| M. Takahashi, R. Akama, S. Hirokawa: "Normal Proofs and Their Grammar", Information and Computation 125,pp.144--153, 1996. |
....model with GUI of Java. Chapter 9 is the conclusion. In the chapter we present some further works. 5 Chapter 2 Natural Deduction System and Typed calculus In this chapter we review some basic notion and theorems on Natural Deduction system and an typed calculus. Most of them are found in [4, 5, 6, 10, 14, 15]. We include them in this chapter so that the thesis is self contained. 2.1 Natural deduction system For the simplicity of the presentation, we treat only implication. Conjunction, disjunction and negation can be treated similarly. Formulas are constructed from atomic formulas P 1 ; P 2 ; ....
M. Takahashi, R. Akama, S. Hirokawa: "Normal Proofs and Their Grammar", Information and Computation 125,pp.144--153, 1996.
....on the number of long principal normal inhabitants of with depth in [0; dp ( and on the number of those with depth in [dp ( Dp( where depth is a measure on the structure of a term in normal form. Finally, Ben Yelles de ned in [2] a Search Algorithm (others are in [6] 7] and [5]) that given a type and any integer d 0 can be used to compute all (a nite number) long normal inhabitants of a type with depth d. Thus, using any principal type checking algorithm (for example in [3] it is possible to compute the long principal normal inhabitants of with depth d. ....
M. Takahashi, Y. Akama, and S. Hirokawa. Normal proofs and their grammar. In M. Hagiya and J. Mitchell, editors, Theoretical Aspects of Computer Software (TACS'94), volume 789 of LNCS, pages 465-493. Springer Verlag, 1994.
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