2}, S be non empty many sorted signatures and f_{1}, g_{1}, f_{2}, g_{2} be functions. Suppose f_{1} ≈ f_{2} and f_{1} and g_{1} form morphism between S_{1} and S and f_{2} and g_{2} form morphism between S_{2} and S. Then f_{1}+·f_{2} and g_{1}+·g_{2} form morphism between S_{1}+·S_{2} and S.

(52) Let S_{1}, S_{2}, E be non empty signatures. Then E is an extension of S_{1} and an extension of S_{2} if and only if S_{1} ≈ S_{2} and E is an extension of S_{1}+·S_{2}.

Let S be a non empty signature. One can check that every extension of S is non empty.

Let S be a non void signature. One can check that every extension of S is non void.

The following proposition is true

(53) For all signatures S, T such that S is empty holds T is an extension of

S. Let S be a signature. Observe that there exists an extension of S which is

non empty, non void, and strict.

We now state three propositions:

(54) Let S be a non void signature and E be an extension of S. Suppose f and g form a replacement in E. Then f and g form a replacement in S.

(55) Let S be a non void signature and E be an extension of S. Suppose f and g form a replacement in E. Then E with-replacement(f, g) is an extension of S with-replacement(f, g).

(56) Let S_{1}, S_{2} be non void signatures. Suppose S_{1} ≈ S_{2}. Let f, g be functions. If f and g form a replacement in S_{1}+·S_{2}, then (S_{1}+·S_{2}) with-replacement(f, g) = (S_{1} with-replacement(f, g))+·(S_{2} with-replacement(f, g)).

4. ?????????????????????

Algebra is defined by: (Def. 6) There exists a non void signature S such that it is a feasible algebra over

S. Let S be a signature. An algebra is called an algebra of S if:

(Def. 7) There exists a non void extension E of S such that it is a feasible algebra

over E. Next we state a number of propositions:

(57) For every non void signature S holds every feasible algebra over S is an algebra of S.

(58) For every signature S and for every extension E of S holds every algebra of E is an algebra of S.

(59) Let S be a signature, E be a non empty signature, and A be an algebra over E. Suppose A is an algebra of S. Then the carrier of S ⊆ the carrier of E and the operation symbols of S ⊆ the operation symbols of E.

(60) Let S be a non void signature, E be a non empty signature, and A be an algebra over E. Suppose A is an algebra of S. Let o be an operation symbol of S. Then (the characteristics of A)(o) is a function from (the sorts of A)^{#}(Arity(o)) into (the sorts of A)(the result sort of o).

(61) Let S be a non empty signature, A be an algebra of S, and E be a non empty many sorted signature. If A is an algebra over E, then A is an algebra over E+·S.

(62) Let S_{1}, S_{2} be non empty signatures and A be an algebra over S_{1}. Suppose A is an algebra over S_{2}. Then the carrier of S_{1} = the carrier of S_{2} and the operation symbols of S_{1} = the operation symbols of S_{2}.

(63) For every non void signature S and for every non-empty disjoint algebra A over S holds the sorts of A are one-to-one.

(64) Let S be a non void signature, A be a disjoint algebra over S, and C_{1}, C_{2} be components of the sorts of A. Then C_{1} = C_{2} or C_{1} misses C_{2}.

technical preliminaries to algebraic … 331

(65) Let S, S′ be non void signatures and A be a non-empty disjoint algebra over S. Suppose A is an algebra over S′. Then the many sorted signature of S = the many sorted signature of S′.

(66) Let S′ be a non void signature and A be a non-empty disjoint algebra over S. If A is an algebra of S′, then S is an extension of S′.

References

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

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

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

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

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

[6] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

[8] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part 1. Formalized Mathematics, 2(5):683–687, 1991.

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

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

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

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

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

332 grzegorz bancerek

Multivariate Polynomials with Arbitrary Number of Variables^{1}

Piotr Rudnicki University of Alberta Edmonton

Andrzej Trybulec University of Bialystok

Summary. The goal of this article is to define multivariate polynomials in arbitrary number of indeterminates and then to prove that they constitute a ring (over appropriate structure of coefficients).

The introductory section includes quite a number of auxiliary lemmas related to many different parts of the MML. The second section characterizes the sequence flattening operation, introduced in [8], but so far lacking theorems about its fundamental properties.

We first define formal power series in arbitrary number of variables. The auxiliary concept on which the construction of formal power series is based is the notion of a bag. A bag of a set X is a natural function on X which is zero almost everywhere. The elements of X play the role of formal variables and a bag gives their exponents thus forming a power product. Series are defined for an ordered set of variables (we use ordinal numbers). A series in o variables over a structure S is a function assigning an element of the carrier of S (coefficient) to each bag of o.

We define the operations of addition, complement and multiplication for formal power series and prove their properties which depend on assumed properties of the structure from which the coefficients are taken. (We would like to note that proving associativity of multiplication turned out to be technically complicated.)

Polynomial is defined as a formal power series with finite number of non zero coefficients. In conclusion, the ring of polynomials is defined.

MML Identifier: POLYNOM1.

The papers [33], [42], [41], [18], [43], [44], [45], [14], [19], [35], [32], [4], [34], [17], [15], [10], [11], [16], [21], [2], [9], [27], [29], [39], [37], [40], [7], [13], [5], [22], [1],

1This work has been supported by NSERC Grant OGP9207 and NATO CRG 951368.

[3], [38], [12], [25], [36], [?], [8], [6], [?], [26], [30], [9], [31], [8], [28], [?], [24], [23], [20], and [46] provide the terminology and notation for this paper.

1. Basics

One can prove the following propositions:

(1) For all natural numbers i, j holds ·_{ℕ}(i, j) = i · j.

(2) Let X be a set, A be a non empty set, F be a binary operation on A, f be a function from X into A, and x be an element of A. Then dom(F^{∘}(f, x)) = X.

(3) For all natural numbers a, b, c holds a − ′b − ′c = a − ′(b + c).

(4) For every set X and for every binary relation R such that fieldR ⊆ X holds R is a binary relation on X.

(5) Let K be a non empty loop structure and p_{1}, p_{2} be finite sequences of elements of the carrier of K. If domp_{1} = domp_{2}, then dom(p_{1} + p_{2}) = domp_{1}.

(6) For every function f and for all sets x, y holds rng(f +·(x, y)) ⊆ rngf ∪ {y}.

Let A, B be sets, let f be a function from A into B, let x be a set, and let y be an element of B. Then f +·(x, y) is a function from A into B.

Let X be a set, let f be a many sorted set indexed by X, and let x, y be sets. Then f +·(x, y) is a many sorted set indexed by X.

Next we state the proposition

(7) For every one-to-one function f holds \overline{\overline{(f qua set)}} = \overline{\overline{rngf}} . Let A be a non empty set, let F, G be binary operations on A, and let z, u

be elements of A. Note that ⟨A, F, G, z, u⟩ is non empty.

Let A be a set, let X be a set, let D be a non empty set of finite sequences of A, let p be a partial function from X to D, and let i be a set. Then π_{i}p is an element of D.

Let X be a set and let S be a 1-sorted structure. (Def. 1) A function from X into the carrier of S is said to be a function from X

into S. Let X be a set. Note that there exists an order in X which is linear-order

and well-ordering.

One can prove the following two propositions:

(8) Let X be a non empty set, A be a non empty finite subset of X, R be an order in X, and x be an element of X. Suppose x ∈ A and R linearly orders A and for every element y of X such that y ∈ A holds ⟨⟨x, y⟩⟩∈ R. Then π_{1} SgmX(R, A) = x.

(9) Let X be a non empty set, A be a non empty finite subset of X, R be an order in X, and x be an element of X. Suppose x ∈ A and R linearly orders A and for every element y of X such that y ∈ A holds ⟨⟨y, x⟩⟩∈ R. Then π_{len SgmX(R,A)} SgmX(R, A) = x.

Let X be a non empty set, let A be a non empty finite subset of X, and let R be linear-order order in X. Observe that SgmX(R, A) is non empty and one-to-one.

Let us mention that ∅ is finite sequence yielding. One can verify that there exists a finite sequence which is finite sequence

yielding.

Let F, G be finite sequence yielding finite sequences. Then F ^{⌢} G is a finite sequence yielding finite sequence.

Let D be a set. Note that every finite sequence of elements of D^{∗} is finite sequence yielding.

Let i be a natural number and let f be a finite sequence. Observe that i ↦ f is finite sequence yielding.

Let us observe that every function which is finite sequence yielding is also function yielding.

Let F be a finite sequence yielding finite sequence and let x be a set. Note that F (x) is finite sequence-like.

Let F be a finite sequence. One can check that \overline{\overline{F}} is finite sequence-like. One can check that there exists a finite sequence which is cardinal yielding. One can prove the following proposition

(10) Let f be a function. Then f is cardinal yielding if and only if for every

set y such that y ∈ rngf holds y is a cardinal number. Let F, G be cardinal yielding finite sequences. One can check that F ^{⌢} G is

cardinal yielding.

Let us observe that every finite sequence of elements of ℕ is cardinal yielding. Let us note that there exists a finite sequence of elements of ℕ which is

cardinal yielding.

Let D be a set and let F be a finite sequence of elements of D^{∗}. Then \overline{\overline{F}} is a cardinal yielding finite sequence of elements of ℕ.

Let F be a finite sequence of elements of ℕ and let i be a natural number. One can check that F ↾i is cardinal yielding.

We now state the proposition

(11) For every function F and for every set X holds \overline{\overline{F ↾X}} = \overline{\overline{F}} ↾X.

Let F be an empty function. One can verify that \overline{\overline{F}} is empty. Next we state two propositions:

(12) For every set p holds \overline{\overline{⟨p⟩}}= ⟨\overline{\overline{p}}⟩.

(13) For all finite sequences F, G holds \overline{\overline{F ^{⌢} G}} = \overline{\overline{F}} ^{⌢} \overline{\overline{G}}.

Let X be a set. Note that ε_{X} is finite sequence yielding. Let f be a finite sequence. Note that ⟨f⟩ is finite sequence yielding. Next we state the proposition

(14) Let f be a function. Then f is finite sequence yielding if and only if for

every set y such that y ∈ rngf holds y is a finite sequence. Let F, G be finite sequence yielding finite sequences. Observe that F ^{⌢} G is

finite sequence yielding.

The following propositions are true:

(15) Let L be a non empty loop structure and F be a finite sequence of elements of (the carrier of L)^{∗}. Then dom∑ F = domF.

(16) Let L be a non empty loop structure and F be a finite sequence of elements of (the carrier of L)^{∗}. Then ∑(ε_{(the carrier of L)^{∗}}) = ε_{(the carrier of L)}.

(17) For every non empty loop structure L and for every element p of (the carrier of L)^{∗} holds ⟨∑ p⟩ = ∑⟨p⟩.

(18) Let L be a non empty loop structure and F, G be finite sequences of

elements of (the carrier of L)^{∗}. Then ∑(F ^{⌢} G) = (∑ F)^{⌢} ∑ G. Let L be a non empty groupoid, let a be an element of the carrier of L, and

let p be a finite sequence of elements of the carrier of L. The functor a · p yields a finite sequence of elements of the carrier of L and is defined as follows:

(Def. 2) dom(a·p) = domp and for every set i such that i ∈ domp holds π_{i}(a·p) =

a · π_{i}p. The functor p · a yields a finite sequence of elements of the carrier of L and is defined as follows:

(Def. 3) dom(p·a) = domp and for every set i such that i ∈ domp holds π_{i}(p·a) =

π_{i}p · a. The following propositions are true:

(19) Let L be a non empty groupoid and a be an element of the carrier of L. Then a · ε_{(the carrier of L)} = ε_{(the carrier of L)}.

(20) Let L be a non empty groupoid and a be an element of the carrier of L. Then ε_{(the carrier of L)} · a = ε_{(the carrier of L)}.

(21) For every non empty groupoid L and for all elements a, b of the carrier of L holds a · ⟨b⟩ = ⟨a · b⟩.

(22) For every non empty groupoid L and for all elements a, b of the carrier of L holds ⟨b⟩ · a = ⟨b · a⟩.

(23) Let L be a non empty groupoid, a be an element of the carrier of L, and p, q be finite sequences of elements of the carrier of L. Then a · (p^{⌢} q) = (a · p)^{⌢} (a · q).

(24) Let L be a non empty groupoid, a be an element of the carrier of L, and p, q be finite sequences of elements of the carrier of L. Then (p^{⌢} q) · a =

(p · a)^{⌢} (q · a). Next we state two propositions:

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

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

Let us note that every non empty multiplicative loop with zero structure which is non degenerated is also non trivial.

Let us note that there exists a non empty strict multiplicative loop with zero structure which is unital.

Let us note that there exists a non empty strict double loop structure which is Abelian, add-associative, right zeroed, right complementable, associative, commutative, distributive, unital, and non trivial.

Next we state three propositions:

(27) Let L be an add-associative right zeroed right complementable unital right-distributive non empty double loop structure. If 0_{L} = 1_{L}, then L is trivial.

(28) Let L be an add-associative right zeroed right complementable unital distributive non empty double loop structure, a be an element of the carrier of L, and p be a finite sequence of elements of the carrier of L. Then ∑(a · p) = a · ∑ p.

(29) Let L be an add-associative right zeroed right complementable unital distributive non empty double loop structure, a be an element of the carrier of L, and p be a finite sequence of elements of the carrier of L. Then ∑(p · a) = ∑ p · a.

2. Sequence Flattening

Let D be a set and let F be an empty finite sequence of elements of D^{∗}. Observe that Flat(F) is empty.

One can prove the following propositions:

(30) For every set D and for every finite sequence F of elements of D^{∗} holds lenFlat(F) = ∑ \overline{\overline{F}} .

(31) Let D, E be sets, F be a finite sequence of elements of D^{∗}, and G be a finite sequence of elements of E^{∗}. If \overline{\overline{F}} = \overline{\overline{G}}, then lenFlat(F) = lenFlat(G).

(32) Let D be a set, F be a finite sequence of elements of D^{∗}, and k be a set. Suppose k ∈ domFlat(F). Then there exist natural numbers i, j such that i ∈ domF and j ∈ domF (i) and k = ∑ \overline{\overline{F ↾(i − ′1)}} + j and F (i)(j) = Flat(F)(k).

(33) Let D be a set, F be a finite sequence of elements of D^{∗}, and i, j be natural numbers. If i ∈ domF and j ∈ domF (i), then ∑ \overline{\overline{F ↾(i − ′1)}} + j ∈ domFlat(F) and F (i)(j) = Flat(F)(∑ \overline{\overline{F ↾(i − ′1)}} + j).

(34) Let L be an add-associative right zeroed right complementable non empty loop structure and F be a finite sequence of elements of (the carrier of L)^{∗}. Then ∑ Flat(F) = ∑ ∑ F.

(35) Let X, Y be non empty sets, f be a finite sequence of elements of X^{∗}, and v be a function from X into Y. Then (domf ⟼→ v) ∘ f is a finite sequence of elements of Y^{∗}.

(36) Let X, Y be non empty sets, f be a finite sequence of elements of X^{∗}, and v be a function from X into Y. Then there exists a finite sequence F of elements of Y^{∗} such that F = (domf ⟼→ v) ∘ f and v · Flat(f) = Flat(F).

3. Functions Yielding Natural Numbers

Let us note that ∅ is natural-yielding. Let us mention that there exists a function which is natural-yielding. Let f be a natural-yielding function and let x be a set. Then f(x) is a natural

number.

Let f be a natural-yielding function, let x be a set, and let n be a natural number. Observe that f +·(x, n) is natural-yielding.

Let X be a set. One can check that every function from X into ℕ is naturalyielding.

Let X be a set. Note that there exists a many sorted set indexed by X which is natural-yielding.

Let X be a set and let b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. The functor b_{1} + b_{2} yields a many sorted set indexed by X and is defined by:

(Def. 5)^{2} For every set x holds (b_{1} + b_{2})(x) = b_{1}(x) + b_{2}(x). Let us note that the functor b_{1} + b_{2} is commutative. The functor b_{1} − ′b_{2} yields a many sorted set indexed by X and is defined as follows:

(Def. 6) For every set x holds (b_{1} − ′b_{2})(x) = b_{1}(x) − ′b_{2}(x). Next we state two propositions:

2The definition (Def. 4) has been removed.

(37) Let X be a set and b, b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. If for every set x such that x ∈ X holds b(x) = b_{1}(x) + b_{2}(x), then b = b_{1} + b_{2}.

(38) Let X be a set and b, b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. If for every set x such that x ∈ X holds b(x) = b_{1}(x) − ′b_{2}(x), then b = b_{1} − ′b_{2}.

Let X be a set and let b_{1}, b_{2} be natural-yielding many sorted sets indexed by X. Observe that b_{1} + b_{2} is natural-yielding and b_{1} − ′b_{2} is natural-yielding.

The following two propositions are true:

(39) For every set X and for all natural-yielding many sorted sets b_{1}, b_{2}, b_{3} indexed by X holds (b_{1} + b_{2}) + b_{3} = b_{1} + (b_{2} + b_{3}).

(40) For every set X and for all natural-yielding many sorted sets b, c, d indexed by X holds b − ′c − ′d = b − ′(c + d).

4. The Support of a Function

Let f be a function. The functor supportf is defined as follows: (Def. 7) For every set x holds x ∈ supportf iff f(x) ≠ 0.

We now state the proposition

(41) For every function f holds supportf ⊆ domf.

Let f be a function. We say that f is finite-support if and only if: (Def. 8) supportf is finite.

We introduce f has finite-support as a synonym of f is finite-support. Let us note that ∅ is finite-support. Let us mention that every function which is finite is also finite-support. Let us observe that there exists a function which is natural-yielding, finite-

support, and non empty.

Let f be a finite-support function. Observe that supportf is finite. Let X be a set. One can verify that there exists a function from X into ℕ

which is finite-support.

Let f be a finite-support function and let x, y be sets. Note that f +·(x, y) is finite-support.

Let X be a set. Note that there exists a many sorted set indexed by X which is natural-yielding and finite-support.

One can prove the following two propositions:

(42) For every set X and for all natural-yielding many sorted sets b_{1}, b_{2} indexed by X holds support(b_{1} + b_{2}) = supportb_{1} ∪ supportb_{2}.

(43) For every set X and for all natural-yielding many sorted sets b_{1}, b_{2} indexed by X holds support(b_{1} − ′b_{2}) ⊆ supportb_{1}.

Let X be a non empty set, let S be a zero structure, and let f be a function from X into S. The functor Supportf yields a subset of X and is defined as follows:

(Def. 9) For every element x of X holds x ∈ Supportf iff f(x) ≠ 0_{S}.

Let X be a non empty set, let S be a zero structure, and let p be a function from X into S. We say that p is finite-Support if and only if:

(Def. 10) Supportp is finite.

We introduce p has finite-Support as a synonym of p is finite-Support.

5. Bags

Let X be a set. A bag of X is a natural-yielding finite-support many sorted set indexed by X.

Let X be a finite set. Observe that every many sorted set indexed by X is finite-support.

Let X be a set and let b_{1}, b_{2} be bag of X. Note that b_{1} + b_{2} is finite-support and b_{1} − ′b_{2} is finite-support.

Next we state the proposition

(44) For every set X holds X ⟼→ 0 is a bag of X.

Let n be an ordinal number and let p, q be bag of n. The predicate p < q is defined by:

(Def. 11) There exists an ordinal number k such that p(k) < q(k) and for every

ordinal number l such that l ∈ k holds p(l) = q(l). Let us note that the predicate p < q is antisymmetric.

Next we state the proposition

(45) For every ordinal number n and for all bag p, q, r of n such that p < q

and q < r holds p < r. Let n be an ordinal number and let p, q be bag of n. The predicate p q is

defined as follows: (Def. 12) p < q or p = q.

Let us note that the predicate p q is reflexive.

Next we state four propositions:

(46) For every ordinal number n and for all bag p, q, r of n such that p q and q r holds p r.

(47) For every ordinal number n and for all bag p, q, r of n such that p < q and q r holds p < r.

(48) For every ordinal number n and for all bag p, q, r of n such that p q and q < r holds p < r.

(49) For every ordinal number n and for all bag p, q of n holds p q or q p. Let X be a set and let d, b be bag of X. We say that d divides b if and only

if: (Def. 13) For every set k holds d(k) b(k).

Let us note that the predicate d divides b is reflexive.

Next we state several propositions:

(50) For every set n and for all bag d, b of n such that for every set k such that k ∈ n holds d(k) b(k) holds d divides b.

(51) For every ordinal number n and for all bag b_{1}, b_{2} of n such that b_{1} divides b_{2} holds (b_{2} − ′b_{1}) + b_{1} = b_{2}.

(52) For every set X and for all bag b_{1}, b_{2} of X holds (b_{2} + b_{1}) − ′b_{1} = b_{2}.

(53) For every ordinal number n and for all bag d, b of n such that d divides b holds d b.

(54) For every set n and for all bag b, b_{1}, b_{2} of n such that b = b_{1} + b_{2} holds

b_{1} divides b. Let X be a set. The functor BagsX is defined by:

(Def. 14) For every set x holds x ∈ BagsX iff x is a bag of X. Let X be a set. Then BagsX is a subset of BagsX. One can prove the following proposition

(55) Bags∅ = {∅}.

Let X be a set. Note that BagsX is non empty.

Let X be a set and let B be a non empty subset of BagsX. We see that the element of B is a bag of X.

Let n be a set, let L be a non empty 1-sorted structure, let p be a function from Bagsn into L, and let x be a bag of n. Then p(x) is an element of L.

Let X be a set. The functor EmptyBagX yielding an element of BagsX is defined as follows:

(Def. 15) EmptyBagX = X ⟼→ 0.

We now state several propositions:

(56) For all sets X, x holds (EmptyBagX)(x) = 0.

(57) For every set X and for every bag b of X holds b + EmptyBagX = b.

(58) For every set X and for every bag b of X holds b − ′EmptyBagX = b.

(59) For every set X and for every bag b of X holds EmptyBagX − ′b = EmptyBagX.

(60) For every set X and for every bag b of X holds b − ′b = EmptyBagX.

(61) For every set n and for all bag b_{1}, b_{2} of n such that b_{1} divides b_{2} and b_{2} − ′b_{1} = EmptyBagn holds b_{2} = b_{1}.

(62) For every set n and for every bag b of n such that b divides EmptyBagn holds EmptyBagn = b.

(63) For every set n and for every bag b of n holds EmptyBagn divides b.

(64) For every ordinal number n and for every bag b of n holds EmptyBagn

b. Let n be an ordinal number. The functor BagOrdern yielding an order in

Bagsn is defined as follows: (Def. 16) For all bag p, q of n holds ⟨⟨p, q⟩⟩∈ BagOrdern iff p q.

Let n be an ordinal number. One can verify that BagOrdern is linear-order. Let X be a set and let f be a function from X into ℕ. The functor NatMinorf

yields a subset of ℕ^{X} and is defined by the condition (Def. 17). (Def. 17) Let g be a natural-yielding many sorted set indexed by X. Then g ∈

NatMinorf if and only if for every set x such that x ∈ X holds g(x) f(x).

The following proposition is true

(65) For every set X and for every function f from X into ℕ holds f ∈

NatMinorf. Let X be a set and let f be a function from X into ℕ. Observe that

NatMinorf is non empty and functional.

Let X be a set and let f be a function from X into ℕ. Note that every element of NatMinorf is natural-yielding.

The following proposition is true

(66) For every set X and for every finite-support function f from X into ℕ

holds NatMinorf ⊆ BagsX. Let X be a set and let f be a finite-support function from X into ℕ. Then

supportf is an element of FinX.

