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{"type":"node","id":"2973","labels":["a_node","relation"],"properties":{"name_latex":"proportional to","reference_latex":"","author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","description_latex":"","id":"0002179795","latex":"\\propto","created_datetime":"2026-02-03_20-10-42-459034"}}
{"type":"node","id":"2974","labels":["a_node","inference_rule"],"properties":{"name_latex":"LHS of expr 1 equals LHS of expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111355","latex":"LHS of Eq.~\\ref{eq:#1} is equal to LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2975","labels":["a_node","inference_rule"],"properties":{"name_latex":"RHS of expr 1 equals RHS of expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111863","latex":"RHS of Eq.~\\ref{eq:#1} is equal to RHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2976","labels":["a_node","inference_rule"],"properties":{"name_latex":"X cross both sides by","assumptions_latex":"linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111469","latex":"Take cross product of $#1$ and Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2977","labels":["a_node","inference_rule"],"properties":{"name_latex":"X dot both sides","assumptions_latex":"linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111145","latex":"Take inner product of $#1$ with Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2978","labels":["a_node","inference_rule"],"properties":{"name_latex":"add X to both sides","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111530","latex":"Add $#1$ to both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2979","labels":["a_node","inference_rule"],"properties":{"name_latex":"add expr 1 to expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111980","latex":"Add Eq.~\\ref{eq:#1} to Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"2980","labels":["a_node","inference_rule"],"properties":{"name_latex":"add zero to LHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111242","latex":"Add zero to LHS of Eq.~\\ref{eq:#2}, where $0=#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2981","labels":["a_node","inference_rule"],"properties":{"name_latex":"add zero to RHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111717","latex":"Add zero to RHS of Eq.~\\ref{eq:#2}, where $0=#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"2982","labels":["a_node","inference_rule"],"properties":{"name_latex":"apply divergence","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111463","latex":"Apply divergence to both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"2983","labels":["a_node","inference_rule"],"properties":{"name_latex":"apply function to both sides of expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":2,"notes_latex":"Example: apply sin(x) to 'a = b + c'","id":"0000111490","latex":"Apply function $#1$ with argument $#2$ to Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}"}}
{"type":"node","id":"2984","labels":["a_node","inference_rule"],"properties":{"name_latex":"apply gradient to scalar function","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111531","latex":"Apply gradient to both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"2985","labels":["a_node","inference_rule"],"properties":{"name_latex":"apply operator to bra","assumptions_latex":"quantum: Dirac notation","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111390","latex":"Apply operator in Eq.~\\ref{eq:#1} to bra; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"2986","labels":["a_node","inference_rule"],"properties":{"name_latex":"apply operator to ket","assumptions_latex":"quantum: Dirac notation","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111946","latex":"Apply operator in Eq.~\\ref{eq:#1} to ket; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"2987","labels":["a_node","inference_rule"],"properties":{"name_latex":"assume N dimensions","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":0,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111791","latex":"Assume $#1$ dimensions; decompose vector to be Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"2988","labels":["a_node","inference_rule"],"properties":{"name_latex":"both sides cross X","assumptions_latex":"linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111368","latex":"Take cross product of Eq.~\\ref{eq:#2} and $#1$; yields Eq.~\\ref{eq:#3}"}}
{"type":"node","id":"2989","labels":["a_node","inference_rule"],"properties":{"name_latex":"both sides dot X","assumptions_latex":"linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111567","latex":"Take inner product of Eq.~\\ref{eq:#2} with $#1$; yields Eq.~\\ref{eq:#3}"}}
{"type":"node","id":"2990","labels":["a_node","inference_rule"],"properties":{"name_latex":"boundary condition","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111278","latex":"Boundary condition: Eq.~\\ref{eq:#2} when Eq.~\\ref{eq#1}."}}
{"type":"node","id":"2991","labels":["a_node","inference_rule"],"properties":{"name_latex":"boundary condition for expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111802","latex":"A boundary condition for Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2}"}}
