Shorten SageMath lines and correct alternative expressions in 6.1

main.tex

@@ -167,7 +167,11 @@ $\chern(F)$, and so we can write:
 \begin{sagesilent}
 	ts = stability.Tilt
 	beta_min = var("beta", domain="real")
-	c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
+
+	c_lower_bound = -(
+		ts(beta=beta_min).rank(u)
+		/ts().alpha
+	).expand() + c
 
 	var("q", domain="real")
 	c_in_terms_of_q = c_lower_bound + q
@@ -205,7 +209,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
 we get the following:
 
 \begin{sagesilent}
-	bgmlv1_with_q = bgmlv1.expand().subs(c == c_in_terms_of_q)
+	bgmlv1_with_q = (
+		bgmlv1
+		.expand()
+		.subs(c == c_in_terms_of_q)
+	)
 \end{sagesilent}
 
 \begin{equation}
@@ -220,6 +228,7 @@ This can be rearranged to express a bound on $d$ as follows:
 	var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
 
 	bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()
+
 	bgmlv1_d_ineq = (
 		((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
 		.subs(r_alt == r - R/2) # Resubstitute r back in
@@ -236,45 +245,47 @@ This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
 	# Seperate out the terms of the lower bound for d
-	bgmlv1_d_lowerbound_exp_term = (bgmlv1_d_lowerbound*(R-2*r)).expand().subs(r==2*R)/(R-2*r)
-	bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0).subs(r==0)
-	bgmlv1_d_lowerbound_linear_term = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0).subs(r==1)*r
+	bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
 
-	# 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, "fail"
-	assert bgmlv1_d_lowerbound_linear_term == (v.twist(beta_min).ch[2]/2 + beta_min^2/2 + beta_min*q)*r, "fail"
+	bgmlv1_d_lowerbound_exp_term = (
+		bgmlv1_d_lowerbound
+		- bgmlv1_d_lowerbound_without_hyp
+	).expand()
 
-	assert bgmlv1_d_lowerbound_exp_term == (
-		- 2*R*v.twist(beta_min).ch[2]
-		- 3*R^2*beta_min^2
-		- 4*R*beta_min*q
-		+ C*q
-		- q^2
-	).expand()/(R-2*r), "fail"
+	bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
 
-	assert bgmlv1_d_lowerbound_exp_term == (
-		- 2*D*R + (C^2)/4
-		- ((C - 4*R*beta_min)/2 - q)^2
-	).expand()/(R-2*r), "fail"
+	bgmlv1_d_lowerbound_linear_term = (
+		bgmlv1_d_lowerbound_without_hyp
+		- bgmlv1_d_lowerbound_const_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
+	)
 
 	assert bgmlv1_d_lowerbound_exp_term == (
-		(2*R*beta_min + q)
-		*(2*R*beta_min + q - C)
-		+ 2*D*R
-	).expand()/(2*r - R), "fail"
+		(
+			- R*v.twist(beta_min).ch[2]/2
+			- R*beta_min*q
+			+ C*q
+			- q^2
+		)/(R-2*r)
+	).expand()
 \end{sagesilent}
 
+
 \noindent
 Viewing equation \ref{eqn-bgmlv1_d_lowerbound} as a lower bound for $d$ given
 as a function of $r$, the terms can be rewritten as follows.
 The constant term in $r$ is
-$\chern^{\beta}_2(F) + \beta q$.
+$\chern^{\beta}_2(F)/2 + \beta q$.
 The linear term in $r$ is
-$(\chern^{\beta}_2(F)/2 + \beta^2/2 + \beta q)r$.
+$\sage{bgmlv1_d_lowerbound_linear_term}$.
 Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
 and can be written:
-$\frac{-2R\chern^{\beta}_2(F) - 3R^2\beta^2 - 4Rq\beta + Cq - q^2}{R-2r}$ or
-$\frac{(2R\beta + q)(2R\beta + q - C) + 2DR}{2r-R}$.
+$\frac{R\chern^{\beta}_2(F)/2 + R\beta q - Cq + q^2  }{2r-R}$.
 In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
 $\chern^{\beta}_2(F) = 0$,
 so some of these expressions simplify.
@@ -298,7 +309,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
 we get the following:
 
 \begin{sagesilent}
-	bgmlv2_with_q = bgmlv2.expand().subs(c == c_in_terms_of_q)
+	bgmlv2_with_q = (
+		bgmlv2
+		.expand()
+		.subs(c == c_in_terms_of_q)
+	)
 \end{sagesilent}
 
 \begin{equation}
@@ -311,11 +326,12 @@ This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
 	bgmlv2_d_ineq = (
-		((0 <= bgmlv2_with_q)/2/r + d) # Rearrange for d
-		.expand()
-	)
+		(0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
+		+ d # Rearrange for d
+	).expand()
 