We now state the proposition

(67) For every non empty set X and for every finite-support function f from

X into ℕ holds \overline{\overline{NatMinorf}} = ·_{ℕ}-∑_{support f}(+_{ℕ})^{∘}(f, 1). Let X be a set and let f be a finite-support function from X into ℕ. Observe

that NatMinorf is finite.

Let n be an ordinal number and let b be a bag of n. The functor divisorsb yields a finite sequence of elements of Bagsn and is defined by the condition (Def. 18).

(Def. 18) There exists a non empty finite subset S of Bagsn such that divisorsb =

SgmX(BagOrdern, S) and for every bag p of n holds p ∈ S iff p divides b. Let n be an ordinal number and let b be a bag of n. Observe that divisorsb

is non empty and one-to-one.

We now state four propositions:

(68) Let n be an ordinal number, i be a natural number, and b be a bag of n. If i ∈ domdivisorsb, then π_{i} divisorsb qua element of Bagsn divides b.

(69) For every ordinal number n and for every bag b of n holds π_{1} divisorsb = EmptyBagn and π_{len divisors b} divisorsb = b.

(70) Let n be an ordinal number, i be a natural number, and b, b_{1}, b_{2} be bag of n. If i > 1 and i < lendivisorsb, then π_{i} divisorsb ≠ EmptyBagn and π_{i} divisorsb ≠ b.

(71) For every ordinal number n holds divisorsEmptyBagn = ⟨EmptyBagn⟩. Let n be an ordinal number and let b be a bag of n. The functor decompb

yielding a finite sequence of elements of (Bagsn)^{2} is defined by: (Def. 19) domdecompb = domdivisorsb and for every natural number i and for

every bag p of n such that i ∈ domdecompb and p = π_{i} divisorsb holds π_{i} decompb = ⟨p, b − ′p⟩.

The following three propositions are true:

(72) Let n be an ordinal number, i be a natural number, and b be a bag of n. If i ∈ domdecompb, then there exist bag b_{1}, b_{2} of n such that π_{i} decompb = ⟨b_{1}, b_{2}⟩ and b = b_{1} + b_{2}.

(73) Let n be an ordinal number and b, b_{1}, b_{2} be bag of n. If b = b_{1} + b_{2}, then there exists a natural number i such that i ∈ domdecompb and π_{i} decompb = ⟨b_{1}, b_{2}⟩.

(74) Let n be an ordinal number, i be a natural number, and b, b_{1}, b_{2} be bag of

n. If i ∈ domdecompb and π_{i} decompb = ⟨b_{1}, b_{2}⟩, then b_{1} = π_{i} divisorsb. Let n be an ordinal number and let b be a bag of n. Note that decompb is

non empty one-to-one and finite sequence yielding.

Let n be an ordinal number and let b be an element of Bagsn. One can check that decompb is non empty one-to-one and finite sequence yielding.

One can prove the following four propositions:

(75) For every ordinal number n and for every bag b of n holds π_{1} decompb = ⟨EmptyBagn, b⟩ and π_{len decomp b} decompb = ⟨b, EmptyBagn⟩.

(76) Let n be an ordinal number, i be a natural number, and b, b_{1}, b_{2} be bag of n. If i > 1 and i < lendecompb and π_{i} decompb = ⟨b_{1}, b_{2}⟩, then b_{1} ≠ EmptyBagn and b_{2} ≠ EmptyBagn.

(77) For every ordinal number n holds decompEmptyBagn = ⟨⟨EmptyBagn, EmptyBagn⟩⟩.

(78) Let n be an ordinal number, b be a bag of n, and f, g be finite sequences

of elements of ((Bagsn)^{3})^{∗}. Suppose that

(i) domf = domdecompb, (ii) domg = domdecompb, (iii) for every natural number k such that k ∈ domf holds f(k) =

(decomp(π_{1}π_{k} decompb qua element of Bagsn))^{⌢}(lendecomp(π_{1}π_{k} decompb qua element of Bagsn) ↦ ⟨π_{2}π_{k} decompb⟩), and (iv) for every natural number k such that k ∈ domg holds g(k) =

(lendecomp(π_{2}π_{k} decompb qua element of Bagsn) ↦ ⟨π_{1}π_{k} decompb⟩)^{⌢} decomp(π_{2}π_{k} decompb qua element of Bagsn). Then there exists a permutation p of domFlat(f) such that Flat(g) = Flat(f) · p.

6. Formal Power Series

Let X be a set and let S be a 1-sorted structure. (Def. 20) A function from BagsX into S is said to be a Series of X, S.

Let n be a set, let L be a right zeroed non empty loop structure, and let p, q be Series of n, L. The functor p + q yields a Series of n, L and is defined by:

(Def. 21) For every bag x of n holds (p + q)(x) = p(x) + q(x).

Next we state the proposition

(79) Let n be a set, L be a right zeroed non empty loop structure, and p, q

be Series of n, L. Then Supportp + q ⊆ Supportp ∪ Supportq. Let n be a set, let L be an Abelian right zeroed non empty loop structure,

and let p, q be Series of n, L. Let us notice that the functor p+q is commutative.

Next we state the proposition

(80) Let n be a set, L be an add-associative right zeroed non empty double

loop structure, and p, q, r be Series of n, L. Then (p + q) + r = p + (q + r). Let n be a set, let L be an add-associative right zeroed right complementable

non empty loop structure, and let p be a Series of n, L. The functor − p yielding a Series of n, L is defined as follows:

(Def. 22) For every bag x of n holds (− p)(x) = − p(x).

Let n be a set, let L be an add-associative right zeroed right complementable non empty loop structure, and let p, q be Series of n, L. The functor p−q yielding a Series of n, L is defined by:

(Def. 23) p − q = p + − q.

Let n be a set and let S be a non empty zero structure. The functor 0 (n, S) yields a Series of n, S and is defined by:

(Def. 24) 0 (n, S) = Bagsn ⟼→ 0_{S}.

One can prove the following propositions:

(81) For every set n and for every non empty zero structure S and for every bag b of n holds (0 (n, S))(b) = 0_{S}.

(82) For every set n and for every right zeroed non empty loop structure L and for every Series p of n, L holds p + 0 (n, L) = p.

Let n be a set and let L be a unital non empty multiplicative loop with zero structure. The functor 1 (n, L) yielding a Series of n, L is defined by:

(Def. 25) 1 (n, L) = 0 (n, L) +·(EmptyBagn, 1_{L}).

The following two propositions are true:

(83) Let n be a set, L be an add-associative right zeroed right complementable non empty loop structure, and p be a Series of n, L. Then p−p = 0 (n, L).

(84) Let n be a set and L be a unital non empty multiplicative loop with zero structure. Then (1 (n, L))(EmptyBagn) = 1_{L} and for every bag b of n such that b ≠ EmptyBagn holds (1 (n, L))(b) = 0_{L}.

Let n be an ordinal number, let L be an add-associative right complementable right zeroed non empty double loop structure, and let p, q be Series of n, L. The functor p ∗ q yielding a Series of n, L is defined by the condition (Def. 26).

(Def. 26) Let b be a bag of n. Then there exists a finite sequence s of elements of

the carrier of L such that

(i) (p ∗ q)(b) = ∑ s, (ii) lens = lendecompb, and (iii) for every natural number k such that k ∈ doms there exist bag b_{1}, b_{2}

of n such that π_{k} decompb = ⟨b_{1}, b_{2}⟩ and π_{k}s = p(b_{1}) · q(b_{2}). The following two propositions are true:

(85) Let n be an ordinal number, L be an Abelian add-associative right zeroed right complementable distributive associative non empty double loop structure, and p, q, r be Series of n, L. Then p ∗ (q + r) = p ∗ q + p ∗ r.

(86) Let n be an ordinal number, L be an Abelian add-associative right zeroed right complementable unital distributive associative non empty double loop structure, and p, q, r be Series of n, L. Then (p ∗ q) ∗ r = p ∗ (q ∗ r).

Let n be an ordinal number, let L be an Abelian add-associative right zeroed right complementable commutative non empty double loop structure, and let p, q be Series of n, L. Let us observe that the functor p ∗ q is commutative.

The following three propositions are true:

(87) Let n be an ordinal number, L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and p be a Series of n, L. Then p ∗ 0 (n, L) = 0 (n, L).

(88) Let n be an ordinal number, L be an add-associative right complementable right zeroed distributive unital non trivial non empty double loop structure, and p be a Series of n, L. Then p ∗ 1 (n, L) = p.

(89) Let n be an ordinal number, L be an add-associative right complementable right zeroed distributive unital non trivial non empty double loop structure, and p be a Series of n, L. Then 1 (n, L) ∗ p = p.

7. Polynomials

Let n be a set and let S be a non empty zero structure. Observe that there exists a Series of n, S which is finite-Support.

Let n be an ordinal number and let S be a non empty zero structure. A Polynomial of n, S is a finite-Support Series of n, S.

Let n be an ordinal number, let L be a right zeroed non empty loop structure, and let p, q be Polynomial of n, L. Observe that p + q is finite-Support.

Let n be an ordinal number, let L be an add-associative right zeroed right complementable non empty loop structure, and let p be a Polynomial of n, L. Note that − p is finite-Support.

Let n be a natural number, let L be an add-associative right zeroed right complementable non empty loop structure, and let p, q be Polynomial of n, L. Note that p − q is finite-Support.

Let n be an ordinal number and let S be a non empty zero structure. One can verify that 0 (n, S) is finite-Support.

Let n be an ordinal number and let L be an add-associative right zeroed right complementable unital right-distributive non trivial non empty double loop structure. Note that 1 (n, L) is finite-Support.

Let n be an ordinal number, let L be an add-associative right complementable right zeroed unital distributive non empty double loop structure, and let p, q be Polynomial of n, L. One can verify that p ∗ q is finite-Support.

8. The Ring of Polynomials

Let n be an ordinal number and let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure. The functor Polynom-Ring(n, L) yields a strict non empty double loop structure and is defined by the conditions (Def. 27).

(Def. 27)(i) For every set x holds x ∈ the carrier of Polynom-Ring(n, L) iff x is a

Polynomial of n, L, (ii) for all elements x, y of Polynom-Ring(n, L) and for all Polynomial p, q

of n, L such that x = p and y = q holds x + y = p + q, (iii) for all elements x, y of Polynom-Ring(n, L) and for all Polynomial p, q

of n, L such that x = p and y = q holds x · y = p ∗ q, (iv) 0_{Polynom-Ring (n,L)} = 0 (n, L), and

(v) 1_{Polynom-Ring (n,L)} = 1 (n, L).

Let n be an ordinal number and let L be an Abelian right zeroed addassociative right complementable unital distributive non trivial non empty double loop structure. One can check that Polynom-Ring(n, L) is Abelian.

multivariate polynomials … 347

Let n be an ordinal number and let L be an add-associative right zeroed right complementable unital distributive non trivial non empty double loop structure. One can verify that Polynom-Ring(n, L) is add-associative.

Let n be an ordinal number and let L be a right zeroed add-associative right complementable unital distributive non trivial non empty double loop structure. One can check that Polynom-Ring(n, L) is right zeroed.

Let n be an ordinal number and let L be a right complementable right zeroed add-associative unital distributive non trivial non empty double loop structure. Observe that Polynom-Ring(n, L) is right complementable.

Let n be an ordinal number and let L be an Abelian add-associative right zeroed right complementable commutative unital distributive non trivial non empty double loop structure. Observe that Polynom-Ring(n, L) is commutative.

Let n be an ordinal number and let L be an Abelian add-associative right zeroed right complementable unital distributive associative non trivial non empty double loop structure. Observe that Polynom-Ring(n, L) is associative.

Let n be an ordinal number and let L be a right zeroed Abelian addassociative right complementable unital distributive associative non trivial non empty double loop structure. Note that Polynom-Ring(n, L) is unital and rightdistributive.

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Continuous Lattices between T_{0} Spaces^{1}

Grzegorz Bancerek University of Bialystok

Summary. Formalization of [14, pp. 128–130], chapter II, section 4 (4.1 – 4.9).

MML Identifier: WAYBEL26.

The notation and terminology used in this paper have been introduced in the following articles: [22], [28], [27], [10], [23], [13], [21], [29], [8], [30], [9], [7], [11], [1], [19], [6], [18], [2], [15], [25], [5], [16], [26], [?], [23], [9], [3], [26], [4], [17], [20], [14], [12], [9], [9], [34], [24], [4], [5], [6], [18], [33], [15], [8], [7], [24], [36], [19], [17], [?], and [?].

Let I be a set and let J be a relational structure yielding many sorted set indexed by I. We introduce I -prod_{POS} J as a synonym of ∏ J.

Let I be a set and let J be a relational structure yielding nonempty many sorted set indexed by I. Note that I -prod_{POS} J is constituted functions.

Let I be a set and let J be a topological space yielding nonempty many sorted set indexed by I. We introduce I -prod_{TOP} J as a synonym of ∏ J.

Let X, Y be non empty topological spaces. The functor [X → Y] yielding a non empty strict relational structure is defined as follows:

(Def. 1) [X → Y] = [X → ΩY].

Let X, Y be non empty topological spaces. Observe that [X → Y] is reflexive transitive and constituted functions.

Let X be a non empty topological space and let Y be a non empty T_{0} topological space. Observe that [X → Y] is antisymmetric.

The following propositions are true:

(1) Let X, Y be non empty topological spaces and a be a set. Then a is an element of [X → Y] if and only if a is a continuous map from X into ΩY.

1This work has been supported by KBN Grant 8 T11C 018 12.

(2) Let X, Y be non empty topological spaces and a be a set. Then a is an element of [X → Y] if and only if a is a continuous map from X into Y.

(3) Let X, Y be non empty topological spaces, a, b be elements of [X → Y], and f, g be maps from X into ΩY. If a = f and b = g, then a b iff f g.

Let X, Y be non empty topological spaces, let x be a point of X, and let A be a subset of the carrier of ([X → Y]). Then π_{x}A is a subset of ΩY.

Let X, Y be non empty topological spaces, let x be a set, and let A be a non empty subset of the carrier of ([X → Y]). Observe that π_{x}A is non empty.

We now state three propositions:

(4) Ω(the Sierpiński space) is a topological augmentation of 2_{⊆}^{1} .

(5) Let X be a non empty topological space. Then there exists a map f from ⟨the topology of X, ⊆⟩ into [X → the Sierpiński space] such that f is isomorphic and for every open subset V of X holds f(V) = χ_{V,the carrier of X}.

(6) Let X be a non empty topological space. Then ⟨the topology of X, ⊆⟩ and [X → the Sierpiński space] are isomorphic.

Let X, Y, Z be non empty topological spaces and let f be a continuous map from Y into Z. The functor [X → f] yielding a map from [X → Y] into [X → Z] is defined as follows:

(Def. 2) For every continuous map g from X into Y holds ([X → f])(g) = f · g. The functor [f → X] yielding a map from [Z → X] into [Y → X] is defined as follows:

(Def. 3) For every continuous map g from Z into X holds ([f → X])(g) = g · f. Next we state a number of propositions:

(7) Let X be a non empty topological space and Y be a monotone convergence T_{0}-space. Then [X → Y] is a directed-sups-inheriting relational substructure of (ΩY)^{the carrier of X}.

(8) For every non empty topological space X and for every monotone convergence T_{0}-space Y holds [X → Y] is up-complete.

(9) For all non empty topological spaces X, Y, Z and for every continuous map f from Y into Z holds [X → f] is monotone.

(10) Let X, Y be non empty topological spaces and f be a continuous map from Y into Y. If f is idempotent, then [X → f] is idempotent.

(11) For all non empty topological spaces X, Y, Z and for every continuous map f from Y into Z holds [f → X] is monotone.

(12) Let X, Y be non empty topological spaces and f be a continuous map from Y into Y. If f is idempotent, then [f → X] is idempotent.

(13) Let X, Y, Z be non empty topological spaces, f be a continuous map from Y into Z, x be an element of X, and A be a subset of [X → Y].

continuous lattices between t_{0} spaces 351

Then π_{x}([X → f])^{∘}A = f^{∘}π_{x}A.

(14) Let X be a non empty topological space, Y, Z be monotone convergence T_{0}-spaces, and f be a continuous map from Y into Z. Then [X → f] is directed-sups-preserving.

(15) Let X, Y, Z be non empty topological spaces, f be a continuous map from Y into Z, x be an element of Y, and A be a subset of [Z → X]. Then π_{x}([f → X])^{∘}A = π_{f(x)}A.

(16) Let Y, Z be non empty topological spaces, X be a monotone convergence T_{0}-space, and f be a continuous map from Y into Z. Then [f → X] is directed-sups-preserving.

(17) Let X, Z be non empty topological spaces and Y be a non empty subspace of Z. Then [X → Y] is a full relational substructure of [X → Z].

(18) Let Z be a monotone convergence T_{0}-space, Y be a non empty subspace of Z, and f be a continuous map from Z into Y. Suppose f is a retraction. Then ΩY is a directed-sups-inheriting relational substructure of ΩZ.

(19) Let X be a non empty topological space, Z be a monotone convergence T_{0}-space, Y be a non empty subspace of Z, and f be a continuous map from Z into Y. If f is a retraction, then [X → f] is a retraction of [X → Z] into [X → Y].

(20) Let X be a non empty topological space, Z be a monotone convergence T_{0}-space, and Y be a non empty subspace of Z. If Y is a retract of Z, then [X → Y] is a retract of [X → Z].

(21) Let X, Y, Z be non empty topological spaces and f be a continuous map from Y into Z. If f is a homeomorphism, then [X → f] is isomorphic.

(22) Let X, Y, Z be non empty topological spaces. If Y and Z are homeomorphic, then [X → Y] and [X → Z] are isomorphic.

(23) Let X be a non empty topological space, Z be a monotone convergence T_{0}-space, and Y be a non empty subspace of Z. Suppose Y is a retract of Z and [X → Z] is complete and continuous. Then [X → Y] is complete and continuous.

(24) Let X be a non empty topological space and Y, Z be monotone convergence T_{0}-spaces. Suppose Y is a topological retract of Z and [X → Z] is complete and continuous. Then [X → Y] is complete and continuous.

(25) Let Y be a non trivial T_{0}-space. Suppose Y is not a T_{1} space. Then the Sierpiński space is a topological retract of Y.

(26) Let X be a non empty topological space and Y be a non trivial T_{0}-space. If [X → Y] has l.u.b.’s, then Y is not a T_{1} space.

Let us mention that the Sierpiński space is non trivial and monotone convergence.

One can verify that there exists a T_{0}-space which is injective, monotone convergence, and non trivial.

One can prove the following propositions:

(27) Let X be a non empty topological space and Y be a monotone convergence non trivial T_{0}-space. If [X → Y] is complete and continuous, then ⟨the topology of X, ⊆⟩ is continuous.

(28) Let X be a non empty topological space, x be a point of X, and Y be a monotone convergence T_{0}-space. Then there exists a directed-sups-preserving projection map F from [X → Y] into [X → Y] such that

(i) for every continuous map f from X into Y holds F (f) = X ⟼→ f(x), and

(ii) there exists a continuous map h from X into X such that h = X ⟼→ x

and F = [h → Y].

(29) Let X be a non empty topological space and Y be a monotone convergence T_{0}-space. If [X → Y] is complete and continuous, then ΩY is complete and continuous.

(30) Let X be a non empty 1-sorted structure, I be a non empty set, J be a topological space yielding nonempty many sorted set indexed by I, f be a map from X into I -prod_{TOP} J, and i be an element of I. Then (commute(f))(i) = proj(J, i) · f.

(31) For every 1-sorted structure S and for every set M holds the support of M ⟼→ S = M ⟼→ the carrier of S.

(32) Let X, Y be non empty topological spaces, M be a non empty set, and f be a continuous map from X into M -prod_{TOP}(M ⟼→ Y). Then commute(f) is a function from M into the carrier of ([X → Y]).

(33) For all non empty topological spaces X, Y holds the carrier of ([X → Y]) ⊆ (the carrier of Y)^{the carrier of X}.

(34) Let X, Y be non empty topological spaces, M be a non empty set, and f be a function from M into the carrier of ([X → Y]). Then commute(f) is a continuous map from X into M -prod_{TOP}(M ⟼→ Y).

(35) Let X be a non empty topological space and M be a non empty set. Then there exists a map F from [X → M -prod_{TOP}(M ⟼→ the Sierpiński space)] into M -prod_{POS}(M ⟼→ ([X → the Sierpiński space])) such that F is isomorphic and for every continuous map f from X into M -prod_{TOP}(M ⟼→ the Sierpiński space) holds F (f) = commute(f).

(36) Let X be a non empty topological space and M be a non empty set. Then [X → M -prod_{TOP}(M ⟼→ the Sierpiński space)] and M -prod_{POS}(M ⟼→ ([X → the Sierpiński space])) are isomorphic.

(37) Let X be a non empty topological space. Suppose ⟨the topology of X, ⊆⟩ is continuous. Let Y be an injective T_{0}-space. Then [X → Y] is complete

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

and Scott.

The following proposition is true

(38) Let X be a non empty topological space and L be a non trivial complete Scott top-lattice. Then [X → L] is complete and continuous if and only if ⟨the topology of X, ⊆⟩ is continuous and L is continuous.

Let f be a function. Note that Uniondisjointf is relation-like. Let f be a function. The functor G_{f} yields a binary relation and is defined

as follows: (Def. 4) G_{f} = (Uniondisjointf)^{⌣}.

In the sequel x, y denote sets and f denotes a function. The following propositions are true:

(39) ⟨⟨x, y⟩⟩∈ G_{f} iff x ∈ domf and y ∈ f(x).

(40) For every finite set X holds π_{1}(X) is finite and π_{2}(X) is finite.

(41) Let X, Y be non empty topological spaces, S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩, and f be a map from X into S. If G_{f} is an open subset of [: X, Y :], then f is continuous.

Let W be a binary relation and let X be a set. The functor Θ_{X}(W) yields a function and is defined as follows:

(Def. 5) domΘ_{X}(W) = X and for every x such that x ∈ X holds (Θ_{X}(W))(x) =

W^{∘}{x}. We now state the proposition

(42) For every binary relation W and for every set X such that domW ⊆ X

holds G_{Θ_{X}(W)} = W. Let X, Y be topological spaces. Observe that every subset of the carrier of

[: X, Y :] is relation-like and every element of the topology of [: X, Y :] is relationlike.

Next we state four propositions:

(43) Let X, Y be non empty topological spaces, W be an open subset of [: X, Y :], and x be a point of X. Then W^{∘}{x} is an open subset of Y.

(44) Let X, Y be non empty topological spaces, S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩, and W be an open subset of [: X, Y :]. Then Θ_{the carrier of X}(W) is a continuous map from X into S.

(45) Let X, Y be non empty topological spaces, S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩, and W_{1}, W_{2} be open subsets of [: X, Y :]. Suppose W_{1} ⊆ W_{2}. Let f_{1}, f_{2} be elements of [X → S]. If f_{1} = Θ_{the carrier of X}(W_{1}) and f_{2} = Θ_{the carrier of X}(W_{2}), then f_{1} f_{2}.

354 grzegorz bancerek

(46) Let X, Y be non empty topological spaces and S be a Scott topological augmentation of ⟨the topology of Y, ⊆⟩. Then there exists a map F from ⟨the topology of [: X, Y :], ⊆⟩ into [X → S] such that F is monotone and for every open subset W of [: X, Y :] holds F (W) = Θ_{the carrier of X}(W).

References

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

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

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

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

[5] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.

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

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

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

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

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

[11] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

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

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

[14] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.

[15] Andrzej Kondracki. Mostowski’s fundamental operations - Part I. Formalized Mathematics, 2(3):371–375, 1991.

[16] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

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

[18] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

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

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

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

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

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

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

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

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

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

continuous lattices between t_{0} spaces 355 Received September 24, 1999 356 grzegorz bancerek

Predicate Calculus for Boolean Valued Functions. Part VI

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC14.

