Help := proc()

print(` Test() `);
print(``);

end:

Test := proc() local st, AU, AD, AUUS, ADDS, S, i, flag, a;

st := [{[0]}, {[0]}];
AU  := [{[0]},{st}];
AD  := [{st},{[0]}];
AUUS := [{[0]},{AU}];
ADDS := [{AD},{[0]}];
flag := 1;
S := [0$63];

# Surreal number
S[1] := VG([{[-5]},{}]) = [0];

# some simple cases of VG
S[2]:= VG([{st},{st}]) =[0];
S[3]:= VG([{st},{}]) =[0]; 
S[4]:= VG([{},{st}]) = [0];
S[5]:= VG([{},{AU}]) =[0]; 
S[6]:= VG([{AD},{}]) =[0];
S[7]:= VG([{AU},{AD}]) = st;
S[8]:= VG([{AU},{[0]}]) =st ;
S[9]:= VG([{[0]},{AD}]) = st ;
S[10]:= VG([{st},{[1]}]) = [0];

#Bypass Rule
S[11]:= VG([{[3]},{[{[5]},{[1]}]}]) =[4];
S[12]:= VG([{[2]},{[{[{[2]},{[2]}]},{[1]}]}]) =[{[2]},{[2]}];
S[13]:= VG([{[{[5]},{[5]}]},{}]) =[5];
S[14]:= VG([{},{[{[3]},{[3]}]}]) =[0];

#page 15,65
S[15]:= ToFr([F,B,B,T]) = [0];
S[16]:= ToFr([F,B,T,B]) = [1];
S[17]:= ToFr([F,T,B,B]) = [2]; 
S[18]:= ToFr([T,F,B,B]) = [1];  
S[19]:= ToFr([B,T,B,F]) = [-1/2];
S[20]:= ToFr([T,B,B,F]) = [0]; 
S[21]:= ToFr([T,T,B,F,F]) = st;

#page 128,129
S[22]:= ToFr([T,T,F,B,F,F]) = [{st},{[-1]}]; 
S[23]:= ToFr([B,F,T,T,B,F]) = [{[-1/2]},{[-1]}];
S[24]:= ToFr([T,F,T,B,F,F]) = [{[0]},{[-1/4]}];
S[25]:= ToFr([T,T,B,F,F,B]) = [{[1/4]},{AD}];
S[26]:= ToFr([F,T,T,B,T,F]) = [{[1]},{[1]}];
S[27]:= ToFr([T,B,T,F,F,T]) = [{[0]},{st}];
S[28]:= ToFr([F,T,B,F,F,F]) = [{[0]},{[{[-1]},{[-1]}]}];
S[29]:= ToFr([T,B,F,B,F,T]) = [{[-1]},{[-1]}];
S[30]:= ToFr([T,T,T,B,T,F]) = [{[2]},{[1]}];
S[31]:= ToFr([T,B,T,F,F,F]) = [{[0]},{[{[0]},{[-1/4]}]}];
S[32]:= ToFr([T,F,T,B,F,F]) = [{[0]},{[-1/4]}];

#page 133
S[33]:= ToFr([T,T,B,T,F,F]) = [{[1]},{[{[0]},{[0]}]}];
S[34]:= ToFr([T,T,T,F,B,F]) = [{[{[1/4]},{[0]}]},{[0]}];
S[35]:= ToFr([T,T,T,B,F,F]) = st;

#page 134
S[36]:= ToFr([T,T,T,B,F,T,F]) = [{[{[1]},{[1/2]}]},{[0]}];
S[37]:= ToFr([T,T,T,T,B,F,F]) = st;
S[38]:= ToFr([T,T,F,T,F,B,F]) = ADDS;
S[39]:= ToFr([T,T,T,B,F,F,F]) = st;

#page 135
S[40]:= ToFr([T,B,F,T,B]) = [3/2];
S[41]:= ToFr([B,T,T,B,F]) = [{[1/2]},{[0]}];
S[42]:= ToFr([T,T,B,B,F]) = [1];
S[43]:= ToFr([B,T,T,B,F,F]) = [{AU},{[-1/4]}];
S[44]:= ToFr([T,B,F,F,T,B]) = [{[1]},{[1/2]}];
S[45]:= ToFr([T,T,B,B,F,F]) = st;

#page 136
S[46]:= ToFr([T,B,T,T,F,B]) = [1];
S[47]:= ToFr([T,T,T,B,F,B]) = [5/2];
S[48]:= ToFr([T,T,T,B,B,F]) = [2];
##seq(seq(ToFr([T$a,B,B,F$b]),a=3),b=3);
S[49]:= ToFr([T,T,T,B,B,F,F,F]) = [{[1/8]},{[-1/8]}];

##***************************************************##
#Erickson section3
#3.1
S[50]:= ToFr([T,F,T,B,T,F,T,F]) = [1/4];

#3.3
S[51]:= 
op({seq(evalb(ToFr([T,F,B,F,B$(a+3)])=[{[a]},{[0]}]),a=0..4)});  

#3.5
S[52]:= 
op({seq(evalb(ToFr([T,B,T,F,B,F,B$a])=[{[a+1/2]},{[1]}]),a=6..9)});

#Erickson section4
#4.1
S[53]:= op({seq(evalb(ToFr([B$2,T,F,B$b])=[b-2-1]),b=4..8)});

