Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming. 39  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the theories required for reasoning about safety properties such as arithmetic modulo 2 32 or memory update and access from the underlying mechanism of checking proofs. It is also simple, since it is based on a pure, dependently typed # calculus whose properties have been deeply investigated [9,10]. Proofs in a logic designed for reasoning about safety properties are represented as terms in LF. Checking that a proof is valid is reduced to checking that its representation in the logical framework is well typed. This can be carried out e#ectively even for very large proof objects. ....

R. Virga, Higher-order rewriting with dependent types, Ph.D. thesis, Department of Mathematical Sciences, Carnegie Mellon University, available as Technical Report CMU-CS-99-167 (Sep. 1999).

....other two cases that use new are, respectively, of fix and of case . Note, that in the case case T must be nat, which is directly incorporated into the pattern matching. A necessary and su#cient criterion that implies strengthening theorems for all type families has been developed elsewhere [18]. It is e#ectively computable, and it follows from a static analysis of the dependency relation check : world (L) all fun check nat nat = check (arr T1 T2) arr T # 1 T # 2 ) let end Figure 5: Equivalence of types among objects of di#erent types. From this criterion, it is ....

R. Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....if each of the subgoals # i U i : V i are impossible. However, in general this would not terminate (and cannot, because inhabitation is undecidable) Instead, we only continue to split further if all of the new variables u k : A k in # i have a type that is strictly subordinate to the type A [23, 21]. Otherwise, we fail and report the immediate supergoal as a potential counterexample. 13 Theorem 5. Finitary splitting terminates, either with an indication that the given coverage goal has no ground instances, or failure. Proof. There are only a finite number of variables in a given coverage ....

R. Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, Sept. 1999. Available as Technical Report CMU-CS-99-167. 16

....if each of the subgoals # i U i : V i are impossible. However, in general this would not terminate (and cannot, because inhabitation is undecidable) Instead, we only continue to split further if all of the new variables u k : A k in # i have a type that is strictly subordinate to the type A [27, 24]. Otherwise, we fail and report the immediate supergoal as a potential counterexample. Theorem 5. Finitary splitting terminates, either with an indication that the given coverage goal has no ground instances, or failure. Proof. There are only a finite number of variables in a given coverage ....

R. Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, Sept. 1999. Available as Technical Report CMU-CS-99-167. 18

....normalization when expansion is not restricted [JG95, Gha97] In the dependently typed case, even the definition of long normal form is not obvious [DHW93] and the technical development is fraught with di#culties. We have not been able to reconstruct the proofs in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treatments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91] Coquand s approach is based on ....

....type labels on # abstractions. We show that quasi canonical forms of a given type are su#cient to determine the meaning of an object, since the type labels can be reconstructed (up to definitional equality) from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. In some of the original work on Martin Lof type theory [NPS90] and some subsequent studies [Str91] type theories without type labels have been studied, but to our knowledge they have not ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, September 1999. Available as Technical Report CMU-CS-99-167. 40

....xes of in e. By deriving such annotations, we make all computed classi ers available at every point in the derivation. This sets the stage for reusing computed classi ers in Section 4. Similar technical machinery is used in [8] There is also some resemblance to the marked calculus for LF of [22]. 6 Notation 1 (annotations) Suppose a is an annotation of e 2Exp with Def(a) f 1 ; n g. Then a may be denoted f 1 : a( 1 ) n : a( n )g. Figure 2 presents the calculus. The primary derivable objects are sequents j e a, where a is an annotation and e 2Exp . It is ....

R. Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Carnegie Mellon University, October 1999.

....normalization when expansion is not restricted [JG95, Gha97] In the dependently typed case, even the definition of long normal form is not obvious [DHW93] and the technical development is fraught with difficulties. We have not been able to reconstruct the proofs in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treatments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91] Coquand s approach is based on ....