-	bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs() # Keep hold of lower bound for d
+	# Keep hold of lower bound for d
+	bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
 \end{sagesilent}
 
 \begin{equation}
@@ -325,9 +341,16 @@ This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
 	# Seperate out the terms of the lower bound for d
-	bgmlv2_d_lowerbound_exp_term = (bgmlv2_d_lowerbound*r).expand().subs(r==0)/r
-	bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound.subs(1/r == 0).subs(r==0)
-	bgmlv2_d_lowerbound_linear_term = bgmlv2_d_lowerbound.subs(1/r == 0).subs(r==1)*r
+	bgmlv2_d_lowerbound_without_hyp = bgmlv2_d_lowerbound.subs(1/r == 0)
+	bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound_without_hyp.subs(r==0)
+	bgmlv2_d_lowerbound_linear_term = (
+		bgmlv2_d_lowerbound_without_hyp
+		- bgmlv2_d_lowerbound_const_term
+	).expand()
+	bgmlv2_d_lowerbound_exp_term = (
+		bgmlv2_d_lowerbound
+		- bgmlv2_d_lowerbound_without_hyp
+	).expand()
 \end{sagesilent}
 
 Viewing equation \ref{eqn-bgmlv2_d_lowerbound} as a lower bound for $d$ in term
@@ -361,7 +384,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
 we get the following:
 
 \begin{sagesilent}
-	bgmlv3_with_q = bgmlv3.expand().subs(c == c_in_terms_of_q)
+	bgmlv3_with_q = (
+		bgmlv3
+		.expand()
+		.subs(c == c_in_terms_of_q)
+	)
 \end{sagesilent}
 
 \begin{equation}
@@ -376,14 +403,17 @@ This can be rearranged to express a bound on $d$ as follows:
 	var("r_alt",domain="real") # r_alt = r - R temporary substitution
 
 	bgmlv3_with_q_reparam = (
-		bgmlv3_with_q.subs(r == r_alt + R)
+		bgmlv3_with_q
+		.subs(r == r_alt + R)
 		/r_alt # This operation assumes r_alt > 0
 	).expand()
+
 	bgmlv3_d_ineq = (
 		((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d
 		.subs(r_alt == r - R) # Resubstitute r back in
 		.expand()
 	)
+
 	# Check that this equation represents a bound for d
 	assert bgmlv3_d_ineq.lhs() == d, f"Inequality is of the form: {bgmlv3_d_ineq}"
 	bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
@@ -398,21 +428,26 @@ This can be rearranged to express a bound on $d$ as follows:
 	# Seperate out the terms of the lower bound for d
 
 	bgmlv3_d_upperbound_without_hyp = bgmlv3_d_upperbound.subs(1/(R-r) == 0)
+
 	bgmlv3_d_upperbound_const_term = bgmlv3_d_upperbound_without_hyp.subs(r==0)
+
 	bgmlv3_d_upperbound_linear_term = (
 		bgmlv3_d_upperbound_without_hyp
 		- bgmlv3_d_upperbound_const_term
 	).expand()
+
 	bgmlv3_d_upperbound_exp_term = (
 		bgmlv3_d_upperbound
 		- bgmlv3_d_upperbound_without_hyp
 	).expand()
 
 	# Verify the simplified forms of the terms that will be mentioned in text
+
 	assert bgmlv3_d_upperbound_const_term == ( 
 		v.twist(beta_min).ch[2]
 		+ beta_min*q
 	).expand()
+
 	assert bgmlv3_d_upperbound_exp_term == (
 			R*v.twist(beta_min).ch[2]
 			+ (C - q)^2/2