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Review equation of motion for a spring

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.			5345738321 $F=m a$	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.			4428528271 $F_{\rm spring}=-k x$	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.			6831694380 $a=\frac{d^2 x}{dt^2}$	no validation is available for declarations
4	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}.	4428528271 $F_{\rm spring}=-k x$ 5345738321 $F=m a$		2334518266 $m a=-k x$	input diff is pdg0004183 - pdg0004202 diff is -pdg0001356pdg0004037 - pdg0005156pdg0009140 diff is pdg0001356pdg0004037 + pdg0005156pdg0009140
5	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}.	2334518266 $m a=-k x$	3634715785 $m$	8655294002 $a=-\frac{k}{m}x$	valid
6	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}.	6831694380 $a=\frac{d^2 x}{dt^2}$ 8655294002 $a=-\frac{k}{m}x$		8991236357 $\frac{d^2 x}{dt^2}=-\frac{k}{m} x$	input diff is a - pdg0009140 diff is d*2(-pdg0004037 + x)/dt**2 diff is 0
7	111237: declare guess solution number of inputs: 1; feeds: 0; outputs: 1 Judicious choice as a guessed solution to Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2},	8991236357 $\frac{d^2 x}{dt^2}=-\frac{k}{m} x$		5415824175 $x(t)=A \cos(\omega t)$	no validation is available for declarations
8	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}.	5415824175 $x(t)=A \cos(\omega t)$	5846177002 $t$	7652131521 $\frac{dx}{dt}=-A \omega \sin (\omega t)$	recognized infrule but not yet supported
9	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}.	7652131521 $\frac{dx}{dt}=-A \omega \sin (\omega t)$	1451839362 $t$	5945893986 $\frac{d^2 x}{dt^2}=-A \omega^2 \cos(\omega t)$	recognized infrule but not yet supported
10	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}.	5945893986 $\frac{d^2 x}{dt^2}=-A \omega^2 \cos(\omega t)$ 8991236357 $\frac{d^2 x}{dt^2}=-\frac{k}{m} x$		1772973171 $-\frac{k}{m} x=-A \omega^2 \cos(\omega t)$	input diff is d*2(-pdg0004037 + x)/dt*2 diff is kx/pdg0005156 - pdg0002321*2pdg0009885cos(pdg0001467pdg0002321) diff is Apdg00023212cos(pdg0002321pdg0009491) - pdg0001356pdg0004037/pdg0005156
11	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}.	5415824175 $x(t)=A \cos(\omega t)$ 1772973171 $-\frac{k}{m} x=-A \omega^2 \cos(\omega t)$		2148049269 $-\frac{k}{m} A \cos(\omega t)=-A \omega^2 \cos(\omega t)$	LHS diff is k(Acos(pdg0002321*pdg0009491) - x)/pdg0005156 RHS diff is 0
12	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}.	2148049269 $-\frac{k}{m} A \cos(\omega t)=-A \omega^2 \cos(\omega t)$	7473576008 $\frac{-1}{A \cos(\omega t)}$	1931103031 $\frac{k}{m}=\omega^2$	LHS arithmetic error. Diff: -(Akpdg0001467cos(pdg0002321pdg0009491) + pdg0001356)/pdg0005156
13	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}.	1931103031 $\frac{k}{m}=\omega^2$		1784114349 $\sqrt{\frac{k}{m}}=\omega$ 1888494137 $-\sqrt{\frac{k}{m}}=\omega$	recognized infrule but not yet supported
14	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}.	5415824175 $x(t)=A \cos(\omega t)$ 1784114349 $\sqrt{\frac{k}{m}}=\omega$		6908055431 $x(t)=A \cos\left(\frac{k}{m} t\right)$	LHS diff is sqrt(pdg0001356/pdg0005156) - x(pdg0001467) RHS diff is pdg0002321 - pdg0009885cos(kpdg0001467/pdg0005156)
