Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the rules for case by the rules case(inl(P ) M;N) MP; case(inr(P ) M;N) NP; case(5A B (P ) M;N) 5C (P ) A fundamental result about natural deduction is the fact that every proof (term) reduces to a normal form, which is unique up to renaming. This result was rst proved by Prawitz [24] for i . speci ed in Definition 3.3) is con uent. Equivalently, conversion in is Church Rosser. parallel reduction (see also Barendregt [2] Hindley and Seldin [15] or Stenlund [27] 13 (as in Definition 3.3) is strongly normalizing. 1971) 11] 1972) see also Gallier [7] ....

....P ) M) M )P; or casex inx( P ) of inx(t: x: A) N N [ t; P=x] casex(5 9tA (P ) M) 5C (P ) A fundamental result about natural deduction is the fact that every proof (term) reduces to a normal form, which is unique up to renaming. This result was rst proved by Prawitz [24] for i . speci ed in De nition 7.3) is con uent. Equivalently, conversion in is Church Rosser. parallel reduction (see also Barendregt [2] Hindley and Seldin [15] or Stenlund [27] strongly normalizing. 1971) 11] 1972) see also Gallier [7] If one looks carefully at the ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2nd Scand. Log. Symp., pages 235-307. North-Holland, 1971.

.... have been established quite early by Turing (around 1941, published in [Gan80] and Sanchis [San67] This article employs a proof method that allows to show strong normalization for all typed terms without recourse to inclusive predicates such as strong computability [Tai67] or validity [Pra71] that are not formalizable in primitive recursive arithmetic. A simple induction on types veri es that (strong) normalizability is closed under application while closure under substitution has to be shown simultaneously. Separating concerns. With the prospect of computer checkable formalization ....

.... written Rec s t r) case analysis rR with R (x:s; y:t) commonly denoted by case r of inj 0 x ) s, of inj 1 y ) t) and generalized application rR with R (s; x:t) for r y written as t x fr; sg in [Sch99] The vector notation turns out to replace concepts like branches and endsegments [Pra71] which have been important elements of normalization arguments in systems with permutative conversions. Simply typed calculus. The basic system is dealt with in the rst four sections: In section 1 untyped calculus, function types, reduction and weak and strong normalizability are de ned. In ....

[Article contains additional citation context not shown here]

Dag Prawitz. Ideas and results in proof theory. In Jens E. Fenstad, editor, Proceedings of the Second Scandianvian Logic Symposium, pages 235-307. North{Holland, Amsterdam, 1971.

....a proof # from the relativized premises: # For then we get a proof of ZF # # , which is just ZF # #. So how we obtain # from # To be concrete, suppose we are working with a natural deduction formalization of first order logic. By the normal form theorem [17], since the conclusion of the proof is atomic, we can assume that # applies only elimination rules. We must modify # so that it accepts relativized versions of its premises and delivers a relativized version of its conclusion. The only hard cases involve quantifiers. Where # applies the ....

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Second Scandinavian Logic Symposium, pages 235--308. North-Holland, 1971.

....over P(WFF WFF . Thus the expression # A is a sequent where # (the antecedent) is a finite (possibly empty) set of formulas and A is a formula. As usual in Gentzen systems the meaning of operators and connectives is given by the rules for their introduction and elimination (cf. e.g. [17]) More precisely, this is true in the presence of the structural rules of exchange, duplication and contraction. Otherwise, the introduction and elimination rules have to be supplemented by rules for the structural meaning of the operators involved [8, 18] According to Definition 1 the usual ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307, Amsterdam, 1971. North Holland.

.... x: x:fx ) f Similar problems arise if we try to enrich the calculus with extra rewrite rules which may be confluent by themselves, but which when taken in conjunction with j contraction fail to be confluent [16] Recently several researchers [2,15,20,19,22, 49] have adopted older proposals [41,62,68] that j conversion be interpreted as an expansion: t ) x:tx if t : A B and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as: f ) x:fx ) x: y:fy)x ) 1.2) are prohibited by imposing syntactic restrictions to limit the ....