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

1. Preliminaries

In this paper Y denotes a non empty set. We now state several propositions:

(1) For every element z of Y and for all partitions P_{1}, P_{2} of Y holds EqClass(z, P_{1} ∧ P_{2}) = EqClass(z, P_{1}) ∩ EqClass(z, P_{2}).

(2) Let G be a subset of PARTITIONS(Y) and A, B be partitions of Y. If G is a coordinate and G = {A, B} and A ≠ B, then ⋀ G = A ∧ B.

(3) Let G be a subset of PARTITIONS(Y) and B, C, D be partitions of Y. Suppose G is a coordinate and G = {B, C, D} and B ≠ C and C ≠ D and D ≠ B. Then ⋀ G = B ∧ C ∧ D.

(4) Let G be a subset of PARTITIONS(Y) and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then CompF(A, G) = B ∧ C and CompF(B, G) = C ∧ A and CompF(C, G) = A ∧ B.

(5) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(A, G) = B ∧ C ∧ D.

(6) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(B, G) = A ∧ C ∧ D.

(7) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(C, G) = A ∧ B ∧ D.

(8) Let G be a subset of PARTITIONS(Y) and A, B, C, D be partitions of Y. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then CompF(D, G) = A ∧ C ∧ B.

2. Predicate Calculus

We follow the rules: a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C are partitions of Y.

One can prove the following four propositions:

(9) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(10) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{∀_{a,C} G,A}G,B}G = ∀_{∀_{∀_{a,C} G,B}G,A}G.

(11) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{∃_{a,C} G,A}G,B}G = ∀_{∀_{∃_{a,C} G,B}G,A}G.

(12) Let G be a subset of PARTITIONS(Y), B, C, D be partitions of Y, h be a function, and b, c, d be sets. Suppose B ≠ C and C ≠ D and D ≠ B and h = (B⟼→ . b)+·(C⟼→ . c)+·(D⟼→ . d). Then domh = {B, C, D} and h(B) = b and h(C) = c and h(D) = d and rngh = {h(B), h(C), h(D)}.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[4] Czesław Byliński. Products and coproducts in categories. Formalized Mathematics, 2(5):701–709, 1991.

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

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

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

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

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

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

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

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

Received October 19, 1999 360 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part VII

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC15.

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

In this paper Y is a non empty set. We now state a number of propositions:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C be partitions of Y, and z, u be elements of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A and EqClass(z, C) = EqClass(u, C). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∀_{a,C} G,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∃_{a,C} G,B}G,A}G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{∀_{a,C} G,A}G,B}G = ∃_{∃_{∀_{a,C} G,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{∃_{a,C} G,A}G,B}G = ∃_{∃_{∃_{a,C} G,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∀_{∀_{a,C} G,B}G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∀_{∃_{a,C} G,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∀_{a,C} G,B}G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∃_{a,C} G,B}G,A}G.

(17) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∀_{a,C} G,B}G,A}G.

(18) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∀_{∃_{∃_{a,C} G,B}G,A}G.

(19) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∀_{a,C} G,B}G,A}G.

(20) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∃_{a,C} G,B}G,A}G.

(21) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∀_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∀_{a,C} G,B}G,A}G.

(22) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{∃_{a,C} G,A}G,B}G ⋐ ∃_{∃_{∃_{a,C} G,B}G,A}G.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

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

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

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

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

Received October 19, 1999 364 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part VIII

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC16.

The terminology and notation used in this paper are introduced in the following articles: [3], [5], [6], [4], [1], [2], [5], [1], and [2].

In this paper Y is a non empty set. The following propositions are true:

(1) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B of Y holds ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(7) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B, C of Y holds ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(8) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B, C of Y holds ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(16) For every element a of BVF(Y) and for every subset G of PARTITIONS(Y) and for all partitions A, B, C of Y holds ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

References

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

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

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

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

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

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

368 shunichi kobayashi

Predicate Calculus for Boolean Valued Functions. Part IX

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC17.

The articles [3], [5], [6], [4], [1], [2], [5], [1], and [2] provide the terminology and notation for this paper.

In this paper Y is a non empty set. We now state a number of propositions:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(15) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(16) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

References

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

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

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

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

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

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

372 shunichi kobayashi

Asymptotic Notation. Part I^{1}

Richard Krueger University of Alberta Edmonton

Piotr Rudnicki University of Alberta Edmonton

Paul Shelley University of Alberta Edmonton

Summary. The widely used textbook by Brassard and Bratley [?] includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79–97). We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety. A more detailed report on this experiment will be available separately. This article presents the development of notions and a follow-up article [?] includes examples and solutions to problems. The preliminaries introduce a number of properties of real sequences, some operations on real sequences, and a characterization of convergence. The remaining sections in this article correspond to sections of Chapter 3 of [?]. Section 2 defines the O notation and proves the threshold, maximum, and limit rules. Section 3 introduces the Ω and Θ notations and their analogous rules. Conditional asymptotic notation is defined in Section 4 where smooth functions are also discussed. Section 5 defines some operations on asymptotic notation (we have decided not to introduce the asymptotic notation for functions of several variables as it is a straightforward generalization of notions for unary functions).

MML Identifier: ASYMPT 0.

The notation and terminology used in this paper are introduced in the following papers: [16], [15], [5], [2], [3], [12], [14], [1], [7], [6], [9], [11], [23], [4], [13], [8], [10], [?], and [17].

1This work has been supported by NSERC Grant OGP9207.

1. Preliminaries

In this paper c, d denote real numbers and n, N denote natural numbers. In this article we present several logical schemes. The scheme FinSegRng1

deals with natural numbers A, B, a non empty set C, and a unary functor F yielding an element of C, and states that:

{F(i); i ranges over natural numbers: A i ∧ i B} is a finite non empty subset of C

provided the following condition is satisfied:

• A B.

The scheme FinImInit1 deals with a natural number A, a non empty set B, and a unary functor F yielding an element of B, and states that:

{F(n); n ranges over natural numbers: n A} is a finite non empty subset of B

for all values of the parameters.

The scheme FinImInit2 deals with a natural number A, a non empty set B, and a unary functor F yielding an element of B, and states that:

{F(n); n ranges over natural numbers: n < A} is a finite non empty subset of B

provided the following condition is met:

• A > 0.

Let c be a real number. We say that c is positive if and only if: (Def. 1) c > 0.

We say that c is negative if and only if: (Def. 2) c < 0.

We say that c is logbase if and only if: (Def. 3) c > 0 and c ≠ 1.

One can check the following observations:

∗ there exists a real number which is positive,

∗ there exists a real number which is negative,

∗ there exists a real number which is logbase,

∗ there exists a real number which is non negative,

∗ there exists a real number which is non positive, and

∗ there exists a real number which is non logbase.

Let f be a sequence of real numbers. We say that f is eventually-nonnegative if and only if:

(Def. 4) There exists N such that for every n such that n N holds f(n) 0. We say that f is positive if and only if:

(Def. 5) For every n holds f(n) > 0.

We say that f is eventually-positive if and only if: (Def. 6) There exists N such that for every n such that n N holds f(n) > 0.

We say that f is eventually-nonzero if and only if: (Def. 7) There exists N such that for every n such that n N holds f(n) ≠ 0.

We say that f is eventually-nondecreasing if and only if: (Def. 8) There exists N such that for every n such that n N holds f(n)

f(n + 1). Let us observe that there exists a sequence of real numbers which is

eventually-nonnegative, eventually-nonzero, positive, eventually-positive, and eventually-nondecreasing.

One can check the following observations:

∗ every sequence of real numbers which is positive is also eventuallypositive,

∗ every sequence of real numbers which is eventually-positive is also eventually-nonnegative and eventually-nonzero, and

∗ every sequence of real numbers which is eventually-nonnegative and eventually-nonzero is also eventually-positive.

Let f, g be eventually-nonnegative sequences of real numbers. Observe that f + g is eventually-nonnegative.

Let f be a sequence of real numbers and let c be a real number. The functor c + f yields a sequence of real numbers and is defined by:

(Def. 9) For every n holds (c + f)(n) = c + f(n).

We introduce f + c as a synonym of c + f.

Let f be an eventually-nonnegative sequence of real numbers and let c be a positive real number. One can verify that c f is eventually-nonnegative.

Let f be an eventually-nonnegative sequence of real numbers and let c be a non negative real number. Note that c + f is eventually-nonnegative.

Let f be an eventually-nonnegative sequence of real numbers and let c be a positive real number. One can check that c + f is eventually-positive.

Let f, g be sequences of real numbers. The functor max(f, g) yielding a sequence of real numbers is defined by:

(Def. 10) For every n holds (max(f, g))(n) = max(f(n), g(n)).

Let us observe that the functor max(f, g) is commutative.

Let f be a sequence of real numbers and let g be an eventually-nonnegative sequence of real numbers. Note that max(f, g) is eventually-nonnegative.

Let f be a sequence of real numbers and let g be an eventually-positive sequence of real numbers. Observe that max(f, g) is eventually-positive.

Let f, g be sequences of real numbers. We say that g majorizes f if and only if:

(Def. 11) There exists N such that for every n such that n N holds f(n) g(n). We now state several propositions:

(1) Let f be a sequence of real numbers and N be a natural number. Suppose that for every n such that n N holds f(n) f(n + 1). Let n, m be natural numbers. If N n and n m, then f(n) f(m).

(2) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) ≠ 0, then g/f is convergent and lim(g/f) = (lim(f/g))^{−1} .

(3) For every eventually-nonnegative sequence f of real numbers such that f is convergent holds 0 limf.

(4) Let f, g be sequences of real numbers. If f is convergent and g is convergent and g majorizes f, then limf limg.

(5) Let f be a sequence of real numbers and g be an eventually-nonzero sequence of real numbers. If f/g is divergent to +∞, then g/f is convergent and lim(g/f) = 0.

2. A Notation for ”the order of”

Let f be an eventually-nonnegative sequence of real numbers. The functor Big Ohf yields a non empty set of functions from ℕ to ℝ and is defined by:

(Def. 12) Big Ohf = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,N} (c > 0 ∧ ⋀_{n} (n

N ⇒ t(n) c · f(n) ∧ t(n) 0))}. We now state a number of propositions:

(6) Let x be a set and f be an eventually-nonnegative sequence of real numbers. Suppose x ∈ Big Ohf. Then x is an eventually-nonnegative sequence of real numbers.

(7) Let f be a positive sequence of real numbers and t be an eventuallynonnegative sequence of real numbers. Then t ∈ Big Ohf if and only if there exists c such that c > 0 and for every n holds t(n) c · f(n).

(8) Let f be an eventually-positive sequence of real numbers, t be an eventually-nonnegative sequence of real numbers, and N be a natural number. Suppose t ∈ Big Ohf and for every n such that n N holds f(n) > 0. Then there exists c such that c > 0 and for every n such that n N holds t(n) c · f(n).

(9) For all eventually-nonnegative sequences f, g of real numbers holds Big Ohf + g = Big Ohmax(f, g).

(10) For every eventually-nonnegative sequence f of real numbers holds f ∈ Big Ohf.

(11) For all eventually-nonnegative sequences f, g of real numbers such that f ∈ Big Ohg holds Big Ohf ⊆ Big Ohg.

(12) For all eventually-nonnegative sequences f, g, h of real numbers such that f ∈ Big Ohg and g ∈ Big Ohh holds f ∈ Big Ohh.

(13) Let f be an eventually-nonnegative sequence of real numbers and c be a positive real number. Then Big Ohf = Big Ohc f.

(14) Let c be a non negative real number and x, f be eventually-nonnegative sequences of real numbers. If x ∈ Big Ohf, then x ∈ Big Ohc + f.

(15) For all eventually-positive sequences f, g of real numbers such that f/g is convergent and lim(f/g) > 0 holds Big Ohf = Big Ohg.

(16) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) = 0, then f ∈ Big Ohg and g ∉ Big Ohf.

(17) Let f, g be eventually-positive sequences of real numbers. If f/g is divergent to +∞, then f ∉ Big Ohg and g ∈ Big Ohf.

3. Other Asymptotic Notation

Let f be an eventually-nonnegative sequence of real numbers. The functor Big Omegaf yields a non empty set of functions from ℕ to ℝ and is defined by:

(Def. 13) Big Omegaf = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{d,N} (d > 0 ∧ ⋀_{n} (n

N ⇒ t(n) d · f(n) ∧ t(n) 0))}. One can prove the following propositions:

(18) Let x be a set and f be an eventually-nonnegative sequence of real numbers. Suppose x ∈ Big Omegaf. Then x is an eventually-nonnegative sequence of real numbers.

(19) For all eventually-nonnegative sequences f, g of real numbers holds f ∈ Big Omegag iff g ∈ Big Ohf.

(20) For every eventually-nonnegative sequence f of real numbers holds f ∈ Big Omegaf.

(21) For all eventually-nonnegative sequences f, g, h of real numbers such that f ∈ Big Omegag and g ∈ Big Omegah holds f ∈ Big Omegah.

(22) For all eventually-positive sequences f, g of real numbers such that f/g is convergent and lim(f/g) > 0 holds Big Omegaf = Big Omegag.

(23) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) = 0, then g ∈ Big Omegaf and f ∉ Big Omegag.

(24) Let f, g be eventually-positive sequences of real numbers. If f/g is divergent to +∞, then g ∉ Big Omegaf and f ∈ Big Omegag.

(25) Let f, t be positive sequences of real numbers. Then t ∈ Big Omegaf if and only if there exists d such that d > 0 and for every n holds d · f(n) t(n).

(26) For all eventually-nonnegative sequences f, g of real numbers holds

Big Omegaf + g = Big Omegamax(f, g). Let f be an eventually-nonnegative sequence of real numbers. The functor

Big Thetaf yields a non empty set of functions from ℕ to ℝ and is defined as follows:

(Def. 14) Big Thetaf = Big Ohf ∩ Big Omegaf.

One can prove the following propositions:

(27) Let f be an eventually-nonnegative sequence of real numbers. Then Big Thetaf = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,d,N} (c > 0 ∧ d > 0 ∧ ⋀_{n} (n N ⇒ d · f(n) t(n) ∧ t(n) c · f(n)))}.

(28) For every eventually-nonnegative sequence f of real numbers holds f ∈ Big Thetaf.

(29) For all eventually-nonnegative sequences f, g of real numbers such that f ∈ Big Thetag holds g ∈ Big Thetaf.

(30) For all eventually-nonnegative sequences f, g, h of real numbers such that f ∈ Big Thetag and g ∈ Big Thetah holds f ∈ Big Thetah.

(31) Let f, t be positive sequences of real numbers. Then t ∈ Big Thetaf if and only if there exist c, d such that c > 0 and d > 0 and for every n holds d · f(n) t(n) and t(n) c · f(n).

(32) For all eventually-nonnegative sequences f, g of real numbers holds Big Thetaf + g = Big Thetamax(f, g).

(33) For all eventually-positive sequences f, g of real numbers such that f/g is convergent and lim(f/g) > 0 holds f ∈ Big Thetag.

(34) Let f, g be eventually-positive sequences of real numbers. If f/g is convergent and lim(f/g) = 0, then f ∈ Big Ohg and f ∉ Big Thetag.

(35) Let f, g be eventually-positive sequences of real numbers. If f/g is divergent to +∞, then f ∈ Big Omegag and f ∉ Big Thetag.

4. Conditional Asymptotic Notation

Let f be an eventually-nonnegative sequence of real numbers and let X be a set. The functor Big Oh(f, X) yielding a non empty set of functions from ℕ to ℝ is defined as follows:

(Def. 15) Big Oh(f, X) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,N} (c > 0 ∧ ⋀_{n} (n N ∧ n ∈ X ⇒ t(n) c · f(n) ∧ t(n) 0))}.

Let f be an eventually-nonnegative sequence of real numbers and let X be a set. The functor Big Omega(f, X) yielding a non empty set of functions from ℕ to ℝ is defined as follows:

(Def. 16) Big Omega(f, X) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{d,N} (d > 0 ∧

⋀_{n} (n N ∧ n ∈ X ⇒ t(n) d · f(n) ∧ t(n) 0))}. Let f be an eventually-nonnegative sequence of real numbers and let X be

a set. The functor Big Theta(f, X) yields a non empty set of functions from ℕ to ℝ and is defined by the condition (Def. 17).

(Def. 17) Big Theta(f, X) = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{c,d,N} (c > 0 ∧ d >

0 ∧ ⋀_{n} (n N ∧ n ∈ X ⇒ d · f(n) t(n) ∧ t(n) c · f(n)))}. One can prove the following proposition

(36) For every eventually-nonnegative sequence f of real numbers and for

every set X holds Big Theta(f, X) = Big Oh(f, X) ∩ Big Omega(f, X). Let f be a sequence of real numbers and let b be a natural number. The

functor f taken everyb yields a sequence of real numbers and is defined by: (Def. 18) For every n holds (f taken everyb)(n) = f(b · n).

Let f be an eventually-nonnegative sequence of real numbers and let b be a natural number. We say that f is smooth wrt b if and only if:

(Def. 19) f is eventually-nondecreasing and f taken everyb ∈ Big Ohf.

Let f be an eventually-nonnegative sequence of real numbers. We say that f is smooth if and only if:

(Def. 20) For every natural number b such that b 2 holds f is smooth wrt b.

The following propositions are true:

(37) Let f be an eventually-nonnegative sequence of real numbers. Given a natural number b such that b 2 and f is smooth wrt b. Then f is smooth.

(38) Let f be an eventually-nonnegative sequence of real numbers, t be an eventually-nonnegative eventually-nondecreasing sequence of real numbers, and b be a natural number. Suppose f is smooth and b 2 and t ∈ Big Oh(f, {b^{n} : n ranges over natural numbers}). Then t ∈ Big Ohf.

(39) Let f be an eventually-nonnegative sequence of real numbers, t be an eventually-nonnegative eventually-nondecreasing sequence of real numbers, and b be a natural number. Suppose f is smooth and b 2 and t ∈ Big Omega(f, {b^{n} : n ranges over natural numbers}). Then t ∈ Big Omegaf.

(40) Let f be an eventually-nonnegative sequence of real numbers, t be an eventually-nonnegative eventually-nondecreasing sequence of real numbers, and b be a natural number. Suppose f is smooth and b 2 and t ∈ Big Theta(f, {b^{n} : n ranges over natural numbers}). Then t ∈ Big Thetaf.

380 richard krueger et al.

5. Operations on Asymptotic Notation

Let X be a non empty set and let F, G be non empty sets of functions from X to ℝ. The functor F + G yields a non empty set of functions from X to ℝ and is defined by the condition (Def. 21).

(Def. 21) F +G = {t; t ranges over elements of ℝ^{X}: ⋁_{f,g: element of ℝ^{X}} (f ∈ F ∧ g ∈

G ∧ ⋀_{n: element of X} t(n) = f(n) + g(n))}. Let X be a non empty set and let F, G be non empty sets of functions from

X to ℝ. The functor max(F, G) yielding a non empty set of functions from X to ℝ is defined by the condition (Def. 22).

(Def. 22) max(F, G) = {t; t ranges over elements of ℝ^{X}: ⋁_{f,g: element of ℝ^{X}} (f ∈

F ∧ g ∈ G ∧ ⋀_{n: element of X} t(n) = max(f(n), g(n)))}. The following propositions are true:

(41) For all eventually-nonnegative sequences f, g of real numbers holds Big Ohf + Big Ohg = Big Ohf + g.

(42) For all eventually-nonnegative sequences f, g of real numbers holds

max(Big Ohf, Big Ohg) = Big Ohmax(f, g). Let F, G be non empty sets of functions from ℕ to ℝ. The functor F^{G}

yields a non empty set of functions from ℕ to ℝ and is defined by the condition (Def. 23).

(Def. 23) F^{G} = {t; t ranges over elements of ℝ^{ℕ}: ⋁_{f,g: element of ℝ^{ℕ}} ⋁_{N : element of ℕ} (f ∈ F ∧ g ∈ G ∧ ⋀_{n: element of ℕ} (n N ⇒ t(n) = f(n)^{g(n)}))}.

References

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

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

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

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

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

[8] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

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

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

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

asymptotic notation. part i 381

[12] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

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

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

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

382 richard krueger et al.

Asymptotic Notation. Part II^{1}

Richard Krueger University of Alberta Edmonton

Piotr Rudnicki University of Alberta Edmonton

Paul Shelley University of Alberta Edmonton

Summary. The widely used textbook by Brassard and Bratley [?] includes a chapter devoted to asymptotic notation (Chapter 3, pp. 79–97). We have attempted to test how suitable the current version of Mizar is for recording this type of material in its entirety. This article is a follow-up to [?] in which we introduced the basic notions and general theory. This article presents a Mizar formalization of examples and solutions to problems from Chapter 3 of [?]. (Some of the examples and solved problems are also in [?].) Not all problems have been solved as some required solutions not amenable for formalization.

MML Identifier: ASYMPT 1.

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

1. Examples from the Text

We adopt the following rules: c, e are real numbers, k, n, m, N, n_{1}, M are natural numbers, and x is a set.

We now state two propositions:

1This work has been supported by NSERC Grant OGP9207.

(1) Let t, t_{1} be sequences of real numbers. Suppose that

(i) t(0) = 0, (ii) for every n such that n > 0 holds t(n) = (12 · n^{3} · log_{2} n − 5 · n^{2}) +

(log_{2} n)^{2} + 36, (iii) t_{1}(0) = 0, and (iv) for every n such that n > 0 holds t_{1}(n) = n^{3} · log_{2} n.

Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = t and s_{1} = t_{1} and s ∈ Big Ohs_{1}.

(2) Let a, b be logbase real numbers and f, g be sequences of real numbers. Suppose a > 1 and b > 1 and f(0) = 0 and for every n such that n > 0 holds f(n) = log_{a} n and g(0) = 0 and for every n such that n > 0 holds g(n) = log_{b} n. Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = f and s_{1} = g and Big Ohs = Big Ohs_{1}.

Let a, b, c be real numbers. The functor seq a^{(a,b,c)} yielding a sequence of real numbers is defined by:

(Def. 1) (seq a^{(a,b,c)})(n) = a^{b·n+c}.

Let a be a positive real number and let b, c be real numbers. Note that seq a^{(a,b,c)} is eventually-positive.

The following proposition is true

(3) For all positive real numbers a, b such that a < b holds seq a^{(b,1,0)} ∉ Big Ohseq a^{(a,1,0)}.

The sequence seq logn of real numbers is defined by:

(Def. 2) (seq logn)(0) = 0 and for every n such that n > 0 holds (seq logn)(n) =

log_{2} n. Let a be a real number. The functor seq n^{a} yielding a sequence of real

numbers is defined as follows:

(Def. 3) (seq n^{a})(0) = 0 and for every n such that n > 0 holds (seq n^{a})(n) = n^{a}. Let us note that seq logn is eventually-positive. Let a be a real number. Observe that seq n^{a} is eventually-positive. Next we state several propositions:

(4) Let f, g be eventually-nonnegative sequences of real numbers. Then Big Ohf ⊆ Big Ohg and Big Ohf ≠ Big Ohg if and only if f ∈ Big Ohg and f ∉ Big Omegag.

(5) Big Ohseq logn ⊆ Big Ohseq n^{\frac{1}{2}} and Big Ohseq logn ≠ Big Ohseq n^{\frac{1}{2}} .

(6) seq n^{\frac{1}{2}} ∈ Big Omegaseq logn and seq logn ∉ Big Omegaseq n^{\frac{1}{2}} .

(7) For every sequence f of real numbers and for every natural number k such that for every n holds f(n) = ∑_{κ=0}^{n} seq n^{k}(κ) holds f ∈ Big Thetaseq n^{k} + 1.

