Pensieve header: Experiments with tder.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2013-04"];
<< CheatSheetJ-Verification.m;
$SeriesCompareDegree=10;
{α=MakeLieSeries[u], γ=MakeLieSeries[u+v+b[u,v]]}

lhs= Module[{αk},(αk=α)+(∑_(k=1)^10)(((αk=(αk // (Subscript[ad, u])[γ])))/(k!))];
lhs@{6}

t[1]=γ;
t[k_] /; k>1 := t[k]=t[k-1] // (Subscript[ad, u])[γ];
Table[t[k]@k, {k,7}]

β[1]=t[1]

(α // (Subscript[CC, u])[β[1]])-lhs

β[2]=β[1]+(1/2)t[2];
((α // (Subscript[CC, u])[β[2]])-lhs)≡MakeLieSeries[0]

β[3]=β[2]+a t[3]+b b[t[1], t[2]];
e[3]=Collect[First[((α // (Subscript[CC, u])[β[3]])-lhs)≡MakeLieSeries[0]], _LW]

β[3]

β[3]=β[2]+(1/6) t[3]-(1/12) b[t[1], t[2]];
e[3]=Collect[First[((α // (Subscript[CC, u])[β[3]])-lhs)≡MakeLieSeries[0]], _LW]

β[4]=β[3]+a t[4]+b b[t[1], t[3]];
e[4]=Collect[First[((α // (Subscript[CC, u])[β[4]])-lhs)≡MakeLieSeries[0]], _LW]

β[4]=β[3]+(1/24) t[4]-(1/24) b[t[1], t[3]];
e[4]=Collect[First[((α // (Subscript[CC, u])[β[4]])-lhs)≡MakeLieSeries[0]], _LW]

β[5]=β[4]+(1/(5!))(a t[5]+b b[t[1], t[4]]+ c b[t[2],t[3]]+d b[t[1],b[t[1],t[3]]]+e b[t[1],b[t[1],b[t[1], t[2]]]]+f b[t[2], b[t[1], t[2]]]);
e[5]=Collect[6! First[((α // (Subscript[CC, u])[β[5]])-lhs)≡MakeLieSeries[0]], _LW, Expand]

β[5]=β[4]+(1/(5!))(t[5]-(3/2)b[t[1], t[4]]-b[t[2],t[3]]+(1/6) b[t[1],b[t[1],t[3]]]+(1/6) b[t[1],b[t[1],b[t[1], t[2]]]]-(1/2) b[t[2], b[t[1], t[2]]]);
e[5]=Collect[6! First[((α // (Subscript[CC, u])[β[5]])-lhs)≡MakeLieSeries[0]], _LW, Expand]