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1 Introduction 1

1.1 Types in Computer Science 1

1.2 What Type Systems Are Good For 4

1.3 Type Systems and Language Design 9

1.4 Capsule History 10

1.5 Related Reading 12

2 Mathematical Preliminaries 15

2.1 Sets， Relations， and Functions 15

2.2 Ordered Sets 16

2.3 Sequences 18

2.4 Induction 19

2.5 Background Reading 20

Ⅰ Untyped Systems 21

3 Untyped Arithmetic Expressions 23

3.1 Introduction 23

3.2 Syntax 26

3.3 Induction on Terms 29

3.4 Semantic Styles 32

3.5 Evaluation 34

3.6 Notes 43

4 An ML Implementation of Arithmetic Expressions 45

4.1 Syntax 46

4.2 Evaluation 47

4.3 The Rest of the Story 49

5 The Untyped Lambda-Calculus 51

5.1 Basics 52

5.2 Programming in the Lambda-Calculus 58

5.3 Formalities 68

5.4 Notes 73

6 Nameless Representation of Terms 75

6.1 Terms and Contexts 76

6.2 Shifting and Substitution 78

6.3 Evaluation 80

7 An ML Implementation of the Lambda-Calculus 83

7.1 Terms and Contexts 83

7.2 Shifting and Substitution 85

7.3 Evaluation 87

7.4 Notes 88

Ⅱ Simple Types 89

8 Typed Arithmetic Expressions 91

8.1 Types 91

8.2 The Typing Relation 92

8.3 Safety=Progress ＋ Preservation 95

9 Simply Typed Lambda-Calculus 99

9.1 Function Types 99

9.2 The Typing Relation 100

9.3 Properties of Typing 104

9.4 The Curry-Howard Correspondence 108

9.5 Erasure and Typability 109

9.6 Curry-Style vs.Church-Style 111

9.7 Notes 111

10 An ML Implementation of Simple Types 113

10.1 Contexts 113

10.2 Terms and Types 115

10.3 Typechecking 115

11 Simple Extensions 117

11.1 Base Types 117

11.2 The Unit Type 118

11.3 Derived Forms： Sequencing and Wildcards 119

11.4 Ascription 121

11.5 Let Bindings 124

11.6 Pairs 126

11.7 Tuples 128

11.8 Records 129

11.9 Sums 132

11.10 Variants 136

11.11 General Recursion 142

11.12 Lists 146

12 Normalization 149

12.1 Normalization for Simple Types 149

12.2 Notes 152

13 References 153

13.1 Introduction 153

13.2 Typing 159

13.3 Evaluation 159

13.4 Store Typings 162

13.5 Safety 165

13.6 Notes 170

14 Exceptions 171

14.1 Raising Exceptions 172

14.2 Handling Exceptions 173

14.3 Exceptions Carrying Values 175

Ⅲ Subtyping 179

15 Subtyping 181

15.1 Subsumption 181

15.2 The Subtype Relation 182

15.3 Properties of Subtyping and Typing 188

15.4 The Top and Bottom Types 191

15.5 Subtyping and Other Features 193

15.6 Coercion Semantics for Subtyping 200

15.7 Intersection and Union Types 206

15.8 Notes 207

16 Metatheory of Subtyping 209

16.1 Algorithmic Subtyping 210

16.2 Algorithmic Typing 213

16.3 Joins and Meets 218

16.4 Algorithmic Typing and the Bottom Type 220

17 An ML Implementation of Subtyping 221

17.1 Syntax 221

17.2 Subtyping 221

17.3 Typing 222

18 Case Study： Imperative Objects 225

18.1 What Is Object-Oriented Programming？ 225

18.2 Objects 228

18.3 Object Generators 229

18.4 Subtyping 229

18.5 Grouping Instance Variables 230

18.6 Simple Classes 231

18.7 Adding Instance Variables 233

18.8 Calling Superclass Methods 234

18.9 Classes with Self 234

18.10 Open Recursion through Self 235

18.11 Open Recursion and Evaluation Order 237

18.12 A More Efficient Implementation 241

