Analysis in Univalent Type Theory Auke Bart Booij A thesis submitted to the School of Computer Science University of Birmingham College of Engineering and Physical Sciences for the degree of University of Birmingham DOCTOR OF PHILOSOPHY January 2020 ii ABSTRACT Some constructive real analysis is developed in univalent type theory (UTT). We develop var- ious types of real numbers, and prove several equivalences between those types. We then study computation with real numbers. It is well known how to compute with real numbers in intensional formalizations of mathematics, where equality of real numbers is specied by an imposed equivalence relation on representations such as Cauchy sequences. However, be- cause in UTT equality of real numbers is captured directly by identity types, we are prevented from making any nontrivial discrete observations of arbitrary real numbers. For instance, there is no function which converts real numbers to decimal expansions, as this would not be continuous. To avoid breaking extensionality, we thus restrict our attention to real numbers that have been equipped with a simple structure called a locator. In order to compute, we modify existing constructions in analysis to work with locators, including Riemann integrals, intermediate value theorems and dierential equations. Hence many of the proofs involving locators look familiar, showing that the use of locators is not a conceptual burden. We discuss the possibility of implementing the work in proof assistants and present a Haskell prototype. iii iv ACKNOWLEDGEMENTS This dissertation exists thanks to the detailed, patient and sympathetic supervision of Martín Escardó and Benedikt Ahrens. Thanks to the Birmingham Theory group, and in particular the CARGO category theory reading group for many opportunities to have (often opinionated) discussions on constructive mathematics. Thanks to Thorsten Altenkirch, Andrej Bauer, Mark Bickford, Ulrik Buchholtz, Achim Jung, Milly Maietti, Peter Schuster, Bas Spitters, Steve Vickers and various anonymous re- viewers for their input and inspiring discussions. Thanks to Hajime Ishihara and Tomoaki Hashizaki for their care and company during my stay at the Japan Advanced Institute of Sci- ence and Technology. Thanks to Sebastian Faust for welcoming me in his group without ques- tion after my move to Darmstadt. Thanks to my parents, family, and friends for continued support throughout my studies. Kristina, thank you for your love and support. Thanks to the computing facilities team at the School of Computer Science for providing the most thankless service in IT: fast and stable Wi-Fi. This project has received funding from the School of Computer Science at the University of Birmingham, and from the European Union’s Horizon 2020 research and innovation pro- gramme under the Marie Skłodowska-Curie grant agreement No 731143. v vi CONTENTS 1 Introduction1 1.1 Summary of contributions.............................4 2 Univalent mathematics7 2.1 Case study: propositional logic...........................8 2.1.1 Formalizing the theorem statement....................8 2.1.2 Formalizing the proof........................... 10 2.2 Proofs versus derivations.............................. 11 2.3 Case study: every natural is either even or odd.................. 12 2.3.1 Formalizing the theorem statement.................... 13 2.3.2 Formalizing the proof........................... 22 2.4 Case study: surjective maps, images of maps................... 27 2.4.1 Formalizing the theorem statement.................... 28 2.4.2 Formalizing the proof........................... 36 2.5 Extensionality.................................... 36 2.5.1 Function extensionality.......................... 37 2.5.2 Contractibility............................... 39 2.5.3 Computation rules for higher-inductive types.............. 41 2.5.4 Univalence................................. 42 2.5.5 Consequences of extensionality...................... 45 vii viii CONTENTS 2.6 Subtypes and embeddings............................. 46 2.6.1 Lattice-like structure of HProp ...................... 50 2.6.2 Quantication over subtypes....................... 52 2.6.3 Identications in subtypes......................... 53 2.7 Case study: quotient types............................. 54 2.8 Classical principles................................. 56 2.9 Notes......................................... 58 3 Fixpoints in dcpos 61 3.1 Dcpos......................................... 62 3.2 Fixpoints....................................... 64 3.3 Notes......................................... 66 4 Real numbers 67 4.1 Algebraic structure of numbers.......................... 69 4.2 Rationals....................................... 75 4.3 Archimedean property............................... 