L. Paulson. Constructing recursion operators in intuitionistic type theory. J. Symbolic Comput., 2:325-355, 1986.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....formal derivation, in Nuprl, of a number of induction principles having e#cient general recursion schemes as their computational content. The development described here supports the larger Nuprl project goals of developing practical proof based programming methodologies. In a widely cited paper [11], Paulson derived recursion schemes from a theory of well founded relations. In the final section of that paper, Paulson suggests some alternative approaches which would accomplish the following goals. i. To eliminate non computational content from the programs extracted from proofs that use the ....

L. C. Paulson. Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation, 2(4):325--355, 1986.

....formal derivation, in Nuprl, of a number of induction principles having ecient general recursion schemes as their computational content. The development described here supports the larger Nuprl project goals of developing practical proof based programming methodologies. In a widely cited paper [11], Paulson derived recursion schemes from a theory of well founded relations. In the nal section of that paper, Paulson suggests some alternative approaches which would accomplish the following goals. i. To eliminate non computational content from the programs extracted from proofs that use the ....

L. C. Paulson. Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation, 2(4):325-355, 1986.

.... studied in the 1980 decade, for instance by Constable and Mendler in [3, 2] The AEavor we use in this paper is mostly described by Coquand, Pfenning, and Paulin Mohring in [14, 4, 11] General use of well founded recursion in Martin L#f s intuitionistic type theory was studied by Paulson in [12], who shows that reduction rules can be obtained for each of several means to construct well founded relations from previously known well founded relations. By comparison with Paulson s work, our technique is to obtain reduction rules that are specic to each recursive function. The introduction of ....

Lawrence C. Paulson. Constructing recursion operators in intuitionistic type theory. Technical report 57, University of Cambridge, Computer Laboratory, October 1984.

....scheme. However, we will see that this is not the case, indeed it is straightforward to optimise the resulting program such that the gain of efficiency is maintained. This can be exemplified by using the inductive presentation of general well founded recursion in Type Theory as proposed by [Nor88, Pau86]. We define the predicate Acc which defines the accessible subset of a relation 5 . We can define Acc in ALF inductively: Acc ( A; A)Set; A) Set acc (a A; b A; R(b, a) Acc(R, b) Acc(R, a) Le (Nat; Nat) Set From these it is possible to derive a typed fix point combinator which given ....

Lawrence C. Paulson. Constructing Recursion Operators in Intuitionistic Type Theory. Journal of Symbolic Computation, 2:325--355, 1986.

....the number of theorems, the number of lemmas, and nally the ratio between the number of lines and the dioeerent objects dened or proved. Note that these gures do not include two important contributions that we have been using in the proof. A theory of lexicographic exponentiation derived from [17] is provided within the Coq system. It contains the main result needed for proving that reductions always terminate. A contribution of Lo#c Pottier [19] gave us a non constructive proof of the Dixon s lemma 2 . As explained before, this gives us indirectly the termination of the algorithm. The ....

Lawrence C. Paulson. Constructing Recursion Operators in Intuitionistic Type Theory. Journal of Symbolic Computation, 2(4):325355, December 1986.

....where z=fact x in snd(power m f (n,0) The reader should carefully study these examples, and try to derive general algebraic laws relating the various combinators introduced. More examples of recursive list operators are given in [2, 5] A good introduction to recursive programming is Burge [6]. Functional programming application and implementation is explained in Henderson [15] A good textbook is AbelsonSussman [3] 1.2.4 Defining an object language in CAML We now give a simple example where we define an object language in CAML . Our goal is to define a small calculator of ....

....[4] A. W. Appel. Semantics Directed Code Generation. Twelfth ACM POPL Symposium, NewOrleans (Jan. 1985) 315 324. 5] R. Bird. An Introduction to the Theory of Lists. Course Notes, International Summer School on Logic of Programming and Calculi of Discrete Design, Marktoberdorf, Aug. 86. [6] W. H. Burge. Recursive Programming Techniques. Addison Wesley (1975) 7] L. Cardelli. ML under UNIX. Bell Laboratories, Murray Hill, New Jersey (1982) 8] L. Cardelli. Amber. Bell Laboratories Technical Memorandum TM 11271 840924 10 (1984) 9] W. Clinger, D. P. Friedman and M. Wand. A ....

