Definition of Flat Poset and Existence Theorems for Recursive Call

Summary This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set [12] in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like BaseFunc or RecFunc. The remaining part builds the background for information engineering approach for lists, namely recursive call for posets [21].We formalized some facts from Chapter 8 of this book as an introduction to the next two sections where we concentrate on binary product of posets rather than on a more general case.

[1]  Grzegorz Bancerek,et al.  Bounds in Posets and Relational Substructures 1 , 1997 .

[2]  Bartłomiej Skorulski,et al.  First-countable, Sequential, and Frechet Spaces , 1998 .

[3]  Wojciech A. Trybulec.  Vectors in Real Linear Space , 1990 .

[4]  Glynn Winskel,et al.  The formal semantics of programming languages - an introduction , 1993, Foundation of computing series.

[5]  Yasunari Shidama,et al.  The Lebesgue Monotone Convergence Theorem , 2008, Formaliz. Math..

[6]  Bartłomiej Skorulski.  The Sequential Closure Operator in Sequential and Frechet Spaces , 1999 .

[7]  Konrad Raczkowski,et al.  Topological Properties of Subsets in Real Numbers , 1990 .

[8]  Yoshinori Isomichi.  New concepts in the theory of topological space---supercondensed set, subcondensed set, and condensed set. , 1971 .

[9]  A. Milewska.  The Hahn Banach Theorem in the Vector Space over the Field of Complex Numbers , 2004 .

[10]  Józef Białas.  Properties of the Intervals of Real Numbers , 1993 .

[11]  Czeslaw Bylinski.  Functions from a Set to a Set , 2004 .

[12]  Adam Naumowicz,et al.  Conjugate Sequences , Bounded Complex Sequences and Convergent Complex Sequences , 1996 .

[13]  Andrzej Trybulec,et al.  Miscellaneous Facts about Functions , 1996 .

[14]  Czesław Bylí.  Finite Sequences and Tuples of Elements of a Non-empty Sets , 1990 .

[15]  Beata Madras,et al.  Basic Notation of Universal Algebra , 1992 .

[16]  Y. Shidama.  Banach Space of Bounded Linear Operators , 2003 .

[17]  Artur Korniłowicz,et al.  Journal of Formalized Mathematics on the Topological Properties of Meet-continuous Lattices , 2022 .

[18]  Eugeniusz Kusak.  Abelian Groups, Fields and Vector Spaces 1 , 1990 .

[19]  Konrad Raczkowski,et al.  Equivalence Relations and Classes of Abstraction 1 , 1990 .

[20]  Zbigniew Karno,et al.  On Discrete and Almost Discrete Topological Spaces , 1992 .

[21]  Andrzej Trybulec,et al.  Function Domains and Frænkel Operator , 1990 .

[22]  Karol Pak,et al.  Basic Properties of Metrizable Topological Spaces , 2009, Formaliz. Math..

[23]  Czeslaw Bylinski,et al.  Basic Functions and Operations on Functions , 1989 .

[24]  A. Trybulec.  Domains and Their Cartesian Products , 1990 .

[25]  Adam Grabowski,et al.  On the Category of Posets , 1996 .

[26]  Wojciech A. Trybulec.  Subspaces and Cosets of Subspaces in Real Linear Space , 1990 .

[27]  Properties of Relational Structures , Posets , Lattices and Maps , 2022 .

[28]  Marek Chmur.  The Lattice of Natural Numbers and The Sublattice of it. The Set of Prime Numbers , 1991 .

[29]  Adam Grabowski.  Basic Properties of Rough Sets and Rough Membership Function 1 , 2000 .

[30]  Jozef Bia,et al.  Group and Field Definitions , 1990 .

[31]  G. Bancerek,et al.  Ordinal Numbers , 2003 .

[32]  Adam Grabowski,et al.  Automated Discovery of Properties of Rough Sets , 2013, Fundam. Informaticae.

[33]  Beata Padlewska,et al.  Families of Sets , 1990 .

[34]  J. Harrison.  Formalized Mathematics , 1996 .

[35]  Edmund Woronowicz.  Relations Defined on Sets , 1990 .

[36]  Andrzej Trybulec,et al.  Tuples, Projections and Cartesian Products , 1990 .

[37]  Adam Grabowski.  Relational Formal Characterization of Rough Sets , 2013, Formaliz. Math..

[38]  Yasunari Shidama,et al.  Fixpoint Theorem for Continuous Functions on Chain-Complete Posets , 2010, Formaliz. Math..