76 4.4 Cauchy completeness of real numbers....................... 77 4.5 HoTT book reals................................... 81 4.6 Dedekind reals.................................... 89 4.7 Notes......................................... 92 5 Universal properties of real numbers 95 5.1 Subsets of the Dedekind reals............................ 97 5.1.1 Minimality of the HoTT book reals.................... 98 5.1.2 Euclidean reals and interval objects.................... 100 5.2 Homotopy-initiality of the Euclidean reals.................... 101 5.3 Interval objects................................... 111 5.4 Notes......................................... 116 CONTENTS ix 6 Locators 119 6.1 Denition...................................... 120 6.2 Terminology for locators.............................. 122 6.3 Locators for rationals................................ 122 6.4 The logic of locators................................. 123 6.5 Bounded search................................... 127 6.6 Computing bounds................................. 128 6.7 Locators for algebraic operations.......................... 130 6.8 Locators for limits.................................. 137 6.9 Calculating digits.................................. 138 6.10 Dedekind cuts structure............................... 141 6.11 Notes......................................... 145 7 Some constructive analysis with locators 147 7.1 Preliminaries..................................... 147 7.2 Integrals....................................... 150 7.3 Intermediate value theorems............................ 151 7.4 Notes......................................... 153 8 Metric spaces 155 8.1 Basic denitions in pseudometric spaces..................... 156 8.2 Function spaces................................... 159 8.3 Banach xed point theorem............................ 163 8.4 Picard-Lindelöf.................................... 167 8.5 Notes......................................... 170 9 Computation in proof assistants 173 9.1 From inference rules to proof assistants...................... 173 9.2 Locators....................................... 174 9.3 Metric spaces..................................... 176 x CONTENTS 9.4 Haskell prototype.................................. 177 10 Closing remarks 181 Index 185 Bibliography 189 Chapter 1 INTRODUCTION Traditionally important parts of mathematics can be developed in a constructive setting, as Bishop showed in his self-proclaimed constructivist propaganda [18]. Like most mathemati- cians before Bishop, including constructive mathematicians such as Brouwer, Bishop worked informally, although he did envision formal systems that allowed for computationally execut- ing the nite routines he described [17, 19]. Martin-Löf [75] proposed a formal system in which Bishop’s work was supposed to be formalizable. This formal system and several of its variations are now known as Martin-Löf Type Theory (MLTT). Interpretation of Bishop’s mathematics in MLTT has been described in e.g. Coquand and Spiwack [35, Section 3]. It is well known how to compute with real numbers in such an intensional formalization, with equality of real numbers specied by an imposed equivalence relation on representa- tions [20, 36, 77], such as Cauchy sequences, sequences of nested intervals, or streams of digits. It has to be checked explicitly that functions on the representations preserve these equivalence relations. Discrete observations, such as nite decimal approximations, can be made because representations are given, but a dierent representation of the same real number can result in a dierent observation, and hence discrete observations are necessarily non-extensional. Interpretation of the so-called identity types remained an issue in MLTT. Such identity types represent the logical notion of equality, and so mirroring rst-order logic, were not given any structure beyond being thought of as containing at most one element, despite this seem- ingly not being provable in MLTT. The genie was let out of the bottle by the development of 1 2 a model of MLTT in groupoids, where identity types can contain more than one element [53]. The potential non-uniqueness of identications was exploited in an explanation of identity types originating from homotopy theory [46, 11, 59]. The development of homotopy theory in terms of type theory led to the univalence axiom which reinforces this explanation and makes non-uniqueness of identications provable. The resulting univalent type theory (UTT), which we discuss in Chapter2, allows to for- malize mathematics with the identity types in a much more central role than in plain MLTT. For example, we can dene types of real numbers whose identity types directly capture the intended equality of real numbers, rather than this equality