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Type Arithmetics: Computation based on the theory of types

Oleg Kiselyov
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Abstract

The present paper shows meta-programming turn programming, which is rich enough to express arbitrary arithmetic computations. We demonstrate a type system that implements Peano arithmetics, slightly generalized to negative numbers. Certain types in this system denote numerals. Arithmetic operations on such types-numerals - addition, subtraction, and even division - are expressed as type reduction rules executed by a compiler. A remarkable trait is that division by zero becomes a type error - and reported as such by a compiler.

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Oleg Kiselyov. 2001-04-03. Type Arithmetics: Computation based on the theory of types. https://arxiv.org/abs/cs/0104010

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