Formal libraries for Algebraic Topology: status report 1

Transcription

1 Formal libraries for Algebraic Topology: status report 1 ForMath La Rioja node (Jónathan Heras) Departamento de Matemáticas y Computación Universidad de La Rioja Spain Mathematics, Algorithms and Proofs 2010 November 10, Partially supported by Ministerio de Educación y Ciencia, project MTM C02-01, and by European Commission FP7, STREP project ForMath ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 1/46

2 Contributors Local Contributors: Jesús Aransay César Domínguez Jónathan Heras Laureano Lambán Vico Pascual María Poza Julio Rubio ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 2/46

3 Contributors Local Contributors: Jesús Aransay César Domínguez Jónathan Heras Laureano Lambán Vico Pascual María Poza Julio Rubio Contributors from INRIA - Sophia: Yves Bertot Maxime Dénès Laurence Rideau Contributors from Universidad de Sevilla: Francisco Jesús Martín Mateos José Luis Ruiz Reina ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 2/46

4 Goal Formalization of libraries for Algebraic Topology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 3/46

5 Goal Formalization of libraries for Algebraic Topology Application: Study of digital images ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 3/46

6 Applying topological concepts to analyze images F. Ségonne, E. Grimson, and B. Fischl. Topological Correction of Subcortical Segmentation. International Conference on Medical Image Computing and Computer Assisted Intervention, MICCAI 2003, LNCS 2879, Part 2, pp ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 4/46

7 Table of Contents 1 Mathematical concepts 2 Computing in Algebraic Topology 3 Formalizing Algebraic Topology 4 Incidence simplicial matrices formalized in SSReflect 5 Conclusions and Further Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 5/46

8 Table of Contents Mathematical concepts 1 Mathematical concepts 2 Computing in Algebraic Topology 3 Formalizing Algebraic Topology 4 Incidence simplicial matrices formalized in SSReflect 5 Conclusions and Further Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 6/46

9 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

10 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

11 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

12 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

13 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

14 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

15 Mathematical concepts From General Topology to Algebraic Topology Topological Space Digital Image triangulation Simplicial Complex interpreting algebraic structure Chain Complex computing simplification interpreting Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 7/46

16 Simplicial Complexes Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 8/46

17 Simplicial Complexes Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V. Definition Let α and β be simplices over V, we say α is a face of β if α is a subset of β. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 8/46

18 Simplicial Complexes Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V. Definition Let α and β be simplices over V, we say α is a face of β if α is a subset of β. Definition An ordered (abstract) simplicial complex over V is a set of simplices K over V satisfying the property: α K, if β α β K Let K be a simplicial complex. Then the set S n(k) of n-simplices of K is the set made of the simplices of cardinality n + 1. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 8/46

19 Simplicial Complexes Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification V = (0, 1, 2, 3, 4, 5, 6) K = {, (0), (1), (2), (3), (4), (5), (6), (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (3, 4), (4, 5), (4, 6), (5, 6), (0, 1, 2), (4, 5, 6)} ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 9/46

20 Simplicial Complexes Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification Definition The facets of a simplicial complex K are the maximal simplices of the simplicial complex The facets are: {(0, 3), (1, 3), (2, 3), (3, 4), (0, 1, 2), (4, 5, 6)} ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 10/46

21 Mathematical concepts Chain Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition A chain complex C is a pair of sequences C = (C q, d q) q Z where: For every q Z, the component C q is an R-module, the chain group of degree q For every q Z, the component d q is a module morphism d q : C q C q 1, the differential map For every q Z, the composition d qd q+1 is null: d qd q+1 = 0 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 11/46

22 Mathematical concepts Homology Digital Image Simplicial Complex Chain Complex Homology simplification Definition If C = (C q, d q) q Z is a chain complex: The image B q = im d q+1 C q is the (sub)module of q-boundaries The kernel Z q = ker d q C q is the (sub)module of q-cycles Given a chain complex C = (C q, d q) q Z : d q 1 d q = 0 B q Z q Every boundary is a cycle The converse is not generally true ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 12/46

