Use walrus op in asserts to keep track of alternative exprs pt 3 (final)

main.tex

@@ -261,19 +261,28 @@ bgmlv1_d_lowerbound_linear_term = (
 ).expand()
 
 # Verify the simplified forms of the terms that will be mentioned in text
+
 assert bgmlv1_d_lowerbound_const_term == (
 	v.twist(beta_min).ch[2]/2
 	+ beta_min*q
 )
 
+var("chbv",domain="real") # symbol to represent ch_1^\beta(v)
+
 assert bgmlv1_d_lowerbound_exp_term == (
 	(
-		- R*v.twist(beta_min).ch[2]/2
-		- R*beta_min*q
-		+ C*q
-		- q^2
-	)/(R-2*r)
-).expand()
+		# Keep hold of this alternative expression:
+		bgmlv1_d_lowerbound_exp_term_alt :=
+		(
+			- R*chbv/2
+			- R*beta_min*q
+			+ C*q
+			- q^2
+		)/(R-2*r)
+	)
+	.subs(chbv == v.twist(beta_min).ch[2])
+	.expand()
+)
 \end{sagesilent}
 
 
@@ -476,13 +485,18 @@ assert bgmlv3_d_upperbound_const_term == (
 )
 
 assert bgmlv3_d_upperbound_exp_term == (
-  bgmlv3_d_upperbound_const_term_alt2 :=
 	(
-		R*v.twist(beta_min).ch[2]
-		+ (C - q)^2/2
-		+ R*beta_min*q
-		- D*R
-	)/(r-R)
+		# Keep hold of this alternative expression:
+		bgmlv3_d_upperbound_const_term_alt2 :=
+		(
+			R*chbv
+			+ (C - q)^2/2
+			+ R*beta_min*q
+			- D*R
+		)/(r-R)
+	)
+	.subs(chbv == v.twist(beta_min).ch[2]) # subs real val of ch_1^\beta(v)
+	.expand()
 )
 \end{sagesilent}
 
@@ -514,6 +528,13 @@ $(r,c,d)$ that satisfy all inequalities to give a pseudowall.
 Suppose we take $\beta = \beta_{-}$ in the previous subsections, to find all
 circular walls to the left of the vertical wall (TODO as discussed in ref).
 
+\begin{equation*}
+	\sage{ bgmlv3_d_upperbound_const_term }
+\end{equation*}
+\begin{equation*}
+	\sage{bgmlv3_d_upperbound_const_term_alt1.subs(chbv == 0)}
+\end{equation*}
+
 
 \section{Conclusion}