....may be calculated by first contracting all fi redexes and then performing any remaining j expansions or, vice versa, by performing all j expansions and then contracting any remaining fi redexes. Historically the use of j expansions, as opposed to j contractions, can be traced back to Prawitz [68] and Huet [41] The formulation of the restrictions on expansion required to recover strong normalisation were originally proposed by Mints [62] although it is only recently that several researchers [2,15,19,22,49] using different proof strategies, have proved confluence and strong ....

[Article contains additional citation context not shown here]

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2nd Scandinavian Logic Symposium, pages 235--307. North Holland, 1971.

.... have been established quite early by Turing (around 1941, published in [Gan80] and Sanchis [San67] This paper employs a proof method that allows to show strong normalization for all typed terms without recourse to inclusive predicates such as strong computability [Tai67] or validity [Pra71] that are not formalizable in primitive recursive arithmetic. A simple induction on types veri es that (strong) normalizability is closed under application while closure under substitution has to be shown simultaneously. Separating concerns. With the prospect of computer checkable formalization ....

.... written Rec s t r) case analysis rR with R (x:s; y:t) commonly denoted by case r of inj 0 x : s, of inj 1 y : t) and generalized application rR with R (s; x:t) for r y written as t x fr; sg in [Sch99] The vector notation turns out to replace concepts like branches and endsegments [Pra71] which have been important elements of normalization arguments in systems with permutative conversions. Simply typed calculus. The basic system is dealt with in the rst four sections: In section 1 untyped calculus, function types, reduction and weak and strong normalizability are de ned. In ....

[Article contains additional citation context not shown here]

Dag Prawitz. Ideas and results in proof theory. In Jens E. Fenstad, editor, Proceedings of the Second Scandianvian Logic Symposium, pages 235-307. North{Holland, Amsterdam, 1971.

....induction on the structure of the given derivations, using extensive case analysis. From right to left the theorem follows by case analysis on local or commuting reductions. 24 The calculus further satisfies a normalization theorem. This can be proven either directly via Tait s method as in [Pra71], by a detour via cut elimination as in [Pfe99] via CPS translation as in [dG99] or via an interpretation into a simply typed lambda calculus with disjunction [BBdP98] The latter is in many ways the simplest and easily extends to additional connectives. We map both 2A and 3A as A so that ....

Dag Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

....of that type A. This constructor T corresponds to the modality # just as the constructors in the ordinary typed lambda calculus correspond to in propositional formulas. They gave a natural deduction system for PLL and prove a strong normalization theorem by using the method in Prawitz [Pra97] (see also Tait [Tai67] and Troelstra [Tro73] Gol81] argued for an application of the logic in Grothendieck s topology. He extracted the principle (#) A is locally true at # i# A is true at all points close to # For instance, two functions f and g are said to be equivalent, or to have the same ....

D. Prawitz, Ideas and results in proof-theory, Proceedings of the second Scandinavian logic symposium, North-Holland, 1971, pp. 235-307.

....] n h is encoded without side conditions, in a similar way as v i [t] n (see Definition 3.3) Define d[d # ] h = d[d # ] 0 h . 13 PROOF REDUCTION 20 13 Proof reduction To illustrate how the defined machinery can be used to manipulate proof objects, we define Prawitz proof reduction rules [Pra71]. 15 The goal is to remove detours, as in the following tree. #; # # d [ # # # # (#, d) # # # # # e [ # # # # (# (#, d) e) # Instead of first assuming # to build a proof d of #, introduce the implication # # # and then eliminate it immediately by plugging ....

D. Prawitz. Ideas and results in proof theory. In Jens Erik Fenstad, editor, Proceedings of the Scandinavian Logic Symposium, pages 235307, Amsterdam, 1971, North-Holland.

....acts, in addition to the connectives of classical logic. As indicated above, the pragmatic connectives 1 The philosophical position adopted here seems to us in agreement with the point of view expressed by Professor Solomon Feferman in many occasions, in particular in his review of Prawitz [12] in the J.S.L. 5] A PRAGMATIC INTERPRETATION OF SUBSTRUCTURAL LOGICS 5 are given Heyting s semantics of proofs . A formula is called pragmatically valid (or p valid) if it represents relations between illocutionary acts which hold in all circumstances. Like the notion of provability in ....

D. Prawitz. Ideas and Results in Proof Theory. In Proceedings of the Second Scandinavian Logic Symposium, ed. Fenstad, North-Holland, 1971.

