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Review upper limit on velocity in condensed matter

abstract:
reference: https://arxiv.org/pdf/2004.04818.pdf

step	inference rule	input	feed	output	validity (as per SymPy)
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8106885760 $\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$	no validation is available for declarations
12	ID: 111182; multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	8106885760 $\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$	8857931498 $c$	5838268428 $\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}$	valid
9	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1556389363 $E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$	no validation is available for declarations
7	ID: 111634; substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	8090924099 $v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }$ 8688588981 $a^3 \rho=m$		7837519722 $v=\sqrt{f} \sqrt{\frac{E}{m}}$	LHS diff is -pdg0002077 + pdg0003935pdg00058543 RHS diff is -sqrt(pdg0006235)sqrt(pdg0002241/pdg0009863) + pdg0009863
19	ID: 111773; maximum of expression number of inputs: 1; feeds: 1; outputs: 1 The maximum of Eq.~\\ref{eq:#2} with respect to $#1$ is Eq.~\\ref{eq:#3}	2897612567 $v=\alpha c \sqrt{ \frac{m_e}{A m_p} }$	6259833695 $A$	7701249282 $v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }$	recognized infrule but not yet supported
18	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5789289057 $v=\alpha c \sqrt{ \frac{m_e}{2 m} }$ 5646314683 $m=A m_p$		2897612567 $v=\alpha c \sqrt{ \frac{m_e}{A m_p} }$	LHS diff is -pdg0002077 + pdg0009863 RHS diff is -pdg0001370pdg0004567sqrt(pdg0002515/(pdg0003285pdg0005916)) + pdg0003285pdg0005916
20	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	7701249282 $v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }$			no validation is available for declarations
13	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	1556389363 $E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$ 4107032818 $E_{\rm Rydberg}=E$		3291685884 $E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$	LHS diff is -pdg0002241 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg00079402) RHS diff is pdg0002241 - pdg00019994pdg0002515/(32pdg0001054*2pdg0003141*2pdg0007940**2)
10	ID: 111104; declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			4107032818 $E_{\rm Rydberg}=E$	no validation is available for declarations
15	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3935058307 $v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }$		9640720571 $v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}$	LHS diff is 0 RHS diff is sqrt(2)(pdg0001054pdg0003141pdg0007940sqrt(pdg0001999*4pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863)) - pdg0001999*2sqrt(pdg0002515/pdg0009863))/(8pdg0001054pdg0003141*pdg0007940)
6	ID: 111182; multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	8908736791 $\rho=\frac{m}{a^3}$	2397692197 $a^3$	8688588981 $a^3 \rho=m$	valid
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9376481176 $K=f \frac{E}{a^3}$	no validation is available for declarations
5	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8908736791 $\rho=\frac{m}{a^3}$	no validation is available for declarations
4	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	6504442697 $v=\sqrt{ \frac{K}{\rho} }$ 9376481176 $K=f \frac{E}{a^3}$		8090924099 $v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }$	LHS diff is K - pdg0002077 RHS diff is pdg0002241pdg0006235/pdg00058543 - sqrt(pdg0002241pdg0006235/(pdg0003935pdg0005854*3))
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4560648264 $v=\sqrt{ \frac{K + (4/3) G}{\rho} }$	no validation is available for declarations
14	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9854442418 $v=\sqrt{\frac{E}{m}}$ 3291685884 $E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$		3935058307 $v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }$	LHS diff is -pdg0002077 + pdg0002241 RHS diff is -sqrt(2)sqrt(pdg00019994pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863))/8 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg0007940**2)
17	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5646314683 $m=A m_p$	no validation is available for declarations
8	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	7837519722 $v=\sqrt{f} \sqrt{\frac{E}{m}}$	3685779219 $\sqrt{f} \approx 2$	9854442418 $v=\sqrt{\frac{E}{m}}$	recognized infrule but not yet supported
3	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	4560648264 $v=\sqrt{ \frac{K + (4/3) G}{\rho} }$	9674924517 $K >> G$	6504442697 $v=\sqrt{ \frac{K}{\rho} }$	recognized infrule but not yet supported
16	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9640720571 $v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}$ 5838268428 $\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}$		5789289057 $v=\alpha c \sqrt{ \frac{m_e}{2 m} }$	LHS diff is pdg0001370pdg0004567 - pdg0002077 RHS diff is -sqrt(2)pdg0001370pdg0004567sqrt(pdg0002515/pdg0009863)/2 + pdg0001999*2/(4pdg0001054pdg0003141pdg0007940)

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timing of Neo4j queries:

