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- Page 10: CAMBRIDGE UNIVERSITY PRESS Cambridg
- Page 14: Contents Preface page ix 1 The λ-c
- Page 18: viii Contents 13C Basic generalized
- Page 22: x Preface References for further re
- Page 28: 1 The λ-calculus 1A Introduction W
- Page 32: 1A Introduction 3 Thus h ⋆ can be
- Page 36: 1B Terms and substitution 5 In gene
- Page 40: 1B Terms and substitution 7 • bou
- Page 44: 1B Terms and substitution 9 Proof E
- Page 48: 1C β-reduction 11 cases of simulta
- Page 52: 1C β-reduction 13 (i) P ≡ (λu.v
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1C β-reduction 15 As mentioned bef
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P 1D β-equality 17 Pn ✉ ✉ ✉
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1D β-equality 19 (i) xx(xxx)x, (ii
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2 Combinatory logic 2A Introduction
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2A Introduction to CL 23 Definition
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2B Weak reduction 25 Example 2.12 D
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2C Abstraction in CL 27 Warning 2.2
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2D Weak equality 29 Lemma 2.28 (Sub
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2D Weak equality 31 These are norma
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3 The power of λ and combinators 3
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≡ x(Yx). 3B The fixed-point theor
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3C Böhm’s theorem 37 To prepare
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3C Böhm’s theorem 39 exist m ≥
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3D The quasi-leftmost reduction the
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3E History and interpretation 43 Ch
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3E History and interpretation 45 ga
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4 Representing the computable funct
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4A Introduction 49 SB would not rep
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y (V) with 4B Primitive recursion 5
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4B Primitive recursion 53 Thus QY w
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or equivalently, the equation 4B Pr
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4C Recursive functions 57 A suitabl
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4C Recursive functions 59 φ m1 ...
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4D Abstract numerals 61 Instead of
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5 The undecidability theorem The ai
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Undecidability 65 Definition 5.5 In
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Undecidability 67 Corollary 5.6.3 T
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6 The formal theories λβ and CLw
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6A The theories λβ and CLw 71 Lem
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6C Equivalence of theories 73 compo
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6C Equivalence of theories 75 T ′
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(ζ) (ext) 7A Extensional equality
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7B βη-reduction 79 7B βη-reduct
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Proof By 7.13 and 7.16, like the pr
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(ext) 8A Extensional equality 83 XZ
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8B Extensionality axioms 85 First w
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8B Extensionality axioms 87 for all
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8C Strong reduction 89 Warning The
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8C Strong reduction 91 Theorem 8.20
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9A Introduction 93 called ‘[x] η
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9B The extensional equalities 95 9B
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9B The extensional equalities 97 (c
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9B The extensional equalities 99 Di
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9C Abstraction algorithms in CL 101
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9D CLβ-equality 103 (a) X =w Y =
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9D CLβ-equality 105 Theorem 9.36 (
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10 Simple typing, Church-style 10A
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10B Typed λ-calculus 109 10B Typed
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where 10B Typed λ-calculus 111 δ
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10B Typed λ-calculus 113 The forma
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10C Typed CL 115 The reduction-theo
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10C Typed CL 117 The formal theory
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11 Simple typing, Curry-style in CL
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11A Introduction 121 Notation 11.2
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11B The system TA → C Example 11.
