E. Ritter, D. Pym, and L.A. Wallen. Proof-terms for classical and intuitionistic logic. Theoretical Computer Science, 232(1--2):299-- 333, 2000. 15  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the same as a construct rst introduced by Bierman in his thesis [4] which considers term assignment for intuitionistic linear logic, among other things. There have been several proposals to use expressions to witness the Sequent Calculus implication left introduction rule. For example, in [20], Ritter et al. use a calculus, which contains the standard calculus as a subsystem, to represent classical proofs. Their system has the following implication left introduction rule s : A y : B; t : C z : A B; t[ z A B s) y B ] C Here we avoid the application construct ....

E. Ritter, D. Pym, and L. A. Wallen. Proof-terms for classical and intuitionistic logic (extended abstract). In Proc of the 13th International Conference on Automated Deduction, Lecture Notes in Articial Intelligence. Springer-Verlag, 1996. To appear.

....construct hz A(B ; s=y B it, which binds the free occurrence of y in t, departs significantly from the usual programming construct, and from the expressions (or variants thereof) witnessing the Sequent Calculus implication left introduction rule proposed in the literature. For example, in [25], Ritter et al. have used a calculus that has the calculus constructs of abstraction and application to represent classical proofs. They have the following implication leftintroduction rule Gamma s : A y : B; Delta t : C z : A ( B; Gamma; Delta t[ z A(B s) y B ] C In contrast ....

E. Ritter, D. Pym, and L. A. Wallen. Proof-terms for classical and intuitionistic logic (extended abstract). In Proc of the 13th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1996. To appear.

....it binds the free occurrence of y in t. It turns out that the lambda let is essentially the same as a construct first introduced by Bierman in his thesis [4] There have been several proposals to use expressions to witness the Sequent Calculus implication left introduction rule. For example, in [20], Ritter et al. use a calculus, which contains the standard calculus as a subsystem, to represent classical proofs. Their system has the following implication left introduction rule Gamma s : A y : B; Delta t : C z : A B; Gamma; Delta t[ z A B s) y B ] C In contrast neither ....

E. Ritter, D. Pym, and L. A. Wallen. Proof-terms for classical and intuitionistic logic (extended abstract). In Proc of the 13th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1996. To appear.

....construct hz A(B ; s=y B it, which binds the free occurrence of y in t, looks quite different from the usual programming construct, and from the expressions (or variants thereof) witnessing the Sequent Calculus implication left introduction rule proposed in the literature. For example, in [19], Ritter et al. use a calculus, which contains the standard calculus as a subsystem, to represent classical proofs. Their system has the following implication left introduction rule Gamma s : A y : B; Delta t : C z : A ( B; Gamma; Delta t[ z A(B s) y B ] C In contrast neither ....

E. Ritter, D. Pym, and L. A. Wallen. Proof-terms for classical and intuitionistic logic (extended abstract). In Proc of the 13th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1996. To appear. Definition of the Autonomous and -Autonomous Theories 21

.... confluent; different strategies for reduction gives different proofs in normal form, i.e. proof normalization involves an element of non determinism [32] 3 Other calculi which have been considered in the literature include PCF [70] linear calculus [14] calculus with explicit substitutions [39,84]. Werner also considered in unpublished work, 1992 a classical variant of non dependent logical pure type systems; however his notion of reduction is extremely weak. Barthe, Hatcliff, S rensen ing classical calculi to the issue of uniformity. As a result, the analysis of a classical ....

E. Ritter, D. Pym, and L. Wallen. Proof-terms for classical and intuitionistic logic (extended abstract). In M. McRobbie and J. Slaney, editors, Proceeedings of CADE'96, volume 1104 of Lecture Notes in Artificial Intelligence, pages ??-- ??, 1996.

.... of cases see for example [8, 9, 15, 22, 34, 48, 49, 50, 51, 52, 56, 57, 64] the calculus considered is essentially the simply typed or polymorphic calculus; other calculi considered include higher order calculus [31] ML [23] linear calculus [14] calculus with explicit substitutions [32, 68] or proof irrelevant logical pure type systems [79] In all cases, the calculi considered are non dependent. The question naturally arises whether the above mentioned results scale up to systems of dependent types. Apart from its intrinsic interest, the question has direct implications in ....

E. Ritter, D. Pym, and L. Wallen. Proof-terms for classical and intuitionistic logic (extended abstract). In M. McRobbie and J. Slaney, editors, Proceeedings of CADE'96, volume 1104 of Lecture Notes in Artificial Intelligence, pages ??--??, 1996.

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E. Ritter, D. Pym, and L.A. Wallen. Proof-terms for classical and intuitionistic logic. Theoretical Computer Science, 232(1--2):299-- 333, 2000. 15

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