(8) Let f be a sequence of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{log_{2} n}. Then there exists an eventuallypositive sequence s of real numbers such that s = f and s is not smooth.

Let b be a real number. The functor seq constb yields a sequence of real numbers and is defined as follows:

(Def. 4) seq constb = ℕ ⟼→ b. Let us note that seq const1 is eventually-nonnegative. The following proposition is true

(9) Let f be an eventually-nonnegative sequence of real numbers. Then there exists a non empty set F of functions from ℕ to ℝ such that F = {seq n^{1}} and f ∈ F^{Big Oh seq const 1} iff there exist N, c, k such that c > 0 and for every n such that n N holds 1 f(n) and f(n) c · (seq n^{k})(n).

2. Problem 3.1

We now state the proposition

(10) For every sequence f of real numbers such that for every n holds f(n) = (3 · 10^{6} − 18 · 10^{3} · n) + 27 · n^{2} holds f ∈ Big Ohseq n^{2}.

3. Problem 3.5

The following propositions are true:

(11) seq n^{2} ∈ Big Ohseq n^{3}.

(12) seq n^{2} ∉ Big Omegaseq n^{3}.

(13) There exists an eventually-positive sequence s of real numbers such that s = seq a^{(2,1,1)} and seq a^{(2,1,0)} ∈ Big Thetas.

Let a be a natural number. The functor seq n!a yielding a sequence of real numbers is defined as follows:

(Def. 5) (seq n!a)(n) = (n + a)!. Let a be a natural number. Note that seq n!a is eventually-positive. The following proposition is true

(14) seq n!0 ∉ Big Thetaseq n!1.

4. Problem 3.6

Next we state the proposition

(15) For every sequence f of real numbers such that f ∈ Big Ohseq n^{1} holds f f ∈ Big Ohseq n^{2}.

5. Problem 3.7

One can prove the following proposition

(16) There exists an eventually-positive sequence s of real numbers such that s = seq a^{(2,1,0)} and 2 seq n^{1} ∈ Big Ohseq n^{1} and seq a^{(2,2,0)} ∉ Big Ohs.

6. Problem 3.8

Next we state the proposition

(17) If log_{2} 3 < \frac{159}{100}, then seq n^{log_{2} 3} ∈ Big Ohseq n^{\frac{159}{100}} and seq n^{log_{2} 3} ∉ Big Omegaseq n^{\frac{159}{100}} and seq n^{log_{2} 3} ∉ Big Thetaseq n^{\frac{159}{100}} .

7. Problem 3.11

One can prove the following proposition

(18) Let f, g be sequences of real numbers. Suppose for every n holds f(n) = n mod 2 and for every n holds g(n) = (n + 1) mod 2. Then there exist eventually-nonnegative sequences s, s_{1} of real numbers such that s = f and s_{1} = g and s ∉ Big Ohs_{1} and s_{1} ∉ Big Ohs.

8. Problem 3.19

Next we state two propositions:

(19) For all eventually-nonnegative sequences f, g of real numbers holds Big Ohf = Big Ohg iff f ∈ Big Thetag.

(20) For all eventually-nonnegative sequences f, g of real numbers holds f ∈ Big Thetag iff Big Thetaf = Big Thetag.

9. Problem 3.21

Next we state several propositions:

(21) Let e be a real number and f be a sequence of real numbers. Suppose 0 < e and f(0) = 0 and for every n such that n > 0 holds f(n) = n·log_{2} n. Then there exists an eventually-positive sequence s of real numbers such that s = f and Big Ohs ⊆ Big Ohseq n^{1}+e and Big Ohs ≠ Big Ohseq n^{1}+e.

(22) Let e be a real number and g be a sequence of real numbers. Suppose 0 < e and e < 1 and g(0) = 0 and g(1) = 0 and for every n such that n > 1 holds g(n) = \frac{n^{2}}{log_{2} n}. Then there exists an eventually-positive sequence s of real numbers such that s = g and Big Ohseq n^{1} + e ⊆ Big Ohs and Big Ohseq n^{1} + e ≠ Big Ohs.

(23) Let f be a sequence of real numbers. Suppose f(0) = 0 and f(1) = 0 and for every n such that n > 1 holds f(n) = \frac{n^{2}}{log_{2} n}. Then there exists an eventually-positive sequence s of real numbers such that s = f and Big Ohs ⊆ Big Ohseq n^{8} and Big Ohs ≠ Big Ohseq n^{8}.

(24) Let g be a sequence of real numbers. Suppose that for every n holds g(n) = ((n^{2} − n) + 1)^{4}. Then there exists an eventually-positive sequence s of real numbers such that s = g and Big Ohseq n^{8} = Big Ohs.

(25) Let e be a real number. Suppose 0 < e and e < 1. Then there exists an eventually-positive sequence s of real numbers such that s = seq a^{(1+e,1,0)} and Big Ohseq n^{8} ⊆ Big Ohs and Big Ohseq n^{8} ≠ Big Ohs.

10. Problem 3.22

Next we state several propositions:

(26) Let f, g be sequences of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{log_{2} n} and g(0) = 0 and for every n such that n > 0 holds g(n) = n^{\sqrt{n}}. Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = f and s_{1} = g and Big Ohs ⊆ Big Ohs_{1} and Big Ohs ≠ Big Ohs_{1}.

(27) Let f be a sequence of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{\sqrt{n}}. Then there exist eventually-positive sequences s, s_{1} of real numbers such that s = f and s_{1} = seq a^{(2,1,0)} and Big Ohs ⊆ Big Ohs_{1} and Big Ohs ≠ Big Ohs_{1}.

(28) There exist eventually-positive sequences s, s_{1} of real numbers such that s = seq a^{(2,1,0)} and s_{1} = seq a^{(2,1,1)} and Big Ohs = Big Ohs_{1}.

(29) There exist eventually-positive sequences s, s_{1} of real numbers such that s = seq a^{(2,1,0)} and s_{1} = seq a^{(2,2,0)} and Big Ohs ⊆ Big Ohs_{1} and Big Ohs ≠ Big Ohs_{1}.

(30) There exists an eventually-positive sequence s of real numbers such that s = seq a^{(2,2,0)} and Big Ohs ⊆ Big Ohseq n!0 and Big Ohs ≠ Big Ohseq n!0.

(31) Big Ohseq n!0 ⊆ Big Ohseq n!1 and Big Ohseq n!0 ≠ Big Ohseq n!1.

(32) Let g be a sequence of real numbers. Suppose g(0) = 0 and for every n such that n > 0 holds g(n) = n^{n}. Then there exists an eventually-positive sequence s of real numbers such that s = g and Big Ohseq n!1 ⊆ Big Ohs and Big Ohseq n!1 ≠ Big Ohs.

11. Problem 3.23

Next we state the proposition

(33) Let given n. Suppose n 1. Let f be a sequence of real numbers and k be a natural number. If for every n holds f(n) = ∑_{κ=0}^{n} seq n^{k}(κ), then f(n) \frac{n^{k+1}}{k+1} .

12. Problem 3.24

Next we state the proposition

(34) Let f, g be sequences of real numbers. Suppose g(0) = 0 and for every n such that n > 0 holds g(n) = n · log_{2} n and for every n holds f(n) = log_{2}(n!). Then there exists an eventually-nonnegative sequence s of real numbers such that s = g and f ∈ Big Thetas.

13. Problem 3.26

Next we state the proposition

(35) Let f be an eventually-nondecreasing eventually-nonnegative sequence of real numbers and t be a sequence of real numbers. Suppose that for every n holds if n mod 2 = 0, then t(n) = 1 and if n mod 2 = 1, then t(n) = n. Then t ∉ Big Thetaf.

14. Problem 3.28

Let f be a function from ℕ into ℝ^{∗} and let n be a natural number. Then f(n) is a finite sequence of elements of ℝ.

Let n be a natural number and let a, b be positive real numbers. The functor Prob28(n, a, b) yielding a real number is defined as follows:

(Def. 6)(i) Prob28(n, a, b) = 0 if n = 0,

(ii) there exists a natural number l and there exists a function p_{28} from ℕ into ℝ^{∗} such that l+1 = n and Prob28(n, a, b) = π_{n}p_{28}(l) and p_{28}(0) = ⟨a⟩ and for every natural number n there exists a natural number n_{1} such that n_{1} = ⌈\frac{n+1+1}{2}⌉ and p_{28}(n + 1) = p_{28}(n)^{⌢} ⟨4 · π_{n_{1}}p_{28}(n) + b · (n + 1 + 1)⟩, otherwise.

Let a, b be positive real numbers. The functor seq prob28(a, b) yields a sequence of real numbers and is defined as follows:

(Def. 7) (seq prob28(a, b))(n) = Prob28(n, a, b).

The following proposition is true

(36) For all positive real numbers a, b holds seq prob28(a, b) is eventuallynondecreasing.

15. Problem 3.30

The non empty subset POWEROF2SET of ℕ is defined by: (Def. 8) POWEROF2SET = {2^{n} : n ranges over natural numbers}.

The following propositions are true:

(37) Let f be a sequence of real numbers. Suppose that for every n holds if n ∈ POWEROF2SET, then f(n) = n and if n ∉ POWEROF2SET, then f(n) = 2^{n}. Then f ∈ Big Theta(seq n^{1}, POWEROF2SET) and f ∉ Big Thetaseq n^{1} and seq n^{1} is smooth and f is not eventuallynondecreasing.

(38) Let f, g be sequences of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = n^{2^{^{⌊}^{log _{2} n}^{⌋}}} and g(0) = 0 and for every n such that n > 0 holds g(n) = n^{n}. Then there exists an eventually-positive sequence s of real numbers such that

(i) s = g, (ii) f ∈ Big Theta(s, POWEROF2SET), (iii) f ∉ Big Thetas, (iv) f is eventually-nondecreasing,

(v) s is eventually-nondecreasing, and (vi) s not is smooth wrt 2.

(39) Let g be a sequence of real numbers. Suppose that for every n holds if n ∈ POWEROF2SET, then g(n) = n and if n ∉ POWEROF2SET, then g(n) = n^{2}. Then there exists an eventually-positive sequence s of real numbers such that s = g and seq n^{1} ∈ Big Theta(s, POWEROF2SET) and seq n^{1} ∉ Big Thetas and s taken every2 ∈ Big Ohs and seq n^{1} is eventually-nondecreasing and s is not eventually-nondecreasing.

16. Problem 3.31

Let x be a natural number. The functor Step1x yielding a natural number is defined as follows:

(Def. 9)(i) There exists n such that n! x and x < (n + 1)! and Step1x = n! if

x ≠ 0, (ii) Step1x = 0, otherwise. Next we state the proposition

(40) Let f be a sequence of real numbers. Suppose that for every n holds f(n) = Step1n. Then there exists an eventually-positive sequence s of real numbers such that s = f and f is eventually-nondecreasing and for every n holds f(n) (seq n^{1})(n) and s is not smooth.

17. Problem 3.34

Let us note that seq n^{1} − seq const1 is eventually-positive. Next we state the proposition

(41) Big Thetaseq n^{1} − seq const1 + Big Thetaseq n^{1} = Big Thetaseq n^{1}.

18. Problem 3.35

The following proposition is true

(42) There exists a non empty set F of functions from ℕ to ℝ such that F = {seq n^{1}} and for every n holds (seq n^{− 1})(n) (seq n^{1})(n) and seq n^{− 1} ∉ F^{Big Oh seq const 1}.

19. Addition

We now state the proposition

(43) Let c be a non negative real number and x, f be eventually-nonnegative sequences of real numbers. Given e, N such that e > 0 and for every n such that n N holds f(n) e. If x ∈ Big Ohc + f, then x ∈ Big Ohf.

20. Potentatially Useful

One can prove the following propositions:

(44) 2^{2} = 4.

(45) 2^{3} = 8.

(46) 2^{4} = 16.

(47) 2^{5} = 32.

(48) 2^{6} = 64.

(49) 2^{12} = 4096.

(50) For every n such that n 3 holds n^{2} > 2 · n + 1.

(51) For every n such that n 10 holds 2^{n−1} > (2 · n)^{2}.

(52) For every n such that n 9 holds (n + 1)^{6} < 2 · n^{6}.

(53) For every n such that n 30 holds 2^{n} > n^{6}.

(54) For every real number x such that x > 9 holds 2^{x} > (2 · x)^{2}.

(55) There exists N such that for every n such that n N holds \sqrt{n}−log_{2} n > 1.

(56) For all real numbers a, b, c such that a > 0 and c > 0 and c ≠ 1 holds a^{b} = c^{b·log_{c} a}.

(57) (4 + 1)! = 120.

(58) 5^{5} = 3125.

(59) 4^{4} = 256.

(60) For every n holds (n^{2} − n) + 1 > 0.

(61) For every n such that n 2 holds n! > 1.

(62) For all n_{1}, n such that n n_{1} holds n! n_{1}!.

(63) For every k such that k 1 there exists n such that n! k and k < (n + 1)! and for every m such that m! k and k < (m + 1)! holds m = n.

(64) For every n such that n 2 holds ⌈\frac{n}{2}⌉ < n.

(65) For every n such that n 3 holds n! > n.

(66) For all natural numbers m, n such that m > 0 holds m^{n} is a natural number.

(67) For every n such that n 2 holds 2^{n} > n + 1.

(68) Let a be a logbase real number and f be a sequence of real numbers. Suppose a > 1 and f(0) = 0 and for every n such that n > 0 holds f(n) = log_{a} n. Then f is eventually-positive.

(69) For all eventually-nonnegative sequences f, g of real numbers holds f ∈ Big Ohg and g ∈ Big Ohf iff Big Ohf = Big Ohg.

(70) For all real numbers a, b, c such that 0 < a and a b and c 0 holds a^{c} b^{c}.

(71) For every n such that n 4 holds 2 · n + 3 < 2^{n}.

(72) For every n such that n 6 holds (n + 1)^{2} < 2^{n}.

(73) For every real number c such that c > 6 holds c^{2} < 2^{c}.

(74) Let e be a positive real number and f be a sequence of real numbers. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = log_{2}(n^{e}). Then f/seq n^{e} is convergent and lim(f/seq n^{e}) = 0.

(75) For every real number e such that e > 0 holds seq logn/seq n^{e} is convergent and lim(seq logn/seq n^{e}) = 0.

(76) For every sequence f of real numbers and for every N such that for every n such that n N holds f(n) 0 holds ∑_{κ=0}^{N} f(κ) 0.

(77) For all sequences f, g of real numbers and for every N such that for every n such that n N holds f(n) g(n) holds ∑_{κ=0}^{N} f(κ) ∑_{κ=0}^{N} g(κ).

(78) Let f be a sequence of real numbers and b be a real number. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = b. Let N be a natural number. Then ∑_{κ=0}^{N} f(κ) = b · N.

(79) For all sequences f, g of real numbers and for all natural numbers N, M holds ∑_{κ=N+1}^{M} f(κ) + f(N + 1) = ∑_{κ=N+1+1}^{M} f(κ).

(80) Let f, g be sequences of real numbers, M be a natural number, and given N. Suppose N M + 1. If for every n such that M + 1 n and n N holds f(n) g(n), then ∑_{κ=N+1}^{M} f(κ) ∑_{κ=N+1}^{M} g(κ).

(81) For every n holds ⌈\frac{n}{2}⌉ n.

(82) Let f be a sequence of real numbers, b be a real number, and N be a natural number. Suppose f(0) = 0 and for every n such that n > 0 holds f(n) = b. Let M be a natural number. Then ∑_{κ=N+1}^{M} f(κ) = b · (N − M).

(83) Let f, g be sequences of real numbers, N be a natural number, and c be a real number. Suppose f is convergent and limf = c and for every n such that n N holds f(n) = g(n). Then g is convergent and limg = c.

(84) For every n such that n 1 holds (n^{2} − n) + 1 n^{2}.

(85) For every n such that n 1 holds n^{2} 2 · ((n^{2} − n) + 1).

(86) For every real number e such that 0 < e and e < 1 there exists N such that for every n such that n N holds n·log_{2}(1 + e)−8·log_{2} n > 8·log_{2} n.

(87) For every n such that n 10 holds \frac{2^{2·n}}{n!} < \frac{1}{2^{n−9}} .

(88) For every n such that n 3 holds 2 · (n − 2) n − 1.

(89) For every real number c such that c 0 holds c^{\frac{1}{2}} = \sqrt{c}.

(90) There exists N such that for every n such that n N holds n − \sqrt{n} · log_{2} n > \frac{n}{2}.

(91) For every sequence s of real numbers such that for every n holds s(n) = (1 + \frac{1}{n+1})^{n+1} holds s is non-decreasing.

(92) For every n such that n 1 holds (\frac{n+1}{n})^{n} (\frac{n+2}{n+1})^{n+1}.

(93) For all k, n such that k n holds \binom{n}{k} \frac{\binom{n+1}{k}}{n+1}.

(94) For every sequence f of real numbers such that for every n holds f(n) = log_{2}(n!) and for every n holds f(n) = ∑_{κ=0}^{n} seq logn(κ).

(95) For every n such that n 4 holds n · log_{2} n 2 · n.

(96) Let a, b be positive real numbers. Then Prob28(0, a, b) = 0 and Prob28(1, a, b) = a and for every n such that n 2 there exists n_{1} such that n_{1} = ⌈\frac{n}{2}⌉ and Prob28(n, a, b) = 4 · Prob28(n_{1}, a, b) + b · n.

(97) For every n such that n 2 holds n^{2} > n + 1.

(98) For every n such that n 1 holds 2^{n+1} − 2^{n} > 1.

(99) For every n such that n 2 holds 2^{n} − 1 ∉ POWEROF2SET.

(100) For all n, k such that k 1 and n! k and k < (n + 1)! holds Step1k = n!.

(101) For all real numbers a, b, c such that a > 1 and b a and c 1 holds log_{a} c log_{b} c.

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

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

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

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

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

Predicate Calculus for Boolean Valued Functions. Part X

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC18.

The articles [3], [5], [6], [4], [1], [2], [5], [1], and [2] provide the terminology and notation for this paper.

In this paper Y is a non empty set. We now state a number of propositions:

(1) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(2) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

(3) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(4) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(5) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∃_{a,B}G,A}G.

(6) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∃_{a,B}G,A}G.

(7) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(8) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(9) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(10) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(11) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(12) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(13) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∃_{a,B}G,A} G.

(14) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), and A, B, C be partitions of Y. Suppose G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A. Then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∃_{a,B}G,A} G.

References

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

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

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

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

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

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

Predicate Calculus for Boolean Valued Functions. Part XI

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of usual predicate logic.

MML Identifier: BVFUNC19.

The notation and terminology used here have been introduced in the following papers: [3], [5], [6], [4], [1], [2], [5], [1], and [2].

For simplicity, we use the following convention: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C are partitions of Y.

We now state a number of propositions:

(1) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(2) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(3) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(4) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(5) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∀_{¬a,B} G,A}G.

(6) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∀_{¬a,B} G,A}G.

(7) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(8) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(9) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(10) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(11) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(12) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(13) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(14) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(15) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(16) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(17) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(18) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(19) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(21)^{1} If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and

C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(22) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(23) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(24) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(25) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(26) If G is a coordinate and G = {A, B, C} and A ≠ B and B ≠ C and C ≠ A, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

1The proposition (20) has been removed.

References

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

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

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

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

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

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

400 shunichi kobayashi

Four Variable Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC20.

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

1. Preliminaries

For simplicity, we follow the rules: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C, D are partitions of Y.

We now state several propositions:

(1) Let h be a function and A′, B′, C′, D′be sets. Suppose G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′.

(2) Let A, B, C, D be sets, h be a function, and A′, B′, C′, D′ be sets. If h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(A⟼→ . A′), then domh = {A, B, C, D}.

(3) For every function h and for all sets A′, B′, C′, D′ such that G = {A, B, C, D} and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(A⟼→ . A′) holds rngh = {h(A), h(B), h(C), h(D)}.

(4) Let z, u be elements of Y and h be a function. Suppose G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D. Then EqClass(u, B ∧ C ∧ D) ∩ EqClass(z, A) ≠ ∅.

(5) Let z, u be elements of Y. Suppose G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D and EqClass(z, C ∧ D) = EqClass(u, C ∧ D). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Four Variable Predicate Calculus

One can prove the following propositions:

(6) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(7) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(8) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(9) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(10) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{a,B}G,A}G = ∃_{∃_{a,A}G,B}G.

(11) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(12) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(13) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(14) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(15) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(16) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(17) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(18) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(19) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(20) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(21) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(22) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(23) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{¬a,A} G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(24) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(25) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(26) If G is a coordinate and G = {A, B, C, D} and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(27) If G is a coordinate and G = {A, B, C, D} and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(28) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∀_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(29) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(30) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ¬∀_{∀_{a,B}G,A}G.

(31) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∀_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(32) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(33) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(34) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(35) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(36) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{¬∃_{a,B}G,A} G.

(37) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(38) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(39) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(40) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∀_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(41) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

four variable predicate calculus for boolean … 405

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

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

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

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

406 shunichi kobayashi

Four Variable Predicate Calculus for Boolean Valued Functions. Part II

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC21.

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

For simplicity, we adopt the following rules: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C, D are partitions of Y.

The following propositions are true:

(1) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(2) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ¬∃_{∃_{a,A}G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

(3) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

(4) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∀_{a,B}G,A}G.

408 shunichi kobayashi

(5) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∀_{∃_{a,B}G,A}G.

(6) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ¬∃_{∃_{a,B}G,A}G.

(7) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(8) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(9) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(10) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∀_{a,B}G,A} G.

(11) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(12) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∀_{a,B}G,A} G.

(13) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{¬∃_{a,B}G,A} G.

(14) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{¬∃_{a,B}G,A} G.

(15) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(16) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(17) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(18) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∃_{¬a,B} G,A}G.

(19) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∃_{∀_{¬a,B} G,A}G.

(20) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{¬∃_{a,A}G,B} G ⋐ ∀_{∀_{¬a,B} G,A}G.

(21) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(22) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∀_{a,B}G,A}G.

(23) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∀_{∃_{a,B}G,A}G.

(24) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ¬∃_{∃_{a,B}G,A}G.

(25) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(26) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(27) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(28) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{¬∀_{a,B}G,A} G.

(29) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(30) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∀_{a,B}G,A} G.

(31) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐

∃_{¬∃_{a,B}G,A} G.

(32) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{¬∃_{a,B}G,A} G.

(33) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(34) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∃_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(35) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(36) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∃_{¬a,B} G,A}G.

(37) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∃_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(38) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∃_{¬a,B} G,A}G.

(39) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∃_{∀_{¬a,B} G,A}G.

(40) If G is a coordinate and G = {A, B, C, D} and A ≠ B and A ≠ C and A ≠ D and B ≠ C and B ≠ D and C ≠ D, then ∀_{∀_{¬a,A} G,B}G ⋐ ∀_{∀_{¬a,B} G,A}G.

References

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

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

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

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

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

Function Spaces in the Category of Directed Suprema Preserving Maps^{1}

Grzegorz Bancerek University of Białystok

Adam Naumowicz University of Białystok

Summary. Formalization of [12, pp. 115-117], chapter II, section 2 (2.5 - 2.10).

MML Identifier: WAYBEL27.

The terminology and notation used here are introduced in the following articles: [13], [18], [17], [22], [23], [10], [19], [24], [6], [7], [5], [9], [8], [1], [2], [4], [14], [20], [23], [4], [26], [16], [15], [11], [21], [9], [3], [5], [6], [18], [33], [15], [15], [7], [15], [24], [36], [8], [15], [19], [17], [?], [?], and [?].

1. Currying, Uncurrying and Commuting Functions

Let F be a function. We say that F is uncurrying if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) For every set x such that x ∈ domF holds x is a function yielding

function, and (ii) for every function f such that f ∈ domF holds F (f) = uncurryf.

