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Type Tuple cannot be instantiated; use tuple() instead
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list index out of range
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Review velocity at distance r of object dropped from infinity

abstract: https://www.youtube.com/watch?v=5F1XcTjpJs4 - Derivation of Gravitational Potential Energy by Rhett Allain
reference:

step	inference rule	input	feed	output	validity (as per SymPy)
5	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}.	5902985919 $\vec{F}=G \frac{m_1 m_2}{x^2} \hat{x}$ 7882872592 $W_{\rm to\ system}=\int_{\infty}^r \vec{F}\cdot d\vec{r}$		3566149658 $W_{\rm to\ system}=\int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx$	Type Tuple cannot be instantiated; use tuple() instead
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5902985919 $\vec{F}=G \frac{m_1 m_2}{x^2} \hat{x}$	no validation is available for declarations
12	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}.	8049905441 $\Delta KE=KE_{\rm final} - KE_{\rm initial}$ 1114820451 $W_{\rm by\ system}=\Delta KE$		5779256336 $W_{\rm by\ system}=KE_{\rm final} - KE_{\rm initial}$	valid
15	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2924222857 $v_{\rm initial}=v(r=\infty)$	no validation is available for declarations
8	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	5596822289 $W_{\rm to\ system}=-G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right)$		2061086175 $W_{\rm to\ system}=-G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)$	LHS diff is 0 RHS diff is pdg0005022pdg0006277(-pdg0004851 + pdg0004851(-1/pdg0002530))
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8357234146 $KE=\frac{1}{2} m v^2$	no validation is available for declarations
26	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	2005061870 $v(r)=\sqrt{\frac{2 G m_2}{r}}$			no validation is available for declarations
7	ID: 111662; evaluate definite integral number of inputs: 1; feeds: 0; outputs: 1 Evaluate definite integral Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	8405272745 $W_{\rm to\ system}=-G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx$		5596822289 $W_{\rm to\ system}=-G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right)$	Type Tuple cannot be instantiated; use tuple() instead
9	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	2061086175 $W_{\rm to\ system}=-G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)$		4393670960 $W_{\rm to\ system}=\frac{G m_1 m_2}{r}$	LHS diff is 0 RHS diff is pdg0005022pdg0006277(-pdg0002530*pdg0004851(-1/pdg0002530) - pdg0004851)/pdg0002530
25	ID: 111886; change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	5846639423 $v_{\rm final}=\sqrt{\frac{2 G m_2}{r}}$	3531380618 $v(r)$ 6599829782 $v_{\rm final}$	2005061870 $v(r)=\sqrt{\frac{2 G m_2}{r}}$	LHS diff is pdg0008909 - pdg0001357(pdg0002530) RHS diff is 0
6	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3566149658 $W_{\rm to\ system}=\int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx$		8405272745 $W_{\rm to\ system}=-G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx$	Type Tuple cannot be instantiated; use tuple() instead
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}.	2924222857 $v_{\rm initial}=v(r=\infty)$ 3214170322 $v(r=\infty)=0$		2998709778 $v_{\rm initial}=0$	invalid syntax (<string>, line 0)
3	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1114820451 $W_{\rm by\ system}=\Delta KE$	no validation is available for declarations
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}.	5779256336 $W_{\rm by\ system}=KE_{\rm final} - KE_{\rm initial}$ 9510328252 $KE_{\rm initial}=0$		5850144586 $W_{\rm by\ system}=KE_{\rm final}$	LHS diff is pdg0004121 - pdg0006191 RHS diff is -pdg0005340
10	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8049905441 $\Delta KE=KE_{\rm final} - KE_{\rm initial}$	no validation is available for declarations
13	ID: 111236; change three variables in expression number of inputs: 1; feeds: 6; outputs: 1 Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\\ref{eq:#7}; yields Eq.~\\ref{eq:#8}.	8357234146 $KE=\frac{1}{2} m v^2$	6281834543 $m_1$ 3350802342 $KE_{\rm initial}$ 8066819515 $v$ 3731774096 $KE$ 3274176452 $v_{\rm initial}$ 5904227750 $m$	6091977310 $KE_{\rm initial}=\frac{1}{2} m_1 v_{\rm initial}^2$	list index out of range