....type labels on abstractions. We show that quasi canonical forms of a given type are sufficient to determine the meaning of an object, since the type labels can be reconstructed (up to definitional equality) from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. While it is beyond the scope of this paper, we believe our construction is robust with respect to extension of the type theory with products, unit, linearity, subtyping and similar ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming. 13

.... ) followed by an appeal to the induction hypothesis to pG 2 q and (D u) nd pG 2 q. By Lemma 2 and a process called abstraction de ned below we conclude ; 9x : hyp G 1 conc G 2 : 4) For the general case, dependency relations pose an answer on how to strengthen in a logical framework [Vir99] We say that one type is dependent on another if objects of the former are build in terms of objects of the latter. For example, conc G 2 is dependent on hyp G 1 by rule init, but not nd G 1 by Lemma 2. Formally we write hyp G 1 conc G 2 , but nd G 1 6 conc G 2 . The process of ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Carnegie Mellon University, 1999. CMU-CS-99-167.

....to be discharged. In the second case of ndseq, for example, executing (D 0 u) yields an E which may only depend on h but not on u. That E does not depend on u is justi ed by the way sequent derivations are constructed. They are simply not de ned in terms of natural deduction derivations. Virga [Vir99] proposed a technique that decides if a parameter can possibly occur in an object of a given type or not. His procedure inspects the LF signature and derives the so called subordination relation . A type A is said to be subordinate to A 0 (A A 0 ) if objects of type A may occur within ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....variables T 1 and T 2 in node 3 and R in node 5 might depend on the assumptions in Gamma , i.e. they might depend on the new parameter x and the assumption u. However, a variable T can never depend on expressions or on the typing relation of . Using type dependency analysis based on subordination [50] we can detect and eliminate such dependencies. This allows us to detect more loops in the search tree, i.e. more nodes can be suspended because a variant of it is already in the table. Another optimization concerns the handling of assumptions in context Gamma . When storing a goal A together ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 2000.

....of type i from objects of type o. A term M is less than a term (x:N ) if there exists an instantiation T for x s.t. M is less than [T=x]N and the type of T is a subordinate to N . For a more detailed development of mutual recursion and subordination we refer the reader to R. Virga s PhD thesis [Vir99]. Delta; x:M j hN1 : Nn Gamma P L j 3 Delta; a j hN1 : Nn Gamma P L j 4 Delta Gamma [a=x]M j [a=x]N Delta Gamma x:M j x:N R j Delta; a=x]M j [a=x]N Gamma P Delta; x:M j x:N Gamma P L j Delta Gamma [a=x]M OE N Delta Gamma x : A:M OE N RL OE a ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....shifted type, because natural deduction derivations can never occur as subderivations to sequent derivations. Lemma 2 (Strengthening) If D is a canonical derivation in LF of judgment ; u : nd G 1 ; h : hyp G 1 M : conc G 2 then ; h : hyp G 1 M : conc G 2 . In previous work Virga [Vir99] has studied the possibility to strengthen in LF carefully, and has introduced the notion of dependency relation. We say that the type conc G 2 subordinates hyp G 1 (by init) but not nd G 1 (because of Lemma 2) Formally we write hyp G 1 conc G 2 , but nd G 1 6 conc G 2 . In the ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....the theories required for reasoning about safety properties such as arithmetic modulo 2 32 or memory update and access from the underlying mechanism of checking proofs. It is also simple, since it is based on a pure, dependently typed calculus whose properties have been deeply investigated [9,10]. Proofs in a logic designed for reasoning about safety properties are represented as terms in LF. Checking that a proof is valid is reduced to checking that its representation in the logical framework is well typed. This can be carried out effectively even for very large proof objects. ....

R. Virga, Higher-order rewriting with dependent types, Ph.D. thesis, Department of Mathematical Sciences, Carnegie Mellon University, available as Technical Report CMU-CS-99-167 (Sep. 1999).