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11B The system TA → C and this is
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(c) U W UW 11D Abstraction 127 UW(V
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11D Abstraction 129 Case 4: X ≡ X
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11E Subject-reduction 131 Definitio
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11E Subject-reduction 133 Therefore
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11F Typable CL-terms 135 there exis
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11G Link with Church’s approach 1
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11H Principal types 139 (i) Γ is a
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11H Principal types 141 Also, if x
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11I Adding new axioms 143 S (a→b
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11I Adding new axioms 145 Definitio
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11J Propositions-as-types 147 easil
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11J Propositions-as-types 149 Discu
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11J Propositions-as-types 151 possi
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11J Propositions-as-types 153 Remar
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11K The equality-rule Eq ′ 155 11
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11K The equality-rule Eq ′ 157 Co
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12 Simple typing, Curry-style in λ
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12A The system TA → λ 161 Deduct
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12A The system TA → λ 163 such t
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12B Basic properties 165 Term Type
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12B Basic properties 167 under redu
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12B Basic properties 169 then follo
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12C Typable λ-terms 171 Example 12
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12D Propositions-as-types 173 (b) L
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12D Propositions-as-types 175 to ea
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12E Rule Eq ′ If this replacement
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12E Rule Eq ′ atypetoλx.xx). The
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13B Dependent function types 181 13
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(G i) [x : σ] M : τ (λx.M) :Gσ(
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13C Basic generalized typing 185 De
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13D Deductive rules 187 (axiom) ⊢
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13D Deductive rules 189 If this rul
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and the modified form of rule (Π i
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(axiom) ⊢ ⋆ : ✷ (start) Γ
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13E Church typing in λ 195 proofs
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13E Church typing in λ 197 The sys
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13E Church typing in λ 199 and (λ
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13E Church typing in λ 201 Remark
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13F Normalization in PTSs 203 Defin
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13F Normalization in PTSs 205 Remar
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13F Normalization in PTSs 207 Lemma
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13G Propositions-as-types 209 13G P
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13G Propositions-as-types 211 It is
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13G Propositions-as-types 213 We us
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and It is also easy to show that an
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13H PTSs with equality 217 Hence, b
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13H PTSs with equality 219 Furtherm
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14A Applicative structures 221 of C
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14B Combinatory algebras 223 For ev
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14B Combinatory algebras 225 Warnin
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14B Combinatory algebras 227 Defini
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15 Models of λ-calculus 15A The de
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15A The definition of λ-model 231
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15A The definition of λ-model 233
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15A The definition of λ-model 235
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15B Syntax-free definitions 237 Dis
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15B Syntax-free definitions 239 Cas
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15B Syntax-free definitions 241 Con
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15C General properties of λ-models
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15C General properties of λ-models
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16 Scott’s D∞ and other models
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16A C.p.o.s 249 λd ∈ D.φ(d) (
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16A C.p.o.s 251 computable function
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16B Continuous functions 253 Exerci
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Proof Straightforward. 16B Continuo
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16C The construction of D∞ That i
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16C The construction of D∞ But φ
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16D Properties of D∞ 16D Basic pr
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16D Properties of D∞ Thus a0, a1,
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16D Properties of D∞ Lemma 16.47
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16E D∞ is a λ-model 267 16E D∞
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16E D∞ is a λ-model 269 = ψr
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16F Other models 271 16F Some other
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16F Other models 273 versions of P
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16F Other models 275 algebras to pr
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α-conversion 277 P ⊲1β (λx.(λ
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α-conversion 279 Lemma A1.7 For al
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α-conversion 281 Corollary A1.14.1
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A2A β-reduction 283 By Appendix A1
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A2A β-reduction 285 making [N/x]M,
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A2A β-reduction 287 Case 4: M ≡
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A2B Other reductions 289 where the
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A2B Other reductions 291 Next, defi
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Appendix A3 Strong normalization pr
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A3A SN for λ 295 Lemma A3.10 Let
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A3B SN for CLw 297 M ⋆ N ≡ (λx
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A3C SN for CLZ → KXY U1 ...Un ⊲
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A3C SN for CLZ → term Zτ m, and
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A3C SN for CLZ → Basis (m =0and m
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Appendix A4 Care of your pet combin
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Appendix A5 Answers to starred exer
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Answers to starred exercises 309 (
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Finally, for 1 ≤ i ≤ k, Answers
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Answers to starred exercises 313 as
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Answers to starred exercises 315 11
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(c) (d) 1 [x : σ →σ →τ] Answ
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Answers to starred exercises 319 Th
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Answers to starred exercises 321 To
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324 References [Bar84] H. P. Barend
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326 References [Chu41] A. Church. T
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328 References [HS80] J. R. Hindley
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330 References [ML75] P. Martin-Lö
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332 References [Sco76] D. S. Scott.
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0, numeral in typing system, 214 0,
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336 List of symbols λ-model, 231 a
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338 Index B, 21 in λ, 34 assigning
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340 Index extensionality axioms, se
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342 Index numerals abstract, 61 of
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344 Index standardization, 42 (star