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Review Compton's equation for scattering

abstract:
reference:

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.			1638282134 $\vec{p}_{\rm before}=\vec{p}_{\rm after}$	no validation is available for declarations
2	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8257621077 $\vec{p}_{\rm before}=\vec{p}_{1}$	no validation is available for declarations
3	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8311458118 $\vec{p}_{\rm after}=\vec{p}_{2}+\vec{p}_{electron}$	no validation is available for declarations
4	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}.	8257621077 $\vec{p}_{\rm before}=\vec{p}_{1}$ 1638282134 $\vec{p}_{\rm before}=\vec{p}_{\rm after}$		3951205425 $\vec{p}_{\rm after}=\vec{p}_{1}$	LHS diff is -pdg0005493 + pdg0006029 RHS diff is pdg0005493 - pdg0006029
5	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}.	8311458118 $\vec{p}_{\rm after}=\vec{p}_{2}+\vec{p}_{electron}$ 3951205425 $\vec{p}_{\rm after}=\vec{p}_{1}$		8139187332 $\vec{p}_{1}=\vec{p}_{2}+\vec{p}_{electron}$	LHS diff is pdg0002097 + pdg0004299 - pdg0006029 RHS diff is -pdg0002097 - pdg0004299 + pdg0006029
6	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	8139187332 $\vec{p}_{1}=\vec{p}_{2}+\vec{p}_{electron}$	0002338514 $\vec{p}_{2}$	5530148480 $\vec{p}_{1}-\vec{p}_{2}=\vec{p}_{electron}$	valid
7	111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\\ref{eq:#1} with RHS; yields Eq.~\\ref{eq:#2}.	5530148480 $\vec{p}_{1}-\vec{p}_{2}=\vec{p}_{electron}$		7917051060 $\vec{p}_{electron}=\vec{p}_{1}-\vec{p}_{2}$	valid
8	111253: multiply expr 1 by expr 2 number of inputs: 2; feeds: 0; outputs: 1 Multiply Eq.~\\ref{eq:#1} by Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	7917051060 $\vec{p}_{electron}=\vec{p}_{1}-\vec{p}_{2}$ 7917051060 $\vec{p}_{electron}=\vec{p}_{1}-\vec{p}_{2}$		6742123016 $\vec{p}_{electron}\cdot\vec{p}_{electron}=( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})$	invalid syntax (<string>, line 0)

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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.			1638282134 \(\vec{p}_{\rm before}=\vec{p}_{\rm after}\)	no validation is available for declarations
2	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8257621077 \(\vec{p}_{\rm before}=\vec{p}_{1}\)	no validation is available for declarations
3	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8311458118 \(\vec{p}_{\rm after}=\vec{p}_{2}+\vec{p}_{electron}\)	no validation is available for declarations
4	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}.	8257621077 \(\vec{p}_{\rm before}=\vec{p}_{1}\) 1638282134 \(\vec{p}_{\rm before}=\vec{p}_{\rm after}\)		3951205425 \(\vec{p}_{\rm after}=\vec{p}_{1}\)	LHS diff is -pdg0005493 + pdg0006029 RHS diff is pdg0005493 - pdg0006029
5	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}.	8311458118 \(\vec{p}_{\rm after}=\vec{p}_{2}+\vec{p}_{electron}\) 3951205425 \(\vec{p}_{\rm after}=\vec{p}_{1}\)		8139187332 \(\vec{p}_{1}=\vec{p}_{2}+\vec{p}_{electron}\)	LHS diff is pdg0002097 + pdg0004299 - pdg0006029 RHS diff is -pdg0002097 - pdg0004299 + pdg0006029
6	111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	8139187332 \(\vec{p}_{1}=\vec{p}_{2}+\vec{p}_{electron}\)	0002338514 \(\vec{p}_{2}\)	5530148480 \(\vec{p}_{1}-\vec{p}_{2}=\vec{p}_{electron}\)	valid
7	111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\\ref{eq:#1} with RHS; yields Eq.~\\ref{eq:#2}.	5530148480 \(\vec{p}_{1}-\vec{p}_{2}=\vec{p}_{electron}\)		7917051060 \(\vec{p}_{electron}=\vec{p}_{1}-\vec{p}_{2}\)	valid
8	111253: multiply expr 1 by expr 2 number of inputs: 2; feeds: 0; outputs: 1 Multiply Eq.~\\ref{eq:#1} by Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	7917051060 \(\vec{p}_{electron}=\vec{p}_{1}-\vec{p}_{2}\) 7917051060 \(\vec{p}_{electron}=\vec{p}_{1}-\vec{p}_{2}\)		6742123016 \(\vec{p}_{electron}\cdot\vec{p}_{electron}=( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})\)	invalid syntax (<string>, line 0)