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Review angle of maximum distance for projectile motion

abstract:
reference:

step	inference rule	input	feed	output	validity (as per SymPy)
7	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}.	1087417579 $0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)$	4829590294 $t_f$	2086924031 $0=- \frac{1}{2} g t_f + v_0 \sin(\theta)$	valid
4	ID: 111355; LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\\ref{eq:#1} is equal to LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9112191201 $y_f=0$ 5379546684 $y_f=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$		8198310977 $0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$	valid
22	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	5353282496 $d=\frac{v_0^2}{g}$			no validation is available for declarations
16	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2405307372 $\sin(2 x)=2 \sin(x) \cos(x)$	no validation is available for declarations
10	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5438722682 $x=v_0 t \cos(\theta) + x_0$	no validation is available for declarations
21	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3607070319 $d=\frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)$		5353282496 $d=\frac{v_0^2}{g}$	LHS diff is 0 RHS diff is pdg0005153*2(sin(pdg0003141/2) - 1)/pdg0001649
12	ID: 111278; boundary condition number of inputs: 1; feeds: 0; outputs: 1 Boundary condition: Eq.~\\ref{eq:#2} when Eq.~\\ref{eq#1}.	4370074654 $t=t_f$		2378095808 $x_f=x_0 + d$	no validation is available for assumptions
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}.	3485125659 $x_f=v_0 t_f \cos(\theta) + x_0$ 2378095808 $x_f=x_0 + d$		4268085801 $x_0 + d=v_0 t_f \cos(\theta) + x_0$	LHS diff is -pdg0001943 + pdg0002467pdg0005153cos(pdg0001575) RHS diff is pdg0001943 - pdg0002467pdg0005153cos(pdg0001575)
14	ID: 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}.	4268085801 $x_0 + d=v_0 t_f \cos(\theta) + x_0$	8072682558 $x_0$	7233558441 $d=v_0 t_f \cos(\theta)$	valid
11	ID: 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}.	5438722682 $x=v_0 t \cos(\theta) + x_0$	6732786762 $t$ 5194141542 $x_f$ 6463266449 $t_f$ 3273630811 $x$	3485125659 $x_f=v_0 t_f \cos(\theta) + x_0$	list index out of range
2	ID: 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}.	9862900242 $y=- \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$	4162188238 $t_f$ 8120663858 $y_f$ 8406170337 $y$ 2403773761 $t$	5379546684 $y_f=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$	list index out of range
6	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}.	8198310977 $0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$ 1650441634 $y_0=0$		1087417579 $0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)$	LHS diff is pdg0001469 RHS diff is pdg0001469
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}.	2519058903 $\sin(2 \theta)=2 \sin(\theta) \cos(\theta)$ 2297105551 $d=v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)$		8922441655 $d=\frac{v_0^2}{g} \sin(2 \theta)$	valid
3	ID: 111278; boundary condition number of inputs: 1; feeds: 0; outputs: 1 Boundary condition: Eq.~\\ref{eq:#2} when Eq.~\\ref{eq#1}.	5373931751 $t=t_f$		9112191201 $y_f=0$	no validation is available for assumptions
23	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	1541916015 $\theta=\frac{\pi}{4}$			no validation is available for declarations
9	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}.	1191796961 $\frac{1}{2} g t_f=v_0 \sin(\theta)$	2510804451 $2/g$	4778077984 $t_f=\frac{2 v_0 \sin(\theta)}{g}$	valid
15	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}.	7233558441 $d=v_0 t_f \cos(\theta)$ 4778077984 $t_f=\frac{2 v_0 \sin(\theta)}{g}$		2297105551 $d=v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)$	LHS diff is -pdg0001943 + pdg0002467 RHS diff is 2pdg0005153(-pdg0005153cos(pdg0001575) + 1)sin(pdg0001575)/pdg0001649
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9862900242 $y=- \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$	no validation is available for declarations
8	ID: 111530; add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	2086924031 $0=- \frac{1}{2} g t_f + v_0 \sin(\theta)$	6974054946 $\frac{1}{2} g t_f$	1191796961 $\frac{1}{2} g t_f=v_0 \sin(\theta)$	valid
20	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}.	8922441655 $d=\frac{v_0^2}{g} \sin(2 \theta)$ 1541916015 $\theta=\frac{\pi}{4}$		3607070319 $d=\frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)$	LHS diff is pdg0001575 - pdg0001943 RHS diff is pdg0003141/4 - pdg0005153*2sin(pdg0003141/2)/pdg0001649
5	ID: 111104; declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			1650441634 $y_0=0$	no validation is available for declarations
17	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}.	2405307372 $\sin(2 x)=2 \sin(x) \cos(x)$	7214442790 $x$ 7587034465 $\theta$	2519058903 $\sin(2 \theta)=2 \sin(\theta) \cos(\theta)$	valid
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}	8922441655 $d=\frac{v_0^2}{g} \sin(2 \theta)$	5667870149 $\theta$	1541916015 $\theta=\frac{\pi}{4}$	recognized infrule but not yet supported

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timing of Neo4j queries:

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step	inference rule	input	feed	output	validity (as per SymPy)
7	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}.	1087417579 \(0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)\)	4829590294 \(t_f\)	2086924031 \(0=- \frac{1}{2} g t_f + v_0 \sin(\theta)\)	valid
4	ID: 111355; LHS of expr 1 equals LHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 LHS of Eq.~\\ref{eq:#1} is equal to LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9112191201 \(y_f=0\) 5379546684 \(y_f=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)		8198310977 \(0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)	valid
22	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	5353282496 \(d=\frac{v_0^2}{g}\)			no validation is available for declarations
16	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2405307372 \(\sin(2 x)=2 \sin(x) \cos(x)\)	no validation is available for declarations
10	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5438722682 \(x=v_0 t \cos(\theta) + x_0\)	no validation is available for declarations
21	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3607070319 \(d=\frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)\)		5353282496 \(d=\frac{v_0^2}{g}\)	LHS diff is 0 RHS diff is pdg0005153*2(sin(pdg0003141/2) - 1)/pdg0001649
12	ID: 111278; boundary condition number of inputs: 1; feeds: 0; outputs: 1 Boundary condition: Eq.~\\ref{eq:#2} when Eq.~\\ref{eq#1}.	4370074654 \(t=t_f\)		2378095808 \(x_f=x_0 + d\)	no validation is available for assumptions
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}.	3485125659 \(x_f=v_0 t_f \cos(\theta) + x_0\) 2378095808 \(x_f=x_0 + d\)		4268085801 \(x_0 + d=v_0 t_f \cos(\theta) + x_0\)	LHS diff is -pdg0001943 + pdg0002467pdg0005153cos(pdg0001575) RHS diff is pdg0001943 - pdg0002467pdg0005153cos(pdg0001575)
14	ID: 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}.	4268085801 \(x_0 + d=v_0 t_f \cos(\theta) + x_0\)	8072682558 \(x_0\)	7233558441 \(d=v_0 t_f \cos(\theta)\)	valid
11	ID: 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}.	5438722682 \(x=v_0 t \cos(\theta) + x_0\)	6732786762 \(t\) 5194141542 \(x_f\) 6463266449 \(t_f\) 3273630811 \(x\)	3485125659 \(x_f=v_0 t_f \cos(\theta) + x_0\)	list index out of range
2	ID: 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}.	9862900242 \(y=- \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)	4162188238 \(t_f\) 8120663858 \(y_f\) 8406170337 \(y\) 2403773761 \(t\)	5379546684 \(y_f=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\)	list index out of range
6	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}.	8198310977 \(0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0\) 1650441634 \(y_0=0\)		1087417579 \(0=- \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)\)	LHS diff is pdg0001469 RHS diff is pdg0001469
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}.	2519058903 \(\sin(2 \theta)=2 \sin(\theta) \cos(\theta)\) 2297105551 \(d=v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)\)		8922441655 \(d=\frac{v_0^2}{g} \sin(2 \theta)\)	valid
3	ID: 111278; boundary condition number of inputs: 1; feeds: 0; outputs: 1 Boundary condition: Eq.~\\ref{eq:#2} when Eq.~\\ref{eq#1}.	5373931751 \(t=t_f\)		9112191201 \(y_f=0\)	no validation is available for assumptions
23	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	1541916015 \(\theta=\frac{\pi}{4}\)			no validation is available for declarations
9	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}.	1191796961 \(\frac{1}{2} g t_f=v_0 \sin(\theta)\)	2510804451 \(2/g\)	4778077984 \(t_f=\frac{2 v_0 \sin(\theta)}{g}\)	valid
15	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}.	7233558441 \(d=v_0 t_f \cos(\theta)\) 4778077984 \(t_f=\frac{2 v_0 \sin(\theta)}{g}\)		2297105551 \(d=v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)\)	LHS diff is -pdg0001943 + pdg0002467 RHS diff is 2pdg0005153(-pdg0005153cos(pdg0001575) + 1)sin(pdg0001575)/pdg0001649
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9862900242 \(y=- \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0\)	no validation is available for declarations
8	ID: 111530; add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	2086924031 \(0=- \frac{1}{2} g t_f + v_0 \sin(\theta)\)	6974054946 \(\frac{1}{2} g t_f\)	1191796961 \(\frac{1}{2} g t_f=v_0 \sin(\theta)\)	valid
20	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}.	8922441655 \(d=\frac{v_0^2}{g} \sin(2 \theta)\) 1541916015 \(\theta=\frac{\pi}{4}\)		3607070319 \(d=\frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)\)	LHS diff is pdg0001575 - pdg0001943 RHS diff is pdg0003141/4 - pdg0005153*2sin(pdg0003141/2)/pdg0001649
5	ID: 111104; declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			1650441634 \(y_0=0\)	no validation is available for declarations
17	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}.	2405307372 \(\sin(2 x)=2 \sin(x) \cos(x)\)	7214442790 \(x\) 7587034465 \(\theta\)	2519058903 \(\sin(2 \theta)=2 \sin(\theta) \cos(\theta)\)	valid
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}	8922441655 \(d=\frac{v_0^2}{g} \sin(2 \theta)\)	5667870149 \(\theta\)	1541916015 \(\theta=\frac{\pi}{4}\)	recognized infrule but not yet supported