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L. C. Paulson. "Constructing Recursion Operators in Intuitionistic Type Theory." Tech. Report 57, Computer Laboratory, University of Cambridge (Oct. 1984).

....Acc(R) Note that all r normal forms, i.e. all x such that no R(y; x) exists are in Acc(R) because the precondition is vacuously true. We can now implement a refined version of fix which realizes general well founded recursion via structural recursion over the proof of Acc(R) 17 Due to Paulson [Pau86] and Nordstrom [Nor88] fix (A, B Set; R (A; A)Set; a A; b A;R(b, a) B) B; a A; Acc(R, a) B fix (A, B, R, h, a, acc( h 2 ) h(a, b, h 1 ]fix (A, B, R, h, b, h 2 (b, h 1 ) A source for well founded recursion are the internal versions of the structural orderings on ....

Lawrence C. Paulson. Constructing Recursion Operators in Intuitionistic Type Theory. Journal of Symbolic Computation, 2:325--355, 1986.

....about type equality. Remark 5.4 Even it is apparent that a relation is well founded, proving this may be diOEcult and well founded relations are most easily constructed from simpler ones, using rules that preserve the well founded property. Some rules can be described for well founded relations [21]. 5.3.2 The type of accessible elements To include general recursion inside the theory, B. Nordstr#m has proposed a new type in the theory [19] Let be a binary relation on a set A (i.e, x; y) proposition for x; y : A) The set Acc(A; of accessible elements of in A is the set of ....

....as types allows a compact logical system but some uninteresting propositional constructions can complicate the computational ones. For instance, treating the proposition x OE x as a type can be complicated, and even the nal term can be computed without computations of elements of x OE x [21]. The alternative of [6] is to reason about programs in an untyped logical theory of propositions where propositions are primitive and with types as predicate. Here the xed point Y exists and can de ne recursive functions. Well founded induction is required but Y takes the place of well founded ....

L.C. Paulson. Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation, 2:325355, 1986.

....of the arcane theory of idempotent substitutions is needed solely for proving the Tail Ordering Proposition. MW s formulation is much neater: t 2 SUBST s, u 2 SUBST s) un (COMB t 1 t 2 , COMBu 1 u 2 ) 22 Induction over certain well founded relations can be reduced to structural induction [25]. Note how un is built up from simpler relations. Induction over the lexicographic combination of two relations gives rise to nested inductions. Induction over the immediate occurs in relation (of which OCCS is the transitive closure) is simply structural induction. The complication is ....

....substitutions. On the other hand, every LCF proof teaches us something about methods. The unification proof requires an understanding of many kinds of structural induction [24] Deriving its well founded induction rule involves a new set of techniques that has become one of my research interests [25]. Soko#lowski s proof of the soundness of Hoare rules uses powerful generalizations of goals and tactics [26] The constant accumulation of techniques means that future proofs can be more ambitious than this one. Acknowledgements: In such a large project, it is hard to remember everyone who ....

L. C. Paulson, Constructing recursion operators in Intuitionistic Type Theory, Report 57, Computer Lab., University of Cambridge (1984).

....the equality rule is then proved. This work follows Suppes s treatment of transfinite recursion in set theory [36] Operator wfrec is defined once and for all, and its properties proved, for all wellfounded relations in the logic. It is far stronger than my work in Martin Lof s Type Theory [28], which considers certain ways of constructing well founded relations and their corresponding recursion operators. 13 Conclusions Programs are typically verified within a special logic of computation. Although several such logics have been successful, they sometimes restrict abstract ....

Lawrence C. Paulson. Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation, 2:325--355, 1986.