15	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	6908055431 $x(t)=A \cos\left(\frac{k}{m} t\right)$			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.			5345738321 \(F=m a\)	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.			4428528271 \(F_{\rm spring}=-k x\)	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.			6831694380 \(a=\frac{d^2 x}{dt^2}\)	no validation is available for declarations
4	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}.	4428528271 \(F_{\rm spring}=-k x\) 5345738321 \(F=m a\)		2334518266 \(m a=-k x\)	input diff is pdg0004183 - pdg0004202 diff is -pdg0001356pdg0004037 - pdg0005156pdg0009140 diff is pdg0001356pdg0004037 + pdg0005156pdg0009140
5	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}.	2334518266 \(m a=-k x\)	3634715785 \(m\)	8655294002 \(a=-\frac{k}{m}x\)	valid
6	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}.	6831694380 \(a=\frac{d^2 x}{dt^2}\) 8655294002 \(a=-\frac{k}{m}x\)		8991236357 \(\frac{d^2 x}{dt^2}=-\frac{k}{m} x\)	input diff is a - pdg0009140 diff is d*2(-pdg0004037 + x)/dt**2 diff is 0
7	111237: declare guess solution number of inputs: 1; feeds: 0; outputs: 1 Judicious choice as a guessed solution to Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2},	8991236357 \(\frac{d^2 x}{dt^2}=-\frac{k}{m} x\)		5415824175 \(x(t)=A \cos(\omega t)\)	no validation is available for declarations
8	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}.	5415824175 \(x(t)=A \cos(\omega t)\)	5846177002 \(t\)	7652131521 \(\frac{dx}{dt}=-A \omega \sin (\omega t)\)	recognized infrule but not yet supported
9	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}.	7652131521 \(\frac{dx}{dt}=-A \omega \sin (\omega t)\)	1451839362 \(t\)	5945893986 \(\frac{d^2 x}{dt^2}=-A \omega^2 \cos(\omega t)\)	recognized infrule but not yet supported
10	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}.	5945893986 \(\frac{d^2 x}{dt^2}=-A \omega^2 \cos(\omega t)\) 8991236357 \(\frac{d^2 x}{dt^2}=-\frac{k}{m} x\)		1772973171 \(-\frac{k}{m} x=-A \omega^2 \cos(\omega t)\)	input diff is d*2(-pdg0004037 + x)/dt*2 diff is kx/pdg0005156 - pdg0002321*2pdg0009885cos(pdg0001467pdg0002321) diff is Apdg00023212cos(pdg0002321pdg0009491) - pdg0001356pdg0004037/pdg0005156
11	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}.	5415824175 \(x(t)=A \cos(\omega t)\) 1772973171 \(-\frac{k}{m} x=-A \omega^2 \cos(\omega t)\)		2148049269 \(-\frac{k}{m} A \cos(\omega t)=-A \omega^2 \cos(\omega t)\)	LHS diff is k(Acos(pdg0002321*pdg0009491) - x)/pdg0005156 RHS diff is 0
12	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}.	2148049269 \(-\frac{k}{m} A \cos(\omega t)=-A \omega^2 \cos(\omega t)\)	7473576008 \(\frac{-1}{A \cos(\omega t)}\)	1931103031 \(\frac{k}{m}=\omega^2\)	LHS arithmetic error. Diff: -(Akpdg0001467cos(pdg0002321pdg0009491) + pdg0001356)/pdg0005156
13	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}.	1931103031 \(\frac{k}{m}=\omega^2\)		1784114349 \(\sqrt{\frac{k}{m}}=\omega\) 1888494137 \(-\sqrt{\frac{k}{m}}=\omega\)	recognized infrule but not yet supported
14	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}.	5415824175 \(x(t)=A \cos(\omega t)\) 1784114349 \(\sqrt{\frac{k}{m}}=\omega\)		6908055431 \(x(t)=A \cos\left(\frac{k}{m} t\right)\)	LHS diff is sqrt(pdg0001356/pdg0005156) - x(pdg0001467) RHS diff is pdg0002321 - pdg0009885cos(kpdg0001467/pdg0005156)
15	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	6908055431 \(x(t)=A \cos\left(\frac{k}{m} t\right)\)			no validation is available for declarations