{"type":"node","id":"2992","labels":["a_node","inference_rule"],"properties":{"name_latex":"change five variables in expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":10,"notes_latex":"","id":"0000111410","latex":"Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ and $#9$ to $#10$ in Eq.~\\ref{eq:#11}; yields Eq.~\\ref{eq:#12}."}}
{"type":"node","id":"2993","labels":["a_node","inference_rule"],"properties":{"name_latex":"change four variables in expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":8,"notes_latex":"","id":"0000111777","latex":"Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ in Eq.~\\ref{eq:#9}; yields Eq.~\\ref{eq:#10}."}}
{"type":"node","id":"2994","labels":["a_node","inference_rule"],"properties":{"name_latex":"change six variables in expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":12,"notes_latex":"","id":"0000111471","latex":"Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ and $#7$ to $#8$ and $#9$ to $#10$ and $#11$ to $#12$ in Eq.~\\ref{eq:#13}; yields Eq.~\\ref{eq:#14}."}}
{"type":"node","id":"2995","labels":["a_node","inference_rule"],"properties":{"name_latex":"change three variables in expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":6,"notes_latex":"","id":"0000111236","latex":"Change of variable $#1$ to $#2$ and $#3$ to $#4$ and $#5$ to $#6$ in Eq.~\\ref{eq:#7}; yields Eq.~\\ref{eq:#8}."}}
{"type":"node","id":"2996","labels":["a_node","inference_rule"],"properties":{"name_latex":"change two variables in expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":4,"notes_latex":"","id":"0000111984","latex":"Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}."}}
{"type":"node","id":"2997","labels":["a_node","inference_rule"],"properties":{"name_latex":"change variable X to Y","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":2,"notes_latex":"","id":"0000111886","latex":"Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}."}}
{"type":"node","id":"2998","labels":["a_node","inference_rule"],"properties":{"name_latex":"claim LHS equals RHS","assumptions_latex":"","number_of_outputs":0,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111345","latex":"Thus we see that LHS of Eq.~\\ref{eq:#1} is equal to RHS."}}
{"type":"node","id":"2999","labels":["a_node","inference_rule"],"properties":{"name_latex":"claim expr 1 equals expr 2","assumptions_latex":"","number_of_outputs":0,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111550","latex":"Thus we see that Eq.~\\ref{eq:#1} is equivalent to Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3000","labels":["a_node","inference_rule"],"properties":{"name_latex":"combine like terms","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111728","latex":"Combine like terms in Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3001","labels":["a_node","inference_rule"],"properties":{"name_latex":"conjugate both sides","assumptions_latex":"complex values","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111484","latex":"Conjugate both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3002","labels":["a_node","inference_rule"],"properties":{"name_latex":"conjugate function X","assumptions_latex":"complex values","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111996","latex":"Conjugate $#1$ in Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3003","labels":["a_node","inference_rule"],"properties":{"name_latex":"conjugate transpose both sides","assumptions_latex":"complex valued linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111696","latex":"Conjugate transpose of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3004","labels":["a_node","inference_rule"],"properties":{"name_latex":"declare assumption","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":0,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111104","latex":"Eq.~\\ref{eq:#1} is an assumption."}}
{"type":"node","id":"3005","labels":["a_node","inference_rule"],"properties":{"name_latex":"declare final expression","assumptions_latex":"","number_of_outputs":0,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111341","latex":"Eq.~\\ref{eq:#1} is one of the final equations."}}
{"type":"node","id":"3006","labels":["a_node","inference_rule"],"properties":{"name_latex":"declare guess solution","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"https://en.wikipedia.org/wiki/Ansatz","id":"0000111237","latex":"Judicious choice as a guessed solution to Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2},"}}
{"type":"node","id":"3007","labels":["a_node","inference_rule"],"properties":{"name_latex":"declare identity","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":0,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111299","latex":"Eq.~\\ref{eq:#1} is an identity."}}
{"type":"node","id":"3008","labels":["a_node","inference_rule"],"properties":{"name_latex":"declare initial expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":0,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111981","latex":"Eq.~\\ref{eq:#1} is an initial equation."}}
