# Mizar Analysis of Algorithms: Preliminaries

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## Mizar Analysis of Algorithms: Preliminaries

Algorithms and its parts - instructions - are formalized as elements of if-while algebras. An if-while algebra is a (1-sorted) universal algebra which has 4 operations: a constant - the empty instruction, a binary catenation of instructions, a ternary conditional instruction, and a binary while instruction. An execution function is defined on pairs (s, I), where s is a state (an element of certain set of states) and I is an instruction, and results in states. The execution function obeys control structures using the set of distinguished true states, i.e. a condition instruction is executed and the continuation of execution depends on if the resulting state is in true states or not. Termination is also defined for pairs (s, I) and depends on the execution function. The existence of execution function determined on elementary instructions and its uniqueness for terminating instructions are shown.

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

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

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

• [4] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.

• [5] Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.

• [6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

• [7] Grzegorz Bancerek. König's lemma. Formalized Mathematics, 2(3):397-402, 1991.

• [8] Grzegorz Bancerek. Sets and functions of trees and joining operations of trees. Formalized Mathematics, 3(2):195-204, 1992.

• [9] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.

• [10] Grzegorz Bancerek. Minimal signature for partial algebra. Formalized Mathematics, 5(3):405-414, 1996.

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

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

• [13] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91-101, 1993.

• [14] Grzegorz Bancerek and Piotr Rudnicki. The set of primitive recursive functions. Formalized Mathematics, 9(4):705-720, 2001.

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

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

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

• [18] Ewa Burakowska. Subalgebras of the universal algebra. Lattices of subalgebras. Formalized Mathematics, 4(1):23-27, 1993.

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

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

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

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

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

• [24] 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.

• [25] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

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

• [27] Czesław Byliński. Subcategories and products of categories. Formalized Mathematics, 1(4):725-732, 1990.

• [28] Patricia L. Carlson and Grzegorz Bancerek. Context-free grammar - part 1. Formalized Mathematics, 2(5):683-687, 1991.

• [29] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

• [30] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.

• [31] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573-577, 1997.

• [32] Małgorzata Korolkiewicz. Homomorphisms of algebras. Quotient universal algebra. Formalized Mathematics, 4(1):109-113, 1993.

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

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

• [35] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra. Formalized Mathematics, 3(2):251-253, 1992.

• [36] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

• [37] Beata Perkowska. Free universal algebra construction. Formalized Mathematics, 4(1):115-120, 1993.

• [38] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67-74, 1996.

• [39] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

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

• [41] Andrzej Trybulec. Function domains and Fránkel operator. Formalized Mathematics, 1(3):495-500, 1990.

• [42] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.

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

• [44] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.

• [45] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.

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

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

• [50] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

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