Citations: Implementing Mathematics with the Nuprl Development System - Constable, Allen, Bromley, Cleaveland, Cremer, Harper, Howe, Knoblock, Mendler, Panangaden, Sasaki, Smith (ResearchIndex)

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R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D. J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the Nuprl Development System. PrenticeHall, 1986.

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Formalising Strong Normalisation Proofs of Explicit.. - Kamareddine, Qiao (2002) (Correct)

....Why formalise proofs in a proof checker The past thirty years have seen much work on formalising proofs from paper into a proof checker (e.g. 9, 27, 31, 3] Pioneering work on this started in 1967 with de Bruijn s in uential proof checker Automath. Since, many proof checkers have been built [7, 15, 24, 29, 8] into which many proofs have been formalised. Formalisation in a proof checker is useful even if the proof on paper is fully trusted and correct. Reasons for this include: Some complex proofs may be unconvincing unless they are checked by a proof checker. Formalisation in a proof checker ....

R. L. Constable, S. Allen, H. Bromely, W. Cleveland, et al. Implementing Mathematics with the Nuprl Development System. Prentice-Hall, Inc., Englewood Clis, NJ, 1986.

Program Extraction in simply-typed Higher Order Logic - Berghofer (2002) (1 citation) (Correct)

....One of the most fascinating properties of constructive logic is that a proof of a speci cation contains an algorithm which, by construction, satis es this speci cation. This idea forms the basis for program extraction mechanisms, which can be found in theorem provers such as Coq [3] or Nuprl [11]. To date, program extraction has mainly been restricted to theorem provers based on expressive dependent type theories such as the Calculus of Constructions [12] A notable exception is the Minlog System by Schwichtenberg [5] which is based on minimal rst order logic. Although Isabelle is based ....

.... in x4 yields the following correctness theorem, which is automatically derived from the above proof: case warshall r i j k of None ) 8 x : is path r x i j k j Some q ) is path r q i j k 6 Related work The rst theorem provers to support program extraction were Constable s Nuprl system [11], which is based on Martin L of type theory, and the PX system by Hayashi [14] The Coq system [3] which is based on the Calculus of Inductive Constructions (CIC) can extract programs to OCaml [19] and Haskell. Paulin Mohring [18, 17] has given a realizability interpretation for the Calculus of ....

R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D. J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the Nuprl Development System. Prentice-Hall, NJ, 1986.

Bridging the lambda sigma- and lambda s-Styles of.. - Kamareddine, Ríos (1997) (Correct)

....in a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] In other words, one only substitutes one occurrence of x by y and continues the substitution later. Theorem provers like Nuprl [Con86] and HOL [GM93] implement substitution which allows the local replacement of some abbreviated term. This avoids a size explosion when it is necessary to replace a variable by a huge term only in specific places to prove a certain theorem. Formalisation helps in studying the termination and ....

R. Constable et al. Implementing Mathematics with the NUPRL Development System. Prentice-Hall, 1986.

Calculi of Generalized beta-Reduction and Explicit.. - Kamareddine, Ríos, Wells (1998) (Correct)

....a formal manner. Local substitutions are needed as follows. Given xx[x: y] one may not be interested in having yy as the result of xx[x: y] but rather only yx[x: y] In other words, one only substitutes one occurrence of x by y, and continues the substitution later. Theorem provers such as Nuprl [CABC86] and HOL [GM93] implement substitution that allows the local replacement of some abbreviated term. This avoids a size explosion when it is necessary to replace a variable by a huge term only in specific places to prove a certain theorem. Formalization helps in studying the termination and ....

R. L. Constable, S. Allen, H. Bromely, and W. Cleveland. Implementing Mathematics with the NUPRL Development System. Prentice-Hall, Englewood Cliffs, NJ, 1986. 40

Quotient Types: A Modular Approach - Nogin (2002) (Correct)

....proposition of the form A = B C Ui where Ui is the i th universe of types. However in this paper we will often omit C Ui for simplicity. MetaPRL system uses the unit element 0 or it as a , NuPRL uses Ax and [27] uses Triv. The squash operator (sometimes also called hide) was introduced in [9]. It is also used in MetaPR[ 11,12,14] 4 In the next section we will present the axiornatization we chose for the squash operator and we will explain our choices in Section 3. 2.2 Squash Operator: Axioms First, whenever A is non empty, A] must be non empty as well: FA ( Squashlntro ) ....