23 Mathematical concepts Homology Digital Image Simplicial Complex Chain Complex Homology simplification Definition If C = (C q, d q) q Z is a chain complex: The image B q = im d q+1 C q is the (sub)module of q-boundaries The kernel Z q = ker d q C q is the (sub)module of q-cycles Definition Given a chain complex C = (C q, d q) q Z : d q 1 d q = 0 B q Z q Every boundary is a cycle The converse is not generally true Let C = (C q, d q) q Z be a chain complex. For each degree n Z, the n-homology module of C is defined as the quotient module H n(c ) = Zn B n ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 12/46

24 Mathematical concepts From Simplicial Complexes to Chain Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Definition Let K be an (ordered abstract) simplicial complex. Let n 1 and 0 i n be two integers n and i. Then the face operator n i is the linear map n i : S n(k) S n 1 (K) defined by: n i ((v 0,..., v n)) = (v 0,..., v i 1, v i+1,..., v n). The i-th vertex of the simplex is removed, so that an (n 1)-simplex is obtained. Definition Let K be a simplicial complex. Then the chain complex C (K) canonically associated with K is defined as follows. The chain group C n(k) is the free Z module generated by the n-simplices of K. In addition, let (v 0,..., v n 1 ) be a n-simplex of K, the differential of this simplex is defined as: n d n := ( 1) i i n i=0 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 13/46

25 Mathematical concepts Simplification: Perturbation techniques Digital Image Simplicial Complex Chain Complex Homology simplification Definition A reduction ρ between two chain complexes C y D (denoted by ρ : C D ) is a triple ρ = (f, g, h) satisfying the following relations: 1) fg = Id D ; 2) d C h + hd C = Id C gf ; 3) fh = 0; hg = 0; hh = 0. h C f g D Theorem If C D, then C = D A, with A acyclic, which implies that H n(c ) = H n(d ) for all n. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 14/46

26 Mathematical concepts Simplification: Perturbation techniques Digital Image Simplicial Complex Chain Complex Homology Reduction h (C, d C ) simplification g f (D, d D + δ 1 ) (D, d D ) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 15/46

27 Mathematical concepts Simplification: Perturbation techniques Digital Image Simplicial Complex Chain Complex Homology simplification h1 EPL (C, d C + δ 1 ) Reduction h (C, d C ) g1 f 1 g f (D, d D + δ 1 ) δ 1 (D, d D ) Easy Perturbation Lemma (EPL) Basic Perturbation Lemma (BPL) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 15/46

28 Mathematical concepts Simplification: Perturbation techniques Digital Image Simplicial Complex Chain Complex Homology simplification EPL Reduction BPL h1 (C, d C + δ 1 ) h (C, d C ) h2 δ 2 (C, d C + δ 2 ) g1 f 1 g f g2 f 2 (D, d D + δ 1 ) (D, d D ) (D, d + δ D 2 ) δ 1 Easy Perturbation Lemma (EPL) Basic Perturbation Lemma (BPL) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 15/46

29 Mathematical concepts Simplification: Discrete Morse Theory Digital Image Simplicial Complex Chain Complex Homology Discrete Morse Theory: Discrete vector fields DVF simplification ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 16/46

30 Mathematical concepts Simplification: Discrete Morse Theory Digital Image Simplicial Complex Chain Complex Homology Discrete Morse Theory: Discrete vector fields DVF Critical cells simplification ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 16/46

31 Mathematical concepts Simplification: Discrete Morse Theory Digital Image Simplicial Complex Chain Complex Homology simplification Discrete Morse Theory: Discrete vector fields DVF Critical cells From a chain complex C and a DVF V on C constructs a reduction from C to C c where generators of C c are the critical cells of C with respect to V ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 16/46

32 Mathematical concepts Simplification: Discrete Morse Theory Digital Image Simplicial Complex Chain Complex Homology simplification Discrete Morse Theory: Discrete vector fields DVF Critical cells From a chain complex C and a DVF V on C constructs a reduction from C to C c where generators of C c are the critical cells of C with respect to V R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90-145, A. Romero, F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 16/46

33 Computing Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification Computing Homology groups: From a Chain Complex (C n, d n ) n Z of finite type d n can be expressed as matrices Homology groups are obtained from a diagonalization process ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 17/46

