Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Carreno et al. [CMT02], pages 281--297.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....r In] r In] SquashMemIntro ) In MetaPRL squash was first introduced by J.Hickey as a replacement for NuPRL s hidden hypotheses mechanism, but eventually it became clear that it gives a mechanism substantially widely useful than NuPRL s hidden hypotheses. We use the sequent schema syntax of [23] for specifying rules. Essentially, variables that are explicitly mentioned may occur free only where they are explicitly men tioned. Second, using (SquashMemElim) we can prove ( Sq uashElim ) Third, we can prove that squashed equality implies equality: a stronger version of ( SquashElim2 ....

....it might become more likely to fail) But if a tactic is only meant to propel the proof further without necessarily completing it (such as for example NuPRL s uto and MetaPRL s autoT) then allowing such tactic to use irreversible rules can make things substantially less pleasant to the user. See [23] for a description of MetaPRL s derived rules mechanism. More specifically, elimination rules should be locally complete and locally sound with respect to the introduction rules, as described in [26] But since we believe that this third guideline is not as crucial as the first two, we chose not ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. Accepted to TPHOLs 2002, 2002.

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Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Carreno et al. [CMT02], pages 281--297.

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Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Victor A. Carre~no, Cezar A. Mu~noz, and Sophiene Tahar, editors, Proceedings of the 15 LNCS, pages 281-297. Springer-Verlag, 2002.

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Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Victor A. Carre~no, Cezar A. Mu~noz, and Sophiene Tahar, editors, Proceedings of the 15 International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002), volume 2410 of Lecture Notes in Computer Science, pages 281-297. SpringerVerlag, 2002.

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Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Victor A. Carreno, Cezar A. Munoz, and Sophiene Tahar, editors, Proceedings of the 15 International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002.

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Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Carre~no et al. [15], pages 281-297.

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A. Nogin and J. Hickey. Sequent schema for derived rules. In Theorem Proving in Higher-Order Logics (TPHOLs '02), 2002.

....are passed to MetaPRL for execution. The final term is then converted to a specified compiler representation (in Figure 2 this is the functional IR) and compilation proceeds to generate executable code. 3. 1 Term language The term rewriting engine we use belongs to the MetaPRL logical framework [10, 14]. All logical terms, including goals and subgoals, are expressed in the language of terms. The general syntax of all terms has three parts. Each term has 1) an operator name, which is a unique name identifying the term; 2) a list of parameters representing constant values; and 3) a list of ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Theorem Proving in Higher-Order Logics (TPHOLs '02), 2002.

....programming languages that appear well suited for compiler implementation, like ML [19] still do not address other issues, such as substitution and preservation of scoping in the compiled program. In this paper, we present an alternative approach, based on the use of higher order abstract syntax [15, 16] and term rewriting in a general purpose logical framework. All program transformations, from parsing to code generation, are cleanly isolated and speci ed as term rewrites. In our system, term This work was supported in part by the DoD Multidisciplinary University Research Initiative (MURI) ....

....y sum[ f x; y g A few examples are shown in the table. Numbers have an integer parameter. The lambda term contains a binding occurrence: the variable x is bound in the subterm b. Term rewrites are speci ed in MetaPRL using second order variables, which explicitly de ne scoping and substitution [15]. A second order variable pattern has the form v[v 1 ; v n ] which represents an arbitrary term that may have free variables v 1 ; v n . The corresponding substitution has the form v[t 1 ; t n ] which speci es the simultaneous, capture avoiding substitution of terms t 1 ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Victor A. Carre~no, Cezar A. Mu~noz, and Sophiene Tahar, editors, Proceedings of the 15 International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002.

....on a daily basis. To get the latest version, go to http: metaprl.org theories.pdf. To learn more about the MetaPRL system, see http: metaprl.org , Hic01] and [Nog02c] The syntax used for expressing MetaPRL rules and rewrites is a variation of the sequent schemata language described in [NH02] Base Meta Theory 2.1 Base theory module The Base theory theory is not a theory in the strict sense. It defines only two rules (in the Base rewrite module; Section 2.10) Instead, it serves as the connection to the primitive MetaPRL prover, and it defines several resources that are useful for ....

.... 8 failwithT, 8 firstT, 9 genAssumT, 80 genUnivCDT, 79 idT, 8 ifLabT, 11 instHypT, 79 keepingLabelT, 11 memberOfT, 240 moveToConclT, 78 natIndT, 262 nthAssumT, 8 nthHypT, 37 onAllAssumT, 11 onAllClausesT, 11 onAllHypsT, 11 onClausesT, 11 onClauseT, 10 onConclT, 10 onHypsT, 11 onHypT, 10 onSomeHypT, 11 325 orelseT, 9 powerT, 264 progressT, 9 quotientT, 142 removeHiddenLabelT, 11 repeatForT, 9 repeatMForT, 11 repeatMT, 11 repeatT, 9 selT, 10 seqOnSameConclT, 9 seqT, 9 setExtT, 240 setOfT, 240 setSubstT, 237 splitBoolT, 88 splitITE, 88 sqsquashT, 42 squashT, 42 ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Carreno et al. [CMT02], pages 281--297. 331

....programming languages that appear well suited for compiler implementation, like ML [19] still do not address other issues, such as substitution and preservation of scoping in the compiled program. In this paper, we present an alternative approach, based on the use of higher order abstract syntax [15, 16] and term rewriting in a general purpose logical framework. All program transformations, from parsing to code generation, are cleanly isolated and speci ed as term rewrites. In our system, term rewrites specify an equivalence between two code fragments that is valid in any context. Rewrites are ....

