Formalized Mathematics最新文献

{"title":"Stability of the 7-3 Compressor Circuit for Wallace Tree. Part I","authors":"K. Wasaki","doi":"10.2478/forma-2020-0005","DOIUrl":"https://doi.org/10.2478/forma-2020-0005","url":null,"abstract":"Summary To evaluate our formal verification method on a real-size calculation circuit, in this article, we continue to formalize the concept of the 7-3 Compressor (STC) Circuit [6] for Wallace Tree [11], to define the structures of calculation units for a very fast multiplication algorithm for VLSI implementation [10]. We define the circuit structure of the tree constructions of the Generalized Full Adder Circuits (GFAs). We then successfully prove its circuit stability of the calculation outputs after four and six steps. The motivation for this research is to establish a technique based on formalized mathematics and its applications for calculation circuits with high reliability, and to implement the applications of the reliable logic synthesizer and hardware compiler [5].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72712719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Klein-Beltrami model. Part III","authors":"Roland Coghetto","doi":"10.2478/forma-2020-0001","DOIUrl":"https://doi.org/10.2478/forma-2020-0001","url":null,"abstract":"Summary Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4],[5]. With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. In this article we prove that our constructed model (we prefer “Beltrami-Klein” name over “Klein-Beltrami”, which can be seen in the naming convention for Mizar functors, and even MML identifiers) satisfies the congruence symmetry, the congruence equivalence relation, and the congruence identity axioms formulated by Tarski (and formalized in Mizar as described briefly in [8]).","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87428305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"About Graph Complements","authors":"Sebastian Koch","doi":"10.2478/forma-2020-0004","DOIUrl":"https://doi.org/10.2478/forma-2020-0004","url":null,"abstract":"Summary This article formalizes different variants of the complement graph in the Mizar system [3], based on the formalization of graphs in [6].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85565950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"AIM Loops and the AIM Conjecture","authors":"C. Brown, Karol Pąk","doi":"10.2478/forma-2019-0027","DOIUrl":"https://doi.org/10.2478/forma-2019-0027","url":null,"abstract":"Summary In this article, we prove, using the Mizar [2] formalism, a number of properties that correspond to the AIM Conjecture. In the first section, we define division operations on loops, inner mappings T, L and R, commutators and associators and basic attributes of interest. We also consider subloops and homomorphisms. Particular subloops are the nucleus and center of a loop and kernels of homomorphisms. Then in Section 2, we define a set Mlt Q of multiplicative mappings of Q and cosets (mostly following Albert 1943 for cosets [1]). Next, in Section 3 we define the notion of a normal subloop and construct quotients by normal subloops. In the last section we define the set InnAut of inner mappings of Q, define the notion of an AIM loop and relate this to the conditions on T, L, and R defined by satisfies TT, etc. We prove in Theorem (67) that the nucleus of an AIM loop is normal and finally in Theorem (68) that the AIM Conjecture follows from knowing every AIM loop satisfies aa1, aa2, aa3, Ka, aK1, aK2 and aK3. The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky [4] (in [3]) as well as Veroff’s Prover9 files.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72408053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Formal Development of Rough Inclusion Functions","authors":"Adam Grabowski","doi":"10.2478/forma-2019-0028","DOIUrl":"https://doi.org/10.2478/forma-2019-0028","url":null,"abstract":"Summary Rough sets, developed by Pawlak [15], are important tool to describe situation of incomplete or partially unknown information. In this article, continuing the formalization of rough sets [12], we give the formal characterization of three rough inclusion functions (RIFs). We start with the standard one, κ£, connected with Łukasiewicz [14], and extend this research for two additional RIFs: κ1, and κ2, following a paper by Gomolińska [4], [3]. We also define q-RIFs and weak q-RIFs [2]. The paper establishes a formal counterpart of [7] and makes a preliminary step towards rough mereology [16], [17] in Mizar [13].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79527415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"About Vertex Mappings","authors":"Sebastian Koch","doi":"10.2478/forma-2019-0025","DOIUrl":"https://doi.org/10.2478/forma-2019-0025","url":null,"abstract":"Summary In [6] partial graph mappings were formalized in the Mizar system [3]. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in [7], a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized here.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81346371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Field Extensions and Kronecker’s Construction","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2019-0022","DOIUrl":"https://doi.org/10.2478/forma-2019-0022","url":null,"abstract":"Summary This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In this fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]F has a root.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87708727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Intersection of Fields F with F [X]","authors":"Christoph Schwarzweller","doi":"10.2478/forma-2019-0021","DOIUrl":"https://doi.org/10.2478/forma-2019-0021","url":null,"abstract":"Summary This is the third part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/. Because F is not a subfield of F [X]/ we construct in the second part the field (E ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/ and therefore consider F as a subfield of F [X]/”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/ with the canonical monomorphism ϕ: F → F [X]/. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]F has a root.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79299851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"About Graph Mappings","authors":"Sebastian Koch","doi":"10.2478/forma-2019-0024","DOIUrl":"https://doi.org/10.2478/forma-2019-0024","url":null,"abstract":"Summary In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system [7], [2]. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author’s knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as well.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88703893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Underlying Simple Graphs","authors":"Sebastian Koch","doi":"10.2478/forma-2019-0023","DOIUrl":"https://doi.org/10.2478/forma-2019-0023","url":null,"abstract":"Summary In this article the notion of the underlying simple graph of a graph (as defined in [8]) is formalized in the Mizar system [5], along with some convenient variants. The property of a graph to be without decorators (as introduced in [7]) is formalized as well to serve as the base of graph enumerations in the future.","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79364242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}