34 Computing Mathematical concepts Digital Image Simplicial Complex Chain Complex Homology simplification Computing Homology groups: From a Chain Complex (C n, d n ) n Z of finite type d n can be expressed as matrices Homology groups are obtained from a diagonalization process From a Chain Complex (C n, d n ) n Z of non finite type Effective Homology Theory Reductions J. Rubio and F. Sergeraert. Constructive Algebraic Topology, Bulletin des Sciences Mathématiques, 126: , ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 17/46

35 Mathematical concepts Digital Images Digital Image Simplicial Complex Chain Complex Homology simplification 2D digital images: elements are pixels 3D digital images: elements are voxels ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 18/46

36 Mathematical concepts From a digital image to simplicial complexes Digital Image Simplicial Complex Chain Complex Homology simplification R. Ayala, E. Domínguez, A.R. Francés, A. Quintero. Homotopy in digital spaces. Discrete Applied Mathematics 125 (2003) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 19/46

37 Table of Contents Computing in Algebraic Topology 1 Mathematical concepts 2 Computing in Algebraic Topology 3 Formalizing Algebraic Topology 4 Incidence simplicial matrices formalized in SSReflect 5 Conclusions and Further Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 20/46

38 Computing in Algebraic Topology Computing in Algebraic Topology Digital Image triangulation Simplicial Complex interpreting graded structure Chain Complex simplification Homology computing Demonstration fkenzo: user interface for Sergeraert s Kenzo system ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 21/46

39 Table of Contents Formalizing Algebraic Topology 1 Mathematical concepts 2 Computing in Algebraic Topology 3 Formalizing Algebraic Topology 4 Incidence simplicial matrices formalized in SSReflect 5 Conclusions and Further Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 22/46

40 Formalizing Algebraic Topology Formalization of Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology Formalized in ACL2 simplification J. Heras, V. Pascual and J. Rubio. ACL2 verification of Simplicial Complexes programs for the Kenzo system. Preprint. Formalization in Coq/SSReflect Y. Bertot, L. Rideau and ForMath La Rioja node. Technical report on a SSReflect week. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 23/46

41 Formalizing Algebraic Topology Formalization of Chain Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Formalized in ACL2, Isabelle and Coq L. Lambán, F. J. Martín-Mateos, J. L. Ruiz-Reina and J. Rubio. When first order is enough: the case of Simplicial Topology. Preprint. J. Heras and V. Pascual. An ACL2 infrastructure to formalize Kenzo Higher-Order constructors. Preprint. J. Aransay and C. Domínguez. Modelling Differential Structures in Proof Assistants: The Graded Case. In Proceedings 12th International Conference on Computer Aided Systems Theory (EUROCAST 2009), volume 5717 of Lecture Notes in Computer Science, pages , ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 24/46

42 Formalizing Algebraic Topology From Simplicial Complexes to Chain Complexes Digital Image Simplicial Complex Chain Complex Homology Simplicial Complexes Simplicial Sets Formalized in ACL2 simplification J. Heras, V. Pascual and J. Rubio, Proving with ACL2 the correctness of simplicial sets in the Kenzo system. In LOPSTR 2010, Lecture Notes in Computer Science. Springer-Verlag. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 25/46

43 Formalizing Algebraic Topology From Simplicial Complexes to Chain Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Simplicial Complexes Simplicial Sets Chain Complexes Formalized in ACL2 J. Heras, V. Pascual and J. Rubio, Proving with ACL2 the correctness of simplicial sets in the Kenzo system. In LOPSTR 2010, Lecture Notes in Computer Science. Springer-Verlag. Formalized in ACL2 L. Lambán, F. J. Martín-Mateos, J. L. Ruiz-Reina and J. Rubio. When first order is enough: the case of Simplicial Topology. Preprint. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 25/46

44 Formalizing Algebraic Topology Simplification: Reductions Digital Image Simplicial Complex Chain Complex Homology simplification h C(K) f g C N (K) Formalized in ACL2 L. Lambán, F. J. Martín-Mateos, J. L. Ruiz-Reina and J. Rubio. When first order is enough: the case of Simplicial Topology. Preprint. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 26/46