18.13 Recap 244

18.14 Notes 245

19 Case Study： Featherweight Java 247

19.1 Introduction 247

19.2 Overview 249

19.3 Nominal and Structural Type Systems 251

19.4 Definitions 254

19.5 Properties 261

19.6 Encodings vs.Primitive Objects 262

19.7 Notes 263

Ⅳ Recursive Types 265

20 Recursive Types 267

20.1 Examples 268

20.2 Formalities 275

20.3 Subtyping 279

20.4 Notes 279

21 Metatheory of Recursive Types 281

21.1 Induction and Coinduction 282

21.2 Finite and Infinite Types 284

21.3 Subtyping 286

21.4 A Digression on Transitivity 288

21.5 Membership Checking 290

21.6 More Efficient Algorithms 295

21.7 Regular Trees 298

21.8 μ-Types 299

21.9 Counting Subexpressions 304

21.10 Digression： An Exponential Algorithm 309

21.11 Subtyping Iso-Recursive Types 311

21.12 Notes 312

V Polymorphism 315

22 Type Reconstruction 317

22.1 Type Variables and Substitutions 317

22.2 Two Views of Type Variables 319

22.3 Constraint-Based Typing 321

22.4 Unification 326

22.5 Principal Types 329

22.6 Implicit Type Annotations 330

22.7 Let-Polymorphism 331

22.8 Notes 336

23 Universal Types 339

23.1 Motivation 339

23.2 Varieties of Polymorphism 340

23.3 System F 341

23.4 Examples 344

23.5 Basic Properties 353

23.6 Erasure， Typability， and Type Reconstruction 354

23.7 Erasure and Evaluation Order 357

23.8 Fragments of System F 358

23.9 Parametricity 359

23.10 Impredicativity 360

23.11 Notes 361

24 Existential Types 363

24.1 Motivation 363

24.2 Data Abstraction with Existentials 368

24.3 Encoding Existentials 377

24.4 Notes 379

25 An ML Implementation of System F 381

25.1 Nameless Representation of Types 381

25.2 Type Shifting and Substitution 382

25.3 Terms 383

25.4 Evaluation 385

25.5 Typing 386

26 Bounded Quantification 389

26.1 Motivation 389

26.2 Definitions 391

26.3 Examples 396

26.4 Safety 400

26.5 Bounded Existential Types 406

26.6 Notes 408

27 Case Study： Imperative Objects， Redux 411

28 Metatheory of Bounded Quantification 417

28.1 Exposure 417

28.2 Minimal Typing 418

28.3 Subtyping in Kernel F〈： 421

28.4 Subtyping in Full F〈： 424

28.5 Undecidability of Full F〈： 427

28.6 Joins and Meets 432

28.7 Bounded Existentials 435

28.8 Bounded Quantification and the Bottom Type 436

Ⅵ Higher-Order Systems 437

29 Type Operators and Kinding 439

29.1 Intuitions 440

29.2 Definitions 445

30 Higher-Order Polymorphism 449

30.1 Definitions 449

30.2 Example 450

30.3 Properties 453

30.4 Fragments of Fω 461

30.5 Going Further： Dependent Types 462

31 Higher-Order Subtyping 467

31.1 Intuitions 467

31.2 Definitions 469

31.3 Properties 472

31.4 Notes 472

32 Case Study： Purely Functional Objects 475

32.1 Simple Objects 475

32.2 Subtyping 476

32.3 Bounded Quantification 477

32.4 Interface Types 479

32.5 Sending Messages to Objects 480

32.6 Simple Classes 481

32.7 Polymorphic Update 482

32.8 Adding Instance Variables 485

32.9 Classes with “Self” 486

32.10 Notes 488

Appendices 491

A Solutions to Selected Exercises 493

B Notational Conventions 565

B.1 Metavariable Names 565

B.2 Rule Naming Conventions 565

B.3 Naming and Subscripting Conventions 566

References 567

Index 605