....y sum[ f x; y g A few examples are shown in the table. Numbers have an integer parameter. The lambda term contains a binding occurrence: the variable x is bound in the subterm b. Term rewrites are speci ed in MetaPRL using secondorder variables, which explicitly de ne scoping and substitution [15]. A second order variable pattern has the form v[v1 ; vn ] which represents an arbitrary term that may have free variables v1 ; vn . The corresponding substitution has the form v[t1 ; tn ] which speci es the simultaneous, capture avoiding substitution of terms t1 ; ....

A. Nogin and J. Hickey. Sequent schema for derived rules. In V. A. Carre~no, C. A. Mu~noz, and S. Tahar, editors, Proceedings of the 15 International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2002.

....rules can be specified to perform program transformation and optimization. The final term is then converted to a specified compiler representation and compilation proceeds to generate executable code. 3. 1 Term language The term rewriting engine we use belongs to the MetaPRL logical framework [8, 14]. All logical terms, including goals and subgoals, are expressed in the language of terms. The general syntax of all terms has three parts. Each term has 1) an operator name, which is a unique name identifying the term; 2) a list of parameters representing constant values; and 3) a list of ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Theorem Proving in Higher-Order Logics (TPHOLs '02), 2002.

....would be expressed using the following rule. rewrite beta : apply lambda v.e 1 [v] e 2 ## e 1 [e 2 ] This declaration defines a rewrite rule called beta that can be applied to a beta redex, performing the substitution. Note that the statement of the rewrite uses second order substitution [3, 20]. The pattern e 1 [v] matches a term in which the variable v is allowed to be free, and the term e 1 [e 2 ] in the contractum constructs the term matched by e 1 with e 2 substituted for v. One important property of the term rewriting system is that variable binding and scoping is explicit, and ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Theorem Proving in Higher-Order Logics (TPHOLs '02), 2002.

....be expressed using the following rule. rewrite beta : applyflambdafv:e 1 [v]g; e 2 g e 1 [e 2 ] This declaration de nes a conversion called beta that can be applied within a proof to any redex, performing the substitution. Note that the statement of the rewrite uses second order substitution [1, 13]. The pattern e 1 [v] represents a term in which the variable v is allowed to be free, and the term e 1 [e 2 ] represents e 1 with e 2 substituted for v. 4 Brian Aydemir et al. Syntax, and terms All logical terms, including goals and subgoals, are expressed in the language of terms. The general ....

....is then performed by normalizing the program with respect to these rewrites. Note that the expression e in the redex does not mention the variable v, which means that v is not allowed to appear free in e (the second order pattern e[v] would have allowed v to appear in e, and would not be provable [13]) Also, note that the rst order de nition, below, using substitution would not be as useful for dead code elimination because the rule does not specify explicitly that the variable v is dead. 10 Brian Aydemir et al. let v : t = unop a in e) e[ unop a) v] There are two main di erences ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Theorem Proving in Higher-Order Logics (TPHOLs '02), 2002.

....would be expressed using the following rule. rewrite beta : apply lambda v.e 1 [v] e 2 ## e 1 [e 2 ] This declaration defines a rewrite rule called beta that can be applied to a beta redex, performing the substitution. Note that the statement of the rewrite uses second order substitution [2, 18]. The pattern e 1 [v] matches a term in which the variable v is allowed to be free, and the term e 1 [e 2 ] in the contractum constructs the term matched by e 1 with e 2 substituted for v. One important property of the term rewriting system is that variable binding and scoping is explicit, and ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. In Theorem Proving in Higher-Order Logics (TPHOLs '02), 2002.

.... 2 [A] SquashMemIntro) In MetaPRL squash was rst introduced by J.Hickey as a replacement for NuPRL s hidden hypotheses mechanism, but eventually it became clear that it gives a mechanism substantially widely useful than NuPRL s hidden hypotheses. We use the sequent schema syntax of [19] for specifying rules. Essentially, variables that are explicitly mentioned may occur freely only where they are explicitly mentioned. Second, using (SquashMemElim) we can prove a stronger version of (Squash Elim) x : A; t 1 [ t 2 [ 2 B[ x : A] x] t 1 [x] t 2 [x] 2 ....

....might become more likely to fail) But if a tactic is only meant to propel the proof further without necessarily completing it (such as for example NuPRL s Auto and MetaPRL s autoT) then allowing such tactic to use irreversible rules can make things substantially less pleasant to the user. See [19] for a description of MetaPRL s derived rules mechanism. More speci cally, elimination rules should be locally complete and locally sound with respect to the introduction rules, as described in [22] But since we believe that this third guideline is not as crucial as the rst two, we chose not ....

Aleksey Nogin and Jason Hickey. Sequent schema for derived rules. Submitted to TPHOLs

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