45 Formalizing Algebraic Topology Simplification: Perturbation techniques Digital Image Simplicial Complex Chain Complex Homology simplification EPL: Formalized in ACL2, Coq and Isabelle J. Heras and V. Pascual. An ACL2 infrastructure to formalize Kenzo Higher-Order constructors. Preprint. J. Aransay and C. Domínguez. Modelling Differential Structures in Proof Assistants: The Graded Case. In Proceedings 12th International Conference on Computer Aided Systems Theory (EUROCAST 2009), volume 5717 of Lecture Notes in Computer Science, pages , ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 27/46

46 Formalizing Algebraic Topology Simplification: Perturbation techniques Digital Image Simplicial Complex Chain Complex Homology simplification EPL: Formalized in ACL2, Coq and Isabelle J. Heras and V. Pascual. An ACL2 infrastructure to formalize Kenzo Higher-Order constructors. Preprint. J. Aransay and C. Domínguez. Modelling Differential Structures in Proof Assistants: The Graded Case. In Proceedings 12th International Conference on Computer Aided Systems Theory (EUROCAST 2009), volume 5717 of Lecture Notes in Computer Science, pages , BPL: Formalized in Isabelle/HOL J. Aransay, C. Ballarin and J. Rubio. A mechanized proof of the Basic Perturbation Lemma. Journal of Automated Reasoning, 40(4): , Formalization of Bicomplexes in Coq C. Domínguez and J. Rubio. Effective Homology of Bicomplexes, formalized in Coq. To appear in Theoretical Computer Science. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 27/46

47 Formalizing Algebraic Topology Simplification: Discrete Morse Theory Digital Image Simplicial Complex Chain Complex Homology simplification Formalization of Discrete Morse Theory: Work in progress ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 28/46

48 Homology Groups Formalizing Algebraic Topology Digital Image Simplicial Complex Chain Complex Homology simplification Formalization of Discrete Morse Theory: Work in progress Formalization of Homology groups: Future Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 28/46

49 Formalizing Algebraic Topology Formalization of digital images Digital Image Simplicial Complex Chain Complex Homology 2D digital images: Binary matrix simplification t f f t Formalized in Coq: R. O Connor. A Computer Verified Theory of Compact Sets. In SCSS 2008, RISC Linz Report Series. ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 29/46

50 Formalizing Algebraic Topology From Digital Images to Simplicial Complexes Digital Image Simplicial Complex Chain Complex Homology simplification Elements of digital images Facets of a Simplicial Complex Future work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 30/46

51 Incidence simplicial matrices formalized in SSReflect Table of Contents 1 Mathematical concepts 2 Computing in Algebraic Topology 3 Formalizing Algebraic Topology 4 Incidence simplicial matrices formalized in SSReflect 5 Conclusions and Further Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 31/46

52 Incidence simplicial matrices formalized in SSReflect From Simplicial Complexes to Homology Simplicial Complex graded structure Incidence Matrices differential Chain Complex computing Homology ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 32/46

53 Incidence simplicial matrices formalized in SSReflect SSReflect SSReflect: Extension of Coq Developed while formalizing the Four Color Theorem Provides new libraries: ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 33/46

54 Incidence simplicial matrices formalized in SSReflect SSReflect SSReflect: Extension of Coq Developed while formalizing the Four Color Theorem Provides new libraries: matrix.v: matrix theory finset.v and fintype.v: finite set theory and finite types bigops.v: indexed big operations, like n f (i) or f (i) zmodp.v: additive group and ring Z p i=0 i I ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 33/46

55 Incidence simplicial matrices formalized in SSReflect Representation of Simplicial Complexes in SSReflect Definition Let V be a finite ordered set, called the vertex set, a simplex over V is any finite subset of V Variable V : fintype. Definition simplex := {set V} ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 34/46

56 Incidence simplicial matrices formalized in SSReflect Representation of Simplicial Complexes in SSReflect Definition Let V be a finite ordered set, called the vertex set, a simplex over V is any finite subset of V. Definition A finite ordered (abstract) simplicial complex over V is a finite set of simplices K over V satisfying the property: α K, if β α β K Variable V : fintype. Definition simplex := {set V}. Definition good_sc (c : {set simplex}) := forall x, x \in c -> forall y : simplex, y \subset x -> y \in c ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 34/46