.... as found in everyday mathematics, we may ask questions about the identity of proofs: when is it that two formal proof figures correspond to the same informal argument in classical logic What properties of proofs are preserved in the process of cut elimination The conjecture proposed by Prawitz [17] for intuitionistic natural deducton NJ that two natural deduction derivations represent the same informal proof if and only if they are equivalent up to normalization is today regarded as implausible, in the sense that normalization is not expected to preserve the identity of proofs. If our ....

D. Prawitz. Ideas and Results in Proof Theory, in Proceedings of the Second Scandinavian Logic Symposium, ed. Fenstad, North-Holland, 1971.

....programmingand of isomorphisms of types [CDC91] They give a complete (infinite) set of reduction rules for the calculus, which is proved confluent using just weak confluence, weak normalization and some additional properties. Meanwhile, in the field of proof theory, Prawitz was suggesting [Pra71] to turn these extensional equalities into expansion rules, rather than contractions. Building on such ideas, but motivated by the study of coherence problems in category theory, Mints gives a first faulty proof that in the typed framework expansion rules, if handled with care, are weakly ....

D. Prawitz. Ideas and results in proof theory. Proceedings of the 2nd Scandinavian Logic Symposium, pages 235--307, 1971.

....Am ] M L is faithful for L if M L is sound and complete for L. Informally, M ipl is sound for ipl because the additional axioms are true and the rules of M are sound. A better argument is by induction on normal proofs in M. Here is a summary of the proof theoretic concepts of Prawitz [31, 32]. For simplicity, let us ignore equality rules, identifying terms that are equivalent up to # conversions. A branch in a proof traces the construction and destruction of a formula. Each branch is obtained by repeatedly walking downwards from a premise of a rule to its conclusion, but terminates ....

D. Prawitz, Ideas and results in proof theory, in: J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium (North-Holland, 1971), pages 235--308.

....: A Theta B:hx; yi and z : A Theta B:z are in normal form and they are different, so the previous diagram cannot be closed. End of Example 2 2 Fortunately, this inconvenient situation can be overcome by turning the extensional equalities into expansion rules. This idea was suggested by Prawitz [Pra71] in the field of proof theory, and then further studied by Huet [Hue76] in the field of higher order unification, and by Mints [Min79] who was motivated by problems in category theory. Expansions, however, seem to have passed unnoticed for a long time, until recent years, when it has been shown ....

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the 2nd Scandinavian Logic Symposium, pages 235--307, Amsterdam, The Netherlands, 1971. North Holland.

....by Girard, it is not sufficient that a deduction be in normal form for it to satisfy the subformula property. We are forced to introduce a new reduction relation, written ; c (and called c reduction) which Girard calls a commuting conversion (they are called permutative reductions by Prawitz [33]) These deal with what Girard calls the bad elimination rules ( E and E ) A deduction of the form A B [A] C [B] C E C . r E D is reduced to A B [A] C . r E D [B] C . r E D E D ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings of Second Scandinavian Logic Symposium, pages 235--307, 1971.

....Am ] M L is faithful for L if ML is sound and complete for L. Informally, M ipl is sound for ipl because the additional axioms are true and the rules of M are sound. A better argument is by induction on normal proofs in M. Here is a summary of the proof theoretic concepts of Prawitz [5, 6]. For simplicity, let us ignore equality rules, identifying terms that are equivalent up to # conversions. A branch in a proof traces the construction and destruction of a formula. Each branch is obtained by repeatedly walking downwards from a premise of a rule to its conclusion, but terminates ....

Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Second Scandinavian Logic Symposium, pages 235--308. North-Holland, 1971. 10

....and restricted search space for proofs. It also allows us to establish negative results, such as how incompleteness can arise; we show how analysis of normal forms provides a basis for investigating tradeoffs in formalizations. To reduce notational overhead, we follow, where possible, Prawitz [20,21]. Definition 15. A maximal lwff in a derivation is an lwff that is both the conclusion of an introduction rule and the major premise of an elimination rule. A maximal lwff constitutes a detour in a derivation, and we remove it by (finitely many applications of) proper reductions. Three possible ....