#4.2
S[54]:= op({seq(evalb(ToFr([B$2,T,F,T,F,B$b])=[2*(b-2)-1]),b=3..4)});
S[55]:= op({seq(evalb(ToFr([B$3,T,F,T,F,B$b])=[b-1]),b=6..9)});

#4.3
S[56]:= op({seq(evalb(ToFr([B$6,T,T,F,B$b])=[0]),b=0..3)});

#4.5
S[57]:= op({seq(seq(evalb(ToFr([T$a,F,B$b])=[a*(b-1)]),a=0..3),b=1..3)});

#Erickson section5
#5.2
S[58]:= 
op({seq(evalb(ToFr([T$a,B$3,F])=[{[(a-1)*(3-1)]},{[(a-1)*(3-1)]}]),a=1..5)});
S[59]:= op({seq(evalb(ToFr([T$a,B$4,F])=[(a-1)*(4-1)]),a=1..3)});

#Erickson section6
#6.3
S[60]:= op({seq(evalb(ToFr([T$a,B,F,B,F])=[0]),a=2..5)});

#Erickson section7
#7.1
S[61]:= 
op({seq(seq(evalb(ToFr([T$a,B,B,F$b])=[{VG([{[a-3]},{[a-b]}])},
{VG([{st},{[3-b]}])}]),a=b+1..b+3),b=2..3)});

#7.2
S[62]:= op({seq(evalb(ToFr([T$a,B,B,B,F,F])=[{[a-2]},{[a-2]}]),a=2..3)});

#7.3
S[63]:= op({seq(evalb(ToFr([T$a,B,B,B,F,F,F])=[a-7/2]),a=5..5)});

for i from 1 to nops(S) do
     
  if not(evalb(S[i])) then
    print(i,`is not the same`);
    flag :=0;   
  fi:
   
od:

`if` (flag = 1, print(`Things in first part are ok.`),NULL); 

S := [0$15];

#A2
S[1] := SVG([{[-1]},{}])=[0];
assume(a[1]>=2);
S[2] := SVG([{[{[{[a[1]-3]},{[1]}]},{[0]}]},{}])=[0];
#A31
assume(a[1]>=1);
S[3] := SVG([{[a[1]-1/2]},{}])=[a[1]];
assume(a[1]>=1);
S[4] := SVG([{[a[1]-1]},{[a[1]+2]}])=[a[1]];
assume(a[1]>=1);
S[5] := SVG([{[a[1]-1],[{[a[1]]},{[a[1]]}]},{}])=[a[1]];
assume(a[1]>=1, a[2]>=3);
S[6]:= 
SVG([{[a[1]-1],[{[{[2*a[1]+a[2]-3]},{[a[1]+1]}]},{[a[1]]}]},{}])=[a[1]];
assume(a[1]>=1);
S[7] := SVG([{[{[a[1]-1]},{[a[1]-1]}]},{[a[1]-1/2]}])=[a[1]-1];
assume(a[1]>=1);
S[8] := SVG([{[{[{[2*a[1]-2]},{[a[1]]}]},{[a[1]-1]}],[a[1]]},{[a[1]]}])
             =[{[a[1]]},{[a[1]]}];
assume(a[1]>=1, a[2]>=2);
S[9] := 
SVG([{[{[{[2*a[1]+a[2]-3]},{[a[1]]}]},{[a[1]-1]}],[{[2*a[1]+a[2]-2]}
        ,{[a[1]+1]}]},{[a[1]]}]) =  
[{[{[2*a[1]+a[2]-2]},{[a[1]+1]}]},{[a[1]]}] ;
assume(a[1]>=1, a[2]>=1);
S[10] := SVG([{[{[2*a[1]+a[2]-2]},{[a[1]]}],[2*a[1]+a[2]-1]},{[a[1]+1]}])
         =[{[2*a[1]+a[2]-1]},{[a[1]+1]}];

#A32
assume(a[1]>=1);
S[11] := SVG([{[{[{[{[a[1]-3]},[1/2]]},{[0]}]},{[0]}]},{}])=[0];
assume(a[1]>=2);
S[12] := SVG([{[{[{[a[1]-3]},{[1]}]},{[0]}]},{[1]}])=[0];
assume(a[1]>=2);
S[13] := 
SVG([{[{[a[1]-2]},{[{[0]},{[0]}]}]},{[{[{[{[a[1]-3]},{[1/2]}]},{[0]}]},
                      {[0]}]}])= [{[0]},{[0]}];

#B2
assume(a[1]>=0,a[2]>=0);
S[14] := 
SVG([{[{[{[a[2]-a[1]-4]},{[a[2]-a[1]]}]},{[{[a[2]-a[1]-2]},{[a[2]-a[1]+2]}]}]},
                  
{[{[{[a[2]-a[1]-2]},{[a[2]-a[1]+2]}]},{[{[a[2]-a[1]]},{[a[2]-a[1]+4]}]}]}])
		    = [{[a[2]-a[1]-2]},{[a[2]-a[1]+2]}];

#A41
assume(a[1]>=1, a[2]>=2);
S[15] := SVG([{[2*a[1]+a[2]-2],[{[2*a[1]+a[2]-1]},{[2*a[1]+a[2]-1]}]},
              {[{[2*a[1]+a[2]-1]},{[2*a[1]+a[2]-1]}]}])=[2*a[1]+a[2]-1];

for i from 1 to nops(S) do
   if evalb(S[i]) = false then
       print(i, " is bad"); 
   fi:
od:

print(` Everthing in part 2) looks good. End of testing.`);
return();
end: