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• Multivariate Analysis

November 11, 2013

Contents

1 Nat-Bijection: Bijections between natural numbers and other types 12 1.1 Type nat × nat . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Type nat + nat . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Type int . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Type nat list . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Finite sets of naturals . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 From sets to naturals . . . . . . . . . . . . . . . . . . 16 1.5.3 From naturals to sets . . . . . . . . . . . . . . . . . . 17 1.5.4 Proof of isomorphism . . . . . . . . . . . . . . . . . . 17

2 Countable: Encoding (almost) everything into natural num- bers 18 2.1 The class of countable types . . . . . . . . . . . . . . . . . . . 18 2.2 Conversion functions . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Countable types . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Rationals are Countably Infinite . . . . . . . . . . . . . . 19 2.5 Automatically proving countability of datatypes . . . . . . . 20 2.6 Countable datatypes . . . . . . . . . . . . . . . . . . . . . . . 21

3 Infinite-Set: Infinite Sets and Related Concepts 21 3.1 Infinite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Infinitely Many and Almost All . . . . . . . . . . . . . . . . . 24 3.3 Enumeration of an Infinite Set . . . . . . . . . . . . . . . . . 27 3.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Countable-Set: Countable sets 28 4.1 Predicate for countable sets . . . . . . . . . . . . . . . . . . . 28 4.2 Enumerate a countable set . . . . . . . . . . . . . . . . . . . . 29 4.3 Closure properties of countability . . . . . . . . . . . . . . . . 31 4.4 Misc lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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5 Glbs: Definitions of Lower Bounds and Greatest Lower Bounds, analogous to Lubs 33 5.1 Rules about the Operators greatestP, isLb and isGlb . . . . . 33

6 FuncSet: Pi and Function Sets 35 6.1 Basic Properties of Pi . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Composition With a Restricted Domain: compose . . . . . . 37 6.3 Bounded Abstraction: restrict . . . . . . . . . . . . . . . . . . 38 6.4 Bijections Between Sets . . . . . . . . . . . . . . . . . . . . . 38 6.5 Extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.6 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.7 Extensional Function Spaces . . . . . . . . . . . . . . . . . . . 40

6.7.1 Injective Extensional Function Spaces . . . . . . . . . 43 6.7.2 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . 43

7 L2-Norm: Square root of sum of squares 43

8 Inner-Product: Inner Product Spaces and the Gradient Deriva- tive 45 8.1 Real inner product spaces . . . . . . . . . . . . . . . . . . . . 46 8.2 Class instances . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.3 Gradient derivative . . . . . . . . . . . . . . . . . . . . . . . . 48

9 Product-plus: Additive group operations on product types 50 9.1 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 9.2 Class instances . . . . . . . . . . . . . . . . . . . . . . . . . . 51

10 Product-Vector: Cartesian Products as Vector Spaces 52 10.1 Product is a real vector space . . . . . . . . . . . . . . . . . . 52 10.2 Product is a topological space . . . . . . . . . . . . . . . . . . 53

10.2.1 Continuity of operations . . . . . . . . . . . . . . . . . 54 10.2.2 Separation axioms . . . . . . . . . . . . . . . . . . . . 55

10.3 Product is a metric space . . . . . . . . . . . . . . . . . . . . 55 10.4 Product is a complete metric space . . . . . . . . . . . . . . . 56 10.5 Product is a normed vector space . . . . . . . . . . . . . . . . 56

10.5.1 Pair operations are linear . . . . . . . . . . . . . . . . 56 10.5.2 Frechet derivatives involving pairs . . . . . . . . . . . 57

10.6 Product is an inner product space . . . . . . . . . . . . . . . 57

11 Euclidean-Space: Finite-Dimensional Inner Product Spaces 58 11.1 Type class of Euclidean spaces . . . . . . . . . . . . . . . . . 58 11.2 Subclass relationships . . . . . . . . . . . . . . . . . . . . . . 59 11.3 Class instances . . . . . . . . . . . . . . . . . . . . . . . . . . 59

11.3.1 Type real . . . . . . . . . . . . . . . . . . . . . . . . . 59 11.3.2 Type complex . . . . . . . . . . . . . . . . . . . . . . . 60

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11.3.3 Type ′a × ′b . . . . . . . . . . . . . . . . . . . . . . . 60

12 Linear-Algebra: Elementary linear algebra on Euclidean spaces 60 12.1 Orthogonality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.2 Linear functions. . . . . . . . . . . . . . . . . . . . . . . . . . 64 12.3 Bilinear functions. . . . . . . . . . . . . . . . . . . . . . . . . 65 12.4 Adjoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 12.5 Interlude: Some properties of real sets . . . . . . . . . . . . . 67 12.6 A generic notion of ”hull” (convex, affine, conic hull and clo-

sure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12.7 Archimedean properties and useful consequences . . . . . . . 69 12.8 A bit of linear algebra. . . . . . . . . . . . . . . . . . . . . . . 70 12.9 Euclidean Spaces as Typeclass . . . . . . . . . . . . . . . . . 75 12.10Linearity and Bilinearity continued . . . . . . . . . . . . . . . 76 12.11We continue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 12.12Infinity norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 12.13Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 12.14An ordering on euclidean spaces that will allow us to talk

about intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 87

13 Sum-of-Squares: A decision method for universal multivari- ate real arithmetic with addition, multiplication and order- ing using semidefinite programming 88

14 Norm-Arith: General linear decision procedure for normed spaces 91

15 Topology-Euclidean-Space: Elementary topology in Euclidean space. 94 15.1 Topological Basis . . . . . . . . . . . . . . . . . . . . . . . . . 95 15.2 Countable Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 96 15.3 Polish spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 15.4 General notion of a topology as a value . . . . . . . . . . . . 98

15.4.1 Main properties of open sets . . . . . . . . . . . . . . 98 15.4.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . 99 15.4.3 Subspace topology . . . . . . . . . . . . . . . . . . . . 100 15.4.4 The standard Euclidean topology . . . . . . . . . . . . 101

15.5 Open and closed balls . . . . . . . . . . . . . . . . . . . . . . 102 15.6 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 104 15.7 Limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 15.8 Interior of a Set . . . . . . . . . . . . . . . . . . . . . . . . . . 106 15.9 Closure of a Set . . . . . . . . . . . . . . . . . . . . . . . . . . 108 15.10Frontier (aka boundary) . . . . . . . . . . . . . . . . . . . . . 109 15.11Filters and the “eventually true” quantifier . . . . . . . . . . 110

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15.12Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 15.13Infimum Distance . . . . . . . . . . . . . . . . . . . . . . . . . 117 15.14More properties of closed balls . . . . . . . . . . . . . . . . . 118 15.15Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 15.16Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

15.16.1 Bolzano-Weierstrass property . . . . . . . . . . . . . . 122 15.16.2 Sequential compactness . . . . . . . . . . . . . . . . . 126 15.16.3 Total boundedness . . . . . . . . . . . . . . . . . . . . 127 15.16.4 Heine-Borel theorem . . . . . . . . . . . . . . . . . . . 127 15.16.5 Complete the chain of compactness variants . . . . . . 128 15.16.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . 129

15.17Bounded closed nest property (proof does not use Heine-Borel)130 15.18Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

15.18.1 Structural rules for pointwise continuity . . . . . . . . 134 15.18.2 Structural rules for setwise continuity . . . . . . . . . 134 15.18.3 Structural rules for uniform continuity . . . . . . . . . 134

15.19Topological stuff lifted from and dropped to R . . . . . . . . 141 15.20Pasted sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 15.21Separation between points and sets . . . . . . . . . . . . . . . 145 15.22Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 15.23Closure of halfspaces and hyperplanes . . . . . . . . . . . . . 150 15.24Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 153 15.25Some properties of a canonical subspace . . . . . . . . . . . . 156 15.26Affine transformations of intervals . . . . . . . . . . . . . . . 156 15.27Banach fixed point theorem (not really topological...) . . . . . 157 15.28Edelstein fixed point theorem . . . . . . . . . . . . . . . . . . 158

16 Convex: Convexity in real vector spaces 158 16.1 Convexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 16.2 Explicit expressions for convexity in terms of arbitrary sums. 159 16.3 Arithmetic operations on sets preserve convexity. . . . . . . . 161

17 Set-Algebras: Algebraic operations on sets 163

18 Convex-Euclidean-Space: Convex sets, functions and re- lated things. 169 18.1 Affine set and affine hull . . . . .