D. Prawitz. Ideas and results in proof theory. In J. E. Fensted, editor, Proceedings of the 2nd Scandinavian Logic Symposium, pages 235--307. North-Holland, Amsterdam, 1971.

....possible argument, they are equal. The classical extensional axiom in the theory of # calculus is the # equality, written as #x.Mx = # M , if x is not free in M . This equality can be operationally interpreted from left to right, as the classical # contraction rule, or in the other way round [36, 33, 23], as an b # expansion. The traditional contractive way breaks confluence in many # calculi [6, 12] while the b # expansion can be combined with many other higherorder reduction rules [11, 2, 15, 24, 12, 13] by preserving confluence and strong normalization. The main goal of this paper is to ....

D. Prawitz. Ideas and results in proof theory. Proceedings of the 2nd Scandinavian Logic Symposium, pages 235--307, 1971.

.... sequent calculi and good tableau ones have been provided for some of the most prominent (propositional) modal logics [Fitting, 1972; Wallen, 1990] Rather recently, also Intuitionistic Logic and the related ProofTheory (based on such tools as Prawitz s normalizable natural calculus [Prawitz, 1965; Prawitz, 1971], or Gentzen s sequent calculus with eliminable cut [Gentzen, 1969] or Dragalin s improved variant of Gentzen s sequent calculus [Dragalin, 1988] or Fitting s tableau calculus with signed formulas [Fitting, 1969] have been revisited in order to improve the involved proof strategies. Thus, in ....

Prawitz, D. (1971). Ideas and results in proof theory. In Proceedings of the Second Scandinavian Logic Symposium, pages 235-307. North-Holland.

.... proof theoretic results involving Intuitionistic Natural and Sequent Calculi, as well as Normalization and Cut elimination Theorems and their relations to the explicit de nability and the disjunction property, more generally, to the subformula property (the reader is referred to [Prawitz, 1965; Prawitz, 1971; Troelstra, 1973b; Girard et al. 1989] for a discussion about Natural Calculi and Normalization, and to [Takeuti, 1975; Girard, 1987; Girard et al. 1989] for a discussion about Sequent Calculi and Cut elimination) However, several links can be found between the two areas, which give a proof ....

....and : Only calculi conforming to such a paradigm of the proofs as programs are reasonable candidates, we believe, to be used for program synthesis. One of the main results about the computational content of proofs is, without any doubt, Prawitz s Normalization Theorem [Prawitz, 1965; Prawitz, 1971]. In the frame of Intuitionistic Arithmetic this result has a clear computational meaning and can be considered one of the most interesting implementations of the Proof as Programs paradigm. A deeper discussion on this point can be found e.g in [Girard et al. 1989; Miglioli and Ornaghi, 1981a; ....

Prawitz, D. (1971). Ideas and results in proof theory. In Proceedings of the Second Scandinavian Logic Symposium, pages 235-307. North-Holland.

.... the labels might track resources and their use [5] We study this combination in the case of propositional modal logics and show how it can provide a simple and usable implementation of a large collection of logics (including K, D, T , B, S4, S4:2, KD45, S5) in a natural deduction (ND, [16, 17]) setting. We view a proof system for an LDS as consisting of two parts: a base logic for manipulating labelled formulae, and a separate labelling algebra for reasoning about the labels. Our base logic, in which labels represent possible worlds in the Kripke frame, is a labelled ND presentation ....

D. Prawitz. Ideas and Results in Proof Theory. In J. E. Fensted, editor, Proc. 2nd Scandinavian Logic Symp., Amsterdam, 1971. NorthHolland.

....and x:f(x) Gamma T x: where f is of type T T and x of type T. Note that both f and x: cannot be contracted further. This is a serious drawback since it can easily occur when one adds j contraction to algebraic rewriting systems. Prawitz suggested that j expansion be used as a rewrite rule [20]. Mints also realised this option. The following example shows that applying j expansion unconditionally easily leads to an infinite reduction sequence. x A B Gamma j y A :x A B (y A ) Gamma j y A : y A :x A B (y A ) y A ) Gamma j Delta Delta Delta However, this ....