....key property of set type is that when we have a witness w x: A I B[x] we know that w A and we know that B[w] is non empty; but in general we have no way of reconstructing a witness for B[w] 4. 2 Set Type: Traditional Approach Set types were first introduced in [7] and were also formalized in [3,9,12,24]. In those traditional implementations of type theory the rules for set types are somewhat asymmetric. When proving something like r; y: x: n I y x: one was forced to apply the set elimination rule before the set introduction rule. As we will see in a moment, the problem was that the ....

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Robert L. Constable, Stuart F. Allen, H.M. Bromley, W.R. Cleaveland, j.F. Cremer, R.W. Harper, Douglas J. Howe, T.B. Knoblock, N.P. Mentiler, P. Panangaden, James T. Sasaki, and Scott F. Smith. Implementing Mathematics with the NuPRL Development System. Prentice-Hall, N J, 1986.

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MetaPRL - A Modular Logical Environment - Hickey, Nogin, Constable.. (2003) (1 citation) Self-citation (Constable) (Correct)

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R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D. J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the Nuprl Development System. PrenticeHall, 1986.

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R. L. Constable, S. F. Allen, H. Bromley, W. Cleaveland, J. Cremer, R. Harper, D. J. Howe, T. Knoblock, N. Mendler, P. Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the Nuprl Development System. PrenticeHall, 1986.

Linear Logic and Noncommutativity in the Calculus of Structures - Straßburger (2003) (Correct)

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R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D.J.Howe,T.B.Knoblock,N.P.Mendler,P.Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the NuPrl Development System.Prentice Hall, 1986.

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Linear Logic and Noncommutativity in the Calculus of Structures - Straßburger (2003) (21 citations) (Correct)

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R. L. Constable, S. F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, D.J.Howe,T.B.Knoblock,N.P.Mendler,P.Panangaden, J. T. Sasaki, and S. F. Smith. Implementing Mathematics with the NuPrl Development System.Prentice Hall, 1986.

Le Fun: Logic, equations, and Functions - Ait-Kaci, Lincoln, Nasr (1986) (Correct)

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Constable, R.L., et al., Implementing Mathematics with the Nuprl Development System. Prentice-Hall, Englewood Cliffs, NJ. 1986.

Hierarchical Contextual Reasoning - Autexier (2003) (Correct)

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Constable, Robert L., Allen, Stuart F., Bromley, H.M., Cleaveland, W.R., Cremer, J.F., Harper, R.W., Howe, Douglas J., Knoblock, T.B., Mendler, N.P., Panangaden, P., Sasaki, James T. and Smith, Scott F. (1986). Implementing Mathematics with the Nuprl Development System. Prentice-Hall, NJ.

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MMode, a Mizar Mode for the proof assistant Coq - Giero, Wiedijk (2003) (Correct)

Certified Reasoning on Real Numbers and Objects in.. - Ciaffaglione (Correct)

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R. L. Constable. Implementing mathematics with the Nuprl development system. Prentice-Hall, 1986.

C-CoRN, the Constructive Coq Repository at Nijmegen - Cruz-Filipe, Geuvers, Wiedijk (Correct)

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Robert L. Constable et al. Implementing Mathematics with the Nuprl Development System. Prentice-Hall, NJ, 1986.

Logical Aspects of Digital Mathematics Libraries - Allen, Caldwell, Constable (2001) (Correct)

Program Extraction in simply-typed Higher Order Logic - Berghofer (2002) (1 citation) (Correct)

Mathematical Knowledge Management in HELM - Andrea Asperti Asperti (2001) (1 citation) (Correct)

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R. Constable et al. \Implementing Mathematics with the Nuprl Development System". Prentice-Hall, NJ, 1986.

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Generalized Fi-Reduction and Explicit Substitutions - Fairouz Kamareddine And (Correct)

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R. Constable et al. Implementing Mathematics with the NUPRL Development System. PrenticeHall, 1986.

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Robert L. Constable, Stuart F. Allen, H.M. Bromley, W.R. Cleaveland, J.F. Cremer, R.W. Harper, Douglas J. Howe, T.B. Knoblock, N.P. Mendler, P. Panangaden, James T. Sasaki, and Scott F. Smith. Implementing Mathematics with the Nuprl Development System. Prentice-Hall, NJ, 1986.

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