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2D The typable terms 292D8.1 Note The BCKA-terms are terms without multiple occurrences of variables(except possibly for binding occurrences), so the above theorem connects untypabilitywith multiple occurrences of variables. On the other hand not every term withmultiple occurrences is untypable; consider S Axyzxz(yz) in 2A8.7(iii) for example.
3The principal-type algorithmA typable term has in general an infinite set of types in TAR. For example if I =_it is possible to assign to I every type with form 6-*6, by the following deduction:x:a --> x:a-,I)--* o-- a (-I)(like 2A8.3). But all the types in this infinite set are substitution-instances of the onetypea-*a,and it is easy to see that I has no other types than these. (In fact every deductionfor I must have the simple form shown above, by the subject-construction theorem.)The type a---+a is called a principal type for I.The aim of the present chapter is to show that the existence of a principal typeis a property of all typable terms, not just I. In effect the principal type of a termis the most general type it can receive in TAR, and the principal type theorem willsay that every typable term has one. Further, and most important in practice, analgorithm will be described for finding it.This algorithm will decide whether a given term M is typable and, if the answeris "yes", will output a principal type for M. Such algorithms are usually calledtype-checking or principal-type or PT algorithms. The existence of a PT algorithmis what gives TAR and its extensions such as ML their practical value, since ifthe typability of a program is regarded as a safety criterion the programmer willwant to be able to decide effectively whether a newly created program satisfies thiscriterion.The PT algorithm below will be easy to describe and even easier to apply inpractice. But to prove the PT theorem a bald statement of the algorithm will notbe enough; we shall need also a proof that the algorithm is correct, i.e. that it doeswhat it claims to do. In the account below a correctness proof will be included withthe algorithm in the form of comments to each of its steps, explaining the purposeand effect of each step as the reader meets it.A little knowledge of substitution, unification and most general unifiers willbe needed so introductions to these will be included before the statement of thealgorithm.30
Page 2 and 3: BASIC SIMPLE TYPE THEORY
Page 4 and 5: BASIC SIMPLE TYPE THEORYJ. Roger Hi
Page 6: To Carol
Page 9 and 10: VlllContents7C The converse PT proo
Page 11 and 12: xIntroductionhave proved themselves
Page 14 and 15: 1The type-free A-calculusThe R-calc
Page 16 and 17: 1A A-terms and their structure 31A6
Page 18 and 19: I B #-reduction and #-normal forms
Page 20 and 21: IC rl- and firs-reductions 7Proof S
Page 22 and 23: 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 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 74 and 75: 4B Semantics and completeness 61Pro
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
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6B The Curry-Howard isomorphism 79T
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6B The Curry-Howard isomorphism 816
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6B The Curry-Howard isomorphism 83t
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6C Some weaker logics 85for some cl
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6C Some weaker logics 87logic in 6A
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6D Axiom-based versions 89Deduction
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6D Axiom-based versions 916D6.1 Not
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The converse principal-type algorit
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7B Identifications 957A3 Converse P
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7C The converse PT proofNext, suppo
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Since (a-+b)* __ a--+b, we must pro
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7C The converse PT proof 101Conside
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instance, construct an m.g.c.i. V -
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AD-7D Condensed detachment 1057D6 M
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7D Condensed detachment 107Proof By
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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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