Welcome to the Physics Derivation Graph

Return to navigation page or list derivations

Review upper limit on velocity in condensed matter

abstract:
reference: https://arxiv.org/pdf/2004.04818.pdf

step	inference rule	input	feed	output	validity (as per SymPy)
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8106885760 $\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$	no validation is available for declarations
12	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}.	8106885760 $\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$	8857931498 $c$	5838268428 $\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}$	valid
9	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1556389363 $E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$	no validation is available for declarations
7	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}.	8090924099 $v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }$ 8688588981 $a^3 \rho=m$		7837519722 $v=\sqrt{f} \sqrt{\frac{E}{m}}$	LHS diff is -pdg0002077 + pdg0003935pdg00058543 RHS diff is -sqrt(pdg0006235)sqrt(pdg0002241/pdg0009863) + pdg0009863
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}	2897612567 $v=\alpha c \sqrt{ \frac{m_e}{A m_p} }$	6259833695 $A$	7701249282 $v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }$	recognized infrule but not yet supported
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}.	5789289057 $v=\alpha c \sqrt{ \frac{m_e}{2 m} }$ 5646314683 $m=A m_p$		2897612567 $v=\alpha c \sqrt{ \frac{m_e}{A m_p} }$	LHS diff is -pdg0002077 + pdg0009863 RHS diff is -pdg0001370pdg0004567sqrt(pdg0002515/(pdg0003285pdg0005916)) + pdg0003285pdg0005916
20	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	7701249282 $v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }$			no validation is available for declarations
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}.	1556389363 $E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$ 4107032818 $E_{\rm Rydberg}=E$		3291685884 $E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$	LHS diff is -pdg0002241 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg00079402) RHS diff is pdg0002241 - pdg00019994pdg0002515/(32pdg0001054*2pdg0003141*2pdg0007940**2)
10	ID: 111104; declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			4107032818 $E_{\rm Rydberg}=E$	no validation is available for declarations
15	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3935058307 $v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }$		9640720571 $v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}$	LHS diff is 0 RHS diff is sqrt(2)(pdg0001054pdg0003141pdg0007940sqrt(pdg0001999*4pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863)) - pdg0001999*2sqrt(pdg0002515/pdg0009863))/(8pdg0001054pdg0003141*pdg0007940)
6	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}.	8908736791 $\rho=\frac{m}{a^3}$	2397692197 $a^3$	8688588981 $a^3 \rho=m$	valid
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9376481176 $K=f \frac{E}{a^3}$	no validation is available for declarations
5	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8908736791 $\rho=\frac{m}{a^3}$	no validation is available for declarations
4	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}.	6504442697 $v=\sqrt{ \frac{K}{\rho} }$ 9376481176 $K=f \frac{E}{a^3}$		8090924099 $v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }$	LHS diff is K - pdg0002077 RHS diff is pdg0002241pdg0006235/pdg00058543 - sqrt(pdg0002241pdg0006235/(pdg0003935pdg0005854*3))
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4560648264 $v=\sqrt{ \frac{K + (4/3) G}{\rho} }$	no validation is available for declarations
14	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}.	9854442418 $v=\sqrt{\frac{E}{m}}$ 3291685884 $E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$		3935058307 $v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }$	LHS diff is -pdg0002077 + pdg0002241 RHS diff is -sqrt(2)sqrt(pdg00019994pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863))/8 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg0007940**2)
17	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5646314683 $m=A m_p$	no validation is available for declarations
8	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	7837519722 $v=\sqrt{f} \sqrt{\frac{E}{m}}$	3685779219 $\sqrt{f} \approx 2$	9854442418 $v=\sqrt{\frac{E}{m}}$	recognized infrule but not yet supported
3	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	4560648264 $v=\sqrt{ \frac{K + (4/3) G}{\rho} }$	9674924517 $K >> G$	6504442697 $v=\sqrt{ \frac{K}{\rho} }$	recognized infrule but not yet supported
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}.	9640720571 $v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}$ 5838268428 $\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}$		5789289057 $v=\alpha c \sqrt{ \frac{m_e}{2 m} }$	LHS diff is pdg0001370pdg0004567 - pdg0002077 RHS diff is -sqrt(2)pdg0001370pdg0004567sqrt(pdg0002515/pdg0009863)/2 + pdg0001999*2/(4pdg0001054pdg0003141pdg0007940)

Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Actions: Edit Derivation

Generate Tex file or PDF file

Delete Derivation and all associated steps

This does not remove expressions, symbols, and operations.

timing of Neo4j queries:

0.028 seconds for pdg_app/to_review_derivation: node_properties, derivation1240788
0.047 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation3529069
0.019 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7399941
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7399941
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7399941
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7399941
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_sequence_index3529069
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5049784
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5049784
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5049784
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5049784
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3556292
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3556292
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3556292
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3556292
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3911604
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3911604
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3911604
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3911604
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4782186
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4782186
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4782186
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4782186
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7562503
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7562503
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7562503
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7562503
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4762539
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4762539
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4762539
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4762539
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5129253
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5129253
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5129253
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5129253
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6929402
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6929402
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6929402
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6929402
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8378509
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT8378509
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED8378509
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT8378509
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7450034
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7450034
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7450034
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7450034
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2132771
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2132771
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2132771
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2132771
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9362372
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9362372
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9362372
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9362372
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6990281
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6990281
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6990281
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6990281
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3682262
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3682262
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3682262
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3682262
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9470079
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9470079
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9470079
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9470079
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4455856
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4455856
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4455856
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4455856
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule1254434
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT1254434
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED1254434
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT1254434
0.016 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2027055
0.017 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2027055
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2027055
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2027055
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3225241
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3225241
0.015 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3225241
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3225241
0.014 seconds for pdg_app/to_review_derivation: get_list_of_step_dicts_in_this_derivation1240788
0.017 seconds for pdg_app/to_review_derivation: get_list_of_sequence_values_for_derivation_id1240788608598
0.037 seconds for pdg_app/to_review_derivation: get_list_node_dicts_of_type inference_rule1240788
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2109265
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2109265
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2109265
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2109265
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5255784
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5255784
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5255784
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5255784
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9482643
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9482643
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9482643
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9482643
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8931246
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT8931246
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED8931246
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT8931246
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6452898
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6452898
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6452898
0.013 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6452898
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8160842
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT8160842
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED8160842
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT8160842
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9025029
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9025029
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9025029
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9025029
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4468178
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4468178
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4468178
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4468178
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6655092
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6655092
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6655092
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6655092
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5883264
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5883264
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5883264
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5883264
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2141623
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2141623
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2141623
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2141623
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7421907
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7421907
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7421907
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7421907
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4895159
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4895159
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4895159
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4895159
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7807157
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7807157
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7807157
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7807157
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule1828587
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT1828587
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED1828587
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT1828587
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3230740
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3230740
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3230740
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3230740
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2542535
0.013 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2542535
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2542535
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2542535
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6240344
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6240344
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6240344
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6240344
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8479007
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT8479007
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED8479007
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT8479007
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9390892
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9390892
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9390892
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9390892
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5294820
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5294820
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5294820
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5294820
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7059797
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7059797
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7059797
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7059797
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9530699
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9530699
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9530699
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9530699
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3202527
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3202527
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3202527
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3202527
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5479092
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5479092
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5479092
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5479092
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule4128377
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT4128377
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED4128377
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT4128377
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule6843240
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT6843240
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED6843240
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT6843240
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9279012
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9279012
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9279012
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9279012
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2681280
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2681280
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2681280
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2681280
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule1613782
0.003 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT1613782
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED1613782
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT1613782
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7163317
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7163317
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7163317
0.011 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7163317
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3018439
0.011 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3018439
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3018439
0.011 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3018439
0.017 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule5452394
0.01 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT5452394
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED5452394
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT5452394
0.008 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7230860
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7230860
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED7230860
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7230860
0.009 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2498647
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2498647
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2498647
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2498647
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9359490
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9359490
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9359490
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9359490
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule3020432
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT3020432
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED3020432
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT3020432
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9925147
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9925147
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9925147
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9925147
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2629787
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT2629787
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED2629787
0.006 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT2629787
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule9693161
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT9693161
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_FEED9693161
0.004 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT9693161

