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Review Euler equation proof

abstract:
reference:

step	inference rule	input	feed	output	validity (as per SymPy)
10	ID: 111721; make expr power number of inputs: 1; feeds: 1; outputs: 1 Make Eq.~\\ref{eq:#2} the power of $#1$; yields Eq.~\\ref{eq:#3}.	4923339482 $i x=\log(y)$	0006656532 $e$	9482923849 $\exp(i x)=y$	LHS diff is -pdg0002718*(pdg0001464pdg0004621) + exp(pdg0001464pdg0004621) RHS diff is pdg0001452 - pdg0002718*(log(pdg0001452)/log(10))
7	ID: 111132; indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}.	9482113948 $\frac{dy}{y}=i dx$	0004264724 $y$	9482943948 $\log(y)=i dx$	recognized infrule but not yet supported
5	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}.	9848294829 $\frac{d}{dx} y=y i$	0003954314 $dx$	9848292229 $dy=y i dx$	Type Tuple cannot be instantiated; use tuple() instead
3	ID: 111260; factor out X from RHS number of inputs: 1; feeds: 1; outputs: 1 Factor $#1$ from the RHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9429829482 $\frac{d}{dx} y=-\sin(x) + i\cos(x)$	0007563791 $i$	9482984922 $\frac{d}{dx} y=(i\sin(x) + \cos(x)) i$	Type Tuple cannot be instantiated; use tuple() instead
8	ID: 111132; indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}.	9482943948 $\log(y)=i dx$	0006563727 $x$	4928239482 $\log(y)=i x$	recognized infrule but not yet supported
2	ID: 111649; differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\\ref{eq:#2} with respect to $#1$; yields Eq.~\\ref{eq:#3}.	9492920340 $y=\cos(x)+i \sin(x)$	0007636749 $x$	9429829482 $\frac{d}{dx} y=-\sin(x) + i\cos(x)$	recognized infrule but not yet supported
6	ID: 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}.	9848292229 $dy=y i dx$	0009877781 $y$	9482113948 $\frac{dy}{y}=i dx$	invalid syntax (<string>, line 0)
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9492920340 $y=\cos(x)+i \sin(x)$	no validation is available for declarations
4	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}.	9492920340 $y=\cos(x)+i \sin(x)$ 9482984922 $\frac{d}{dx} y=(i\sin(x) + \cos(x)) i$		9848294829 $\frac{d}{dx} y=y i$	Type Tuple cannot be instantiated; use tuple() instead
12	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	4938429483 $\exp(i x)=\cos(x)+i \sin(x)$			no validation is available for declarations
9	ID: 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}.	4928239482 $\log(y)=i x$		4923339482 $i x=\log(y)$	valid
11	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}.	9492920340 $y=\cos(x)+i \sin(x)$ 9482923849 $\exp(i x)=y$		4938429483 $\exp(i x)=\cos(x)+i \sin(x)$	LHS diff is 0 RHS diff is pdg0001452 - pdg0004621*sin(pdg0001464) - cos(pdg0001464)

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timing of Neo4j queries:

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step	inference rule	input	feed	output	validity (as per SymPy)
10	ID: 111721; make expr power number of inputs: 1; feeds: 1; outputs: 1 Make Eq.~\\ref{eq:#2} the power of $#1$; yields Eq.~\\ref{eq:#3}.	4923339482 \(i x=\log(y)\)	0006656532 \(e\)	9482923849 \(\exp(i x)=y\)	LHS diff is -pdg0002718*(pdg0001464pdg0004621) + exp(pdg0001464pdg0004621) RHS diff is pdg0001452 - pdg0002718*(log(pdg0001452)/log(10))
7	ID: 111132; indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}.	9482113948 \(\frac{dy}{y}=i dx\)	0004264724 \(y\)	9482943948 \(\log(y)=i dx\)	recognized infrule but not yet supported
5	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}.	9848294829 \(\frac{d}{dx} y=y i\)	0003954314 \(dx\)	9848292229 \(dy=y i dx\)	Type Tuple cannot be instantiated; use tuple() instead
3	ID: 111260; factor out X from RHS number of inputs: 1; feeds: 1; outputs: 1 Factor $#1$ from the RHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9429829482 \(\frac{d}{dx} y=-\sin(x) + i\cos(x)\)	0007563791 \(i\)	9482984922 \(\frac{d}{dx} y=(i\sin(x) + \cos(x)) i\)	Type Tuple cannot be instantiated; use tuple() instead
8	ID: 111132; indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}.	9482943948 \(\log(y)=i dx\)	0006563727 \(x\)	4928239482 \(\log(y)=i x\)	recognized infrule but not yet supported
2	ID: 111649; differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\\ref{eq:#2} with respect to $#1$; yields Eq.~\\ref{eq:#3}.	9492920340 \(y=\cos(x)+i \sin(x)\)	0007636749 \(x\)	9429829482 \(\frac{d}{dx} y=-\sin(x) + i\cos(x)\)	recognized infrule but not yet supported
6	ID: 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}.	9848292229 \(dy=y i dx\)	0009877781 \(y\)	9482113948 \(\frac{dy}{y}=i dx\)	invalid syntax (<string>, line 0)
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9492920340 \(y=\cos(x)+i \sin(x)\)	no validation is available for declarations
4	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}.	9492920340 \(y=\cos(x)+i \sin(x)\) 9482984922 \(\frac{d}{dx} y=(i\sin(x) + \cos(x)) i\)		9848294829 \(\frac{d}{dx} y=y i\)	Type Tuple cannot be instantiated; use tuple() instead
12	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	4938429483 \(\exp(i x)=\cos(x)+i \sin(x)\)			no validation is available for declarations
9	ID: 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}.	4928239482 \(\log(y)=i x\)		4923339482 \(i x=\log(y)\)	valid
11	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}.	9492920340 \(y=\cos(x)+i \sin(x)\) 9482923849 \(\exp(i x)=y\)		4938429483 \(\exp(i x)=\cos(x)+i \sin(x)\)	LHS diff is 0 RHS diff is pdg0001452 - pdg0004621*sin(pdg0001464) - cos(pdg0001464)