We say that F is currying if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) For every set x such that x ∈ domF holds x is a function and π_{1}(x)

is a binary relation, and (ii) for every function f such that f ∈ domF holds F (f) = curryf.

We say that F is commuting if and only if the conditions (Def. 3) are satisfied.

1This work has been supported by KBN Grant 8 T11C 018 12.

(Def. 3)(i) For every set x such that x ∈ domF holds x is a function yielding

function, and (ii) for every function f such that f ∈ domF holds F (f) = commute(f). Let us observe that every function which is empty is also uncurrying, curry-

ing, and commuting.

One can check that there exists a function which is uncurrying, currying, and commuting.

Let F be an uncurrying function and let X be a set. Note that F ↾X is uncurrying.

Let F be a currying function and let X be a set. One can check that F ↾X is currying.

One can prove the following propositions:

(1) Let X, Y, Z, D be sets. Suppose D ⊆ (Z^{Y})^{X}. Then there exists a many sorted set F indexed by D such that F is uncurrying and rngF ⊆ Z^{[: X, Y :]}.

(2) Let X, Y, Z, D be sets. Suppose D ⊆ Z^{[: X, Y :]}. Then there exists a many sorted set F indexed by D such that F is currying and if if Y = ∅, then X = ∅, then rngF ⊆ (Z^{Y})^{X}.

Let X, Y, Z be sets. Note that there exists a many sorted set indexed by (Z^{Y})^{X} which is uncurrying and there exists a many sorted set indexed by Z^{[: X, Y :]} which is currying.

Next we state several propositions:

(3) Let A, B be non empty sets, C be a set, and f, g be commuting functions. If domf ⊆ (C^{B})^{A} and rngf ⊆ domg, then g · f = id_{dom f}.

(4) Let B be a non empty set, A, C be sets, f be an uncurrying function, and g be a currying function. If domf ⊆ (C^{B})^{A} and rngf ⊆ domg, then g · f = id_{dom f}.

(5) Let A, B, C be sets, f be a currying function, and g be an uncurrying function. If domf ⊆ C^{[: A, B:]} and rngf ⊆ domg, then g · f = id_{dom f}.

(6) For every function yielding function f and for all sets i, A such that i ∈ domcommute(f) holds (commute(f))(i)^{∘}A ⊆ π_{i}f^{∘}A.

(7) Let f be a function yielding function and i, A be sets. If for every function g such that g ∈ f^{∘}A holds i ∈ domg, then π_{i}f^{∘}A ⊆ (commute(f))(i)^{∘}A.

(8) For all sets X, Y and for every function f such that rngf ⊆ Y^{X} and for all sets i, A such that i ∈ X holds (commute(f))(i)^{∘}A = π_{i}f^{∘}A.

(9) For every function f and for all sets i, A such that [: A, {i}:] ⊆ domf holds π_{i}(curryf)^{∘}A = f^{∘}[: A, {i}:].

Let X be a set and let Y be a non empty functional set. Observe that every function from X into Y is function yielding.

Let T be a constituted functions 1-sorted structure. Observe that the carrier of T is functional.

Let X be a set and let L be a non empty relational structure. Observe that L^{X} is constituted functions.

One can verify that there exists a lattice which is constituted functions, complete, and strict and there exists a 1-sorted structure which is constituted functions and non empty.

Let T be a constituted functions non empty relational structure. Note that every non empty relational substructure of T is constituted functions.

Next we state four propositions:

(10) Let S, T be complete lattices, f be an idempotent map from T into T, and h be a map from S into Imf. Then f · h = h.

(11) Let S be a non empty relational structure and T, T_{1} be non empty relational structures. Suppose T is a relational substructure of T_{1}. Let f be a map from S into T and f_{1} be a map from S into T_{1}. If f is monotone and f = f_{1}, then f_{1} is monotone.

(12) Let S be a non empty relational structure and T, T_{1} be non empty relational structures. Suppose T is a full relational substructure of T_{1}. Let f be a map from S into T and f_{1} be a map from S into T_{1}. If f_{1} is monotone and f = f_{1}, then f is monotone.

(13) For every set X and for every subset V of X holds (χ_{V,X})^{−1} ({1}) = V and (χ_{V,X})^{−1} ({0}) = X \ V.

2. Maps of Power Posets

Let X be a non empty set, let T be a non empty relational structure, let f be an element of T^{X}, and let x be an element of X. Then f(x) is an element of T.

Next we state several propositions:

(14) Let X be a non empty set, T be a non empty relational structure, and f, g be elements of T^{X}. Then f g if and only if for every element x of X holds f(x) g(x).

(15) Let X be a set and L, S be non empty relational structures. Suppose the relational structure of L = the relational structure of S. Then L^{X} = S^{X}.

(16) Let S_{1}, S_{2}, T_{1}, T_{2} be non empty topological spaces. Suppose that (i) the topological structure of S_{1} = the topological structure of S_{2}, and (ii) the topological structure of T_{1} = the topological structure of T_{2}.

Then [S_{1} → T_{1}] = [S_{2} → T_{2}].

(17) Let X be a set. Then there exists a map f from 2_{⊆}^{X} into (2_{⊆}^{1} )^{X} such that f is isomorphic and for every subset Y of X holds f(Y) = χ_{Y,X}.

(18) For every set X holds 2_{⊆}^{X} and (2_{⊆}^{1} )^{X} are isomorphic.

(19) Let X, Y be non empty sets, T be a non empty poset, S_{1} be a full non empty relational substructure of (T^{X})^{Y}, S_{2} be a full non empty relational substructure of (T^{Y})^{X}, and F be a map from S_{1} into S_{2}. If F is commuting, then F is monotone.

(20) Let X, Y be non empty sets, T be a non empty poset, S_{1} be a full non empty relational substructure of (T^{Y})^{X}, S_{2} be a full non empty relational substructure of T^{[: X, Y :]}, and F be a map from S_{1} into S_{2}. If F is uncurrying, then F is monotone.

(21) Let X, Y be non empty sets, T be a non empty poset, S_{1} be a full non empty relational substructure of (T^{Y})^{X}, S_{2} be a full non empty relational substructure of T^{[: X, Y :]}, and F be a map from S_{2} into S_{1}. If F is currying, then F is monotone.

3. Posets of Directed Suprema Preserving Maps

Let S be a non empty relational structure and let T be a non empty reflexive antisymmetric relational structure. The functor UPS(S, T) yielding a strict relational structure is defined by the conditions (Def. 4).

(Def. 4)(i) UPS(S, T) is a full relational substructure of T^{the carrier of S}, and

(ii) for every set x holds x ∈ the carrier of UPS(S, T) iff x is a directed-

sups-preserving map from S into T. Let S be a non empty relational structure and let T be a non empty refle-

xive antisymmetric relational structure. Observe that UPS(S, T) is non empty reflexive antisymmetric and constituted functions.

Let S be a non empty relational structure and let T be a non empty poset. Note that UPS(S, T) is transitive.

One can prove the following proposition

(22) Let S be a non empty relational structure and T be a non empty reflexive antisymmetric relational structure. Then the carrier of UPS(S, T) ⊆ (the carrier of T)^{the carrier of S}.

Let S be a non empty relational structure, let T be a non empty reflexive antisymmetric relational structure, let f be an element of UPS(S, T), and let s be an element of S. Then f(s) is an element of T.

The following propositions are true:

(23) Let S be a non empty relational structure, T be a non empty reflexive antisymmetric relational structure, and f, g be elements of UPS(S, T). Then f g if and only if for every element s of S holds f(s) g(s).

(24) For all complete Scott top-lattices S, T holds UPS(S, T) = SCMaps(S, T).

(25) Let S, S′ be non empty relational structures and T, T′ be non empty

reflexive antisymmetric relational structures. Suppose that

(i) the relational structure of S = the relational structure of S′, and (ii) the relational structure of T = the relational structure of T′.

Then UPS(S, T) = UPS(S′, T′). Let S, T be complete lattices. Note that UPS(S, T) is complete. One can prove the following propositions:

(26) Let S, T be complete lattices. Then UPS(S, T) is a sups-inheriting relational substructure of T^{the carrier of S}.

(27) For all complete lattices S, T and for every subset A of UPS(S, T) holds

supA = ⨆_{(T^{the carrier of S})} A. Let S_{1}, S_{2}, T_{1}, T_{2} be non empty reflexive antisymmetric relational structures

and let f be a map from S_{1} into S_{2}. Let us assume that f is directed-sups-preserving. Let g be a map from T_{1} into T_{2}. Let us assume that g is directedsups-preserving. The functor UPS(f, g) yielding a map from UPS(S_{2}, T_{1}) into UPS(S_{1}, T_{2}) is defined as follows:

(Def. 5) For every directed-sups-preserving map h from S_{2} into T_{1} holds

(UPS(f, g))(h) = g · h · f. We now state a number of propositions:

(28) Let S_{1}, S_{2}, S_{3}, T_{1}, T_{2}, T_{3} be non empty posets, f_{1} be a directed-sups-preserving map from S_{2} into S_{3}, f_{2} be a directed-sups-preserving map from S_{1} into S_{2}, g_{1} be a directed-sups-preserving map from T_{1} into T_{2}, and g_{2} be a directed-sups-preserving map from T_{2} into T_{3}. Then UPS(f_{2}, g_{2}) · UPS(f_{1}, g_{1}) = UPS(f_{1} · f_{2}, g_{2} · g_{1}).

(29) For all non empty reflexive antisymmetric relational structures S, T holds UPS(id_{S}, id_{T}) = id_{UPS(S,T)}.

(30) Let S_{1}, S_{2}, T_{1}, T_{2} be complete lattices, f be a directed-sups-preserving map from S_{1} into S_{2}, and g be a directed-sups-preserving map from T_{1} into T_{2}. Then UPS(f, g) is directed-sups-preserving.

(31) Ω(the Sierpiński space) is Scott.

(32) For every complete Scott top-lattice S holds [S → the Sierpiński space] = UPS(S, 2_{⊆}^{1} ).

(33) Let S be a complete lattice. Then there exists a map F from UPS(S, 2_{⊆}^{1} ) into ⟨σ(S), ⊆⟩ such that F is isomorphic and for every directed-sups-preserving map f from S into 2_{⊆}^{1} holds F (f) = f^{−1} ({1}).

(34) For every complete lattice S holds UPS(S, 2_{⊆}^{1} ) and ⟨σ(S), ⊆⟩ are isomorphic.

(35) Let S_{1}, S_{2}, T_{1}, T_{2} be complete lattices, f be a map from S_{1} into S_{2}, and g be a map from T_{1} into T_{2}. If f is isomorphic and g is isomorphic, then UPS(f, g) is isomorphic.

(36) Let S_{1}, S_{2}, T_{1}, T_{2} be complete lattices. Suppose S_{1} and S_{2} are isomorphic and T_{1} and T_{2} are isomorphic. Then UPS(S_{2}, T_{1}) and UPS(S_{1}, T_{2}) are isomorphic.

(37) Let S, T be complete lattices and f be a directed-sups-preserving projection map from T into T. Then ImUPS(id_{S}, f) = UPS(S, Imf).

(38) Let X be a non empty set, S, T be non empty posets, f be a directedsups-preserving map from S into T^{X}, and i be an element of X. Then (commute(f))(i) is a directed-sups-preserving map from S into T.

(39) Let X be a non empty set, S, T be non empty posets, and f be a directedsups-preserving map from S into T^{X}. Then commute(f) is a function from X into the carrier of UPS(S, T).

(40) Let X be a non empty set, S, T be non empty posets, and f be a function from X into the carrier of UPS(S, T). Then commute(f) is a directed-sups-preserving map from S into T^{X}.

(41) For every non empty set X and for all non empty posets S, T holds there exists a map from UPS(S, T^{X}) into UPS(S, T)^{X} which is commuting and isomorphic.

(42) For every non empty set X and for all non empty posets S, T holds UPS(S, T^{X}) and (UPS(S, T))^{X} are isomorphic.

(43) For all continuous complete lattices S, T holds UPS(S, T) is continuous.

(44) For all algebraic complete lattices S, T holds UPS(S, T) is algebraic.

(45) Let R, S, T be complete lattices and f be a directed-sups-preserving map from R into UPS(S, T). Then uncurryf is a directed-sups-preserving map from [: R, S:] into T.

(46) Let R, S, T be complete lattices and f be a directed-sups-preserving map from [: R, S:] into T. Then curryf is a directed-sups-preserving map from R into UPS(S, T).

(47) For all complete lattices R, S, T holds there exists a map from UPS(R, UPS(S, T)) into UPS([: R, S:], T) which is uncurrying and isomorphic.

References

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

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

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

[4] Grzegorz Bancerek. Monoids. Formalized Mathematics, 3(2):213–225, 1992.

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

function spaces in the category of directed … 417

[8] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

[12] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.

[13] Krzysztof Hryniewiecki. Relations of tolerance. Formalized Mathematics, 2(1):105–109, 1991.

[14] Beata Madras. Product of family of universal algebras. Formalized Mathematics, 4(1):103– 108, 1993.

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

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

[17] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

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

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

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

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

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

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

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

418 grzegorz bancerek and adam naumowicz

Property of Complex Functions

Takashi Mitsuishi Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. This article introduces properties of complex function, calculations of them, boundedness and constant.

MML Identifier: CFUNCT 1.

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

1. Definitions of Complex Functions

For simplicity, we adopt the following convention: X, Y are sets, C is a non empty set, c is an element of C, f, f_{1}, f_{2}, f_{3}, g, g_{1} are partial functions from C to ℂ, p is a real number, and r, q are elements of ℂ.

A Complex is an element of ℂ.

Let us consider C, f_{1}, f_{2}. The functor \frac{f_{1}}{f_{2}} yielding a partial function from C to ℂ is defined by:

(Def. 1) dom(\frac{f_{1}}{f_{2}} ) = domf_{1} ∩ (domf_{2} \ f_{2}^{−1} (\{0_{ℂ}\})) and for every c such that c ∈ dom(\frac{f_{1}}{f_{2}} ) holds (\frac{f_{1}}{f_{2}} )_{c} = (f_{1})_{c} · ((f_{2})_{c})^{−1} .

Let us consider C, f. The functor \frac{1}{f} yields a partial function from C to ℂ and is defined by:

(Def. 2) dom(\frac{1}{f}) = domf \f^{−1} (\{0_{ℂ}\}) and for every c such that c ∈ dom(\frac{1}{f}) holds

(\frac{1}{f})_{c} = (f_{c})^{−1} . One can prove the following propositions:

(3)^{1} dom(f_{1} + f_{2}) = domf_{1} ∩ domf_{2} and for every c such that c ∈ dom(f_{1} +

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

(4) dom(f_{1}−f_{2}) = domf_{1}∩domf_{2} and for every c such that c ∈ dom(f_{1}−f_{2}) holds (f_{1} − f_{2})_{c} = (f_{1})_{c} − (f_{2})_{c}.

(5) dom(f_{1} f_{2}) = domf_{1} ∩ domf_{2} and for every c such that c ∈ dom(f_{1} f_{2}) holds (f_{1} f_{2})_{c} = (f_{1})_{c} · (f_{2})_{c}.

(6) dom(\frac{f_{1}}{f_{2}} ) = domf_{1} ∩ (domf_{2} \ f_{2}^{−1} (\{0_{ℂ}\})) and for every c such that c ∈ dom(\frac{f_{1}}{f_{2}} ) holds (\frac{f_{1}}{f_{2}} )_{c} = (f_{1})_{c} · ((f_{2})_{c})^{−1} .

(7) dom(r f) = domf and for every c such that c ∈ dom(r f) holds (r f)_{c} = r · f_{c}.

(9)^{2} dom(− f) = domf and for every c such that c ∈ dom(− f) holds (− f)_{c} =

− f_{c}.

(10) dom(\frac{1}{f}) = domf \f^{−1} (\{0_{ℂ}\}) and for every c such that c ∈ dom(\frac{1}{f}) holds (\frac{1}{f})_{c} = (f_{c})^{−1} .

(15)^{3} dom(\frac{1}{g}) ⊆ domg and domg ∩ (domg \ g^{−1} (\{0_{ℂ}\})) = domg \ g^{−1} (\{0_{ℂ}\}).

(16) dom(f_{1} f_{2}) \ (f_{1} f_{2})^{−1} ({0_{ℂ}}) = (domf_{1} \ f_{1}^{−1} ({0_{ℂ}})) ∩ (domf_{2} \ f_{2}^{−1} ({0_{ℂ}})).

(17) If c ∈ dom(\frac{1}{f}), then f_{c} ≠ 0_{ℂ}.

(18) (\frac{1}{f})^{−1} (\{0_{ℂ}\}) = ∅.

(19) |f|^{−1} ({0}) = f^{−1} ({0_{ℂ}}) and (− f)^{−1} ({0_{ℂ}}) = f^{−1} ({0_{ℂ}}).

(20) dom(_{\frac{1}{\frac{1}{f}}}) = dom(f↾ dom(\frac{1}{f})).

(21) If r ≠ 0_{ℂ}, then (r f)^{−1} ({0_{ℂ}}) = f^{−1} ({0_{ℂ}}).

2. Basic Properties of Operations

The following propositions are true:

(22) (f_{1} + f_{2}) + f_{3} = f_{1} + (f_{2} + f_{3}).

(23) (f_{1} f_{2}) f_{3} = f_{1} (f_{2} f_{3}).

(24) (f_{1} + f_{2}) f_{3} = f_{1} f_{3} + f_{2} f_{3}.

(25) f_{3} (f_{1} + f_{2}) = f_{3} f_{1} + f_{3} f_{2}.

(26) r (f_{1} f_{2}) = (r f_{1}) f_{2}.

(27) r (f_{1} f_{2}) = f_{1} (r f_{2}).

(28) (f_{1} − f_{2}) f_{3} = f_{1} f_{3} − f_{2} f_{3}.

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

2The proposition (8) has been removed.

3The propositions (11)–(14) have been removed.

(29) f_{3} f_{1} − f_{3} f_{2} = f_{3} (f_{1} − f_{2}).

(30) r (f_{1} + f_{2}) = r f_{1} + r f_{2}.

(31) (r · q) f = r (q f).

(32) r (f_{1} − f_{2}) = r f_{1} − r f_{2}.

(33) f_{1} − f_{2} = (− 1_{ℂ}) (f_{2} − f_{1}).

(34) f_{1} − (f_{2} + f_{3}) = f_{1} − f_{2} − f_{3}.

(35) 1_{ℂ} f = f.

(36) f_{1} − (f_{2} − f_{3}) = (f_{1} − f_{2}) + f_{3}.

(37) f_{1} + (f_{2} − f_{3}) = (f_{1} + f_{2}) − f_{3}.

(38) |f_{1} f_{2}| = |f_{1}| |f_{2}|.

(39) |r f| = |r| |f|.

(40) − f = (− 1_{ℂ}) f.

(41) − − f = f.

(42) f_{1} − f_{2} = f_{1} + − f_{2}.

(43) f_{1} − − f_{2} = f_{1} + f_{2}.

(44) _{\frac{1}{\frac{1}{f}}} = f↾ dom(\frac{1}{f}).

(45) \frac{1}{f_{1} f_{2}} = \frac{1}{f_{1}} \frac{1}{f_{2}} .

(46) If r ≠ 0_{ℂ}, then \frac{1}{r f} = r^{−1} \frac{1}{f} .

(47) 1_{ℂ} ≠ 0_{ℂ}.

(48) (− 1_{ℂ})^{−1} = − 1_{ℂ}.

(49) \frac{1}{− f} = (− 1_{ℂ}) \frac{1}{f} .

(50) \frac{1}{|f|} = |\frac{1}{f} |.

(51) \frac{f}{g} = f \frac{1}{g}.

(52) r \frac{g}{f} = \frac{r g}{f} .

(53) \frac{f}{g} g = f↾ dom(\frac{1}{g}).

(54) \frac{f}{g} \frac{f_{1}}{g_{1}} = \frac{f f_{1}}{g g_{1}} .

(55) _{\frac{1}{\frac{f_{1}}{f_{2}}}} = \frac{f_{2}↾ dom( \frac{1}{f_{2}} )}{f_{1}}.

(56) g \frac{f_{1}}{f_{2}} = \frac{g f_{1}}{f_{2}} .

(57) _{\frac{g}{\frac{f_{1}}{f_{2}}}} = \frac{g (f_{2}↾ dom( \frac{1}{f_{2}} ))}{f_{1}}.

(58) − \frac{f}{g} = \frac{− f}{g} and \frac{f}{− g} = − \frac{f}{g} .

(59) \frac{f_{1}}{f} + \frac{f_{2}}{f} = \frac{f_{1}+f_{2}}{f} and \frac{f_{1}}{f} − \frac{f_{2}}{f} = \frac{f_{1}−f_{2}}{f} .

(60) \frac{f_{1}}{f} + \frac{g_{1}}{g} = \frac{f_{1} g+g_{1} f}{f g} .

(61) \frac{\frac{f}{g}}{\frac{f_{1}}{g_{1}}} = \frac{f (g_{1}↾ dom( \frac{1}{g_{1}} ))}{g f_{1}}.

(62) \frac{f_{1}}{f} − \frac{g_{1}}{g} = \frac{f_{1} g−g_{1} f}{f g} .

(63) |\frac{f_{1}}{f_{2}} | = \frac{|f_{1}|}{|f_{2}|}.

(64) (f_{1}+f_{2})↾X = f_{1}↾X+f_{2}↾X and (f_{1}+f_{2})↾X = f_{1}↾X+f_{2} and (f_{1}+f_{2})↾X = f_{1} + f_{2}↾X.

(65) (f_{1} f_{2})↾X = (f_{1}↾X) (f_{2}↾X) and (f_{1} f_{2})↾X = (f_{1}↾X) f_{2} and (f_{1} f_{2})↾X = f_{1} (f_{2}↾X).

(66) (− f)↾X = − f↾X and \frac{1}{f} ↾X = \frac{1}{f↾X} and |f|↾X = |f↾X|.

(67) (f_{1}−f_{2})↾X = f_{1}↾X−f_{2}↾X and (f_{1}−f_{2})↾X = f_{1}↾X−f_{2} and (f_{1}−f_{2})↾X = f_{1} − f_{2}↾X.

(68) \frac{f_{1}}{f_{2}} ↾X = \frac{f_{1}↾X}{f_{2}↾X} and \frac{f_{1}}{f_{2}} ↾X = \frac{f_{1}↾X}{f_{2}} and \frac{f_{1}}{f_{2}} ↾X = \frac{f_{1}}{f_{2}↾X} .

(69) (r f)↾X = r (f↾X).

3. Total Partial Functions from a Domain, to Complex

One can prove the following propositions: (70)(i) f_{1} is total and f_{2} is total iff f_{1} + f_{2} is total, (ii) f_{1} is total and f_{2} is total iff f_{1} − f_{2} is total, and (iii) f_{1} is total and f_{2} is total iff f_{1} f_{2} is total.

(71) f is total iff r f is total.

(72) f is total iff − f is total.

(73) f is total iff |f| is total.

(74) \frac{1}{f} is total iff f^{−1} (\{0_{ℂ}\}) = ∅ and f is total.

(75) f_{1} is total and f_{2}^{−1} (\{0_{ℂ}\}) = ∅ and f_{2} is total iff \frac{f_{1}}{f_{2}} is total.

(76) If f_{1} is total and f_{2} is total, then (f_{1}+f_{2})_{c} = (f_{1})_{c}+(f_{2})_{c} and (f_{1}−f_{2})_{c} = (f_{1})_{c} − (f_{2})_{c} and (f_{1} f_{2})_{c} = (f_{1})_{c} · (f_{2})_{c}.