57 Incidence simplicial matrices formalized in SSReflect Incidence Matrices Definition Let X and Y be two ordered finite sets of simplices, we call incidence matrix to a matrix m n where m = X n = Y Y [1] Y [n] X [1] a 1,1 a 1,n M = X [m] a m,1 a m,n a i,j = { 1 if X [i] is a face of Y [j] 0 if X [i] is not a face of Y [j] ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 35/46

58 Incidence simplicial matrices formalized in SSReflect Incidence Matrices Definition Let X and Y be two ordered finite sets of simplices, we call incidence matrix to a matrix m n where m = X n = Y Y [1] Y [n] X [1] a 1,1 a 1,n M = X [m] a m,1 a m,n a i,j = { 1 if X [i] is a face of Y [j] 0 if X [i] is not a face of Y [j] Lemma lt12 : 1 < 2. Proof. by done. Qed. Definition Z2_ring := (Zp_ring lt12). Lemma p : 0 < 2. Proof. by done. Qed. Definition p_0_2 := inzp p 0. Definition p_1_2 := inzp p ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 35/46

59 Incidence simplicial matrices formalized in SSReflect Incidence Matrices Definition Let X and Y be two ordered finite sets of simplices, we call incidence matrix to a matrix m n where m = X n = Y Y [1] Y [n] X [1] a 1,1 a 1,n M = X [m] a m,1 a m,n a i,j = { 1 if X [i] is a face of Y [j] 0 if X [i] is not a face of Y [j] Variables Top Left:{set simplex}. Definition seq_ss (SS: {set simplex}):= enum (mem SS) : seq simplex ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 35/46

60 Incidence simplicial matrices formalized in SSReflect Incidence Matrices Definition Let X and Y be two ordered finite sets of simplices, we call incidence matrix to a matrix m n where m = X n = Y Y [1] Y [n] X [1] a 1,1 a 1,n M = X [m] a m,1 a m,n a i,j = { 1 if X [i] is a face of Y [j] 0 if X [i] is not a face of Y [j] Definition incidencefunction (i j : nat) := if (nth x0 (seq_ss Left) i) \subset (nth x0 (seq_ss Top) j) then p_1_2 else p_0_2. Definition incidencematrix := matrix_of_fun incidencefunction (m:=size (seq_ss Left)) (n:=size (seq_ss Top)) (R:=Z2_ring) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 35/46

61 Incidence simplicial matrices formalized in SSReflect Incidence Matrices Definition Let C be a finite set of simplices, A be the set of n-simplices of C with an order between its elements and B the set of (n 1)-simplices of C with an order between its elements. We call incidence matrix of dimension n (n 1), to a matrix p q where p = B q = A M i,j = { 1 if B[i] is a face of A[j] 0 if B[i] is not a face of A[j] Variable c: {set simplex}. Variable n:nat. Definition top_n := [set x \in c # x == n+1] : {set simplex}. Definition left_n_1 := [set x \in c # x == n] : {set simplex}. Definition incidence_matrix_n := incidencematrix top_n left_n_ ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 36/46

62 Incidence simplicial matrices formalized in SSReflect Incidence Matrices of Simplicial Complexes (0, 1) (0, 2) (0, 3) (1, 2) (1, 3) (2, 3) (3, 4) (4, 5) (4, 6) (5, 6) (0) (1) (2) (3) (4) (5) (6) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 37/46

63 Incidence simplicial matrices formalized in SSReflect Incidence Matrices of Simplicial Complexes (0, 1, 2) (4, 5, 6) (0, 1) 1 0 (0, 2) 1 0 (0, 3) 0 0 (1, 2) 1 0 (1, 3) 0 0 (2, 3) 0 0 (3, 4) 0 0 (4, 5) 0 1 (4, 6) 0 1 (5, 6) 0 1 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 37/46