21	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}.	2907404069 $W_{\rm by\ system}=W_{\rm to\ system}$ 9081138616 $W_{\rm by\ system}=\frac{1}{2} m_1 v_{\rm final}^2$		4947831649 $\frac{1}{2} m_1 v_{\rm final}^2=W_{\rm to\ system}$	LHS diff is -pdg0005022pdg00089092/2 + pdg0009372 RHS diff is pdg0005022pdg0008909**2/2 - pdg0009372
20	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2907404069 $W_{\rm by\ system}=W_{\rm to\ system}$	no validation is available for declarations
22	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}.	4393670960 $W_{\rm to\ system}=\frac{G m_1 m_2}{r}$ 4947831649 $\frac{1}{2} m_1 v_{\rm final}^2=W_{\rm to\ system}$		6892595652 $\frac{1}{2} m_1 v_{\rm final}^2=\frac{G m_1 m_2}{r}$	valid
17	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}.	6091977310 $KE_{\rm initial}=\frac{1}{2} m_1 v_{\rm initial}^2$ 2998709778 $v_{\rm initial}=0$		9510328252 $KE_{\rm initial}=0$	LHS diff is pdg0001934 - pdg0004121 RHS diff is 0
14	ID: 111236; change three variables in expression number of inputs: 1; feeds: 6; outputs: 1 Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\\ref{eq:#7}; yields Eq.~\\ref{eq:#8}.	8357234146 $KE=\frac{1}{2} m v^2$	1616666229 $v_{\rm final}$ 9350720370 $m$ 3939572542 $KE_{\rm final}$ 3166466250 $m_1$ 6038673136 $v$ 4587046017 $KE$	8552710882 $KE_{\rm final}=\frac{1}{2} m_1 v_{\rm final}^2$	list index out of range
4	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			7882872592 $W_{\rm to\ system}=\int_{\infty}^r \vec{F}\cdot d\vec{r}$	no validation is available for declarations
24	ID: 111524; square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}.	7112646057 $v_{\rm final}^2=\frac{2 G m_2}{r}$		5693047217 $v_{\rm final}=-\sqrt{\frac{2 G m_2}{r}}$ 5846639423 $v_{\rm final}=\sqrt{\frac{2 G m_2}{r}}$	recognized infrule but not yet supported
19	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}.	5850144586 $W_{\rm by\ system}=KE_{\rm final}$ 8552710882 $KE_{\rm final}=\frac{1}{2} m_1 v_{\rm final}^2$		9081138616 $W_{\rm by\ system}=\frac{1}{2} m_1 v_{\rm final}^2$	LHS diff is pdg0005340 - pdg0006191 RHS diff is 0
23	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}.	6892595652 $\frac{1}{2} m_1 v_{\rm final}^2=\frac{G m_1 m_2}{r}$	7410526982 $2/m_1$	7112646057 $v_{\rm final}^2=\frac{2 G m_2}{r}$	valid
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3214170322 $v(r=\infty)=0$	no validation is available for declarations

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timing of Neo4j queries:

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step	inference rule	input	feed	output	validity (as per SymPy)
5	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}.	5902985919 \(\vec{F}=G \frac{m_1 m_2}{x^2} \hat{x}\) 7882872592 \(W_{\rm to\ system}=\int_{\infty}^r \vec{F}\cdot d\vec{r}\)		3566149658 \(W_{\rm to\ system}=\int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)	Type Tuple cannot be instantiated; use tuple() instead
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5902985919 \(\vec{F}=G \frac{m_1 m_2}{x^2} \hat{x}\)	no validation is available for declarations
12	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}.	8049905441 \(\Delta KE=KE_{\rm final} - KE_{\rm initial}\) 1114820451 \(W_{\rm by\ system}=\Delta KE\)		5779256336 \(W_{\rm by\ system}=KE_{\rm final} - KE_{\rm initial}\)	valid
15	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2924222857 \(v_{\rm initial}=v(r=\infty)\)	no validation is available for declarations
8	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	5596822289 \(W_{\rm to\ system}=-G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right)\)		2061086175 \(W_{\rm to\ system}=-G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)	LHS diff is 0 RHS diff is pdg0005022pdg0006277(-pdg0004851 + pdg0004851(-1/pdg0002530))
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8357234146 \(KE=\frac{1}{2} m v^2\)	no validation is available for declarations
26	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	2005061870 \(v(r)=\sqrt{\frac{2 G m_2}{r}}\)			no validation is available for declarations
7	ID: 111662; evaluate definite integral number of inputs: 1; feeds: 0; outputs: 1 Evaluate definite integral Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	8405272745 \(W_{\rm to\ system}=-G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)		5596822289 \(W_{\rm to\ system}=-G m_1 m_2 \left(\left.\frac{-1}{x}\right\|^r_{\infty}\right)\)	Type Tuple cannot be instantiated; use tuple() instead
