Type theory and formal proof: an introduction
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Cambridge
Cambridge Univ. Press
2014
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| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XXV, 436 S. |
| ISBN: | 9781107036505 |
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Record in the Search Index
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| adam_text |
Titel: Type theory and formal proof
Autor: Nederpelt, Rob
Jahr: 2014
Contents
Foreword, by Henk Barendregt page xiii
Preface xv
Acknowledgements xxvii
Greek aiphabet xxviii
1 Untyped lambda calculus 1
1.1 Input Output, behaviour of functions 1
1.2 The essence of functions 2
1.3 Lambda-terms 4
1.4 Free and bound variables 8
1.5 Alpha conversion 9
1.6 Substitution 11
1.7 Lambda-terms modulo a-equivalence 14
1.8 Beta reduction 16
1.9 Normal forms and confluence 19
1.10 Fixed Point Theorem 24
1.11 Conclusions 26
1.12 Further reading 27
Exercises 29
2 Simply typed lambda calculus 33
2.1 Adding types 33
2.2 Simple types 34
2.3 Church-typing and Curry-typing 36
2.4 Derivation rules for Church's A- 39
2.5 Different formats for a derivation in A— 44
2.6 Kinds of problems to be solved in type theory 46
2.7 Well-typedness in A—»• 47
2.8 Type Checking in A— 50
2.9 Term Finding in A— 51
53
59
63
64
65
66
69
69
71
72
73
76
78
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80
82
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85
88
90
93
94
95
97
99
100
100
101
103
103
105
107
109
115
118
119
121
123
123
125
Contents
2.10 General properties of A—»
2.11 Reduction and A—
2.12 Consequences
2.13 Conclusions
2.14 Further reading
Exercises
Second order typed lambda calculus
3.1 Type-abstraction and type-application
3.2 II-types
3.3 Second order abstraction and application rules
3.4 The System A2
3.5 Example of a derivation in A2
3.6 Properties of A2
3.7 Conclusions
3.8 Further reading
Exercises
Types dependent on types
4.1 Type constructors
4.2 Sort-rule and var-rule in Au;
4.3 The weakening rule in Xlü
4.4 The formation rule in Au
4.5 Application and abstraction rules in Au;
4.6 Shortened derivations
4.7 The conversion rule
4.8 Properties of Aa;
4.9 Conclusions
4.10 Further reading
Exercises
Types dependent on terms
5.1 The missing extension
5.2 Derivation rules of AP
5.3 An example derivation in AP
5.4 Minimal predicate logic in AP
5.5 Example of a logical derivation in AP
5.6 Conclusions
5.7 Further reading
Exercises
The Calculus of Constructions
6.1 The system AC
6.2 The A-cube
Contents ix
6.3 Properties of AC 128
6.4 Conclusions 132
6.5 Further reading 133
Exercises I34
7 The encoding of logical notions in AC 137
7.1 Absurdity and negation in type theory 137
7.2 Conjunction and disjunction in type theory 139
7.3 An example of propositional logic in AC 144
7.4 Classical logic in AC 146
7.5 Predicate logic in AC 150
7.6 An example of predicate logic in AC 154
7.7 Conclusions 157
7.8 Further reading 159
Exercises 162
8 Definitions 165
8.1 The nature of definitions 165
8.2 Inductive and recursive definitions 167
8.3 The format of definitions 168
8.4 Instantiations of definitions 170
8.5 A formal format for definitions 172
8.6 Definitions depending on assumptions 174
8.7 Giving names to proofs 175
8.8 A general proof and a specialised version 178
8.9 Mathematical statements as formal definitions 180
8.10 Conclusions 182
8.11 Further reading 183
Exercises 185
9 Extension of AC with definitions 189
9.1 Extension of AC to the system ADo 189
9.2 Judgements extended with definitions 190
9.3 The rule for adding a definition 192
9.4 The rule for instantiating a definition 193
9.5 Definition unfolding and 5-conversion 197
9.6 Examples of 5-conversion 200
9.7 The conversion rule extended with A 202
9.8 The derivation rules for ADo 203
9.9 A closer look at the derivation rules of ADo 204
9.10 Conclusions 206
9.11 Further reading 207
Exercises 208
x Contents
10 Rules and properties of AD 211
10.1 Descriptive versus primitive definitions 211
10.2 Axioms and axiomatic notions 212
10.3 Rules for primitive definitions 214
10.4 Properties of AD 215
10.5 Normalisation and confluence in AD 219
10.6 Conclusions 221
10.7 Further reading 221
Exercises 223
11 Flag-style natural deduction in AD 225
11.1 Formal derivations in AD 225
11.2 Comparing formal and flag-style AD 228
11.3 Conventions about flag-style proofs in AD 229
