Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a cut free proof (that is, it can be computed) As we mentioned above, such direct reasoning on logic specification involves instantiations of eigenvariables. Similarly, focusing on their extensional nature guaranteed by cut elimination, enrichments to the sequent calculus have been proposed by [7, 24, 6, 9] in which eigenvariables are intended as variables to be substituted. This enrichment to proof theory (discussed here in Section 4) holds promise for providing proof systems for the direct reasoning of logic specifications (see, for example, the above mentioned papers as well as [10, 11] These ....

....over the set of constants only (the dependency on can be forgotten) Of course, the type of h will be 1 n 0 instead of simply 0 . In the inference rules of Figure 2, we write (h ) to denote (hx 1 : xn ) For the sake of consistency with a naming convention from the papers [8, 9], we shall refer to the inference system defined with just the rules in Figure 2 as FO (mnemonic for a first order logic for expressions ) The proof system resulting from the addition of the rules for r (Figure 1) Below are some theorems of FO involving r. In these formulas, we use ....

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R. McDowell and D. Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

....conundrum [Gir92] cut elimination does not work in the presence of induction rules of the form above, and it is necessary to cheat, namely to hide some cut inside 44 the induction rules to get cut elimination. We may do this, for example, by adapting McDowell and Miller s induction rule [MM98] let LKc ind 0 be LKc plus the following rules: oe; Gamma 8x 1 11 ; x n 1 1n 1 Delta V 1j = F [x : x j 1j ] oe F [x : c1 (x 1 ; x n 1 ) Delta . oe; Gamma 8x 1 p1 ; x np pnp Delta V pj = F [x : x j pj ] oe F [x : cp(x 1 ....

Raymond McDowell and Dale Miller. Cut elimination for a logic with definitions and induction. In TYPES'96. Springer Verlag, 1998. Available at ftp:// ftp.cis.upenn.edu/pub/papers/miller/cut elim.dvi.gz.

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R. McDowell and D. Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

No context found.

Raymond McDowell and Dale Miller. Cut elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

No context found.

Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

No context found.

Raymond McDowell and Dale Miller. Cut elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

No context found.

R. McDowell and D. Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

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R. McDowell and D. Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

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Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

....in Delta, the subformula property does not hold for proof systems using this formulation of induction. In fact, we can derive the following inference rule from the induction rule: Gamma B B; Delta Gamma Gamma This derived rule resembles the cut rule but requires a nat assumption. In [McD97, MM00], an intuitionistic logic incorporating definitions and inductions, was presented and cut elimination was shown. Although FO does not have the subformula property, the cut elimination theorem still provides a strong basis for reasoning about proofs in FO [McD97, MM97, MM00] In fact, the ....

....a nat assumption. In [McD97, MM00] an intuitionistic logic incorporating definitions and inductions, was presented and cut elimination was shown. Although FO does not have the subformula property, the cut elimination theorem still provides a strong basis for reasoning about proofs in FO [McD97, MM97, MM00]. In fact, the formulation of the natL rule and the failure of the subformula property reflect the fact that in actual mathematical practice, finding the proper induction hypothesis requires insight and creativity; they are not simply rearrangements of the subformulas of the conclusion. As a ....

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Raymond McDowell and Dale Miller. Cut elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

....sequent inference rules that introduce logical connective will not directly help here. One natural extension of the sequent calculus is then to add left and right introduction rules for atoms. Hallnas and Schroeder Heister [HSH91,SH93] Girard [Gir92] and more recently, McDowell and Miller [MM97,MM00] have all considered just such introduction rules for non logical constants, using a notion of definition. 2 A proof theoretic notion of definitions A definition is a finite collection of definition clauses of the form = B] where H is an atomic formula (the one being defined) every free ....

....lvl(#x.B) lvl(#x.B) lvl(B) We now require that for every definition clause #x[p t = B] lvl(B) lvl(p) If a definition is restricted to the Horn case, then this restriction is trivial to satisfy. Cut elimination for this use of definition within intuitionistic logic was proved in [McD97,MM00]. In fact, that proof was for a logic that also included a formulation of induction. Definitions are a way to introduce logical equivalences so that we do not introduce into proof search meaningless cycles: for example, if our specification contains the equivalence H B, then when proving a ....

Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. TCS, 232:91--119, 2000.

....with proof rules incorporating definitions [14, 31] and in the note [12] Girard independently developed a similar use of definitions for linear logic. More recently, McDowell and Miller have incorporated definitions into an intuitionistic proof system that also includes natural number induction [21, 22]. In all of these cases, it can be shown that if certain restrictions are placed on the structure of definitions, defined concepts have left and right introduction rules that enjoy a cut elimination theorem. Some examples of using such a definition mechanism have been given for equality reasoning ....

....of def L with def L unif and adding proofs of the additional premises. Proof The first part is obvious. The second part is based on the observation that the proofs for the additional premises can be built as instances of the proofs of the premises of def L. An analogous result was proved in [20, 22]. The only difference is that in def L unif we consider unifiers instead of pairs of substitutions oe; ae such that Aae = Hoe. However the two formulation can be easily shown to be equivalent: we can always ff convert the variables x in 8x: H 4 = B] so that oe [ ae is a unifier for A and H. In ....

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Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

....with proof rules incorporating definitions [14, 23] and in the note [12] Girard independently developed a similar use of definitions for linear logic. Most recently, McDowell and Miller have incorporated definitions into an intuitionistic proof system that also included natural number induction [18]. In all of these cases, it can be shown that if certain restrictions are placed on the structure of definitions, defined concepts have left and right introduction rules that enjoy a cut elimination theorem. Some examples of using such a definition mechanism have been given for equality reasoning ....

....for conjunction, oe for implication, and 8 and 9 for universal and existential quantification. As was shown in [23] if definitions do not contain implications in clause bodies, then cut elimination can be proved for intuitionistic logic extended with our def L and def R rules. The paper [18] provides a more liberal form of definitions that allow some occurrences of implications. We shall need this stronger form of definitions here. Toward that end we assume that each predicate symbol p in the language has associated with it a natural number lvl(p) the level of the predicate. The ....

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Raymond McDowell and Dale Miller. Cut elimination for a logic with definitions and induction. Draft manuscript submitted to the proceedings of the TYPES'96 workshop, March 1997.

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Raymond McDowell and Dale Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

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R. McDowell and D. Miller. Cut-elimination for a logic with definitions and induction. Theoretical Computer Science, 232:91--119, 2000.

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