#### D.4.16.7 is_nested

Procedure from library `mregular.lib` (see mregular_lib).

Usage:
  is_nested (i); i monomial ideal

Return:
  1 if i is of nested type, 0 otherwise.
  (returns -1 if i=(0) or i=(1)).

Assume:
  i is a nonzero proper monomial ideal.

Notes:
  1. The ideal must be monomial, otherwise the result has no meaning (so check this before using this procedure).
  2. is_nested is used in procedures depthIdeal, regIdeal and satiety.
  3. When i is a monomial ideal of nested type of S=K[x(0)..x(n)], the a-invariant of S/i coincides with the upper bound obtained using the procedure regIdeal with printlevel > 0.

Theory:
  A monomial ideal is of nested type if its associated primes are all of the form (x(0),...,x(i)) for some i<=n.
  (see definition and effective criterion to check this property in the preprint 'Saturation and Castelnuovo-Mumford regularity' by Bermejo-Gimenez, 2004).

Example:

    LIB "mregular.lib";
    ring s=0,(x,y,z,t),dp;
    ideal i1=x2,y3; ideal i2=x3,y2,z2; ideal i3=x3,y2,t2;
    ideal i=intersect(i1,i2,i3);
    is_nested(i);
    ==> 0
    ideal ch=x,y,z,z+t;
    map phi=ch;
    ideal I=lead(std(phi(i)));
    is_nested(I);
    ==> 1

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