9	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	2061086175 \(W_{\rm to\ system}=-G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)\)		4393670960 \(W_{\rm to\ system}=\frac{G m_1 m_2}{r}\)	LHS diff is 0 RHS diff is pdg0005022pdg0006277(-pdg0002530*pdg0004851(-1/pdg0002530) - pdg0004851)/pdg0002530
25	ID: 111886; change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	5846639423 \(v_{\rm final}=\sqrt{\frac{2 G m_2}{r}}\)	3531380618 \(v(r)\) 6599829782 \(v_{\rm final}\)	2005061870 \(v(r)=\sqrt{\frac{2 G m_2}{r}}\)	LHS diff is pdg0008909 - pdg0001357(pdg0002530) RHS diff is 0
6	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3566149658 \(W_{\rm to\ system}=\int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx\)		8405272745 \(W_{\rm to\ system}=-G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx\)	Type Tuple cannot be instantiated; use tuple() instead
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}.	2924222857 \(v_{\rm initial}=v(r=\infty)\) 3214170322 \(v(r=\infty)=0\)		2998709778 \(v_{\rm initial}=0\)	invalid syntax (<string>, line 0)
3	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1114820451 \(W_{\rm by\ system}=\Delta KE\)	no validation is available for declarations
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}.	5779256336 \(W_{\rm by\ system}=KE_{\rm final} - KE_{\rm initial}\) 9510328252 \(KE_{\rm initial}=0\)		5850144586 \(W_{\rm by\ system}=KE_{\rm final}\)	LHS diff is pdg0004121 - pdg0006191 RHS diff is -pdg0005340
10	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8049905441 \(\Delta KE=KE_{\rm final} - KE_{\rm initial}\)	no validation is available for declarations
13	ID: 111236; change three variables in expression number of inputs: 1; feeds: 6; outputs: 1 Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\\ref{eq:#7}; yields Eq.~\\ref{eq:#8}.	8357234146 \(KE=\frac{1}{2} m v^2\)	6281834543 \(m_1\) 3350802342 \(KE_{\rm initial}\) 8066819515 \(v\) 3731774096 \(KE\) 3274176452 \(v_{\rm initial}\) 5904227750 \(m\)	6091977310 \(KE_{\rm initial}=\frac{1}{2} m_1 v_{\rm initial}^2\)	list index out of range
21	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}.	2907404069 \(W_{\rm by\ system}=W_{\rm to\ system}\) 9081138616 \(W_{\rm by\ system}=\frac{1}{2} m_1 v_{\rm final}^2\)		4947831649 \(\frac{1}{2} m_1 v_{\rm final}^2=W_{\rm to\ system}\)	LHS diff is -pdg0005022pdg00089092/2 + pdg0009372 RHS diff is pdg0005022pdg0008909**2/2 - pdg0009372
20	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2907404069 \(W_{\rm by\ system}=W_{\rm to\ system}\)	no validation is available for declarations
22	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}.	4393670960 \(W_{\rm to\ system}=\frac{G m_1 m_2}{r}\) 4947831649 \(\frac{1}{2} m_1 v_{\rm final}^2=W_{\rm to\ system}\)		6892595652 \(\frac{1}{2} m_1 v_{\rm final}^2=\frac{G m_1 m_2}{r}\)	valid
17	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}.	6091977310 \(KE_{\rm initial}=\frac{1}{2} m_1 v_{\rm initial}^2\) 2998709778 \(v_{\rm initial}=0\)		9510328252 \(KE_{\rm initial}=0\)	LHS diff is pdg0001934 - pdg0004121 RHS diff is 0
14	ID: 111236; change three variables in expression number of inputs: 1; feeds: 6; outputs: 1 Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\\ref{eq:#7}; yields Eq.~\\ref{eq:#8}.	8357234146 \(KE=\frac{1}{2} m v^2\)	1616666229 \(v_{\rm final}\) 9350720370 \(m\) 3939572542 \(KE_{\rm final}\) 3166466250 \(m_1\) 6038673136 \(v\) 4587046017 \(KE\)	8552710882 \(KE_{\rm final}=\frac{1}{2} m_1 v_{\rm final}^2\)	list index out of range
4	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			7882872592 \(W_{\rm to\ system}=\int_{\infty}^r \vec{F}\cdot d\vec{r}\)	no validation is available for declarations
24	ID: 111524; square root both sides number of inputs: 1; feeds: 0; outputs: 2 Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}.	7112646057 \(v_{\rm final}^2=\frac{2 G m_2}{r}\)		5693047217 \(v_{\rm final}=-\sqrt{\frac{2 G m_2}{r}}\) 5846639423 \(v_{\rm final}=\sqrt{\frac{2 G m_2}{r}}\)	recognized infrule but not yet supported
19	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}.	5850144586 \(W_{\rm by\ system}=KE_{\rm final}\) 8552710882 \(KE_{\rm final}=\frac{1}{2} m_1 v_{\rm final}^2\)		9081138616 \(W_{\rm by\ system}=\frac{1}{2} m_1 v_{\rm final}^2\)	LHS diff is pdg0005340 - pdg0006191 RHS diff is 0
23	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}.	6892595652 \(\frac{1}{2} m_1 v_{\rm final}^2=\frac{G m_1 m_2}{r}\)	7410526982 \(2/m_1\)	7112646057 \(v_{\rm final}^2=\frac{2 G m_2}{r}\)	valid
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3214170322 \(v(r=\infty)=0\)	no validation is available for declarations