Add conditions on r for bounds

main.tex

@@ -564,17 +564,20 @@ vertical wall (TODO as discussed in ref).
 	d &\geq&
 	\sage{bgmlv1_d_lowerbound_linear_term}
 	&+ \sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
-	+& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
+	+& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)},
+	 &\qquad\text{when\:} r > \frac{R}{2}
 \\
 	d &\leq&
 	\sage{bgmlv2_d_upperbound_linear_term}
 	&+ \sage{bgmlv2_d_upperbound_const_term}
-	+& \sage{bgmlv2_d_upperbound_exp_term}
+	+& \sage{bgmlv2_d_upperbound_exp_term},
+	 &\qquad\text{when\:} r > 0
 \\
 	d &\leq&
 	\sage{bgmlv3_d_upperbound_linear_term}
 	&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
-	+& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
+	+& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)},
+	 &\qquad\text{when\:} r > R
 \end{align}
 
 Furthermore, we get an extra bound for $d$ resulting from the condition that the