64 Incidence simplicial matrices formalized in SSReflect Product of two consecutive incidence matrices in Z 2 Theorem (Product of two consecutive incidence matrices in Z 2 ) Let K be a finite simplicial complex over V with an order between the simplices of the same dimension and let n 1 be a natural number n, then the product of the n-th incidence matrix of K and the (n + 1)-incidence matrix of K over the ring Z/2Z is equal to the null matrix Theorem incidence_matrices_sc_product: forall (V:finType)(n:nat)(sc: {set (simplex V)}), good_sc sc -> n >= 1 -> mulmx (R:=Z2_ring) (incidence_matrix_n sc n) (incidence_matrix_n sc (n+1)) = null_mx Z2_ring (size (seq_ss (left_n_1 sc n))) (size (seq_ss (top_n sc (n+1)))) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 38/46

65 Incidence simplicial matrices formalized in SSReflect Sketch of the proof Let S n+1 be the set of (n + 1)-simplices of K with an order between its elements Let S n be the set of n-simplices of K with an order between its elements Let S n 1 be the set of (n 1)-simplices of K with an order between its elements ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 39/46

66 Incidence simplicial matrices formalized in SSReflect Sketch of the proof Let S n+1 be the set of (n + 1)-simplices of K with an order between its elements Let S n be the set of n-simplices of K with an order between its elements Let S n 1 be the set of (n 1)-simplices of K with an order between its elements S n[1] S n[r1] S n+1 [1] S n+1 [r3] S n 1 [1] a 1,1 a 1,r1 S n[1] b 1,1 b 1,r1 M n(k) = , M n+1(k) = S n 1 [r2] a r2,1 a r2,r1 S n[r1] b r1,1 b r1,r3 where r1 = S n, r2 = S n 1 and r3 = S n+1 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 39/46

67 Incidence simplicial matrices formalized in SSReflect Sketch of the proof where c 1,1 c 1,r3 M n (K) M n+1 (K) =..... c r2,1 c r2,r3 c i, j = 1 j0 r1 a i, j0 b j0, j ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 40/46

68 Incidence simplicial matrices formalized in SSReflect Sketch of the proof where c 1,1 c 1,r3 M n (K) M n+1 (K) =..... c r2,1 c r2,r3 we need to prove that c i, j = 1 j0 r1 i, j, c i, j = 0 in order to prove that M n M n+1 = 0 a i, j0 b j0, j ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 40/46

69 Incidence simplicial matrices formalized in SSReflect Sketch of the proof a i, j0 b j0, j = 1 j0 r a i, j0 b j0, j a i, j0 b j0, j a i, j0 b j0, j a i, j0 b j0, j j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 41/46

70 Incidence simplicial matrices formalized in SSReflect Sketch of the proof a i, j0 b j0, j = 1 j0 r a i, j0 b j0, j a i, j0 b j0, j a i, j0 b j0, j a i, j0 b j0, j j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] a i, j0 b j0, j = ( 1 j0 r1 1) j0 M n 2 [i] M n 1 [j0] M n 1 [j0] M n[j] ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 41/46

71 Incidence simplicial matrices formalized in SSReflect Sketch of the proof a i, j0 b j0, j = 1 j0 r a i, j0 b j0, j a i, j0 b j0, j a i, j0 b j0, j a i, j0 b j0, j j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] j0 S n 1 [i] S n[j0] S n[j0] S n+1 [j] a i, j0 b j0, j = ( 1 j0 r1 1) j0 M n 2 [i] M n 1 [j0] M n 1 [j0] M n[j] a i, j0 b j0, j = {j0 (1 j0 r1) (S n 1 [i] S n[j0]) (S n[j0] S n+1 [j])} 1 j0 r1 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 41/46

72 Incidence simplicial matrices formalized in SSReflect Sketch of the proof {j0 (1 j0 r1) (S n 1 [i] S n [j0]) (S n [j0] S n+1 [j])} = 0 mod 2 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 42/46

73 Incidence simplicial matrices formalized in SSReflect Sketch of the proof {j0 (1 j0 r1) (S n 1 [i] S n [j0]) (S n [j0] S n+1 [j])} = 0 mod 2 S n 1 [i] S n+1 [j] {j0 (1 j0 r1) (S n 1 [i] S n[j0]) (S n[j0] S n+1 [j])} = 0 Otherwise, k such that S n 1 [i] S n [k] S n+1 [j] ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 42/46

