Schurmann, Carsten. (2000). Automating the meta-theory of deductive systems. Tech. rept. CMU-CS-00-146. Department of Computer Science, Carnegie Mellon University.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that are introduced by dependent types complicate this scenario slightly, as we discuss below. One might expect that these dynamic extensions are local to the datatype, which can be viewed as consisting of the traditional static part and the dynamic part. But this is not the case. Interestingly [14, 15], one can use these extensions to express such properties as: Every newly introduced parameter x:exp is well typed, expressed by the related assumption u:of x t for some type t : tp . We will see this in the next section when we show how to program in Delphin (see also Example 1 below) How, ....

....are free to extend datatypes dynamically during evaluation as long as these extensions conform to the rules stipulated by the world in which a function is defined. We say that a function cannot leave the world in which it lives during evaluation. The idea of worlds is not new; it was introduced in [14], studied as a means of defining recursive functions in [15] and applied to reasoning by induction in [16] Worlds have also been implemented in the Twelf system [10] The block from Example 1 is declared in Twelf as follows: block L : some T:tp x:exp u:of x T . 3.2 Language ....

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C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, 2000. CMU-CS-00-146.

....family) between judgments whom the logic programming like interpretation provides an operational semantics to. Finally, external coverage checking (which are, for the time being, limited to LF [22] verifies that the given relation is indeed a realizer for that theorem. Only as we speak, M# [20], the meta logic of LF has been extended to # , a meta logic for LLF [10] In the same vein, Polakow and Pfenning [17] have used an Ordered Logical Framework to formally show that terms resulting from a CPS translation obey stack like ordering properties with respect to intermediate values ....

....of bunches in BI [19] As far as the infrastructure is concerned, note that similarly to [13] in this case study we only needed to induct closed terms, although we reason (typically by inversion) in presence of hypothetical judgments. Inducting HOAS style over open terms is a major challenge [20]; in this setting generic judgments are particularly problematic, but can be dealt with by switching to a more expressive SL, based on a eigenvariable encoding [12] The new theory of terms in infinite context underlying the new version of Hybrid [2] directly supports this syntax. With that in ....

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, CarnegieMellon University, 2000. CMU-CS-00-146.

.... use of the induction principles used in the cut admissibility proof, since this requires a deep embedding [9] The use of such deep embeddings to formalise meta logical results is rare [8, 7] To our knowledge, the only full formalisation of a proof of cut admissibility is that of Schurmann [10], but the calculi used by both Pfenning and Schurmann contain no explicit structural rules, and structural rules like contraction are usually the bane of cut elimination. Here, we use a deep embedding of the display calculus #RA into Isabelle HOL to fully formalise the admissibility of the cut ....

C Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Dept. of Comp. Sci. , Carnegie Mellon University, USA, CMU-CS-00-146, 2000.

....which 10 LF does not (as such a construct would destroy LF s notion of canonical forms) Consequently, SSTP can use primitive recursion on encoded types to support intensional type analysis [9, 5] which LTT cannot. Various proposals have been made for extending LF with primitive recursion [6, 20, 19] and we are exploring integrating one of these into LTT. LTT provides the power for very expressive type systems by allowing operations to demand proofs of arbitrary propositions (thereby escaping the usual restrictions of decidable typechecking) and by allowing operations to demand linear ....

C. Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Carnegie Mellon University, School of Computer Science, Pittsburgh, Pennsylvania, Oct. 2000.

....v : A Fig. 4. A formal system for strictness Theorem 1. Given a coverage goal # A : type and a strict pattern # i A i : type. Then it is decidable if there is a substitution # # : # i such that A i [#] Moreover, if such a substitution exists it is uniquely determined. Proof. See [21]. If such a # exists then the coverage goal in question is covered by one of the patterns. On the other hand, if no such # exist, either there exists a pattern that covers the coverage goal but does not happen to cover it immediately, or none of the patterns will ever cover it, in which case ....

....always a set of substitutions that is non redundant and complete. Theorem 3 (Splitting is non redundant and complete) The set of substitutions generated by splitting is non redundant and complete. Proof. From properties of complete sets of most general unifiers and lemmas about raising (see [21]) # but not those whose results are indeterminate because of residual equations, which are are not permitted 9 3.4 The Coverage Algorithm Recall that a coverage goal # A : type is immediately covered by a collection of terms # i A i : type if there is an i and # A i [# i ] ....

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C. Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Department of Computer Science, Carnegie Mellon University, Aug. 2000. Available as Technical Report CMU-CS-00-146.

....interested in matching and not full unification. Theorem 1. Given a coverage goal # A : type and a strict pattern # i A i : type. Then it is decidable if there is a substitution # # : # i such that A i [#] Moreover, if such a substitution exists it is uniquely determined. Proof. See [24]. This unification becomes matching because the input arguments of the goal are ground at run time. 5 ls ld #; #, y : A M ls lb ls pd #; #, y : A1 ls pb ls c (1 c M1 . Mn ls a (1 a M1 . Mn y : A # #; # ls var (1 y M1 . ....

....of higher order dependently typed unification [6] it is enough to maintain well typedness modulo postponed equations if we eventually solve them from left to right. This means that if we have no candidates from the first two kinds of equations, we call a strict higher order matching algorithm [24] on the residual equations. If this succeeds then A # covers A. Otherwise, A # does not cover A and we suggest no candidate variables for splitting because it would be di#cult to guarantee termination. When considering a particular coverage goal # A : type, we apply the above algorithm with ....

[Article contains additional citation context not shown here]

C. Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Department of Computer Science, Carnegie Mellon University, Aug. 2000. Available as Technical Report CMU-CS-00-146.

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Schurmann, Carsten. (2000). Automating the meta-theory of deductive systems. Tech. rept. CMU-CS-00-146. Department of Computer Science, Carnegie Mellon University.

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Schurmann, Carsten. (2000). Automating the meta-theory of deductive systems. Ph.D. thesis, Carnegie Mellon University.

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C. Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Department of Computer Science, Carnegie Mellon University, Aug. 2000. Available as Technical Report CMU-CS-00-146.

No context found.

C. Schurmann. Automating the Meta-Theory of Deductive Systems. Ph. D. thesis, Carnegie Mellon University, Pittsburgh, PA, 2000.

No context found.

C. Schurmann. Automating the Meta Theory of Deductive Systems. PhD thesis, Department of Computer Science, Carnegie Mellon University, Aug. 2000. Available as Technical Report CMU-CS-00-146.

No context found.

C. Schurmann. Automating the Meta-Theory of Deductive Systems. PhD thesis, CarnegieMellon University, 2000. CMU-CS-00-146.

No context found.

C. Schurmann. Automating the Meta-Theory of Deductive Systems. Ph. D. thesis, Carnegie Mellon University, Pittsburgh, PA, 2000.

No context found.

C. Schurmann. Automating the Meta-Theory of Deductive Systems. Ph. D. thesis, Carnegie Mellon University, Pittsburgh, PA, 2000.

No context found.

Schurmann, C. Automating the Meta Theory of Deductive Systems. PhD thesis, Dept. of Comp. Sci. , Carnegie Mellon University, USA, CMU-CS-00-146, 2000.

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