Pensieve Header: A 2D B-Picture.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\w-Computations"]

ar[i_,j_] := t[i]h[j]

HMultiply[x_,y_,z_][mix_] := Module[
{cox, coy,jx,jy,jxy},
cox = D[mix, h[x]] /. t[s_] :> c[s];
coy = D[mix, h[y]] /. t[s_] :> c[s];
jx=If[cox===0, 1, (((-1+(E^cox)))/cox)];
jy=If[coy===0, 1, (((-1+(E^coy)))/coy)];
jxy=If[cox+coy===0, 1, (((-1+(E^(cox+coy))))/(cox+coy))];
(mix /. {h[x]->0, h[y]->0})+ ((jx h[z]D[mix, h[x]])/jxy) + (E^cox)((jy h[z]D[mix, h[y]])/jxy) 
]

BDisplay[expr_] := Collect[expr, _h, Collect[#, _t, FullSimplify]&]

HMultiply[3,4,5][t[1]h[3]+t[2]h[4]] // BDisplay

a=t[1]h[4]+t[2]h[5]+t[3]h[6]

HMultiply[4,5,7][a] // BDisplay

Expand2[expr_] := Series[expr /. {c[i_]:> z c[i], t[i_] :> z t[i]}, {z,0,2}]

HMultiply[4,5,7][a] // Expand2

HMultiply[7,6,8][HMultiply[4,5,7][a]] // BDisplay

HMultiply[4,7,8][HMultiply[5,6,7][a]] // BDisplay

(HMultiply[7,6,8][HMultiply[4,5,7][a]] - HMultiply[4,7,8][HMultiply[5,6,7][a]]) // BDisplay

From B-Side Operations, II:

THConjugate[y_,x_][mix_] := Module[
{cox, coy, γ, δ, rest, jx},
cox = D[mix, h[x]] /. t[y] -> 0;
coy = D[mix, t[y]] /. h[x] -> 0;
γ = Coefficient[mix, t[y]h[x]];
δ = γ c[y]+(cox /. t->c);
rest = mix /. t[y]->0;
coy ((((E^δ)-1)cox c[y]+t[y]δ)/((E^δ)(cox /. t->c)+γ c[y]))+γ c[y](((1-(E^(-δ))))/δ)W+γ t[y]h[x]+rest
];
HTConjugate[x_, y_][mix_] := ((mix /. h[x] -> -h[x]) // THConjugate[y,x]) /. h[x] -> -h[x]

THConjugate[y,x][t[y]h[1]+t[1]h[x]] // BDisplay

a1=ar[1,3]+ar[4,2]

a2 = a1 // HTConjugate[3,4] // HTConjugate[2,4] // HMultiply[2,3,2]//BDisplay

a3 = a1 // HMultiply[3,2,2] // HTConjugate[2,4]// BDisplay

a2==a3

ComposeList[{HTConjugate[3,4], HTConjugate[2,4], HMultiply[2,3,2]}, a1] // BDisplay // MatrixForm

ComposeList[{HMultiply[3,2,2], HTConjugate[2,4]}, a1] // BDisplay // MatrixForm

ComposeList[{HMultiply[2,3,2], HTConjugate[2,4]}, a1] // BDisplay // MatrixForm

b1=ar[1,2]+ar[4,3]

b2 = b1 // HTConjugate[3,4] // HTConjugate[2,4] // HMultiply[3,2,2]//BDisplay

b3 = b1 // HMultiply[2,3,2] // HTConjugate[2,4]// BDisplay

FullSimplify[b2==b3]