add comments to hnf_cutter

Signed-off-by: Lev Nachmanson <levnach@hotmail.com>

src/math/lp/hnf_cutter.cpp

@@ -159,24 +159,24 @@ namespace lp  {
 
   The algorithm is based on the Simplex algorithm. The solution space
   forms a polyhedron in Q^n . If the solution space is non-empty, the
-  Simplex algorithm returns a solution of Ax≤b . We further assume
+  Simplex algorithm returns a solution of Ax <= b . We further assume
   that the returned solution x0 is a vertex of the polyhedron, i. e.,
   there is a nonsingular square submatrix A′ and a corresponding
-  vector b′ , such that A′x0=b′ . We call A′x≤b′ the defining
+  vector b′ , such that A′x0=b′ . We call A′x <=b′ the defining
   constraints of the vertex. If the returned solution is not on a
   vertex we introduce artificial branches on input variables into A
   and use these branches as defining constraints. These branches are
   rarely needed in practise.
 
-The main idea is to bring the constraint system A′x≤bb′ into a Hermite
+The main idea is to bring the constraint system A′x<=b′ into a Hermite
 normal form H and to compute the unimodular matrix U with A′U=H . The
-Hermite normal form is uniquely defined. The constraint system A′x≤b′
+Hermite normal form is uniquely defined. The constraint system A′x<=b′
-is equivalent to Hy≤b′ with y:=(U−1)x . Since the solution x0 of
+is equivalent to Hy <=b′ with y:=(U−1)x . Since the solution x0 of
 A′x0=b′ is not integral, the corresponding vector y0=(U−1)x0 is not
 integral, either. The cuts from proofs algorithm creates an extended
-branch on one of the components y_i of y , i. e., y_i≤⌊y0_i⌋ or
+branch on one of the components y_i of y , i. e., y_i <= floor(y0_i) or
-y_i≥⌈y0_i⌉.  Further on in the paper there is a lemma showing that
+y_i>=ceil(y0_i).  Further on in the paper there is a lemma showing that
-branch y_i≥⌈y0_i⌉ is impossible.
+branch y_i >= ceil(y0_i) is impossible.
  */
     lia_move hnf_cutter::create_cut(lar_term& t, mpq& k, explanation* ex, bool & upper
 #ifdef Z3DEBUG