step	inference rule	input	feed	output	validity (as per SymPy)
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8106885760 \(\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)	no validation is available for declarations
12	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}.	8106885760 \(\alpha=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)	8857931498 \(c\)	5838268428 \(\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)	valid
9	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1556389363 \(E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)	no validation is available for declarations
7	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}.	8090924099 \(v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\) 8688588981 \(a^3 \rho=m\)		7837519722 \(v=\sqrt{f} \sqrt{\frac{E}{m}}\)	LHS diff is -pdg0002077 + pdg0003935pdg00058543 RHS diff is -sqrt(pdg0006235)sqrt(pdg0002241/pdg0009863) + pdg0009863
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}	2897612567 \(v=\alpha c \sqrt{ \frac{m_e}{A m_p} }\)	6259833695 \(A\)	7701249282 \(v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }\)	recognized infrule but not yet supported
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}.	5789289057 \(v=\alpha c \sqrt{ \frac{m_e}{2 m} }\) 5646314683 \(m=A m_p\)		2897612567 \(v=\alpha c \sqrt{ \frac{m_e}{A m_p} }\)	LHS diff is -pdg0002077 + pdg0009863 RHS diff is -pdg0001370pdg0004567sqrt(pdg0002515/(pdg0003285pdg0005916)) + pdg0003285pdg0005916
20	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	7701249282 \(v_u=\alpha c \sqrt{ \frac{m_e}{m_p} }\)			no validation is available for declarations
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}.	1556389363 \(E_{\rm Rydberg}=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\) 4107032818 \(E_{\rm Rydberg}=E\)		3291685884 \(E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)	LHS diff is -pdg0002241 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg00079402) RHS diff is pdg0002241 - pdg00019994pdg0002515/(32pdg0001054*2pdg0003141*2pdg0007940**2)
10	ID: 111104; declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			4107032818 \(E_{\rm Rydberg}=E\)	no validation is available for declarations
15	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	3935058307 \(v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)		9640720571 \(v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\)	LHS diff is 0 RHS diff is sqrt(2)(pdg0001054pdg0003141pdg0007940sqrt(pdg0001999*4pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863)) - pdg0001999*2sqrt(pdg0002515/pdg0009863))/(8pdg0001054pdg0003141*pdg0007940)
6	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}.	8908736791 \(\rho=\frac{m}{a^3}\)	2397692197 \(a^3\)	8688588981 \(a^3 \rho=m\)	valid
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			9376481176 \(K=f \frac{E}{a^3}\)	no validation is available for declarations
5	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8908736791 \(\rho=\frac{m}{a^3}\)	no validation is available for declarations
4	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}.	6504442697 \(v=\sqrt{ \frac{K}{\rho} }\) 9376481176 \(K=f \frac{E}{a^3}\)		8090924099 \(v=\sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\)	LHS diff is K - pdg0002077 RHS diff is pdg0002241pdg0006235/pdg00058543 - sqrt(pdg0002241pdg0006235/(pdg0003935pdg0005854*3))
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4560648264 \(v=\sqrt{ \frac{K + (4/3) G}{\rho} }\)	no validation is available for declarations
14	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}.	9854442418 \(v=\sqrt{\frac{E}{m}}\) 3291685884 \(E=\frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)		3935058307 \(v=\sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)	LHS diff is -pdg0002077 + pdg0002241 RHS diff is -sqrt(2)sqrt(pdg00019994pdg0002515/(pdg0001054*2pdg0003141*2pdg0007940*2pdg0009863))/8 + pdg0001999*4pdg0002515/(32pdg00010542pdg0003141*2pdg0007940**2)
17	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5646314683 \(m=A m_p\)	no validation is available for declarations
8	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	7837519722 \(v=\sqrt{f} \sqrt{\frac{E}{m}}\)	3685779219 \(\sqrt{f} \approx 2\)	9854442418 \(v=\sqrt{\frac{E}{m}}\)	recognized infrule but not yet supported
3	ID: 111782; drop non-dominant term number of inputs: 1; feeds: 1; outputs: 1 Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}	4560648264 \(v=\sqrt{ \frac{K + (4/3) G}{\rho} }\)	9674924517 \(K >> G\)	6504442697 \(v=\sqrt{ \frac{K}{\rho} }\)	recognized infrule but not yet supported
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}.	9640720571 \(v=\frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\) 5838268428 \(\alpha c=\frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)		5789289057 \(v=\alpha c \sqrt{ \frac{m_e}{2 m} }\)	LHS diff is pdg0001370pdg0004567 - pdg0002077 RHS diff is -sqrt(2)pdg0001370pdg0004567sqrt(pdg0002515/pdg0009863)/2 + pdg0001999*2/(4pdg0001054pdg0003141pdg0007940)