11.4 Introduction and elimination rules 232
11.5 Rules for constructive propositional logic 234
11.6 Examples of logical derivations in AD 237
11.7 Suppressing unaltered parameter lists 239
11.8 Rules for classical propositional logic 240
11.9 Alternative natural deduction rules for V 243
11.10 Rules for constructive predicate logic 246
11.11 Rules for classical predicate logic 249
11.12 Conclusions 252
11.13 Further reading 253
Exercises 254
12 Mathematics in AD: a first attempt 257
12.1 An example to start with 257
12.2 Equality 259
12.3 The congruence property of equality 262
12.4 Orders 264
12.5 A proof about Orders 266
12.6 Unique existence 268
12.7 The descriptor l 271
12.8 Conclusions 274
12.9 Further reading 275
Exercises 276
13 Sets and subsets 279
13.1 Dealing with subsets in AD 279
13.2 Basic set-theoretic notions 282
13.3 Special subsets 287
13.4 Relations 288
Contents xi
13.5 Maps 291
13.6 Representation of mathematical notions 295
13.7 Conclusions 296
13.8 Further reading 297
Exercises 302
14 Numbers and arithmetic in AD 305
14.1 The Peano axioms for natural numbers 305
14.2 Introducing integers the axiomatic way 308
14.3 Basic properties of the 'new' N 313
14.4 Integer addition 316
14.5 An example of a basic computation in AD 320
14.6 Arithmetical laws for addition 322
14.7 Closure under addition for natural and negative numbers 324
14.8 Integer subtraction 327
14.9 The opposite of an integer 330
14.10 Inequality relations on Z 332
14.11 Multiplication of integers 335
14.12 Divisibility 338
14.13 Irrelevance of proof 340
14.14 Conclusions 341
14.15 Further reading 343
Exercises 344
15 An elaborated example 349
15.1 Formalising a proof of Bezout's Lemma 349
15.2 Preparatory work 352
15.3 Part I of the proof of Bezout's Lemma 354
15.4 Part II of the proof 357
15.5 Part III of the proof 360
15.6 The holes in the proof of Bezout's Lemma 363
15.7 The Minimum Theorem for Z 364
15.8 The Division Theorem 369
15.9 Conclusions 371
15.10 Further reading 373
Exercises 376
16 Further perspectives 379
16.1 Useful applications of AD 379
16.2 Proof assistants based on type theory 380
16.3 Future of the field 384
16.4 Conclusions 386
16.5 Further reading 387
xii
Contents
Appendix A Logic in AD 391
A.l Constructive propositional logic 391
A.2 Classical propositional logic 393
A.3 Constructive predicate logic 395
A.4 Classical predicate logic 396
Appendix B Arithmetical axioms, definitions and lemmas 397
Appendix C Two complete example proofs in AD 403
C.l Closure under addition in N 403
C.2 The Minimum Theorem 405
Appendix D Derivation rules for AD 409
References 411
Index of names 419
Index of definitions 421
Index of Symbols 423
Index of subjects 425 |
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| spelling | Nederpelt, Rob 1942- Verfasser (DE-588)115937077X aut Type theory and formal proof an introduction Rob Nederpelt ; Herman Geuvers 1. publ. Cambridge Cambridge Univ. Press 2014 XXV, 436 S. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Type theory Beweistheorie (DE-588)4145177-6 gnd rswk-swf Typentheorie (DE-588)4121795-0 gnd rswk-swf Typentheorie (DE-588)4121795-0 s Beweistheorie (DE-588)4145177-6 s DE-604 Geuvers, Herman 1964- Verfasser (DE-588)1066888620 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027601105&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nederpelt, Rob 1942- Geuvers, Herman 1964- Type theory and formal proof an introduction Type theory Beweistheorie (DE-588)4145177-6 gnd Typentheorie (DE-588)4121795-0 gnd |
| subject_GND | (DE-588)4145177-6 (DE-588)4121795-0 |
| title | Type theory and formal proof an introduction |
| title_auth | Type theory and formal proof an introduction |
| title_exact_search | Type theory and formal proof an introduction |
| title_full | Type theory and formal proof an introduction Rob Nederpelt ; Herman Geuvers |
| title_fullStr | Type theory and formal proof an introduction Rob Nederpelt ; Herman Geuvers |
| title_full_unstemmed | Type theory and formal proof an introduction Rob Nederpelt ; Herman Geuvers |
| title_short | Type theory and formal proof |
| title_sort | type theory and formal proof an introduction |
| title_sub | an introduction |
| topic | Type theory Beweistheorie (DE-588)4145177-6 gnd Typentheorie (DE-588)4121795-0 gnd |
| topic_facet | Type theory Beweistheorie Typentheorie |
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