FUNCTION :  findtype4 - find type 4 identities
                

CALLING SEQUENCE :  findtype4(T)
                    

PARAMETERS :   T - positive integer  
                  -            

GLOBAL VARIABLES : 
        xprint,NEWJACID,RJID,SYMJID,
        TT1,TT2,PROVEDFL1,EBL

SYNOPSIS :   
        Before running the functions G,H,GM,HM,GE,HE must be defined.   

        findtype4(T) cycles through symbolic expressions
        
           _GM(a)_HM(b) + c _GM(b)_HM(a)
        
        where  2 ≤ n ≤ T, ab=n, (a,b)=1,  b < a, c in {-1,1},  and
        
        (*)    GE(a) + HE(b) - (GE(b) + HE(a))  in Z

        and least one of a, b is even,
        using CHECKRAMIDF to check whether the expression corresponds to a 
        likely eta-product.
        If proveit is true then provemodfuncidBATCH (from theataids 
        package) is used to prove it. Condition (*) eliminates the case of fractional 
        powers of q.  The procedure also returns a list of [a,b,c] which 
        give identities.
        NOTE: Output should be assigned myramtype4.

EXAMPLES :   

>  with(qseries):
>  with(thetaids):
>  with(ramarobinsids):
>  xprint:=false: proveit:=true:
>  G:=j->1/GetaL([1],5,j):H:=j->1/GetaL([2],5,j):
>  GM:=j->1/MGetaL([1],5,j): HM:=j->1/MGetaL([2],5,j):
>  GE:=j->-GetaLEXP([1],5,j):HE:=j->-GetaLEXP([2],5,j):
>  findtype4(6);
"n=", 5
*** There were NO errors.  Each term was modular function on
    Gamma1(120). Also -mintotord=128. To prove the identity
    we need to  check up to O(q^(130)).
    To be on the safe side we check up to O(q^(368)).
*** The identity below is PROVED!
[6, 1, -1]
        _GM(6) _HM(1) - _GM(1) _HM(6) = 

                                 3           3          
           eta(24 tau) eta(6 tau)  eta(4 tau)  eta(tau) 
          ----------------------------------------------
                     3                                 3
          eta(12 tau)  eta(8 tau) eta(3 tau) eta(2 tau) 
                          [[6, 1, -1]]


DISCUSSION :
    For G,H defined G*(6) H*(1) - G*(1) H*(6) is an eta-product and
    the identity is proved.

SEE ALSO :  

findtype4, findtype2,
findtype3, findtype4,
findtype5, findtype6,
findtype7, findtype8,
findtype9, findtype40