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Bibliography 175SCEDROV, A. [1990] A guide to polymorphic types, in Odifreddi 1990, pp. 387-420.SCHWICHTENBERG, H. [1991] An upper bound for reduction sequences in typedlambda-calculus, Archiv fur Math. Logik 30 (1991), 405-408.SELDIN, J. P. [1968] Studies in Illative Combinatory Logic, Ph.D. thesis, University ofAmsterdam, Netherlands 1968.SELDIN, J. P. [1977] A sequent calculus for type assignment, J. Symbolic Logic 42 (1977),11-28.SELDIN, J. P. [1978] A sequent calculus formulation of type assignment with equality rulesfor the ,1/3-calculus, J. Symbolic Logic 43 (1978), 643-649.STATMAN, R. [1979a] Intuitionistic propositional logic is polynomial-space complete,Theoretical Computer Science 9 (1979), 67-72.STATMAN, R. [1979b] The typed A-calculus is not elementary recursive, TheoreticalComputer Science 9 (1979), 73-81.STATMAN, R. [1980] On the existence of closed terms in the typed A-calculus; Part 1 in ToH.B. Curry, ed. J.R. Hindley and J.P. Seldin, Academic Press, UK 1980, pp. 511-534;Part 2 in Theoretical Computer Science 15 (1981), 329-338.STOUGHTON, A. [1988] Substitution revisited, Theoretical Computer Science 59 (1988),317-325.TAIT, W. W. [1965] Infinitely long terms of transfinite type, in Formal Systems and RecursiveFunctions, ed. J. N. Crossley, M. A. Dummett, North-Holland Co., Netherlands 1965,pp. 176-185.TAIT, W. W. [1967] Intensional interpretations of functionals of finite type, J. Symbolic Logic32 (1967), 198-212.TAKAHASHI, M. H. [1991] Theory of Computation, Computability and Lambda-calculus,Kindai Kagaku Sha, Tokyo, Japan 1991 (in Japanese).TAKAHASHI, M. H., AKAMA, Y., HIROKAWA, S. [1994] Normal proofs and theirgrammar, in Theoretical Aspects of Computer Software (TACS '94), ed. by M. Hagiya,J. C. Mitchell, Lecture Notes in Computer Science, Springer-Verlag, Germany, No. 789(1994), 465-493.TIURYN, J. [1990] Type inference problems: a survey, in Mathematical Foundations ofComputer Science 1990, ed. by B. Rovan, Lecture Notes in Computer Science,Springer-Verlag, Germany, No. 452 (1990), 105-120.TROELSTRA, A. S. [1973] (editor) Metamathematical Investigation of IntuitionisticArithmetic and Analysis, Lecture Notes in Mathematics, Springer-Verlag, Germany,No. 344 (1973).TROELSTRA, A. S., van DALEN, D. [1988] Constructivism in Mathematics, anIntroduction, Vols. 1 & 2, North-Holland Co., Netherlands 1988.TURING, A. M. [1942] Notes published in Gandy 1980a.TYSZKIEWICZ, J. [1988] Complexity of Type Inference in Finitely Typed LambdaCalculus, Master's thesis, University of Warsaw, Poland 1988.URQUHART, A. [1984] The undecidability of entailment and relevant implication, J.Symbolic Logic 49 (1984), 1059-1073.URQUHART, A. [1995] The complexity of propositional proofs, Bull. Symbolic Logic 1(1995), 425-467.DE VRIJER, R. [1987] Exactly estimating functionals and strong normalization, Proc.Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 90 (1987), 479-493.WAND, M. [1987] A simple algorithm and proof for type inference, Fundamenta Informaticae10 (1987),115-122.ZAIONC, M. [1985] The set of unifiers in typed A-calculus as regular expression, in RewritingTechniques and Applications, ed. by J.-P. Jouannaud, Lecture Notes in ComputerScience, Springer-Verlag, Germany, No. 202 (1985), pp. 430-440.ZAIONC, M. [1987a] The regular expression descriptions of unifier set in the typedA-calculus, Fundamenta Informaticae 10 (1987), 309-322.
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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
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1D Restricted A-terms 11(ii) The BC
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2A The system TAA 132A2 Definition
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2A The system TA, 152A5.1 Notation
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2A The system TA2172A8.3 Example Le
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Proof Trivial from 2A9.2A The syste
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2B The subject-construction theorem
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2B The subject-construction theorem
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2C Subject reduction and expansion2
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2D The typable terms 27However, Cha
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2D The typable terms 292D8.1 Note T
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3A Principal types and their histor
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3A Principal types and their histor
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3B Type-substitutions 353B1 Notatio
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3B Type-substitutions 37Then r U (s
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3C Motivating the PT algorithm 39(i
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3D Unification 41this pair was show
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3D Unification 43If pk * Tk and the
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A' _ §(Ap) for some s; hence in pa
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3E The PT algorithm 47Let r - rl U
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3E The PT algorithm 49for some subs
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3E The PT algorithm 513E4 Further R
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4A The equality rule 53The name "TA
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(M).4A The equality rule 554A3 Weak
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4B Semantics and completeness 574A1
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4B Semantics and completeness 59413
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4B Semantics and completeness 61Pro
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5A version using typed termsIn Chap
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5A Typed terms 655A1.5 Warning If M
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5B Reducing typed terms(ii) if MT E
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5B Reducing typed terms 695B5.1 Not
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5C Normalization theorems 71of rede
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SC Normalization theorems 73in leng
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6A Intuitionist implicational logic
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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
Page 138 and 139: 8D The Counting algorithm 1258D3.1
Page 140 and 141: 8E The structure of a nf-scheme 127
Page 142 and 143: 8E The structure of a nf-scheme129x
Page 144 and 145: 8E The structure of a nf-scheme131T
Page 146 and 147: 8F Stretching, shrinking and comple
Page 154 and 155: 9A The structure of a term 1419A2 D
Page 156 and 157: 9A The structure of a term(ii) if r
Page 158 and 159: 9B Residuals 145Proof-note Two case
Page 160 and 161: 9B Residuals 1479134.1 Lemma Every
Page 162 and 163: 9C The structure of a TAR-deduction
Page 164 and 165: 9D The structure of a type 151below
Page 166 and 167: 9E The condensed structure of a typ
Page 168 and 169: 9E The condensed structure of a typ
Page 170 and 171: 9F Imitating combinatory logic in A
Page 172: [x].N*Before constructing9F Imitati
Page 175 and 176: 162 Answers to starred exerciseswit
Page 177 and 178: 164 Answers to starred exercisesIf
Page 179 and 180: 166 Answers to starred exercisesTo
Page 182 and 183: BibliographyReferences to unpublish
Page 184 and 185: Bibliography 171DOSEN, K. [1992a] M
Page 186 and 187: Bibliography 173KALMAN, J. A. [1983
Page 190 and 191: Table of principal typesThis table
Page 192 and 193: IndexA-logics (see axiom-based logi
Page 194 and 195: Index 181D-incompleteness of BCI, B
Page 196 and 197: Index183PT algorithm, converse (see
Page 198 and 199: Index185#-contraction 4of typed ter
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