{"type":"node","id":"3009","labels":["a_node","inference_rule"],"properties":{"name_latex":"differentiate with respect to","assumptions_latex":"differential equations","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111649","latex":"Differentiate Eq.~\\ref{eq:#2} with respect to $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3010","labels":["a_node","inference_rule"],"properties":{"name_latex":"distribute conjugate to factors","assumptions_latex":"complex values","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111474","latex":"Distribute conjugate to factors in Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3011","labels":["a_node","inference_rule"],"properties":{"name_latex":"distribute conjugate transpose to factors","assumptions_latex":"complex valued linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111890","latex":"Distribute conjugate transpose to factors in Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3012","labels":["a_node","inference_rule"],"properties":{"name_latex":"divide both sides by","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111975","latex":"Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3013","labels":["a_node","inference_rule"],"properties":{"name_latex":"divide expr 1 by expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111421","latex":"Divide Eq.~\\ref{eq:#1} by Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3014","labels":["a_node","inference_rule"],"properties":{"name_latex":"drop non-dominant term","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111782","latex":"Based on the assumption $#1$, drop non-dominant term in Eq.~\\ref{#2}; yeilds Eq.~\\ref{#3}"}}
{"type":"node","id":"3015","labels":["a_node","inference_rule"],"properties":{"name_latex":"evaluate definite integral","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111662","latex":"Evaluate definite integral Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3016","labels":["a_node","inference_rule"],"properties":{"name_latex":"expand LHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111561","latex":"Expand the LHS of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3017","labels":["a_node","inference_rule"],"properties":{"name_latex":"expand RHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111546","latex":"Expand the RHS of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3018","labels":["a_node","inference_rule"],"properties":{"name_latex":"expand integrand","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111581","latex":"Expand integrand of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3019","labels":["a_node","inference_rule"],"properties":{"name_latex":"expand magnitude to conjugate","assumptions_latex":"complex values","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111166","latex":"Expand $#1$ in Eq.~\\ref{eq:#2} with conjugate; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3020","labels":["a_node","inference_rule"],"properties":{"name_latex":"expr 1 is equivalent to expr 2 under the condition","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111768","latex":"Eq.~\\ref{eq:#1} is equivalent to Eq.~\\ref{eq:#2} under the condition in Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3021","labels":["a_node","inference_rule"],"properties":{"name_latex":"expr 1 is true under condition expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111698","latex":"Eq.~\\ref{eq:#1} is valid when Eq.~\\ref{eq:#2} occurs; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3022","labels":["a_node","inference_rule"],"properties":{"name_latex":"factor out X","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111432","latex":"Factor $#1$ from Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3023","labels":["a_node","inference_rule"],"properties":{"name_latex":"factor out X from LHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111613","latex":"Factor $#1$ from the LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3024","labels":["a_node","inference_rule"],"properties":{"name_latex":"factor out X from RHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111260","latex":"Factor $#1$ from the RHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3025","labels":["a_node","inference_rule"],"properties":{"name_latex":"function is even","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":3,"notes_latex":"","id":"0000111329","latex":"$#1$ is even with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}."}}
{"type":"node","id":"3026","labels":["a_node","inference_rule"],"properties":{"name_latex":"function is odd","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":3,"notes_latex":"","id":"0000111522","latex":"$#1$ is odd with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}."}}
{"type":"node","id":"3027","labels":["a_node","inference_rule"],"properties":{"name_latex":"indefinite integral over","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111344","latex":"Indefinite integral of both sides of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3028","labels":["a_node","inference_rule"],"properties":{"name_latex":"indefinite integrate LHS over","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111137","latex":"Indefinite integral of LHS of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3029","labels":["a_node","inference_rule"],"properties":{"name_latex":"indefinite integrate RHS over","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111132","latex":"Indefinite integral of RHS of Eq.~\\ref{eq:#2} over $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3030","labels":["a_node","inference_rule"],"properties":{"name_latex":"indefinite integration","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111608","latex":"Indefinite integral of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3031","labels":["a_node","inference_rule"],"properties":{"name_latex":"integrate","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"not clear how to specify limits for an arbitrary number of integrals","id":"0000111408","latex":"Integrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}."}}