(77) If f is total, then (r f)_{c} = r · f_{c}.

(78) If f is total, then (− f)_{c} = − f_{c} and |f|(c) = |f_{c}|.

(79) If \frac{1}{f} is total, then (\frac{1}{f})_{c} = (f_{c})^{−1} .

(80) If f_{1} is total and \frac{1}{f_{2}} is total, then (\frac{f_{1}}{f_{2}} )_{c} = (f_{1})_{c} · ((f_{2})_{c})^{−1} .

4. Bounded and Constant Partial Functions from a Domain, to Complex

Let us consider C, f, Y. We say that f is bounded on Y if and only if: (Def. 3) |f| is bounded on Y.

One can prove the following propositions:

(81) f is bounded on Y iff there exists a real number p such that for every c such that c ∈ Y ∩ domf holds |f_{c}| p.

(82) If Y ⊆ X and f is bounded on X, then f is bounded on Y.

(83) If X ∩ domf = ∅, then f is bounded on X.

(84) If f is bounded on Y, then r f is bounded on Y.

(85) |f| is lower bounded on X.

(86) If f is bounded on Y, then |f| is bounded on Y and − f is bounded on Y.

(87) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} + f_{2} is bounded on X ∩ Y.

(88) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} f_{2} is bounded on X ∩ Y and f_{1} − f_{2} is bounded on X ∩ Y.

(89) If f is bounded on X and bounded on Y, then f is bounded on X ∪ Y.

(90) Suppose f_{1} is a constant on X and f_{2} is a constant on Y. Then f_{1} + f_{2} is a constant on X ∩ Y and f_{1} − f_{2} is a constant on X ∩ Y and f_{1} f_{2} is a constant on X ∩ Y.

(91) If f is a constant on Y, then q f is a constant on Y.

(92) If f is a constant on Y, then |f| is a constant on Y and − f is a constant on Y.

(93) If f is a constant on Y, then f is bounded on Y.

(94) If f is a constant on Y, then for every r holds r f is bounded on Y and − f is bounded on Y and |f| is bounded on Y.

(95) If f_{1} is bounded on X and f_{2} is a constant on Y, then f_{1} +f_{2} is bounded on X ∩ Y.

(96) Suppose f_{1} is bounded on X and f_{2} is a constant on Y. Then f_{1} − f_{2} is bounded on X ∩ Y and f_{2} − f_{1} is bounded on X ∩ Y and f_{1} f_{2} is bounded on X ∩ Y.

References

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

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

424 takashi mitsuishi et al.

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

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

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

[6] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[7] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

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

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

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

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

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

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

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

Property of Complex Sequence and Continuity of Complex Function

Takashi Mitsuishi Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. This article introduces properties of complex sequence and continuity of complex function. The first section shows convergence of complex sequence and constant complex sequence. In the next section, definition of continuity of complex function and properties of continuous complex function are shown.

MML Identifier: CFCONT 1.

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

1. Complex Sequence

For simplicity, we adopt the following rules: n, m, k are natural numbers, x is a set, X, X_{1} are sets, g, x_{0}, x_{1}, x_{2} are elements of ℂ, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6} are complex sequences, Y is a subset of ℂ, f, f_{1}, f_{2}, h, h_{1}, h_{2} are partial functions from ℂ to ℂ, r, s are real numbers, and N_{1} is an increasing sequence of naturals.

Let us consider h, s_{3}. Let us assume that rngs_{3} ⊆ domh. The functor h · s_{3} yields a complex sequence and is defined as follows:

(Def. 1) h · s_{3} = (h qua function) ·(s_{3}).

Let us consider f, x_{0}. We say that f is continuous in x_{0} if and only if: (Def. 2) x_{0} ∈ domf and for every s_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}).

The following propositions are true: (2)^{1} s_{4} = s_{5} − s_{6} iff for every n holds s_{4}(n) = s_{5}(n) − s_{6}(n).

(3) rng(s_{3} ↑ n) ⊆ rngs_{3}.

(4) If rngs_{3} ⊆ domf, then s_{3}(n) ∈ domf.

(5) x ∈ rngs_{3} iff there exists n such that x = s_{3}(n).

(6) s_{3}(n) ∈ rngs_{3}.

(7) If s_{4} is a subsequence of s_{3}, then rngs_{4} ⊆ rngs_{3}.

(8) If s_{4} is a subsequence of s_{3} and s_{3} is non-zero, then s_{4} is non-zero.

(9) (s_{4} + s_{5}) N_{1} = s_{4} N_{1} + s_{5} N_{1} and (s_{4} − s_{5}) N_{1} = s_{4} N_{1} − s_{5} N_{1} and (s_{4} s_{5}) N_{1} = s_{4} N_{1} (s_{5} N_{1}).

(10) (g s_{3}) N_{1} = g (s_{3} N_{1}).

(11) (− s_{3}) N_{1} = − s_{3} N_{1} and |s_{3}| · N_{1} = |s_{3} N_{1}|.

(12) (s_{3} N_{1})^{−1} = s_{3}^{−1} N_{1}.

(13) (s_{4}/s_{3}) N_{1} = (s_{4} N_{1})/(s_{3} N_{1}).

(14) If for every n holds s_{3}(n) ∈ Y, then rngs_{3} ⊆ Y.

(15) If rngs_{3} ⊆ domh, then h · s_{3} = (h qua function) ·(s_{3}).

(16) If rngs_{3} ⊆ domf, then (f · s_{3})(n) = f_{s_{3}(n)}.

(17) If rngs_{3} ⊆ domf, then (f · s_{3}) ↑ n = f · (s_{3} ↑ n).

(18) If rngs_{3} ⊆ domh_{1} ∩ domh_{2}, then (h_{1} + h_{2}) · s_{3} = h_{1} · s_{3} + h_{2} · s_{3} and (h_{1} − h_{2}) · s_{3} = h_{1} · s_{3} − h_{2} · s_{3} and (h_{1} h_{2}) · s_{3} = (h_{1} · s_{3}) (h_{2} · s_{3}).

(19) If rngs_{3} ⊆ domh, then (g h) · s_{3} = g (h · s_{3}).

(20) If rngs_{3} ⊆ domh, then − h · s_{3}= (− h) · s_{3}.

(21) If rngs_{3} ⊆ dom(\frac{1}{h}), then h · s_{3} is non-zero.

(22) If rngs_{3} ⊆ dom(\frac{1}{h}), then \frac{1}{h} · s_{3} = (h · s_{3})^{−1} .

(23) If rngs_{3} ⊆ domh, then ℜ((h · s_{3}) N_{1}) = ℜ(h · (s_{3} N_{1})).

(24) If rngs_{3} ⊆ domh, then ℑ((h · s_{3}) N_{1}) = ℑ(h · (s_{3} N_{1})).

(25) If rngs_{3} ⊆ domh, then (h · s_{3}) N_{1} = h · (s_{3} N_{1}).

(26) If rngs_{4} ⊆ domh and s_{5} is a subsequence of s_{4}, then h·s_{5} is a subsequence of h · s_{4}.

(27) If h is total, then (h · s_{3})(n) = h_{s_{3}(n)}.

(28) If h is total, then h · (s_{3} ↑ n) = (h · s_{3}) ↑ n.

(29) If h_{1} is total and h_{2} is total, then (h_{1} + h_{2}) · s_{3} = h_{1} · s_{3} + h_{2} · s_{3} and (h_{1} − h_{2}) · s_{3} = h_{1} · s_{3} − h_{2} · s_{3} and (h_{1} h_{2}) · s_{3} = (h_{1} · s_{3}) (h_{2} · s_{3}).

(30) If h is total, then (g h) · s_{3} = g (h · s_{3}).

(31) If rngs_{3} ⊆ dom(h↾X), then (h↾X) · s_{3} = h · s_{3}.

1The proposition (1) has been removed.

(32) If rngs_{3} ⊆ dom(h↾X) and if rngs_{3} ⊆ dom(h↾Y) or X ⊆ Y, then (h↾X) · s_{3} = (h↾Y) · s_{3}.

(33) If rngs_{3} ⊆ dom(h↾X) and h^{−1} (\{0_{ℂ}\}) = ∅, then (\frac{1}{h}↾X) · s_{3} = ((h↾X) ·

s_{3})^{−1} . Let f be a function. We say that f is constant if and only if:

(Def. 3) For all sets n_{1}, n_{2} such that n_{1} ∈ domf and n_{2} ∈ domf holds f(n_{1}) =

f(n_{2}). Let us consider s_{3}. Let us observe that s_{3} is constant if and only if:

(Def. 4) There exists g such that for every n holds s_{3}(n) = g.

Next we state a number of propositions:

(34) s_{3} is constant iff there exists g such that rngs_{3} = {g}.

(35) s_{3} is constant iff for every n holds s_{3}(n) = s_{3}(n + 1).

(36) s_{3} is constant iff for all n, k holds s_{3}(n) = s_{3}(n + k).

(37) s_{3} is constant iff for all n, m holds s_{3}(n) = s_{3}(m).

(38) s_{3} ↑ k is a subsequence of s_{3}.

(39) If s_{4} is a subsequence of s_{3} and s_{3} is convergent, then s_{4} is convergent.

(40) If s_{4} is a subsequence of s_{3} and s_{3} is convergent, then lims_{4} = lims_{3}.

(41) If s_{3} is convergent and there exists k such that for every n such that k n holds s_{4}(n) = s_{3}(n), then s_{4} is convergent.

(42) If s_{3} is convergent and there exists k such that for every n such that k n holds s_{4}(n) = s_{3}(n), then lims_{3} = lims_{4}.

(43) If s_{3} is convergent, then s_{3} ↑ k is convergent and lim(s_{3} ↑ k) = lims_{3}.

(44) If s_{3} is convergent and there exists k such that s_{3} = s_{4} ↑ k, then s_{4} is convergent.

(45) If s_{3} is convergent and there exists k such that s_{3} = s_{4} ↑k, then lims_{4} = lims_{3}.

(46) If s_{3} is convergent and lims_{3} ≠ 0_{ℂ}, then there exists k such that s_{3} ↑ k is non-zero.

(47) If s_{3} is convergent and lims_{3} ≠ 0_{ℂ}, then there exists s_{4} which is a subsequence of s_{3} and non-zero.

(48) If s_{3} is constant, then s_{3} is convergent.

(49) If s_{3} is constant and g ∈ rngs_{3} or s_{3} is constant and there exists n such that s_{3}(n) = g, then lims_{3} = g.

(50) If s_{3} is constant, then for every n holds lims_{3} = s_{3}(n).

(51) If s_{3} is convergent and lims_{3} ≠ 0_{ℂ}, then for every s_{4} such that s_{4} is a subsequence of s_{3} and non-zero holds lim(s_{4}^{−1} ) = (lims_{3})^{−1} .

(52) If s_{3} is constant and s_{4} is convergent, then lim(s_{3} + s_{4}) = s_{3}(0) + lims_{4} and lim(s_{3} − s_{4}) = s_{3}(0) − lims_{4} and lim(s_{4} − s_{3}) = lims_{4} − s_{3}(0) and

lim(s_{3} s_{4}) = s_{3}(0) · lims_{4}. The scheme CompSeqChoice concerns and states that:

There exists s_{1} such that for every n holds P[n, s_{1}(n)] provided the following requirement is met:

• For every n there exists g such that P[n, g].

2. Continuity of Complex Sequence

One can prove the following propositions:

(53) f is continuous in x_{0} if and only if the following conditions are satisfied: (i) x_{0} ∈ domf, and (ii) for every s_{1} such that rngs_{1} ⊆ domf and s_{1} is convergent and lims_{1} =

x_{0} and for every n holds s_{1}(n) ≠ x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}).

(54) f is continuous in x_{0} if and only if the following conditions are satisfied: (i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

x_{1} such that x_{1} ∈ domf and |x_{1} − x_{0}| < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(55) Suppose f_{1} is continuous in x_{0} and f_{2} is continuous in x_{0}. Then f_{1} + f_{2} is continuous in x_{0} and f_{1} − f_{2} is continuous in x_{0} and f_{1} f_{2} is continuous in x_{0}.

(56) If f is continuous in x_{0}, then g f is continuous in x_{0}.

(57) If f is continuous in x_{0}, then − f is continuous in x_{0}.

(58) If f is continuous in x_{0} and f_{x_{0}} ≠ 0_{ℂ}, then \frac{1}{f} is continuous in x_{0}.

(59) If f_{1} is continuous in x_{0} and (f_{1})_{x_{0}} ≠ 0_{ℂ} and f_{2} is continuous in x_{0}, then

\frac{f_{2}}{f_{1}} is continuous in x_{0}. Let us consider f, X. We say that f is continuous on X if and only if:

(Def. 5) X ⊆ domf and for every x_{0} such that x_{0} ∈ X holds f↾X is continuous

in x_{0}. One can prove the following propositions:

(60) Let given X, f. Then f is continuous on X if and only if the following

conditions are satisfied:

(i) X ⊆ domf, and (ii) for every s_{1} such that rngs_{1} ⊆ X and s_{1} is convergent and lims_{1} ∈ X

holds f · s_{1} is convergent and f_{lim s_{1}} = lim(f · s_{1}).

(61) f is continuous on X if and only if the following conditions are satisfied: (i) X ⊆ domf, and (ii) for all x_{0}, r such that x_{0} ∈ X and 0 < r there exists s such that 0 < s

and for every x_{1} such that x_{1} ∈ X and |x_{1} − x_{0}| < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(62) f is continuous on X iff f↾X is continuous on X.

(63) If f is continuous on X and X_{1} ⊆ X, then f is continuous on X_{1}.

(64) If x_{0} ∈ domf, then f is continuous on {x_{0}}.

(65) Let given X, f_{1}, f_{2}. Suppose f_{1} is continuous on X and f_{2} is continuous on X. Then f_{1} + f_{2} is continuous on X and f_{1} − f_{2} is continuous on X and f_{1} f_{2} is continuous on X.

(66) Let given X, X_{1}, f_{1}, f_{2}. Suppose f_{1} is continuous on X and f_{2} is continuous on X_{1}. Then f_{1} + f_{2} is continuous on X ∩ X_{1} and f_{1} − f_{2} is continuous on X ∩ X_{1} and f_{1} f_{2} is continuous on X ∩ X_{1}.

(67) For all g, X, f such that f is continuous on X holds g f is continuous on X.

(68) If f is continuous on X, then − f is continuous on X.

(69) If f is continuous on X and f^{−1} (\{0_{ℂ}\}) = ∅, then \frac{1}{f} is continuous on X.

(70) If f is continuous on X and (f↾X)^{−1} (\{0_{ℂ}\}) = ∅, then \frac{1}{f} is continuous on X.

(71) If f_{1} is continuous on X and f_{1}^{−1} ({0_{ℂ}}) = ∅ and f_{2} is continuous on X, then \frac{f_{2}}{f_{1}} is continuous on X.

(72) If f is total and for all x_{1}, x_{2} holds f_{x_{1}+x_{2}} = f_{x_{1}} + f_{x_{2}} and there exists

x_{0} such that f is continuous in x_{0}, then f is continuous on ℂ. Let us consider X. We say that X is compact if and only if:

(Def. 6) For every s_{1} such that rngs_{1} ⊆ X there exists s_{2} such that s_{2} is a

subsequence of s_{1} and convergent and lims_{2} ∈ X. We now state two propositions:

(73) For every f such that domf is compact and f is continuous on domf holds rngf is compact.

(74) If Y ⊆ domf and Y is compact and f is continuous on Y, then f^{∘}Y is compact.

References

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

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

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

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

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

430 takashi mitsuishi et al.

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

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

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

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

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

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

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

Scalar Multiple of Riemann Definite Integral

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. The goal of this article is to prove a scalar multiplicity of Riemann definite integral. Therefore, we defined a scalar product to the subset of real space, and we proved some relating lemmas. At last, we proved a scalar multiplicity of Riemann definite integral. As a result, a linearity of Riemann definite integral was proven by unifying the previous article [?].

MML Identifier: INTEGRA2.

The notation and terminology used in this paper have been introduced in the following papers: [18], [20], [19], [11], [2], [4], [5], [21], [12], [6], [7], [3], [13], [7], [?], [14], [23], [23], [10], [17], [8], [1], [9], [9], [15], and [16].

1. Lemmas of Finite Sequence

We adopt the following convention: r, x, y denote real numbers, i, j denote natural numbers, and p denotes a finite sequence of elements of ℝ.

One can prove the following proposition

(1) For every closed-interval subset A of ℝ and for every x holds x ∈ A iff inf A x and x supA.

Let I_{1} be a finite sequence of elements of ℝ. We say that I_{1} is non-decreasing if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let n be a natural number. Suppose n ∈ domI_{1} and n + 1 ∈ domI_{1}. Let r, s be real numbers. If r = I_{1}(n) and s = I_{1}(n + 1), then r s.

Let us observe that there exists a finite sequence of elements of ℝ which is non-decreasing.

Next we state three propositions:

(2) Let p be a non-decreasing finite sequence of elements of ℝ and given i, j. If i ∈ domp and j ∈ domp and i j, then p(i) p(j).

(3) Let given p. Then there exists a non-decreasing finite sequence q of elements of ℝ such that p and q are fiberwise equipotent.

(4) Let D be a non empty set, f be a finite sequence of elements of D, and k_{1}, k_{2}, k_{3} be natural numbers. If 1 k_{1} and k_{3} lenf and k_{1} k_{2} and k_{2} < k_{3}, then (mid(f, k_{1}, k_{2}))^{⌢} mid(f, k_{2} + 1, k_{3}) = mid(f, k_{1}, k_{3}).

2. Scalar Product of Real Subset

Let X be a subset of ℝ and let r be a real number. The functor r ·X yielding a subset of ℝ is defined as follows:

(Def. 2) r · X = {r · x : x ∈ X}. The following propositions are true:

(5) Let X, Y be non empty sets and f be a partial function from X to ℝ. If f is upper bounded on X and Y ⊆ X, then f↾Y is upper bounded on Y.

(6) Let X, Y be non empty sets and f be a partial function from X to ℝ. If f is lower bounded on X and Y ⊆ X, then f↾Y is lower bounded on Y.

(7) For every non empty subset X of ℝ holds r · X is non empty.

(8) For every subset X of ℝ holds r · X = {r · x : x ∈ X}.

(9) For every non empty subset X of ℝ such that X is upper bounded and 0 r holds r · X is upper bounded.

(10) For every non empty subset X of ℝ such that X is upper bounded and r 0 holds r · X is lower bounded.

(11) For every non empty subset X of ℝ such that X is lower bounded and 0 r holds r · X is lower bounded.

(12) For every non empty subset X of ℝ such that X is lower bounded and r 0 holds r · X is upper bounded.

(13) For every non empty subset X of ℝ such that X is upper bounded and 0 r holds sup(r · X) = r · supX.

(14) For every non empty subset X of ℝ such that X is upper bounded and r 0 holds inf(r · X) = r · supX.

(15) For every non empty subset X of ℝ such that X is lower bounded and 0 r holds inf(r · X) = r · inf X.

(16) For every non empty subset X of ℝ such that X is lower bounded and r 0 holds sup(r · X) = r · inf X.

3. Scalar Multiple of Integral

The following propositions are true:

(17) For every non empty set X and for every partial function f from X to ℝ such that f is total holds rng(r f) = r · rngf.

(18) For all non empty sets X, Z and for every partial function f from X to ℝ holds rng(r (f↾Z)) = r · rng(f↾Z).

(19) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (upper sum setr f)(D) r · inf rngf · vol(A).

(20) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (upper sum setr f)(D) r · suprngf · vol(A).

(21) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (lower sum setr f)(D) r · suprngf · vol(A).

(22) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and D be an element of divsA. If f is total and bounded on A and r 0, then (lower sum setr f)(D) r · inf rngf · vol(A).

(23) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is upper bounded on A and total and r 0. Then (upper volume(r f, D))(i) = r · (upper volume(f, D))(i).

(24) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is upper bounded on A and total and r 0. Then (lower volume(r f, D))(i) = r · (upper volume(f, D))(i).

(25) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is lower bounded on A and total and r 0. Then (lower volume(r f, D))(i) = r · (lower volume(f, D))(i).

(26) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, D be an element of S, and given i. Suppose i ∈ SeglenD and f is lower bounded on A and total and r 0. Then (upper volume(r f, D))(i) = r · (lower volume(f, D))(i).

(27) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is upper bounded on A and total and r 0, then upper sum(r f, D) = r · upper sum(f, D).

(28) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is upper bounded on A and total and r 0, then lower sum(r f, D) = r · upper sum(f, D).

(29) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is lower bounded on A and total and r 0, then lower sum(r f, D) = r · lower sum(f, D).

(30) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, S be a non empty Division of A, and D be an element of S. If f is lower bounded on A and total and r 0, then upper sum(r f, D) = r · lower sum(f, D).

(31) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. Suppose f is total and bounded on A and f is integrable on A. Then r f is integrable on A and integralr f = r · integralf.

4. Monotoneity of Integral

We now state three propositions:

(32) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. Suppose f is total and bounded on A and f is integrable on A and for every x such that x ∈ A holds f(x) 0. Then integralf 0.

(33) Let A be a closed-interval subset of ℝ and f, g be partial functions from

A to ℝ. Suppose that

(i) f is total and bounded on A, (ii) f is integrable on A, (iii) g is total and bounded on A, and (iv) g is integrable on A.

Then f − g is integrable on A and integralf − g = integralf − integralg.

(34) Let A be a closed-interval subset of ℝ and f, g be partial functions from

A to ℝ. Suppose that

(i) f is total and bounded on A, (ii) f is integrable on A, (iii) g is total and bounded on A, (iv) g is integrable on A, and (v) for every x such that x ∈ A holds f(x) g(x).

Then integralf integralg.

5. Definition of Division Sequence

The following two propositions are true:

(35) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is total and bounded on A, then rngupper sum setf is lower bounded.

(36) Let A be a closed-interval subset of ℝ and f be a partial function from A to ℝ. If f is total and bounded on A, then rnglower sum setf is upper bounded.

Let A be a closed-interval subset of ℝ. A DivSequence of A is a function from ℕ into divsA.

Let A be a closed-interval subset of ℝ and let T be a DivSequence of A. The functor δ_{T} yields a sequence of real numbers and is defined as follows:

(Def. 3) For every i holds δ_{T} (i) = δ_{T (i)}.

Let A be a closed-interval subset of ℝ, let f be a partial function from A to ℝ, and let T be a DivSequence of A. The functor upper sum(f, T) yielding a sequence of real numbers is defined as follows:

(Def. 4) For every i holds (upper sum(f, T))(i) = upper sum(f, T (i)). The functor lower sum(f, T) yields a sequence of real numbers and is defined by:

(Def. 5) For every i holds (lower sum(f, T))(i) = lower sum(f, T (i)).

The following propositions are true:

(37) Let A be a closed-interval subset of ℝ and D_{1}, D_{2} be elements of divsA. If D_{1} D_{2}, then for every j such that j ∈ domD_{2} there exists i such that i ∈ domD_{1} and divset(D_{2}, j) ⊆ divset(D_{1}, i).

(38) For all finite non empty subsets X, Y of ℝ such that X ⊆ Y holds maxX maxY.

(39) For all finite non empty subsets X, Y of ℝ such that there exists y such that y ∈ Y and maxX y holds maxX maxY.

(40) For all closed-interval subsets A, B of ℝ such that A ⊆ B holds vol(A) vol(B).

(41) For every closed-interval subset A of ℝ and for all elements D_{1}, D_{2} of divsA such that D_{1} D_{2} holds δ_{(D_{1})} δ_{(D_{2})}.

436 noboru endou et al.