[39]  Andrzej Trybulec,et al.  A Borsuk Theorem on Homotopy Types , 1991 .

[40]  Wojciech A. Trybulec.  Pigeon Hole Principle , 1990 .

[41]  Adam Grabowski.  The Properties of Supercondensed Sets, Subcondensed Sets and Condensed Sets , 2005 .

[42]  Henryk Oryszczyszyn,et al.  Real Functions Spaces , 1999 .

[43]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

[44]  Edmund Woronowicz.  Relations and Their Basic Properties , 2004 .

[45]  Banach Space of Absolute Summable Real Sequences , 2004 .

[46]  Edmund Woronowicz,et al.  Many-Argument Relations , 1990 .

[47]  Kenneth Halpern August.  The Cardinal Numbers , 1888, Nature.

[48]  Agata Darmochwa,et al.  Topological Spaces and Continuous Functions , 1990 .

[49]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[50]  Jaros law Kotowicz,et al.  Convergent Real Sequences . Upper and Lower Bound of Sets of Real Numbers , 1989 .

[51]  Shunichi Kobayashi,et al.  A Theory of Partitions . Part I , 2004 .

[52]  G. Bancerek.  Countable Sets and Hessenberg's Theorem , 1991 .

[53]  Grzegorz Bancerek,et al.  Complete Lattices , 1992 .

[54]  Andrzej Trybulec.  On the Geometry of a Go-Board , 1996 .

[55]  Grzegorz Bancerek.  Minimal Signature for Partial Algebra , 1996 .

[56]  Grzegorz Bancerek,et al.  Representation Theorem for Stacks , 2011, Formaliz. Math..

[57]  Artur Korniłowicz,et al.  Cartesian Products of Relations and Relational Structures 1 , 1996 .

[58]  C. Kuratowski.  Sur l'opération Ā de l'Analysis Situs , 1922 .

[59]  W. Kellaway,et al.  Complex Numbers , 2019, AMS/MAA Textbooks.

[60]  G. Bancerek.  The Fundamental Properties of Natural Numbers , 1990 .

[61]  M. Dudzicz.  Representation Theorem for Finite Distributive Lattices , 2004 .

[62]  The Sum and Product of Finite Sequences of Real Numbers , 1990 .

[63]  Grzegorz Bancerek,et al.  Tarski's Classes and Ranks , 1990 .

[64]  Jarosław Kotowicz,et al.  Convergent Sequences and the Limit of Sequences , 2004 .

[65]  A. Kondracki.  Basic Properties of Rational Numbers , 1990 .

[66]  Jan Popio,et al.  Real Normed Space , 1991 .

[67]  Adam Grabowski.  On the Kuratowski Closure-Complement Problem , 2004 .

[68]  Czeslaw Bylinski.  Some Basic Properties of Sets , 2004 .

[69]  Wojciech A. Trybulec.  Non-contiguous Substrings and One-to-one Finite Sequences , 1990 .

[70]  Y. Shidama,et al.  Real Linear Space of Real Sequences Noboru Endou Gifu , 2004 .

[71]  Jarosław Kotowicz.  Real Sequences and Basic Operations on Them , 2004 .

[72]  Andrzej Trybulec,et al.  On the Sets Inhabited by Numbers 1 , 2003 .

[73]  Yatsuka Nakamura,et al.  The Definition of Finite Sequences and Matrices of Probability, and Addition of Matrices of Real Elements , 2006 .

[74]  Yatsuka Nakamura,et al.  Bounded Domains and Unbounded Domains , 1999 .

[75]  Grzegorz Bancerek,et al.  Segments of Natural Numbers and Finite Sequences , 1990 .

[76]  Xiquan Liang,et al.  On the Partial Product of Series and Related Basic Inequalities , 2005 .

[77]  G. Bancerek.  Konig's Theorem , 1990 .

[78]  D. F. Goguadze.  About the Notion of Semiring of Sets , 2003 .

[79]  Zbigniew Karno,et al.  The Lattice of Domains of an Extremally Disconnected Space 1 , 1992 .

[80]  G. Winskel.  The formal semantics of programming languages , 1993 .

[81]  Czeslaw Bylinski.  Functions and Their Basic Properties , 2004 .

[82]  Andrzej Trybulec,et al.  Binary Operations Applied to Functions , 1990 .

[83]  Alexander Ostermann,et al.  Real-Valued Functions , 2011 .