74 Incidence simplicial matrices formalized in SSReflect Sketch of the proof {j0 (1 j0 r1) (S n 1 [i] S n [j0]) (S n [j0] S n+1 [j])} = 0 mod 2 S n 1 [i] S n+1 [j] {j0 (1 j0 r1) (S n 1 [i] S n[j0]) (S n[j0] S n+1 [j])} = 0 Otherwise, k such that S n 1 [i] S n [k] S n+1 [j] S n 1 [i] S n+1 [j] {j0 (1 j0 r1) (S n 1 [i] S n[j0]) (S n[j0] S n+1 [j])} = 2 S n+1 [j] = (v 0,..., v n+1 ) S n[k 1 ] = (v 0,..., v j,..., v n+1 ) S n[k 2 ] = (v 0,..., v i,..., v n+1 ) S n 1 [i] = (v 0,..., v i,..., v j,..., v n+1 ) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 42/46

75 Incidence simplicial matrices formalized in SSReflect Sketch of the proof {j0 (1 j0 r1) (S n 1 [i] S n [j0]) (S n [j0] S n+1 [j])} = 0 mod 2 S n 1 [i] S n+1 [j] {j0 (1 j0 r1) (S n 1 [i] S n[j0]) (S n[j0] S n+1 [j])} = 0 Otherwise, k such that S n 1 [i] S n [k] S n+1 [j] S n 1 [i] S n+1 [j] {j0 (1 j0 r1) (S n 1 [i] S n[j0]) (S n[j0] S n+1 [j])} = 2 S n+1 [j] = (v 0,..., v n+1 ) S n[k 1 ] = (v 0,..., v j,..., v n+1 ) S n[k 2 ] = (v 0,..., v i,..., v n+1 ) S n 1 [i] = (v 0,..., v i,..., v j,..., v n+1 ) ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 42/46

76 Incidence simplicial matrices formalized in SSReflect Formalization in SSReflect Summation part: Quite direct ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 43/46

77 Incidence simplicial matrices formalized in SSReflect Formalization in SSReflect Summation part: Quite direct Lemmas from bigops library bigid: F i = F i + i r P i i r P i a i big1: 0 = 0 i r P i i r P i a i F i ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 43/46

78 Incidence simplicial matrices formalized in SSReflect Formalization in SSReflect Summation part: Quite direct Lemmas from bigops library bigid: F i = F i + i r P i i r P i a i big1: 0 = 0 i r P i i r P i a i F i ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 43/46

79 Incidence simplicial matrices formalized in SSReflect Formalization in SSReflect Summation part: Quite direct Lemmas from bigops library bigid: F i = F i + i r P i i r P i a i big1: 0 = 0 i r P i Cardinality part: More details are needed i r P i a i F i ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 43/46

80 Incidence simplicial matrices formalized in SSReflect Formalization in SSReflect Summation part: Quite direct Lemmas from bigops library bigid: F i = F i + F i i r P i i r P i a i i r P i a i big1: 0 = 0 i r P i Cardinality part: More details are needed Auxiliary lemmas Lemmas from finset and fintype libraries ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 43/46

81 Table of Contents Conclusions and Further Work 1 Mathematical concepts 2 Computing in Algebraic Topology 3 Formalizing Algebraic Topology 4 Incidence simplicial matrices formalized in SSReflect 5 Conclusions and Further Work ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 44/46

82 Conclusions and Further Work Conclusions and Further Work Conclusions: Application of Algebraic Topology to the analysis of Digital Images Implemented in a Software System Partially formalized with Theorem Proving tools ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 45/46

83 Conclusions and Further Work Conclusions and Further Work Conclusions: Application of Algebraic Topology to the analysis of Digital Images Implemented in a Software System Partially formalized with Theorem Proving tools Future Work: Formalization: Digital Images to Simplicial Complexes Formalization of Discrete Morse Theory Formalization of computation of Homology groups... ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 45/46

84 The end Thank you for your attention Formal libraries for Algebraic Topology: status report ForMath La Rioja node (Jónathan Heras) Departamento de Matemáticas y Computación Universidad de La Rioja Spain Mathematics, Algorithms and Proofs 2010 November 10, 2010 ForMath La Rioja node (J. Heras) Formal libraries for Algebraic Topology 46/46