{"type":"node","id":"3032","labels":["a_node","inference_rule"],"properties":{"name_latex":"integrate over from to","assumptions_latex":"multi-variable calculus","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":3,"notes_latex":"","id":"0000111937","latex":"Integrate Eq.~\\ref{eq:#4} over $#1$ from lower limit $#2$ to upper limit $#3$; yields Eq.~\\ref{eq:#5}."}}
{"type":"node","id":"3033","labels":["a_node","inference_rule"],"properties":{"name_latex":"make expr power","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111721","latex":"Make Eq.~\\ref{eq:#2} the power of $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3034","labels":["a_node","inference_rule"],"properties":{"name_latex":"maximum of expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111773","latex":"The maximum of Eq.~\\ref{eq:#2} with respect to $#1$ is Eq.~\\ref{eq:#3}"}}
{"type":"node","id":"3035","labels":["a_node","inference_rule"],"properties":{"name_latex":"multiply LHS by unity","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111173","latex":"Multiply LHS of Eq.~\\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\\ref{eq:#3}"}}
{"type":"node","id":"3036","labels":["a_node","inference_rule"],"properties":{"name_latex":"multiply RHS by unity","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111646","latex":"Multiply RHS of Eq.~\\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\\ref{eq:#3}"}}
{"type":"node","id":"3037","labels":["a_node","inference_rule"],"properties":{"name_latex":"multiply both sides by","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111182","latex":"Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3038","labels":["a_node","inference_rule"],"properties":{"name_latex":"multiply expr 1 by expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111253","latex":"Multiply Eq.~\\ref{eq:#1} by Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3039","labels":["a_node","inference_rule"],"properties":{"name_latex":"normalization condition","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":0,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111493","latex":"Normalization condition is Eq.~\\ref{eq:#1}."}}
{"type":"node","id":"3040","labels":["a_node","inference_rule"],"properties":{"name_latex":"partially differentiate with respect to","assumptions_latex":"differential equations","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111680","latex":"Partially differentiate Eq.~\\ref{eq:#2} with respect to $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3041","labels":["a_node","inference_rule"],"properties":{"name_latex":"raise both sides to power","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111483","latex":"Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3042","labels":["a_node","inference_rule"],"properties":{"name_latex":"replace constant with value","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":3,"notes_latex":"","id":"0000111715","latex":"Replace constant $#1$ with value $#2$ and units $#3$ in Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}"}}
{"type":"node","id":"3043","labels":["a_node","inference_rule"],"properties":{"name_latex":"replace curl with LeviCevita summation contravariant","assumptions_latex":"linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111935","latex":"Replace curl in Eq.~\\ref{eq:#1} with Levi-Cevita contravariant; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3044","labels":["a_node","inference_rule"],"properties":{"name_latex":"replace scalar with vector","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111215","latex":"Replace scalar variables in Eq.~\\ref{eq:#1} with equivalent vector variables; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3045","labels":["a_node","inference_rule"],"properties":{"name_latex":"replace summation notation with vector notation","assumptions_latex":"linear algebra","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111894","latex":"Replace summation notation in Eq.~\\ref{eq:#1} with vector notation; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3046","labels":["a_node","inference_rule"],"properties":{"name_latex":"select imaginary parts","assumptions_latex":"complex values","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111891","latex":"Select imaginary parts of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3047","labels":["a_node","inference_rule"],"properties":{"name_latex":"select real parts","assumptions_latex":"complex values","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111198","latex":"Select real parts of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3048","labels":["a_node","inference_rule"],"properties":{"name_latex":"separate three vector components","assumptions_latex":"","number_of_outputs":3,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111552","latex":"Separate three vector components in Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3} and Eq.~\\ref{eq:#4}."}}
