Unindent SageMath lines

main.tex

@@ -150,31 +150,31 @@ $\chern(F) = (R,C,D)$, and consider the possible Chern characters
 $\chern(E) = (r,c,d)$ of some semistabilizer $E$.
  
 \begin{sagesilent}
-  # Requires extra package:
-  #! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
+# Requires extra package:
+#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
 
-	from pseudowalls import *
+from pseudowalls import *
 
-	v = Chern_Char(*var("R C D", domain="real"))
-	u = Chern_Char(*var("r c d", domain="real"))
+v = Chern_Char(*var("R C D", domain="real"))
+u = Chern_Char(*var("r c d", domain="real"))
 
-	Δ = lambda v: v.Q_tilt()
+Δ = lambda v: v.Q_tilt()
 \end{sagesilent}
 
 Recall [ref] that $\chern_1^{\beta_{-}}$ has fixed bounds in terms of
 $\chern(F)$, and so we can write:
 
 \begin{sagesilent}
-	ts = stability.Tilt
-	beta_min = var("beta", domain="real")
+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
+var("q", domain="real")
+c_in_terms_of_q = c_lower_bound + q
 \end{sagesilent}
 
 \begin{equation}
@@ -195,8 +195,8 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
 This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
 
 \begin{sagesilent}
-	# First Bogomolov-Gieseker form expression that must be non-negative:
-	bgmlv1 = Δ(v) - Δ(u) - Δ(v-u)
+# First Bogomolov-Gieseker form expression that must be non-negative:
+bgmlv1 = Δ(v) - Δ(u) - Δ(v-u)
 \end{sagesilent}
 
 \begin{equation}
@@ -209,11 +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}
@@ -225,17 +225,17 @@ we get the following:
 This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
-	var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
+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_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
-		.expand()
-	)
+bgmlv1_d_ineq = (
+	((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
+	.subs(r_alt == r - R/2) # Resubstitute r back in
+	.expand()
+)
 
-	bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
+bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
 \end{sagesilent}
 
 \begin{dmath}
@@ -244,35 +244,35 @@ This can be rearranged to express a bound on $d$ as follows:
 \end{dmath}
 
 \begin{sagesilent}
-	# Seperate out the terms of the lower bound for d
-	bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
-
-	bgmlv1_d_lowerbound_exp_term = (
-		bgmlv1_d_lowerbound
-		- bgmlv1_d_lowerbound_without_hyp
-	).expand()
-
-	bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
-
-	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 == (
-		(
-			- R*v.twist(beta_min).ch[2]/2
-			- R*beta_min*q
-			+ C*q
-			- q^2
-		)/(R-2*r)
-	).expand()
+# Separate out the terms of the lower bound for d
+bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
+
+bgmlv1_d_lowerbound_exp_term = (
+	bgmlv1_d_lowerbound
+	- bgmlv1_d_lowerbound_without_hyp
+).expand()
+
+bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
+
+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 == (
+	(
+		- R*v.twist(beta_min).ch[2]/2
+		- R*beta_min*q
+		+ C*q
+		- q^2
+	)/(R-2*r)
+).expand()
 \end{sagesilent}
 
 
@@ -295,8 +295,8 @@ so some of these expressions simplify.
 This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
 
 \begin{sagesilent}
-	# First Bogomolov-Gieseker form expression that must be non-negative:
-	bgmlv2 = Δ(u)
+# First Bogomolov-Gieseker form expression that must be non-negative:
+bgmlv2 = Δ(u)
 \end{sagesilent}
 
 \begin{equation}
@@ -309,11 +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}
@@ -325,13 +325,13 @@ we get the following:
 This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
-	bgmlv2_d_ineq = (
-		(0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
-		+ d # Rearrange for d
-	).expand()
+bgmlv2_d_ineq = (
+	(0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
+	+ d # Rearrange for d
+).expand()
 
-	# Keep hold of lower bound for d
-	bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
+# Keep hold of lower bound for d
+bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
 \end{sagesilent}
 
 \begin{equation}
@@ -340,17 +340,17 @@ This can be rearranged to express a bound on $d$ as follows:
 \end{equation}
 
 \begin{sagesilent}
-	# Seperate out the terms of the lower bound for d
-	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()
+# Seperate out the terms of the lower bound for d
+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
@@ -370,8 +370,8 @@ for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}.
 This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
 
 \begin{sagesilent}
-	# Third Bogomolov-Gieseker form expression that must be non-negative:
-	bgmlv3 = Δ(v-u)
+# Third Bogomolov-Gieseker form expression that must be non-negative:
+bgmlv3 = Δ(v-u)
 \end{sagesilent}
 
 \begin{equation}
@@ -384,11 +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}
@@ -400,23 +400,23 @@ we get the following:
 This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
-	var("r_alt",domain="real") # r_alt = r - R temporary substitution
-
-	bgmlv3_with_q_reparam = (
-		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
+var("r_alt",domain="real") # r_alt = r - R temporary substitution
+
+bgmlv3_with_q_reparam = (
+	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
 \end{sagesilent}
 
 \begin{dmath}
@@ -425,35 +425,35 @@ This can be rearranged to express a bound on $d$ as follows:
 \end{dmath}
 
 \begin{sagesilent}
-	# Seperate out the terms of the lower bound for d
+# 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_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_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_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()
+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
+# 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_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
-			+ R*beta_min*q
-			- D*R
-		)/(r-R)
+assert bgmlv3_d_upperbound_exp_term == (
+		R*v.twist(beta_min).ch[2]
+		+ (C - q)^2/2
+		+ R*beta_min*q
+		- D*R
+	)/(r-R)
 \end{sagesilent}