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step	inference rule	input	feed	output	validity (as per SymPy)
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8106885760 \(\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)	no validation is available for declarations
12	ID: 111182; multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	8106885760 \(\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)	8857931498 \(c\)	5838268428 \(\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)	valid
9	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1556389363 \(E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)	no validation is available for declarations
7	ID: 111634; substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	8090924099 \(v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\) 8688588981 \(a^3 \rho=m\)		7837519722 \(v=\sqrt{f} \sqrt{\frac{E}{m}}\)	LHS diff is -pdg0002077 + pdg0003935pdg00058543 RHS diff is -sqrt(pdg0006235)sqrt(pdg0002241/pdg0009863) + pdg0009863
19	ID: 111773; maximum of expression number of inputs: 1; feeds: 1; outputs: 1 The maximum of Eq.~\\ref{eq:#2} with respect to $#1$ is Eq.~\\ref{eq:#3}	2897612567 \(v=\alpha c \sqrt{ \frac{m_e}{A m_p} }\)	6259833695 \(A\)	7701249282 \(v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }\)	recognized infrule but not yet supported
18	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	5789289057 \(v=\alpha c \sqrt{ \frac{m_e}{2 m} }\) 5646314683 \(m=A m_p\)		2897612567 \(v=\alpha c \sqrt{ \frac{m_e}{A m_p} }\)	LHS diff is -pdg0002077 + pdg0009863 RHS diff is -pdg0001370pdg0004567sqrt(pdg0002515/(pdg0003285pdg0005916)) + pdg0003285pdg0005916
20	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	7701249282 \(v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }\)			no validation is available for declarations
13	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	1556389363 \(E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\) 4107032818 \(E_{\rm Rydberg}=E\)		3291685884 \(E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)	LHS diff is -pdg0002241 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg00079402) RHS diff is pdg0002241 - pdg00019994pdg0002515/(32pdg0001054*2pdg0003141*2pdg0007940**2)
10	ID: 111104; declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			4107032818 \(E_{\rm Rydberg}=E\)	no validation is available for declarations
15	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3935058307 \(v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)		9640720571 \(v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\)	LHS diff is 0 RHS diff is sqrt(2)(pdg0001054pdg0003141pdg0007940sqrt(pdg0001999*4pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863)) - pdg0001999*2sqrt(pdg0002515/pdg0009863))/(8pdg0001054pdg0003141*pdg0007940)
6	ID: 111182; multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	8908736791 \(\rho=\frac{m}{a^3}\)	2397692197 \(a^3\)	8688588981 \(a^3 \rho=m\)	valid
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9376481176 \(K=f \frac{E}{a^3}\)	no validation is available for declarations
5	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8908736791 \(\rho=\frac{m}{a^3}\)	no validation is available for declarations
4	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	6504442697 \(v=\sqrt{ \frac{K}{\rho} }\) 9376481176 \(K=f \frac{E}{a^3}\)		8090924099 \(v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\)	LHS diff is K - pdg0002077 RHS diff is pdg0002241pdg0006235/pdg00058543 - sqrt(pdg0002241pdg0006235/(pdg0003935pdg0005854*3))
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4560648264 \(v=\sqrt{ \frac{K + (4/3) G}{\rho} }\)	no validation is available for declarations
14	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9854442418 \(v=\sqrt{\frac{E}{m}}\) 3291685884 \(E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)		3935058307 \(v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)	LHS diff is -pdg0002077 + pdg0002241 RHS diff is -sqrt(2)sqrt(pdg00019994pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863))/8 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg0007940**2)
17	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5646314683 \(m=A m_p\)	no validation is available for declarations
8	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	7837519722 \(v=\sqrt{f} \sqrt{\frac{E}{m}}\)	3685779219 \(\sqrt{f} \approx 2\)	9854442418 \(v=\sqrt{\frac{E}{m}}\)	recognized infrule but not yet supported
3	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	4560648264 \(v=\sqrt{ \frac{K + (4/3) G}{\rho} }\)	9674924517 \(K >> G\)	6504442697 \(v=\sqrt{ \frac{K}{\rho} }\)	recognized infrule but not yet supported
16	ID: 111556; substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9640720571 \(v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\) 5838268428 \(\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)		5789289057 \(v=\alpha c \sqrt{ \frac{m_e}{2 m} }\)	LHS diff is pdg0001370pdg0004567 - pdg0002077 RHS diff is -sqrt(2)pdg0001370pdg0004567sqrt(pdg0002515/pdg0009863)/2 + pdg0001999*2/(4pdg0001054pdg0003141pdg0007940)