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

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

Darboux’s Theorem

Noboru Endou Shinshu University

Nagano Katsumi Wasaki Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article, we have proved the Darboux’s theorem. This theorem is important to proof the Riemann integrability. We can replace an upper bound and a lower bound of a function which is the definition of Riemann integration with convergence of sequence by Darboux’s theorem.

MML Identifier: INTEGRA3.

The notation and terminology used in this paper have been introduced in the following papers: [20], [24], [23], [9], [2], [25], [4], [5], [26], [10], [6], [7], [3], [12], [7], [?], [13], [11], [23], [23], [?], [19], [18], [17], [15], [1], [21], [22], [8], [14], [16], and [?].

1. Lemmas of Division

We use the following convention: x, y denote real numbers, i, j, k denote natural numbers, and p, q denote finite sequences of elements of ℝ.

We now state a number of propositions:

(1) Let A be a closed-interval subset of ℝ and D be an element of divsA. If vol(A) ≠ 0, then there exists i such that i ∈ domD and vol(divset(D, i)) > 0.

(2) Let A be a closed-interval subset of ℝ, D be an element of divsA, and given x. If x ∈ A, then there exists j such that j ∈ domD and x ∈ divset(D, j).

(3) Let A be a closed-interval subset of ℝ and D_{1}, D_{2} be elements of divsA. Then there exists an element D of divsA such that D_{1} D and D_{2} D and rngD = rngD_{1} ∪ rngD_{2}.

(4) Let A be a closed-interval subset of ℝ and D, D_{1} be elements of divsA. Suppose δ_{(D_{1})} < minrngupper volume(χ_{A,A}, D). Let given x, y, i. If i ∈ domD_{1} and x ∈ rngD ∩ divset(D_{1}, i) and y ∈ rngD ∩ divset(D_{1}, i), then x = y.

(5) For all p, q such that rngp = rngq and p is increasing and q is increasing holds p = q.

(6) Let A be a closed-interval subset of ℝ, D, D_{1} be elements of divsA, and given i, j. Suppose D D_{1} and i ∈ domD and j ∈ domD and i j. Then indx(D_{1}, D, i) indx(D_{1}, D, j) and indx(D_{1}, D, i) ∈ domD_{1} and indx(D_{1}, D, j) ∈ domD_{1}.

(7) Let A be a closed-interval subset of ℝ, D, D_{1} be elements of divsA, and given i, j. Suppose D D_{1} and i ∈ domD and j ∈ domD and i < j. Then indx(D_{1}, D, i) < indx(D_{1}, D, j) and indx(D_{1}, D, i) ∈ domD_{1} and indx(D_{1}, D, j) ∈ domD_{1}.

(8) For every closed-interval subset A of ℝ and for every element D of divsA holds δ_{D} 0.

(9) Let A be a closed-interval subset of ℝ, g be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and lenD_{1} 2 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪{x} and g is total and bounded on A. Then ∑ lower volume(g, D_{2}) − ∑ lower volume(g, D_{1}) (suprngg − inf rngg) · δ_{(D_{1})}.

(10) Let A be a closed-interval subset of ℝ, g be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and lenD_{1} 2 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪{x} and g is total and bounded on A. Then ∑ upper volume(g, D_{1}) − ∑ upper volume(g, D_{2}) (suprngg − inf rngg) · δ_{(D_{1})}.

(11) Let A be a closed-interval subset of ℝ, D be an element of divsA, r be a real number, and i, j be natural numbers. Suppose i ∈ domD and j ∈ domD and i j and r < (mid(D, i, j))(1). Then there exists a closed-interval subset B of ℝ such that r = inf B and supB = (mid(D, i, j))(lenmid(D, i, j)) and lenmid(D, i, j) = (j − i) + 1 and mid(D, i, j) is a DivisionPoint of B.

(12) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and vol(A) ≠ 0 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪ {x} and f is total and bounded on A and x > inf A. Then ∑ lower volume(f, D_{2})−∑ lower volume(f, D_{1}) (suprngf −inf rngf)· δ_{(D_{1})}.

(13) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D_{1}, D_{2} be elements of divsA, and given x. Suppose x ∈ divset(D_{1}, lenD_{1}) and vol(A) ≠ 0 and D_{1} D_{2} and rngD_{2} = rngD_{1} ∪ {x} and f is total and bounded on A and x > inf A. Then ∑ upper volume(f, D_{1})−∑ upper volume(f, D_{2}) (suprngf−inf rngf)· δ_{(D_{1})}.

(14) Let A be a closed-interval subset of ℝ, D_{1}, D_{2} be elements of divsA, r be a real number, and i, j be natural numbers. Suppose i ∈ domD_{1} and j ∈ domD_{1} and i j and D_{1} D_{2} and r < (mid(D_{2}, indx(D_{2}, D_{1}, i), indx(D_{2}, D_{1}, j)))(1). Then there exists a closed-interval subset B of ℝ and there exist elements M_{1}, M_{2} of divsB such that r = inf B and supB = M_{2}(lenM_{2}) and supB = M_{1}(lenM_{1}) and M_{1} M_{2} and M_{1} = mid(D_{1}, i, j) and M_{2} = mid(D_{2}, indx(D_{2}, D_{1}, i), indx(D_{2}, D_{1}, j)).

(15) Let A be a closed-interval subset of ℝ, D be an element of divsA, and given x. If x ∈ rngD, then D(1) x and x D(lenD).

(16) Let p be a finite sequence of elements of ℝ and given i, j, k. Suppose p is increasing and i ∈ domp and j ∈ domp and k ∈ domp and p(i) p(k) and p(k) p(j). Then p(k) ∈ rngmid(p, i, j).

(17) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D be an element of divsA, and given i. If f is total and bounded on A and i ∈ domD, then inf rng(f↾ divset(D, i)) suprngf.

(18) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, D be an element of divsA, and given i. If f is total and bounded on A and i ∈ domD, then suprng(f↾ divset(D, i)) inf rngf.

2. Darboux’s Theorem

The following two propositions are true:

(19) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and T be a DivSequence of A. Suppose f is total and bounded on A and δ_{T} is convergent to 0 and vol(A) ≠ 0. Then lower sum(f, T) is convergent and limlower sum(f, T) = lower integralf.

(20) Let A be a closed-interval subset of ℝ, f be a partial function from A to ℝ, and T be a DivSequence of A. Suppose f is total and bounded on A and δ_{T} is convergent to 0 and vol(A) ≠ 0. Then upper sum(f, T) is convergent and limupper sum(f, T) = upper integralf.

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Five Variable Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC22.

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

1. Preliminaries

For simplicity, we adopt the following convention: Y denotes a non empty set, a denotes an element of BVF(Y), G denotes a subset of PARTITIONS(Y), and A, B, C, D, E denote partitions of Y.

Next we state a number of propositions:

(1) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(A, G) = B ∧ C ∧ D ∧ E.

(2) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(B, G) = A ∧ C ∧ D ∧ E.

(3) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(C, G) = A ∧ B ∧ D ∧ E.

(4) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(D, G) = A ∧ B ∧ C ∧ E.

(5) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then CompF(E, G) = A ∧ B ∧ C ∧ D.

(6) Let A, B, C, D, E be sets, h be a function, and A′, B′, C′, D′, E′ be sets. Suppose A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′.

(7) Let A, B, C, D, E be sets, h be a function, and A′, B′, C′, D′, E′ be sets. Suppose A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E}.

(8) Let A, B, C, D, E be sets, h be a function, and A′, B′, C′, D′, E′ be sets. Suppose A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E)}.

(9) Let G be a subset of PARTITIONS(Y), A, B, C, D, E be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then EqClass(u, B ∧ C ∧ D ∧ E) ∩ EqClass(z, A) ≠ ∅.

(10) Let G be a subset of PARTITIONS(Y), A, B, C, D, E be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E and EqClass(z, C ∧ D ∧ E) = EqClass(u, C ∧ D ∧ E). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Predicate Calculus

The following propositions are true:

(11) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(12) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(13) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(14) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(15) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(16) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(17) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(18) ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(19) ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(20) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(22)^{1} Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and

1The proposition (21) has been removed.

C ≠ E and D ≠ E. Then ∃_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

(23) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ¬∀_{∀_{a,A}G,B}G = ∃_{¬∀_{a,B}G,A} G.

(24) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ¬∀_{∀_{a,A}G,B}G = ∃_{∃_{¬a,B} G,A}G.

(25) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{¬∀_{a,A}G,B} G ⋐ ¬∀_{∀_{a,B}G,A}G.

(26) Suppose that G is a coordinate and G = {A, B, C, D, E} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and B ≠ C and B ≠ D and B ≠ E and C ≠ D and C ≠ E and D ≠ E. Then ∀_{¬∀_{a,A}G,B} G ⋐ ∃_{∃_{¬a,B} G,A}G.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

Six Variable Predicate Calculus for Boolean Valued Functions. Part I

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC23.

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

1. Preliminaries

For simplicity, we use the following convention: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), and A, B, C, D, E, F are partitions of Y.

We now state a number of propositions:

(1) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(A, G) = B ∧ C ∧ D ∧ E ∧ F.

(2) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(B, G) = A ∧ C ∧ D ∧ E ∧ F.

(3) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(C, G) = A ∧ B ∧ D ∧ E ∧ F.

(4) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(D, G) = A ∧ B ∧ C ∧ E ∧ F.

(5) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(E, G) = A ∧ B ∧ C ∧ D ∧ F.

(6) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then CompF(F, G) = A ∧ B ∧ C ∧ D ∧ E.

(7) Let A, B, C, D, E, F be sets, h be a function, and A′, B′, C′, D′, E′, F′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′ and h(F) = F′.

(8) Let A, B, C, D, E, F be sets, h be a function, and A′, B′, C′, D′, E′, F′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F }.

(9) Let A, B, C, D, E, F be sets, h be a function, and A′, B′, C′, D′, E′, F′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(A⟼→ . A′). Then

rngh = {h(A), h(B), h(C), h(D), h(E), h(F)}.

(10) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then EqClass(u, B ∧ C ∧ D ∧ E ∧ F) ∩ EqClass(z, A) ≠ ∅.

(11) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F and EqClass(z, C ∧ D ∧ E ∧ F) = EqClass(u, C ∧ D ∧ E ∧ F). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Predicate Calculus

One can prove the following propositions:

(12) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(13) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(14) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(15) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(16) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(17) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(18) ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(19) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(20) ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(21) Suppose that G is a coordinate and G = {A, B, C, D, E, F } and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and B ≠ C and B ≠ D and B ≠ E and B ≠ F and C ≠ D and C ≠ E and C ≠ F and D ≠ E and D ≠ F and E ≠ F. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

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

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

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

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

The Construction and Computation of For-loop Programs for SCMPDS^{1}

Jing-Chao Chen Shanghai Jiaotong University

Piotr Rudnicki University of Alberta Edmonton

Summary. This article defines two for-loop statements for SCMPDS. One is called for-up, which corresponds to ”for(i=x;i¡0;i+=n) S” in C language. Another is called for-down, which corresponds to ”for(i=x;i¿0;i-=n) S”. Here, we do not present their unconditional halting (called parahalting) property, because we have not found that there exists a useful for-loop statement with unconditional halting, and the proof of unconditional halting is much simpler than that of conditional halting. It is hard to formalize all halting conditions,but some cases can be formalized. We choose loop invariants as halting conditions to prove halting problem of for-up/down statements.When some variables(except the loop control variable) keep undestroyed on a set for the loop invariant, and the loop body is halting for this condition, the corresponding for-up/down is halting and computable under this condition. The computation of for-loop statements can be realized by evaluating its body.At the end of the article, we verify for-down statements by two examples for summing

MML Identifier: SCMPDS 7.

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

For simplicity, we follow the rules: x denotes a set, n denotes a natural number, a denotes a Int position, i, j, k denote instructions of SCMPDS, s, s_{1}, s_{2} denote states of SCMPDS, l_{1}, l denote instructions-locations of SCMPDS, and I, J, K denote Program-block.

We now state a number of propositions:

1This research is partially supported by the National Natural Science Foundation of China Grant No. 69873033.

(1) For every state s of SCMPDS and for all natural numbers m, n such that IC_{s} = insposm holds ICplusConst(s, n − m) = insposn.

(2) For all finite partial states P, Q of SCMPDS such that P ⊆ Q holds ProgramPart(P) ⊆ ProgramPart(Q).

(3) For all programmed finite partial states P, Q of SCMPDS and for every natural number k such that P ⊆ Q holds Shift(P, k) ⊆ Shift(Q, k).

(4) If IC_{s} = inspos0, then Initialized(s) = s.

(5) If IC_{s} = inspos0, then s+·Initialized(I) = s+·I.

(6) (Computation(s))(n)↾the instruction locations of SCMPDS = s↾the instruction locations of SCMPDS.

(7) Let s_{1}, s_{2} be states of SCMPDS. Suppose IC_{(s_{1})} = IC_{(s_{2})} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM} and s_{1}↾the instruction locations of SCMPDS = s_{2}↾the instruction locations of SCMPDS. Then s_{1} = s_{2}.

(8) l ∈ domI iff l ∈ domInitialized(I).

(9) If x ∈ domI, then I(x) = (s+·(I+·Start-At(l)))(x).

(10) If l_{1} ∈ domI, then (s+·Initialized(I))(l_{1}) = I(l_{1}).

(11) (s+·(I+·Start-At(l)))(a) = s(a).

(12) (s+·Start-At(l_{1}))(IC_{SCMPDS} ) = l_{1}.

(13) card(I;i) = cardI + 1.

(14) (I;i;j)(insposcardI) = i.

(15) (i;I;j);k = i;(I;j;k).

(16) Shift(J, cardI) ⊆ I;J;K.

(17) I ⊆ stopI;J.

(18) If l_{1} ∈ domI, then (Shift(stopI, n))(l_{1} + n) = (Shift(I, n))(l_{1} + n).

(19) If cardI > 0, then (Shift(stopI, n))(insposn) = (Shift(I, n))(insposn).

(20) For every state s of SCMPDS and for every instruction i of SCMPDS such that InsCode(i) ∈ {0, 4, 5, 6} holds Exec(i, s)↾Data-Loc_{SCM} = s↾Data-Loc_{SCM}.

(21) For all states s, s_{3} of SCMPDS holds (s+·s_{3}↾the instruction locations of SCMPDS)↾Data-Loc_{SCM} = s↾Data-Loc_{SCM}.

(22) For every instruction i of SCMPDS holds rngLoad(i) = {i}.

(23) If IC_{(s_{1})} = IC_{(s_{2})} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}, then IC_{Exec(i,s_{1})} = IC_{Exec(i,s_{2})} and Exec(i, s_{1})↾Data-Loc_{SCM} = Exec(i, s_{2})↾Data-Loc_{SCM}.

(24) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and Initialized(stopI) ⊆ s_{1} and Initialized(stopI) ⊆ s_{2} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let i be a natural number. Then IC_{(Computation (s_{1}))(i)} = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

(25) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and halting on s_{1} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let k be a natural number. Then (Computation(s_{1}+·Initialized(stopI)))(k) and (Computation(s_{2}+·Initialized(stopI)))(k) are equal outside the instruction locations of SCMPDS and CurInstr((Computation(s_{1}+·Initialized(stopI)))(k)) = CurInstr((Computation(s_{2}+·Initialized(stopI)))(k)).

(26) Let I be a Program-block. Suppose that (i) I is closed on s_{1} and halting on s_{1}, (ii) Initialized(stopI) ⊆ s_{1}, (iii) Initialized(stopI) ⊆ s_{2}, and (iv) s_{1} and s_{2} are equal outside the instruction locations of SCMPDS.

Let k be a natural number. Then (Computation(s_{1}))(k) and (Computation(s_{2}))(k) are equal outside the instruction locations of SCMPDS and CurInstr((Computation(s_{1}))(k)) = CurInstr((Computation(s_{2}))(k)).

(27) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and halting on s_{1} and Initialized(stopI) ⊆ s_{1} and Initialized(stopI) ⊆ s_{2} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then LifeSpan(s_{1}) = LifeSpan(s_{2}).

(28) Let I be a Program-block. Suppose that (i) I is closed on s_{1} and halting on s_{1}, (ii) Initialized(stopI) ⊆ s_{1}, (iii) Initialized(stopI) ⊆ s_{2}, and (iv) s_{1} and s_{2} are equal outside the instruction locations of SCMPDS.

Then LifeSpan(s_{1}) = LifeSpan(s_{2}) and Result(s_{1}) and Result(s_{2}) are equal outside the instruction locations of SCMPDS.

(29) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose I is closed on s_{1} and halting on s_{1} and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then LifeSpan(s_{1}+·Initialized(stopI)) = LifeSpan(s_{2}+·Initialized(stopI)) and Result(s_{1}+·Initialized(stopI)) and Result(s_{2}+·Initialized(stopI)) are equal outside the instruction locations of SCMPDS.

(30) Let s_{1}, s_{2} be states of SCMPDS and I be a Program-block. Suppose

that

(i) I is closed on s_{1} and halting on s_{1}, (ii) Initialized(stopI) ⊆ s_{1}, (iii) Initialized(stopI) ⊆ s_{2}, and (iv) there exists a natural number k such that (Computation(s_{1}))(k) and

s_{2} are equal outside the instruction locations of SCMPDS. Then Result(s_{1}) and Result(s_{2}) are equal outside the instruction locations of SCMPDS.

Let I be a Program-block. One can verify that Initialized(I) is initial. One can prove the following propositions:

(31) Let s be a state of SCMPDS, I be a Program-block, and a be a Int position. If I is halting on s, then (IExec(I, s))(a) = (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI)))(a).

(32) Let s be a state of SCMPDS, I be a parahalting Program-block, and a be a Int position. Then (IExec(I, s))(a) = (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI)))(a).

(33) Let I be a Program-block and i be a natural number. If Initialized(stopI) ⊆ s and I is closed on s and halting on s and i < LifeSpan(s), then IC_{(Computation (s))(i)} ∈ domI.

(34) Let I be a shiftable Program-block. Suppose Initialized(stopI) ⊆ s_{1} and I is closed on s_{1} and halting on s_{1}. Let n be a natural number. Suppose Shift(I, n) ⊆ s_{2} and cardI > 0 and IC_{(s_{2})} = insposn and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Let i be a natural number. If i < LifeSpan(s_{1}), then IC_{(Computation (s_{1}))(i)} +n = IC_{(Computation (s_{2}))(i)} and CurInstr((Computation(s_{1}))(i)) = CurInstr((Computation(s_{2}))(i)) and (Computation(s_{1}))(i)↾Data-Loc_{SCM} = (Computation(s_{2}))(i)↾Data-Loc_{SCM}.

(35) For every No-StopCode Program-block I such that Initialized(stopI) ⊆ s and I is halting on s and cardI > 0 holds LifeSpan(s) > 0.

(36) Let I be a No-StopCode shiftable Program-block. Suppose Initialized(stopI) ⊆ s_{1} and I is closed on s_{1} and halting on s_{1}. Let n be a natural number. Suppose Shift(I, n) ⊆ s_{2} and cardI > 0 and IC_{(s_{2})} = insposn and s_{1}↾Data-Loc_{SCM} = s_{2}↾Data-Loc_{SCM}. Then IC_{(Computation (s_{2}))(LifeSpan (s_{1}))} = insposcardI+n and (Computation(s_{1}))(LifeSpan(s_{1}))↾Data-Loc_{SCM} = (Computation(s_{2}))(LifeSpan(s_{1}))↾Data-Loc_{SCM}.

(37) Let s be a state of SCMPDS, I be a Program-block, and n be a natural number. If IC_{(Computation (s+·Initialized (I)))(n)} = inspos0, then (Computation(s+·Initialized(I)))(n)+·Initialized(I) = (Computation(s+·Initialized(I)))(n).

(38) Let I be a Program-block, J be a Program-block, and k be a natural number. Suppose I is closed on s and halting on s and k LifeSpan(s+·Initialized(stopI)). Then (Computation(s+·Initialized(stopI)))(k) and (Computation(s+·((I;J)+·Start-At(inspos0))))(k) are equal outside the instruction locations of SCMPDS.

(39) Let I, J be Program-block and k be a natural number. Suppose I ⊆ J and I is closed on s and halting on s and k LifeSpan(s+·Initialized(stopI)). Then (Computation(s+·Initialized(J)))(k) and (Computation(s+·Initialized(stopI)))(k) are equal outside the instruction locations of SCMPDS.

(40) Let I, J be Program-block and k be a natural number. Suppose k LifeSpan(s+·Initialized(stopI)) and I ⊆ J and I is closed on s and halting

on s. Then IC_{(Computation (s+·Initialized (J)))(k)} ∈ domstopI.

(41) Let I, J be Program-block. Suppose I ⊆ J and I is closed on s and halting on s. Then CurInstr((Computation(s+·Initialized(J)))(LifeSpan(s+·Initialized(stopI)))) = halt_{SCMPDS} or IC_{(Computation (s+·Initialized (J)))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(42) Let I, J be Program-block. Suppose I is halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then J is closed on (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI))) and halting on (Computation(s+·Initialized(stopI)))(LifeSpan(s+·Initialized(stopI))).

(43) Let I be a Program-block and J be a shiftable Program-block. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then I;J is closed on s and I;J is halting on s.

(44) Let I be a No-StopCode Program-block and J be a Programblock. If I ⊆ J and I is closed on s and halting on s, then IC_{(Computation (s+·Initialized (J)))(LifeSpan (s+·Initialized (stop I)))} = insposcardI.

(45) Let I be a Program-block, s be a state of SCMPDS, and k be a natural number. If I is halting on s and k < LifeSpan(s+·Initialized(stopI)), then CurInstr((Computation(s+·Initialized(stopI)))(k)) ≠ halt_{SCMPDS} .

(46) Let I, J be Program-block, s be a state of SCMPDS, and k be a natural number. Suppose I is closed on s and halting on s and k < LifeSpan(s+·Initialized(stopI)). Then CurInstr((Computation(s+·Initialized(stopI;J)))(k)) ≠ halt_{SCMPDS} .

(47) Let I be a No-StopCode Program-block and J be a shiftable Program-block. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then LifeSpan(s+·Initialized(stopI;J)) = LifeSpan(s+·Initialized(stopI)) + LifeSpan(Result(s+·Initialized(stopI))+·Initialized(stopJ)).

(48) Let I be a No-StopCode Program-block and J be a shiftable Programblock. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then IExec(I;J, s) = IExec(J, IExec(I, s))+·Start-At(IC_{IExec(J,IExec(I,s))} + cardI).

(49) Let I be a No-StopCode Program-block and J be a shiftable Programblock. Suppose I is closed on s and halting on s and J is closed on IExec(I, s) and halting on IExec(I, s). Then (IExec(I;J, s))(a) = (IExec(J, IExec(I, s)))(a).

(50) Let I be a No-StopCode Program-block and j be a parahalting shiftable instruction of SCMPDS. If I is closed on s and halting on s, then (IExec(I;j, s))(a) = (Exec(j, IExec(I, s)))(a).

2. ?????????????????????

Let a be a Int position, let i be an integer, let n be a natural number, and let I be a Program-block. The functor for-up(a, i, n, I) yields a Program-block and is defined by:

(Def. 1) for-up(a, i, n, I) = ((a, i) >= 0 goto cardI+3);I; AddTo(a, i, n);goto (− (cardI + 2)).

3. ?????????????????????

The following propositions are true:

(51) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then cardfor-up(a, i, n, I) = cardI + 3.

(52) Let a be a Int position, i be an integer, n, m be natural numbers, and I be a Program-block. Then m < cardI + 3 if and only if insposm ∈ domfor-up(a, i, n, I).

(53) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then (for-up(a, i, n, I))(inspos0) = (a, i) >= 0 goto cardI + 3 and (for-up(a, i, n, I))(insposcardI + 1) = AddTo(a, i, n) and (for-up(a, i, n, I))(insposcardI + 2) = goto (− (cardI + 2)).

