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4B Semantics and completeness 61Proof of (2) For ". ", use rule (-*E). To prove "=>", suppose the upper clausein (2) holds and, as a special case, take Q to be a variable z not occurring in Pand such that r+ contains the assignmentz exists.)Then by the upper clause in (2),Hence by rule (-+1),F+-z I-pI"+ Hp Pz:tl.This means that there is a finite subset F* of F+ - z such thathence by 4B9F* I-pF* I-p P: ->rl,which gives the "=>" part of (2). Hence (2) holds and the proof of (1) is complete.Deduction of the theorem from (1) If I' (=p M:T then in particular, for TMp and E0and V O we haveHence by (1),[M]p E ITIVoI" I-p M:z.That is F* I-p M:i for some finite F* c F+. Hence by 4A3.2(vi),(3) F* tM I-p M:r.But F* P M = F P M because F* s r+ and the extra term-variables in F+ werechosen to be distinct from those in M. Hence by the weakening-property of I-,F 1-flM:i.4B10 Remark (Too many completeness-proofs?) In the above completeness-proofthe full strength of the assumption that F --* M:i was valid in every model wasnot used; all we needed was that it was valid in one model, the term-model. Thereis another completeness proof in Barendregt et al. 1983, which uses a more complexkind of term model called a filter model. And there is a third proof in Coppo1984 using the graph model Pw. It looks almost as if the choice of model is quiteirrelevant to the success of the completeness-proof.Furthermore, TAA+p and TAA+p, can be proved sound and complete for at leastthree other variant definitions of the semantics, see Hindley 1983a §5, Hindley 1983b,and Mitchell 1988 §§3-4.This richness of completeness-proofs says in effect that the differences whichundoubtedly exist between these various kinds of models and definitions of semanticscannot be expressed in the rather limited language of TA2+p and TA2+p,.
62 4 Type assignment with equality4B11 Remark By the way, although earlier remarks in this chapter may have giventhe impression that without equality-invariance a type-<strong>theory</strong> has no reasonablesemantics, this is not strictly true. Plotkin 1994 has described a semantics for TA2(and some extensions) based on modelling the concept of reduction rather thanequality. Plotkin's approach seems closer to the "spirit" of TA2; in fact TA2 hasat its heart the notion of reduction rather than conversion, and has always beenseen primarily as a tool for eliminating type-conflicts in practical computations orreductions, rather than as a formal <strong>theory</strong> of equality.
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BASIC SIMPLE TYPE THEORY
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BASIC SIMPLE TYPE THEORYJ. Roger Hi
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To Carol
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VlllContents7C The converse PT proo
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xIntroductionhave proved themselves
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1The type-free A-calculusThe R-calc
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1A A-terms and their structure 31A6
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I B #-reduction and #-normal forms
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IC rl- and firs-reductions 7Proof S
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IC q- and iq-reductions93TFig. lC7a
Page 24 and 25: 1D Restricted A-terms 11(ii) The BC
Page 26 and 27: 2A The system TAA 132A2 Definition
Page 28 and 29: 2A The system TA, 152A5.1 Notation
Page 30 and 31: 2A The system TA2172A8.3 Example Le
Page 32 and 33: Proof Trivial from 2A9.2A The syste
Page 34 and 35: 2B The subject-construction theorem
Page 36 and 37: 2B The subject-construction theorem
Page 38 and 39: 2C Subject reduction and expansion2
Page 40 and 41: 2D The typable terms 27However, Cha
Page 42 and 43: 2D The typable terms 292D8.1 Note T
Page 44 and 45: 3A Principal types and their histor
Page 46 and 47: 3A Principal types and their histor
Page 48 and 49: 3B Type-substitutions 353B1 Notatio
Page 50 and 51: 3B Type-substitutions 37Then r U (s
Page 52 and 53: 3C Motivating the PT algorithm 39(i
Page 54 and 55: 3D Unification 41this pair was show
Page 56 and 57: 3D Unification 43If pk * Tk and the
Page 58 and 59: A' _ §(Ap) for some s; hence in pa
Page 60 and 61: 3E The PT algorithm 47Let r - rl U
Page 62 and 63: 3E The PT algorithm 49for some subs
Page 64 and 65: 3E The PT algorithm 513E4 Further R
Page 66 and 67: 4A The equality rule 53The name "TA
Page 68 and 69: (M).4A The equality rule 554A3 Weak
Page 70 and 71: 4B Semantics and completeness 574A1
Page 72 and 73: 4B Semantics and completeness 59413
Page 76 and 77: 5A version using typed termsIn Chap
Page 78 and 79: 5A Typed terms 655A1.5 Warning If M
Page 80 and 81: 5B Reducing typed terms(ii) if MT E
Page 82 and 83: 5B Reducing typed terms 695B5.1 Not
Page 84 and 85: 5C Normalization theorems 71of rede
Page 86 and 87: SC Normalization theorems 73in leng
Page 88 and 89: 6A Intuitionist implicational logic
Page 90 and 91: 6A Intuitionist implicational logic
Page 92 and 93: 6B The Curry-Howard isomorphism 79T
Page 94 and 95: 6B The Curry-Howard isomorphism 816
Page 96 and 97: 6B The Curry-Howard isomorphism 83t
Page 98 and 99: 6C Some weaker logics 85for some cl
Page 100 and 101: 6C Some weaker logics 87logic in 6A
Page 102 and 103: 6D Axiom-based versions 89Deduction
Page 104 and 105: 6D Axiom-based versions 916D6.1 Not
Page 106 and 107: The converse principal-type algorit
Page 108 and 109: 7B Identifications 957A3 Converse P
Page 110 and 111: 7C The converse PT proofNext, suppo
Page 112 and 113: Since (a-+b)* __ a--+b, we must pro
Page 114 and 115: 7C The converse PT proof 101Conside
Page 116 and 117: instance, construct an m.g.c.i. V -
Page 118 and 119: AD-7D Condensed detachment 1057D6 M
Page 120 and 121: 7D Condensed detachment 107Proof By
Page 122 and 123: 8A Inhabitants 109A Pq-normal inhab
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8A Inhabitants 1118A7.1 Example Let
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8A Inhabitants113Fig. 8A12a.8A11.2
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8B Examples of the search strategy
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8B Examples of the search strategy
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8C The search algorithm 119Long(s)
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8C The search algorithm 121Note. Th
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8C The search algorithm1238C6.1 Exa
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8D The Counting algorithm 1258D3.1
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8E The structure of a nf-scheme 127
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8E The structure of a nf-scheme129x
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8E The structure of a nf-scheme131T
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8F Stretching, shrinking and comple
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9A The structure of a term 1419A2 D
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9A The structure of a term(ii) if r
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9B Residuals 145Proof-note Two case
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9B Residuals 1479134.1 Lemma Every
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9C The structure of a TAR-deduction
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9D The structure of a type 151below
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9E The condensed structure of a typ
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9E The condensed structure of a typ
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9F Imitating combinatory logic in A
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[x].N*Before constructing9F Imitati
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162 Answers to starred exerciseswit
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164 Answers to starred exercisesIf
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166 Answers to starred exercisesTo
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BibliographyReferences to unpublish
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Bibliography 171DOSEN, K. [1992a] M
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Bibliography 173KALMAN, J. A. [1983
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Bibliography 175SCEDROV, A. [1990]
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Table of principal typesThis table
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IndexA-logics (see axiom-based logi
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Index 181D-incompleteness of BCI, B
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Index183PT algorithm, converse (see
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Index185#-contraction 4of typed ter
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