A unary function
f which the Semenov's conditions hold for is
called a
concordant with addition monotone function.
In [5] Semenov proved that for any structure
< N, <, 0, 1,
+, {[/n] | n=1,2,…}, f, f -1
>
of signature
{<, 0, 1,
+, {[/n] | n=1,2,…}, f, f -1
}
where
f is a concordant with addition
monotone function,
f(0)=0,
[x/n] is such a natural number
y, that
ny≤x and
n(y+1)>x, and
f -1(n)=m iff
f(m)≤ n<f(m+1), any
first-order formula is
equivalent to a quantifier-free formula.
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