....of this language and environment, but also from specific contributions which we have used (in a somewhat simplified setting) in this thesis. Let me mention only the most recent ones: schema contexts (Schurmann [Sch00] linear unification (Cervesato and Pfenning [CP96] subordination (Virga [Vir99]) mode and termination analysis (Rohwedder and Pfenning [RP96] This research has been financially supported for seven semesters by the Department of Philosophy at CMU and by a one year scholarship from Consiglio Nazionale delle Ricerche , Italy. Draft of June 22, 2000 14 CHAPTER 1. ....

....simple account of mutually recursion, without an explicit appeal to a call graph. As a matter of fact, we would really need to normalize an atom Q with the trivial clause for every predicate that is mutually recursive to Q and such the type of the parameter in the context is not subordinate [Vir99] to the type of Q. On the other hand Theorems 6.5 and 6.6 ensures, as obvious, that this program transformation is harmless, since it preserves provability and denial. Moreover, we will engineer the rule for clause complementation so that the irrelevant q will be inactive. We will remove those ....

[Article contains additional citation context not shown here]

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....normalization when expansion is not restricted [JG95, Gha97] In the dependently typed case, even the definition of long normal form is not obvious [DHW93] and the technical development is fraught with difficulties. We have not been able to reconstruct the proofs in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treat2 ments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91] Coquand s approach is based ....

....type labels on abstractions. We show that quasi canonical forms of a given type are sufficient to determine the meaning of an object, since the type labels can be reconstructed (up to definitional equality) from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. While it is beyond the scope of this paper, we believe our construction is robust with respect to extension of the type theory with products, unit, linearity, subtyping and similar ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming. 39

....normalization when expansion is not restricted [JG95, Gha97] In the dependently typed case, even the definition of long normal form is not obvious [DHW93] and the technical development is fraught with difficulties. We have not been able to reconstruct the proofs in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treat2 ments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91] Coquand s approach is based on ....

....type labels on abstractions. We show that quasi canonical forms of a given type are sufficient to determine the meaning of an object, since the type labels can be reconstructed (up to definitional equality) from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. While it is beyond the scope of this paper, we believe our construction is robust with respect to extension of the type theory with products, unit, linearity, subtyping and similar ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....normalization when expansion is not restricted [JG95, Gha97] In the dependently typed case, even the definition of long normal form is not obvious [DHW93] and the technical development is fraught with difficulties. We have not been able to reconstruct the proofs in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treatments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91] Coquand s approach is based on ....

....type labels on abstractions. We show that quasi canonical forms of a given type are sufficient to determine the meaning of an object, since the type labels can be reconstructed (up to definitional equality) from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. While it is beyond the scope of this paper, we believe our construction is robust with respect to extension of the type theory with products, unit, linearity, subtyping and similar ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

....normalization when expansion is not restricted [JG95, Gha97] In the dependently typed case, even the definition of long normal form is not obvious [DHW93] and the technical development is fraught with difficulties. We have not been able to reconstruct the proofs in [Gha97] and the development in [Vir99] relies on a complex intermediate system with annotated terms. To address the problems of practicality, Coquand suggested abandoning reduction based treatments of definitional equality in favor of a direct presentation of a practical equivalence algorithm [Coq91] Coquand s approach is based on ....

....type labels on abstractions. We show that quasi canonical forms of a given type are sufficient to determine the meaning of an object, since the type labels can be reconstructed (up to definitional equality) from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. While it is beyond the scope of this paper, we believe our construction is robust with respect to extension of the type theory with products, unit, linearity, subtyping and similar ....

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming.

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Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming. 39

No context found.

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999. Forthcoming. 39

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Virga, Roberto. (1999). Higher-order rewriting with dependent types. Ph.D. thesis, Department of Mathematical Sciences, Carnegie Mellon University.

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Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, September 1999. Available as Technical Report CMU-CS-99-167.

No context found.

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, September 1999. Available as Technical Report CMU-CS-99-167.

No context found.

Roberto Virga. Higher-Order Rewriting with Dependent Types. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999.

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