{"type":"node","id":"3049","labels":["a_node","inference_rule"],"properties":{"name_latex":"separate two vector components","assumptions_latex":"","number_of_outputs":2,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111270","latex":"Separate two vector components in Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}"}}
{"type":"node","id":"3050","labels":["a_node","inference_rule"],"properties":{"name_latex":"separate vector into two trigonometric ratios","assumptions_latex":"","number_of_outputs":2,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"geometry argument in 2D","id":"0000111295","latex":"Separate vector in Eq.~\\ref{eq:#2} into components related by angle $#1$; yields Eq.~\\ref{eq:#3} and Eq.~\\ref{eq:#4}."}}
{"type":"node","id":"3051","labels":["a_node","inference_rule"],"properties":{"name_latex":"simplify","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111457","latex":"Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3052","labels":["a_node","inference_rule"],"properties":{"name_latex":"solve for X","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111543","latex":"Solve Eq.~\\ref{eq:#2} for $#1$; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3053","labels":["a_node","inference_rule"],"properties":{"name_latex":"square root both sides","assumptions_latex":"","number_of_outputs":2,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"related to raise both sides to power","id":"0000111524","latex":"Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3054","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute LHS of expr 1 into expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111556","latex":"Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3055","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute LHS of five expressions into expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":6,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111351","latex":"Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} and LHS of Eq.~\\ref{eq:#4} and LHS of Eq.~\\ref{eq:#5} into Eq.~\\ref{eq:#6}; yields Eq.~\\ref{eq:#7}."}}
{"type":"node","id":"3056","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute LHS of four expressions into expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":5,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111797","latex":"Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} and LHS of Eq.~\\ref{eq:#4} into Eq.~\\ref{eq:#5}; yields Eq.~\\ref{eq:#6}."}}
{"type":"node","id":"3057","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute LHS of six expressions into expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":7,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111381","latex":"Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} and LHS of Eq.~\\ref{eq:#4} and LHS of Eq.~\\ref{eq:#5} and LHS of Eq.~\\ref{eq:#6} into Eq.~\\ref{eq:#7}; yields Eq.~\\ref{eq:#8}."}}
{"type":"node","id":"3058","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute LHS of three expressions into expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":4,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111246","latex":"Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} and LHS of Eq.~\\ref{eq:#3} into Eq.~\\ref{eq:#4}; yields Eq.~\\ref{eq:#5}."}}
{"type":"node","id":"3059","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute LHS of two expressions into expression","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":3,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111732","latex":"Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} into Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}."}}
{"type":"node","id":"3060","labels":["a_node","inference_rule"],"properties":{"name_latex":"substitute RHS of expr 1 into expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111634","latex":"Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3061","labels":["a_node","inference_rule"],"properties":{"name_latex":"subtract X from both sides","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":1,"notes_latex":"","id":"0000111282","latex":"Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}."}}
{"type":"node","id":"3062","labels":["a_node","inference_rule"],"properties":{"name_latex":"subtract expr 1 from expr 2","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":2,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"complement of 'add expr X to expr Y'","id":"0000111222","latex":"Subtract Eq.~\\ref{eq:#1} from Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3063","labels":["a_node","inference_rule"],"properties":{"name_latex":"sum exponents","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111373","latex":"Sum exponents on LHS and RHS of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
{"type":"node","id":"3064","labels":["a_node","inference_rule"],"properties":{"name_latex":"sum exponents LHS","assumptions_latex":"","number_of_outputs":1,"number_of_inputs":1,"author_name_latex":"a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba","number_of_feeds":0,"notes_latex":"","id":"0000111566","latex":"Sum exponents on LHS of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}."}}
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