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Review total electrical resistance for circuit with two resistors in series

abstract:
reference: http://www.dfcd.net/articles/derivations/resistors.html

step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4087145886 $V=I R$	no validation is available for declarations
2	111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}.	4087145886 $V=I R$	9830343096 $V_1$ 7774819339 $R$ 9174439158 $R_1$ 2698469612 $V$	7675171493 $V_1=I R_1$	list index out of range
3	111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}.	4087145886 $V=I R$	7497687256 $V$ 1484794622 $R_2$ 5890617067 $R$ 1608399874 $V_2$	6061695358 $V_2=I R_2$	list index out of range
4	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9063568209 $V_{\rm total}=V_1 + V_2$	no validation is available for declarations
5	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4939880586 $V_{\rm total}=I R_{\rm total}$	no validation is available for declarations
6	111246: substitute LHS of three expressions into expression number of inputs: 4; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} into Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}.	4939880586 $V_{\rm total}=I R_{\rm total}$ 6061695358 $V_2=I R_2$ 7675171493 $V_1=I R_1$ 9063568209 $V_{\rm total}=V_1 + V_2$		2715678478 $I R_{\rm total}=I R_1 + I R_2$	recognized infrule but not yet supported
7	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	2715678478 $I R_{\rm total}=I R_1 + I R_2$	7844317489 $I$	7217021879 $R_{\rm total}=R_1 + R_2$	valid
8	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	7217021879 $R_{\rm total}=R_1 + R_2$			no validation is available for declarations

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timing of Neo4j queries:

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step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4087145886 \(V=I R\)	no validation is available for declarations
2	111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}.	4087145886 \(V=I R\)	9830343096 \(V_1\) 7774819339 \(R\) 9174439158 \(R_1\) 2698469612 \(V\)	7675171493 \(V_1=I R_1\)	list index out of range
3	111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}.	4087145886 \(V=I R\)	7497687256 \(V\) 1484794622 \(R_2\) 5890617067 \(R\) 1608399874 \(V_2\)	6061695358 \(V_2=I R_2\)	list index out of range
4	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9063568209 \(V_{\rm total}=V_1 + V_2\)	no validation is available for declarations
5	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4939880586 \(V_{\rm total}=I R_{\rm total}\)	no validation is available for declarations
6	111246: substitute LHS of three expressions into expression number of inputs: 4; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} into Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}.	4939880586 \(V_{\rm total}=I R_{\rm total}\) 6061695358 \(V_2=I R_2\) 7675171493 \(V_1=I R_1\) 9063568209 \(V_{\rm total}=V_1 + V_2\)		2715678478 \(I R_{\rm total}=I R_1 + I R_2\)	recognized infrule but not yet supported
7	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	2715678478 \(I R_{\rm total}=I R_1 + I R_2\)	7844317489 \(I\)	7217021879 \(R_{\rm total}=R_1 + R_2\)	valid
8	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	7217021879 \(R_{\rm total}=R_1 + R_2\)			no validation is available for declarations