....reducing them to R normal forms. Therefore, with R being left linear, we may have more freedom to design a decision procedure verifying if hM; N i 2 R fifi 2 j j 2 for every pair of 2 ( Sigma) terms M and N . 7 Related Work The use of j expansion as a rewrite rule was suggested in [20]. Mints [19] presented a proof for the confluence and (weak) normalisation of a simply typed calculus with surjective pairing and j expansion. A flaw in Mints s proof was later corrected in [6] Y. Akama [1] proved the confluence and strong normalisation of the above system, which is also given ....

D. Prawitz (1971), Ideas and results of proof theory, Proceedings of the 2nd Scandinavia logic symposium, editor J.E. Fenstad, North-Holland Publishing Company, Amsterdam.

....not satisfied, e.g. Ei without i, or Ec with only c, is out of the scope of this paper) 13 A logic L = B T is the extension of an appropriate base logic B with a Horn relational theory T . Consider the restricted language (with , oe, M u , M e , of page 9. Following Prawitz [31, 32], in Figure 1 we distinguish three families of ND systems according to their treatment of negation: minimal, intuitionistic or classical (we make the distinction by considering the rules, as opposed to Prawitz s rules) The minimal system ML is determined by a base logic containing monl ....

....of b for a in Pi, with a suitable renaming of the variables to avoid clashes, and we use superscripts to associate discharged assumptions with rule applications. Finally, we assume the reader is familiar with the terminology of natural deduction (e.g. major and minor premises in an inference, cf. [31, 32, 45]) Xi 9 Note that, unlike Prawitz s, our minimal system does not satisfy the inversion principle, since it contains monl which is neither an introduction nor an elimination rule. 14 L B T ML rules for ; oe; M u ; M e , monl monR i rules JL rules for ; oe; M u ; M e , monl ....

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D. Prawitz. Ideas and results in proof theory. In J. E. Fensted, editor, Proceedings of the 2nd Scandinavian Logic Symposium. North-Holland, 1971.

....required. Instead, the associated long #(#) normal form is: #x b#b 1 .#x b 2 . x b#b 1 x b 2 Two Aspects of Long #(#) Normal Forms Huet s notion of long #(#) normal forms has had rewriting independent applications in many areas, for example, in the proof theory of both natural deduction [21] and of Gentzen style systems [22] However, as we have indicated, they also arise naturally in rewriting. Theorem 4 (Ghani [13] F #(#) long and the normal forms of # # # # # # coincide. 5 e : x # #x # .e e e #a.e e # # c c : #y # .c #a.c d (if d ....

D. Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland, 1971.

....5 1.2 Search and structure Sequent calculus can be seen as a method for formalizing the process of proof search in natural deduction. Natural deduction systems express the Curry Howard isomorphism most concisely: natural deduction proofs correspond to terms both in syntax and in normalization (Prawitz, 1971). However, natural deduction raises difficulties for describing proof search strategies. Natural deduction involves two kinds of rules, introduction and elimination rules, that should be used in different circumstances in proof search. Elimination rules should be used to decompose assumptions, ....

Prawitz, D. (1971). Ideas and results in proof theory. In Fenstad, J. E., editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307, Amsterdam. North-Holland.

....values in the sense of functional programming, ii) 4 R. Dyckhoff L. F. Pinto natural deductions are well understood and (iii) they lack the redundancy of traditional Gentzen style sequent calculi (arising from the permutations [21] therein) We consider also the expanded normal deductions of [30]. Normal is defined as in [34] Note that this restricts application of E to cases where the conclusion is atomic. in fact causes many problems. Our guiding principle here is that there should be exactly one normal natural deduction proof of = def oe ; there are two possible candidates: ....

....deductions we are interested in and then what sequent calculus like MJ corresponds to it in a bijective fashion. To this end, we may further require that the minimal formulae (of normal natural deductions) are atomic (or ) in this case we have the expanded normal form deductions of Prawitz [30] (equivalently, proof terms in fij long normal form) If we restrict use of axioms in MJ to the matching of atomic formulae (or of ) and the use of S to cases where the goal formula is atomic, and syntactically restrict the programs Delta to hereditary Harrop formulae and goals G to hH goals, ....

D. Prawitz. Ideas and results in proof theory, in: J. E. Fenstad, Proceedings of the Second Scandinavian logic symposium, North-Holland, pp 235--308, 1971.