....involving an immediate component of the argument. This excludes functions that divide by repeated subtraction or that sort by recursively sorting shorter lists. Coding such functions using structural recursion requires ingenuity; consider Smith s treatment of Quicksort [26] Nordstrom [19] and I [21] have attempted to re introduce well founded relations to type theory, with limited success. In ZF set theory, well founded relations reclaim their role as the foundation of induction and recursion. They can express di#cult termination arguments, such as for unification and Quicksort; they include ....

Paulson, L. C., Constructing recursion operators in intuitionistic type theory, J. Symb. Comput. 2 (1986), 325--355

....is wellfounded. This proof will have a remarkable similarity to the LCF proof, which was conducted in domain theory. This section uses a simple mathematical framework with no partial elements. It uses constructive mathematics because this paper is an outgrowth of my study of well founded relations [10] in Martin Lof s Constructive Type Theory [8] Showing that a relation is well founded requires showing the soundness of its rule of well founded induction for an arbitrary predicate P : #x. #x. In constructive reasoning, showing that has no infinite descending chains is ....

....v # and w # .SoIf (u, v # ,w # ) is a predecessor, justifying the final recursive call. By induction hypothesis this returns a normal expression. This reasoning about the various cases of norm 2 can be formalized in my setting of well founded recursion operators in Constructive Type Theory [10]. A function application has type y#exp ISN (y) It returns a pair of results: a normal expression y and a proof object of type ISN (y) Each recursive call on an argument z must be justified by exhibiting a proof object of type z 2 x. This is passed as an additional argument. In the If If ....

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L. C. Paulson, Constructing recursion operators in Intuitionistic Type Theory, Report 57, Computer Lab., University of Cambridge (1984).

....then A is also a well ordering. If f is bijective then obviously f is an isomorphism between the orders #A, A # and #B, B #; it follows that their order types are equal. Sum, product and inverse image are useful for expressing well orderings; this follows Paulson s earlier work [16] within Constructive Type Theory. 4.4. CARDINAL NUMBERS Figure 2 presents the Isabelle ZF definitions of cardinal numbers, following Kunen s 10. The Isabelle theory file extends some Isabelle theories (Order Type and others) with constants, which stand for operators or predicates. The ....

Lawrence C. Paulson. Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation, 2:325--355, 1986.

....involving an immediate component of the argument. This excludes functions that divide by repeated subtraction or that sort by recursively sorting shorter lists. Coding such functions using structural recursion requires ingenuity; consider Smith s treatment of Quicksort [26] Nordstrom [19] and I [21] have attempted to re introduce well founded relations to type theory, with limited success. In ZF set theory, well founded relations reclaim their role as the foundation of induction and recursion. They can express di#cult termination arguments, such as for unification and Quicksort; they include ....

Paulson, L. C., Constructing recursion operators in intuitionistic type theory, J. Symb. Comput. 2 (1986), 325--355

....then A is also a well ordering. If f is bijective then obviously f is an isomorphism between the orders hA; A i and hB; B i; it follows that their order types are equal. Sum, product and inverse image are useful building blocks for well orderings; this follows Paulson s earlier work [16] within Constructive Type Theory. 4.4. CARDINAL NUMBERS Figure 2 presents the Isabelle ZF definitions of cardinal numbers, following Kunen s x10. The Isabelle theory file extends some Isabelle theories (Order Type and others) with constants, which stand for operators or predicates. The constants ....

Lawrence C. Paulson. Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation, 2:325--355, 1986.

....involving an immediate component of the argument. This excludes functions that divide by repeated subtraction or that sort by recursively sorting shorter lists. Coding such functions using structural recursion requires ingenuity; consider Smith s treatment of Quicksort [26] Nordstrom [19] and I [21] have attempted to re introduce well founded relations to type theory, with limited success. In ZF set theory, well founded relations reclaim their role as the foundation of induction and recursion. They can express difficult termination arguments, such as for unification and Quicksort; they ....

Paulson, L. C., Constructing recursion operators in intuitionistic type theory, J. Symb. Comput. 2 (1986), 325--355

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L. Paulson. Constructing recursion operators in intuitionistic type theory. J. Symbolic Comput., 2:325-355, 1986.

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