Proofs and Algorithms by Gilles Dowek

Author:Gilles Dowek
Language: eng
Format: epub
Publisher: Springer London, London

Proof

By induction over the definition of f. If f is a projection F(u 1,…,u n ) reduces to ((((u i &u 1)&…&u i−1)&u i+1)&…&u n ) which under call by name reduces to . The cases corresponding to a zero function, the successor function, addition, multiplication and the characteristic function of the ordering relation are similar.

If the function f is defined by composition using h and g 1,…,g m , then F(u 1,…,u n ) reduces to (H(G 1(u 1,…,u n ),…,G m (u 1,…,u n )))&u 1&…&u n under call by name. By induction hypothesis, this term reduces under call by name to , then to .

If the function f is obtained by minimising the function g, then g(p 1,…,p n ,r) is defined and its value is different from zero for all natural number r strictly less than q, and g(p 1,…,p n ,q)=0. The term F(u 1,…,u n ) reduces under call by name to F′(u 1,…,u n ,0), which reduces to , where v 0 reduces to reduces to , and then to , to , to and finally to . □

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