....informing different areas of computer science, such as symbolic computation and term rewriting systems, abstract data type specifications, functional programming languages and concurrency. The abstract concept of reduction affects different objects: terms [8] combinators and terms [6] proofs [11], graphs [5] polynomials [1] In [8] the concept of Abstract Reduction System is proposed, that, abstracting from the particular objects to which reduction apply, presents a unifying treatment of reduction, where some important concepts, such as confluence, noetherianity, normal form can be ....

D. Prawitz. Ideas and results in proof theory. In Proc. of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland, 1971. 12

....3 below for description. 4 For a more formal description see [Baker 93] A Proof Environment for Arithmetic with the Rule 3 PA by exhibiting a proof tree labelled at the root with the given formula. Syntactical details about this system PA are given in [Lopez Escobar 76, P162] see [Prawitz 71, P266 267] for a natural deduction representation) PA has been described by Schutte as a semi formal system to stress the difference between this and usual formal systems which use finitary rules [Schutte 77, P174] For implementational purposes infinite proofs must be thought of in the ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Studies in Logic and the Foundations of Mathematics: Proceedings of the Second Scandinavian Logic Symposium, volume 63, pages 235--307. North Holland, 1971.

....Corollary 13 If all terms in F are weakly normalisable then they are strongly normalisable. Proof Parallel to the proof of Corollary 11 It is first shown by Girard (1971) that all terms in F are weakly normalisable. Though it is proven later that all terms in F are strongly normalisable by Prawitz (1971), the formulation of 6 reducibility candidates gets complicated accordingly. This comparison can also be done with a sharp weak normalisation proof for the theory of species (Martin Lof, 1971) where the concise and perspicuous formulation of reducibility candidates certainly enhances the ....

Prawitz, D., (1971), Ideas and results of proof theory, Proceedings of the 2nd Scandinavian logic symposium, editor J.E. Fenstad, North-Holland Publishing Company, Amsterdam.

....f : 1 1 and : 1 and with rewrite rule fx ) then ) is confluent. However, as the above counterexample shows shows, the combination of ) with the contractive j rewrite rule fails to be confluent see [5] for further details. Recently several authors [1, 4, 6, 14] have accepted the old proposal [13, 16, 17] that j conversion be interpreted as an expansion f ) x:fx and the resulting rewrite relation has been shown confluent. Infinite reduction sequences such as x:t ) y: x:t)y ) y:t[y=x] j x:t tu ) x:tx)u ) tu (1) are avoided by imposing syntactic restrictions to limit the possibilities for ....

....This restricted expansion relation is strongly normalising, confluent and generates the same equational theory as the unrestricted expansionary rewrite relation thus fij equality can be decided by reduction to normal form in this restricted fragment. In addition, Huet s long fij normal forms [13, 17] are exactly the normal forms of the restricted expansionary rewrite relation and thus j expansions provide a natural mathematical theory of this important class of terms. Perhaps most pleasingly of all, these properties are maintained if one adds other type constructors, base types and ....

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2nd Scandinavian Logic Symposium, pages 235--307. North Holland, 1971.

....and 4. if t Gamma Gamma oe i a then A is computable. Let C T denote the computable terms of type T , and set C to be S fC T j T a type.g Tait [Tai67] originated the strategy of using an inductively defined predicate such as computability to prove termination in the calculus. Prawitz [Pra70] pointed out the possibility of basing a notion of computability, there termed validity, based on I terms rather than on E terms (as Tait s method is) having observed that the latter approach breaks down when sum types are involved. He proves termination of a certain calculus by a method based on ....

D. Prawitz. Ideas and Results in Proof Theory, in J. E. Fenstad, ed., Proceedings of the Second Scandinavian Logic Symposium. North-Holland, Amsterdam, 1971.

.... have been established quite early by Turing (around 1941, published in [Gan80] and Sanchis [San67] This paper employs a proof method that allows to show strong normalization for all typed terms without recourse to inclusive predicates such as strong computability [Tai67] or validity [Pra71] that are not formalizable in primitive recursive arithmetic. A simple induction on types veri es that (strong) normalizability is closed under application while closure under substitution has to be shown simultaneously. Separating concerns. With the prospect of computer checkable formalization ....

.... written Rec s t r) case analysis rR with R (x:s; y:t) commonly denoted by case r of inj 0 x : s, of inj 1 y : t) and generalized application rR with R (s; x:t) for r y written as t x fr; sg in [Sch99] The vector notation turns out to replace concepts like branches and endsegments [Pra71] which have been important elements of normalization arguments in systems with permutative conversions. Simply typed calculus. The basic system is dealt with in the rst four sections: In section 1 untyped calculus, function types, reduction and weak and strong normalizability are de ned. In ....

[Article contains additional citation context not shown here]

Dag Prawitz. Ideas and results in proof theory. In Jens E. Fenstad, editor, Proceedings of the Second Scandianvian Logic Symposium, pages 235-307. North{Holland, Amsterdam, 1971.

....induction on the structure of the given derivations, using extensive case analysis. From right to left the theorem follows by case analysis on local or commuting reductions. 24 The calculus further satisfies a normalization theorem. This can be proven either directly via Tait s method as in [Pra71], by a detour via cut elimination as in [Pfe99] via CPS translation as in [dG99] or via an interpretation into a simply typed lambda calculus with disjunction [BBdP98] The latter is in many ways the simplest and easily extends to additional connectives. We map both 2A and 3A as A so that ....

Dag Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

....proof distribution TUCS Research Group Programming Methodology Research Group 1 Introduction This paper presents a new format for writing proofs, which we call structured calculational proof. Two of the main inspirations we have drawn on when formulating this format are natural deduction [3, 10] and calculational proof [2, 4, 13] The clarity and readability of calculational proof has made it a popular choice among computer scientists. We feel, however, that pure calculational proof provides too little support for the formal decomposition of large proofs into smaller ones. Natural ....

....proof were produced in the format, that this must have been the result of an erroneous proof step made by its author, and not because of the format itself. We do this by showing how the individual steps of a structured calculational proof can be reassembled to form a natural deduction proof [3, 10] of the same result. Since natural deduction is well known to be sound, we will therefore have shown that if the individual steps of the structured calculation proof are sound, then the proof as a whole is also sound. Let us begin by considering how the individual steps of an ordinary ....

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Dag Prawitz. Ideas and results in proof theory. In Jens Erik Fenstad, editor, Proceedings of the 2nd Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 235--307, Oslo, 18--20 June 1970. North Holland, Amsterdam.

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Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

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Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

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D. Prawitz, Ideas and results in proof theory, in: J.E. Fenstad (ed) Proc. 2nd Scandinavian Logic Symp. (North-Holland, 1971) 235--307.

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Dag Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 235--307. North-Holland Publishing Co., 1971.

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Dag Prawitz. Ideas and results in proof theory. In Proc. Second Scandinavian Logic Symposium

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D. Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, nd Scandinavian Logic Symposium, pages 235--307. North Holland, 1971.

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D. Prawitz, Ideas and results in proof theory, in Proceedings of the Second Scandinavian Logic Symposium, 1971, pp. 235--307.

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D. Prawitz. Ideas and results in proof theory. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and The Foundations of Mathematics, pages 235-307. North-Holland, 1971.

No context found.

D. Prawitz. Ideas and Results of Proof Theory. In Proceedings of the 2nd Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 235--307. North-Holland, 1971.

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Dag Prawitz. Ideas and results in proof theory. In Jens Erik Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 235--307, Amsterdam, 1971. North-Holland. Cited on pages 8, 25, 26, and 77.

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Dag Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2 Scandianvian Logic Symposium, pages 235-307. North{ Holland, 1971. 22

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Prawitz, D. Ideas and Results in Proof Theory. in: Proceedings of the Second Scandinavian Logic Symposium, University of Oslo, June 18--20,

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Prawitz, D., Ideas and Results in Proof Theory, Proc. 2nd Scand. Log. Symp., 1970.

No context found.

D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proceedings Second Scandinavian Logic Symposium, pages 235--307. 1971.

No context found.

D. Prawitz. Ideas and results of proof theory. In J.E. Fenstad, editor, The 2nd Scandinavian Logical Symposium, pages 235307. North-Holland, 1970.

No context found.

Prawitz D. Ideas and results in Proof theory, in: J.E. Fenstad (ed.), Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, pp. 235-307, (1973). 47

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