(54) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then for-up(a, i, n, I) is closed on s and for-up(a, i, n, I) is halting on s.

(55) Let s be a state of SCMPDS, I be a Program-block, a, c be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IExec(for-up(a, i, n, I), s) = s+·Start-At(insposcardI + 3).

(56) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IC_{IExec(for-up(a,i,n,I),s)} = insposcardI + 3.

(57) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then (IExec(for-up(a, i, n, I), s))(b) = s(b).

(58) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) < 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then for-up(a, i, n, I) is closed on s and for-up(a, i, n, I) is halting on s.

(59) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) < 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then IExec(for-up(a, i, n, I), s) = IExec(for-up(a, i, n, I), IExec(I; AddTo(a, i, n), s)).

Let I be a shiftable Program-block, let a be a Int position, let i be an integer, and let n be a natural number. One can verify that for-up(a, i, n, I) is shiftable.

Let I be a No-StopCode Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Note that for-up(a, i, n, I) is No-StopCode.

4. ?????????????????????

Let a be a Int position, let i be an integer, let n be a natural number, and let I be a Program-block. The functor for − down (a, i, n, I) yielding a Programblock is defined as follows:

(Def. 2) for − down (a, i, n, I) = ((a, i) <= 0 goto cardI+3);I; AddTo(a, i, − n);goto (− (cardI + 2)).

5. ?????????????????????

Next we state several propositions:

(60) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then cardfor − down (a, i, n, I) = cardI + 3.

(61) Let a be a Int position, i be an integer, n, m be natural numbers, and I be a Program-block. Then m < cardI + 3 if and only if insposm ∈ domfor − down (a, i, n, I).

(62) Let a be a Int position, i be an integer, n be a natural number, and I be a Program-block. Then (for − down (a, i, n, I))(inspos0) = (a, i) <= 0 goto cardI + 3 and (for − down (a, i, n, I))(insposcardI + 1) = AddTo(a, i, − n) and (for − down (a, i, n, I))(insposcardI + 2) = goto (− (cardI + 2)).

(63) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then for − down (a, i, n, I) is closed on s and for − down (a, i, n, I) is halting on s.

(64) Let s be a state of SCMPDS, I be a Program-block, a, c be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IExec(for − down (a, i, n, I), s) = s+·Start-At(insposcardI + 3).

(65) Let s be a state of SCMPDS, I be a Program-block, a be a Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then IC_{IExec(for−down (a,i,n,I),s)} = insposcardI + 3.

(66) Let s be a state of SCMPDS, I be a Program-block, a, b be Int position, i be an integer, and n be a natural number. If s(DataLoc(s(a), i)) 0, then (IExec(for − down (a, i, n, I), s))(b) = s(b).

(67) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) > 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then for − down (a, i, n, I) is closed on s and for − down (a, i, n, I) is halting on s.

(68) Let s be a state of SCMPDS, I be a No-StopCode shiftable Programblock, a be a Int position, i be an integer, n be a natural number, and X be a set. Suppose that

(i) s(DataLoc(s(a), i)) > 0, (ii) DataLoc(s(a), i) ∉ X, (iii) n > 0, (iv) cardI > 0,

(v) a ≠ DataLoc(s(a), i), and (vi) for every state t of SCMPDS such that for every Int position x such

that x ∈ X holds t(x) = s(x) and t(a) = s(a) holds (IExec(I, t))(a) = t(a) and (IExec(I, t))(DataLoc(s(a), i)) = t(DataLoc(s(a), i)) and I is closed on t and halting on t and for every Int position y such that y ∈ X holds (IExec(I, t))(y) = t(y). Then IExec(for − down (a, i, n, I), s) = IExec(for − down (a, i, n, I), IExec(I; AddTo(a, i, − n), s)).

Let I be a shiftable Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Note that for − down (a, i, n, I) is shiftable.

Let I be a No-StopCode Program-block, let a be a Int position, let i be an integer, and let n be a natural number. Note that for − down (a, i, n, I) is No-StopCode.

6. ?????????????????????

Let n be a natural number. The functor sumn yielding a Program-block is defined as follows:

(Def. 3) sumn = (GBP:=0);((GBP)_{2}:=n);((GBP)_{3}:=0); for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))).

The following propositions are true:

(69) For every state s of SCMPDS such that s(GBP) = 0 holds for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))) is closed on s and for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))) is halting on s.

(70) Let s be a state of SCMPDS and n be a natural number. If s(GBP) = 0 and s(intpos2) = n and s(intpos3) = 0, then (IExec(for − down (GBP, 2, 1, Load(AddTo(GBP, 3, 1))), s))(intpos3) = n.

(71) For every state s of SCMPDS and for every natural number n holds

(IExec(sumn, s))(intpos3) = n. Let s_{4}, c_{1}, r_{1}, p_{1}, p_{2} be natural numbers. The functor sum(s_{4}, c_{1}, r_{1}, p_{1}, p_{2})

yields a Program-block and is defined by: (Def. 4) sum(s_{4}, c_{1}, r_{1}, p_{1}, p_{2}) = ((intposs_{4})_{r_{1}}:=0);(intposp_{1}:=p_{2}); for − down (intposs_{4}, c_{1}, 1, AddTo(intposs_{4}, r_{1}, intposp_{2}, 0); AddTo(intposp_{1}, 0, 1)).

One can prove the following three propositions:

(72) Let s be a state of SCMPDS and s_{4}, c_{2}, r_{1}, p_{1}, p_{3} be natural numbers. Suppose s(intposs_{4}) > s_{4} and c_{2} < r_{1} and s(intposp_{1}) = p_{3} and s(intposs_{4}) + r_{1} < p_{1} and p_{1} < p_{3} and p_{3} < s(intposp_{3}). Then for − down (intposs_{4}, c_{2}, 1, AddTo(intposs_{4}, r_{1}, intposp_{3}, 0); AddTo(intposp_{1}, 0, 1))

458 jing-chao chen and piotr rudnicki

is closed on s and for − down (intposs_{4}, c_{2}, 1, AddTo(intposs_{4}, r_{1}, intposp_{3}, 0); AddTo(intposp_{1}, 0, 1)) is halting on s.

(73) Let s be a state of SCMPDS, s_{4}, c_{2}, r_{1}, p_{1}, p_{3} be natural numbers, and f be a finite sequence of elements of ℕ. Suppose that s(intposs_{4}) > s_{4} and c_{2} < r_{1} and s(intposp_{1}) = p_{3} and s(intposs_{4})+r_{1} < p_{1} and p_{1} < p_{3} and p_{3} < s(intposp_{3}) and s(DataLoc(s(intposs_{4}), r_{1})) = 0 and lenf = s(DataLoc(s(intposs_{4}), c_{2})) and for every natural number k such that k < lenf holds f(k + 1) = s(DataLoc(s(intposp_{3}), k)). Then (IExec(for − down (intposs_{4}, c_{2}, 1, AddTo(intposs_{4}, r_{1}, intposp_{3}, 0); AddTo(intposp_{1}, 0, 1)), s))(DataLoc(s(intposs_{4}), r_{1})) = ∑ f.

(74) Let s be a state of SCMPDS, s_{4}, c_{2}, r_{1}, p_{1}, p_{3} be natural numbers, and f be a finite sequence of elements of ℕ. Suppose that s(intposs_{4}) > s_{4} and c_{2} < r_{1} and s(intposs_{4}) + r_{1} < p_{1} and p_{1} < p_{3} and p_{3} < s(intposp_{3}) and lenf = s(DataLoc(s(intposs_{4}), c_{2})) and for every natural number k such that k < lenf holds f(k + 1) = s(DataLoc(s(intposp_{3}), k)). Then (IExec(sum(s_{4}, c_{2}, r_{1}, p_{1}, p_{3}), s))(DataLoc(s(intposs_{4}), r_{1})) = ∑ f.

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 Piotr Rudnicki. Development of terminology for scm. Formalized Mathematics, 4(1):61–67, 1993.

[5] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[7] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

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

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

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

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

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

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

the construction and computation of for-loop … 459 Received December 27, 1999 460 jing-chao chen and piotr rudnicki

Predicate Calculus for Boolean Valued Functions. Part XII

Shunichi Kobayashi Ueda Multimedia Information Center Nagano

Summary. In this paper, we proved some elementary predicate calculus formulae containing the quantifiers of Boolean valued functions with respect to partitions. Such a theory is an analogy of ordinary predicate logic.

MML Identifier: BVFUNC24.

The terminology and notation used in this paper have been introduced in the following papers: [7], [9], [8], [6], [4], [11], [2], [1], [3], [10], [5], [5], [9], [1], and [2].

1. Preliminaries

For simplicity, we follow the rules: Y is a non empty set, a is an element of BVF(Y), G is a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M, N are partitions of Y, and x, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9} are sets.

We now state a number of propositions:

(1) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(A, G) = B ∧ C ∧ D ∧ E ∧ F ∧ J.

(2) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(B, G) = A ∧ C ∧ D ∧ E ∧ F ∧ J.

(3) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(C, G) = A ∧ B ∧ D ∧ E ∧ F ∧ J.

(4) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(D, G) = A ∧ B ∧ C ∧ E ∧ F ∧ J.

(5) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(E, G) = A ∧ B ∧ C ∧ D ∧ F ∧ J.

(6) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(F, G) = A ∧ B ∧ C ∧ D ∧ E ∧ J.

(7) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then CompF(J, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F.

(8) Let A, B, C, D, E, F, J be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′ and h(F) = F′ and h(J) = J′.

(9) Let A, B, C, D, E, F, J be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F, J}.

(10) Let A, B, C, D, E, F, J be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F), h(J)}.

(11) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then EqClass(u, B ∧ C ∧ D ∧ E ∧ F ∧ J) ∩ EqClass(z, A) ≠ ∅.

(12) Let G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J and EqClass(z, C ∧D ∧E ∧F ∧J) = EqClass(u, C ∧D ∧E ∧F ∧J). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

(13) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(A, G) = B ∧C ∧D ∧E ∧F ∧J ∧M.

464 shunichi kobayashi (14) Suppose that

G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(B, G) = A∧C ∧D ∧E ∧F ∧J ∧M.

(15) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(C, G) = A∧B ∧D ∧E ∧F ∧J ∧M.

(16) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(D, G) = A∧B ∧C ∧E ∧F ∧J ∧M.

(17) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(E, G) = A∧B ∧C ∧D ∧F ∧J ∧M.

(18) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(F, G) = A∧B ∧C ∧D ∧E ∧J ∧M.

(19) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J

and F ≠ M and J ≠ M. Then CompF(J, G) = A∧B ∧C ∧D ∧E ∧F ∧M.

(20) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then CompF(M, G) = A∧B ∧C ∧D ∧E ∧F ∧J.

(21) Let A, B, C, D, E, F, J, M be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(M⟼→ . M′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′ and h(F) = F′ and h(J) = J′ and h(M) = M′.

(22) Let A, B, C, D, E, F, J, M be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(M⟼→ . M′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F, J, M}.

(23) Let A, B, C, D, E, F, J, M be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(M⟼→ . M′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F), h(J), h(M)}.

(24) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then EqClass(u, B∧C∧D∧E∧F ∧J∧M)∩EqClass(z, A) ≠ ∅.

(25) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M and EqClass(z, C ∧ D ∧ E ∧ F ∧ J ∧ M) = EqClass(u, C ∧ D ∧ E ∧ F ∧ J ∧ M). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

The scheme UI10 deals with a set A, a set B, a set C, a set D, a set E, a set F, a set G, a set H, a set I, a set J , and and states that:

P[A, B, C, D, E, F, G, H, I, J] provided the parameters meet the following condition:

• For all sets x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10} holds P[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}].

Let us consider x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}. The functor {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} yielding a set is defined by:

(Def. 1) x ∈ {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} iff x = x_{1} or x = x_{2} or x = x_{3} or

x = x_{4} or x = x_{5} or x = x_{6} or x = x_{7} or x = x_{8} or x = x_{9}. Next we state a number of propositions:

(26) x ∈ {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} iff x = x_{1} or x = x_{2} or x = x_{3} or x = x_{4} or x = x_{5} or x = x_{6} or x = x_{7} or x = x_{8} or x = x_{9}.

(27) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}} ∪ {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(28) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}} ∪ {x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(29) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}} ∪ {x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(30) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}} ∪ {x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}.

(31) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}} ∪ {x_{6}, x_{7}, x_{8}, x_{9}}.

(32) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}} ∪ {x_{7}, x_{8}, x_{9}}.

(33) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}} ∪ {x_{8}, x_{9}}.

(34) {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}} ∪ {x_{9}}.

(35) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(A, G) = B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N.

(36) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(B, G) = A ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N.

(37) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(C, G) = A ∧ B ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N.

(38) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(D, G) = A ∧ B ∧ C ∧ E ∧ F ∧ J ∧ M ∧ N.

(39) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(E, G) = A ∧ B ∧ C ∧ D ∧ F ∧ J ∧ M ∧ N.

(40) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(F, G) = A ∧ B ∧ C ∧ D ∧ E ∧ J ∧ M ∧ N.

(41) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(J, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ M ∧ N.

(42) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(M, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ N.

(43) Let G be a subset of PARTITIONS(Y) and A, B, C, D, E, F, J, M, N be partitions of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then CompF(N, G) = A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M.

(44) Let A, B, C, D, E, F, J, M, N be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′, N′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(M⟼→ . M′)+·(N⟼→ . N′)+·(A⟼→ . A′). Then h(A) = A′ and h(B) = B′ and h(C) = C′ and h(D) = D′ and h(E) = E′and h(F) = F′and h(J) = J′and h(M) = M′and h(N) = N′.

(45) Let A, B, C, D, E, F, J, M, N be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′, N′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(M⟼→ . M′)+·(N⟼→ . N′)+·(A⟼→ . A′). Then domh = {A, B, C, D, E, F, J, M, N}.

(46) Let A, B, C, D, E, F, J, M, N be sets, h be a function, and A′, B′, C′, D′, E′, F′, J′, M′, N′ be sets. Suppose that A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and h = (B⟼→ . B′)+·(C⟼→ . C′)+·(D⟼→ . D′)+·(E⟼→ . E′)+·(F ⟼→ . F′)+·(J⟼→ . J′)+·(M⟼→ . M′)+·(N⟼→ . N′)+·(A⟼→ . A′). Then rngh = {h(A), h(B), h(C), h(D), h(E), h(F), h(J), h(M), h(N)}.

(47) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M, N be partitions of Y, z, u be elements of Y, and h be a function. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then EqClass(u, B ∧ C ∧ D ∧ E ∧ F ∧ J ∧ M ∧ N) ∩ EqClass(z, A) ≠ ∅.

(48) Let a be an element of BVF(Y), G be a subset of PARTITIONS(Y), A, B, C, D, E, F, J, M, N be partitions of Y, and z, u be elements of Y. Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N and EqClass(z, C∧D∧E∧F ∧J∧M∧N) = EqClass(u, C∧D∧E∧F ∧J∧M∧N). Then EqClass(u, CompF(A, G)) ∩ EqClass(z, CompF(B, G)) ≠ ∅.

2. Predicate Calculus

We now state a number of propositions:

(49) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(50) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(51) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

predicate calculus for boolean valued … 471

(52) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(53) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(54) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(55) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(56) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(57) ∀_{∃_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(58) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and C ≠ D and C ≠ E and C ≠ F and C ≠ J and D ≠ E and D ≠ F and D ≠ J and E ≠ F and E ≠ J and F ≠ J. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(59) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D 472 shunichi kobayashi

and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(60) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(61) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(62) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(63) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(64) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(66)^{1} Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

1The proposition (65) has been removed.

(67) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and D ≠ E and D ≠ F and D ≠ J and D ≠ M and E ≠ F and E ≠ J and E ≠ M and F ≠ J and F ≠ M and J ≠ M. Then ∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(68) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∀_{a,B}G,A}G.

(69) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G = ∀_{∀_{a,B}G,A}G.

(70) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(71) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∃_{a,B}G,A}G ⋐ ∃_{∃_{a,A}G,B}G.

(72) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∃_{∃_{a,A}G,B}G = ∃_{∃_{a,B}G,A}G.

(73) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G ⋐ ∃_{∀_{a,B}G,A}G.

(74) ∀_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

(75) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then ∀_{∀_{a,A}G,B}G ⋐ ∀_{∃_{a,B}G,A}G.

(76) Suppose that G is a coordinate and G = {A, B, C, D, E, F, J, M, N} and A ≠ B and A ≠ C and A ≠ D and A ≠ E and A ≠ F and A ≠ J and A ≠ M and A ≠ N and B ≠ C and B ≠ D and B ≠ E and B ≠ F and B ≠ J and B ≠ M and B ≠ N and C ≠ D and C ≠ E and C ≠ F and C ≠ J and C ≠ M and C ≠ N and D ≠ E and D ≠ F and D ≠ J and D ≠ M and D ≠ N and E ≠ F and E ≠ J and E ≠ M and E ≠ N and F ≠ J and F ≠ M and F ≠ N and J ≠ M and J ≠ N and M ≠ N. Then

∃_{∀_{a,A}G,B}G ⋐ ∃_{∃_{a,B}G,A}G.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[3] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

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

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

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

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

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

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

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

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

Index of MML Identifiers

BORSUK 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 BVFUNC 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 BVFUNC 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 BVFUNC 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 BVFUNC 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 BVFUNC 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 BVFUNC 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161 FRECHET2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 FSCIRC 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 GATE 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 GATE 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 GATE 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 GATE 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 GENEALG1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 GOBRD13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 GOBRD14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 HILBERT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 HILBERT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 INTEGRA1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 INT 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 JORDAN2C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 JORDAN8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 LATTICE6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 REVROT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 SCMISORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 SCMPDS 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175 SCMPDS 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183 SCMPDS 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 SCMPDS 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 SCMPDS 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211 SCMPDS 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .219 SIN COS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 SPRECT 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 TOPREAL6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 TOPREAL7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Contents Formaliz. Math. 8 (1)

Bounded Domains and Unbounded Domains

By Yatsuka Nakamura et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Rotating and Reversing

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

On the Components of the Complement of a Special Polygonal

Curve

By Andrzej Trybulec and Yatsuka Nakamura  . . . . . . . . . . . 21

Gauges

By Czesław Byliński  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

The Ring of Integers, Euclidean Rings and Modulo Integers

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Logic Gates and Logical Equivalence of Adders

By Yatsuka Nakamura  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

The Sequential Closure Operator in Sequential and Frechet Spa-

ces

By Bartłomiej Skorulski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Properties of the Product of Compact Topological Spaces

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Compactness of the Bounded Closed Subsets of E_{T}^{2}

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Hilbert Positive Propositional Calculus

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Homeomorphism between [:E_{T}^{i}, E_{T}^{j}:] and E_{T}^{i+j}

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Full Subtracter Circuit. Part I

By Katsumi Wasaki and Noboru Endou  . . . . . . . . . . . . . . . . . . . 77

Correctness of Binary Counter Circuits

By Yuguang Yang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Correctness of Johnson Counter Circuits

By Yuguang Yang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

The Definition of the Riemann Definite Integral and some Related

Lemmas

By Noboru Endou and Artur Korniłowicz  . . . . . . . . . . . . . . . 93

Properties of the Trigonometric Function

By Takashi Mitsuishi and Yuguang Yang  . . . . . . . . . . . . . . . . . 103

Predicate Calculus for Boolean Valued Functions. Part II

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 107

Propositional Calculus for Boolean Valued Functions. Part I

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 111

Propositional Calculus for Boolean Valued Functions. Part II

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 115

Insert Sort on SCM _{FSA}

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Correctness of a Cyclic Redundancy Check Code Generator

By Yuguang Yang et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Defining by Structural Induction in the Positive Propositional

Language

By Andrzej Trybulec  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Some Properties of Cells on Go-Board

By Czesław Byliński  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Propositional Calculus for Boolean Valued Functions. Part III

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Propositional Calculus for Boolean Valued Functions. Part IV

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Basic Properties of Genetic Algorithm

By Akihiko Uchibori and Noboru Endou  . . . . . . . . . . . . . . . . . 151

Propositional Calculus for Boolean Valued Functions. Part V

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Properties of Left and Right Components

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Noetherian Lattices

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A Small Computer Model with Push-Down Stack

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

The SCMPDS Computer and the Basic Semantics of its Instruc-

tions

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Computation and Program Shift in the SCMPDS Computer

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

The Construction and Shiftability of Program Blocks for

SCMPDS

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Computation of Two Consecutive Program Blocks for SCMPDS

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

The Construction and Computation of Conditional Statements for

SCMPDS

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Recursive Euclide Algorithm

By Jing-Chao Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Scott-Continuous Functions. Part II

By Adam Grabowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Some Properties of Isomorphism between Relational Structures.

On the Product of Topological Spaces

By Jarosław Gryko and Artur Korniłowicz  . . . . . . . . . . . . . 247

Cages - the External Aproximation of Jordan’s Curve

By Czesław Byliński and Mariusz Żynel  . . . . . . . . . . . . . . . . . . 255

Components and Basis of Topological Spaces

By Robert Milewski  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Properties of the External Approximation of Jordan’s Curve

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Irrationality of e

By Freek Wiedijk  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Injective Spaces. Part II

By Artur Korniłowicz and Jarosław Gryko  . . . . . . . . . . . . . 277

Propositional Calculus for Boolean Valued Functions. Part VI

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Predicate Calculus for Boolean Valued Functions. Part III

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 289

A Characterization of Concept Lattices. Dual Concept Lattices

By Christoph Schwarzweller  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Predicate Calculus for Boolean Valued Functions. Part IV

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 299

Predicate Calculus for Boolean Valued Functions. Part V

By Shunichi Kobayashi and Yatsuka Nakamura  . . . . . . . . . . 303

Definitions of Radix-2^{k} Signed-Digit Number and its Adder Algo-

rithm

By Yoshinori Fujisawa and Yasushi Fuwa  . . . . . . . . . . . . . . . . . 309

Retracts and Inheritance

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Technical Preliminaries to Algebraic Specifications

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Multivariate Polynomials with Arbitrary Number of Variables

By Piotr Rudnicki and Andrzej Trybulec  . . . . . . . . . . . . . . . . 333

Continuous Lattices between T_{0} Spaces

By Grzegorz Bancerek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Predicate Calculus for Boolean Valued Functions. Part VI

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Predicate Calculus for Boolean Valued Functions. Part VII

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Predicate Calculus for Boolean Valued Functions. Part VIII

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Predicate Calculus for Boolean Valued Functions. Part IX

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Asymptotic Notation. Part I

By Richard Krueger et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Asymptotic Notation. Part II

By Richard Krueger et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Predicate Calculus for Boolean Valued Functions. Part X

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Predicate Calculus for Boolean Valued Functions. Part XI

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Four Variable Predicate Calculus for Boolean Valued Functions.

Part I

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Four Variable Predicate Calculus for Boolean Valued Functions.

Part II

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Function Spaces in the Category of Directed Suprema Preserving

Maps

By Grzegorz Bancerek and Adam Naumowicz  . . . . . . . . . . . 411

Property of Complex Functions

By Takashi Mitsuishi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Property of Complex Sequence and Continuity of Complex Func-

tion

By Takashi Mitsuishi et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Scalar Multiple of Riemann Definite Integral

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Darboux’s Theorem

By Noboru Endou et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Five Variable Predicate Calculus for Boolean Valued Functions.

Part I

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Six Variable Predicate Calculus for Boolean Valued Functions.

Part I

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

The Construction and Computation of For-loop Programs for

SCMPDS

By Jing-Chao Chen and Piotr Rudnicki  . . . . . . . . . . . . . . . . . . 449

Predicate Calculus for Boolean Valued Functions. Part XII

By Shunichi Kobayashi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476