MAYBE

Problem:
 0(#()) -> #()
 +(x,#()) -> x
 +(#(),x) -> x
 +(0(x),0(y)) -> 0(+(x,y))
 +(0(x),1(y)) -> 1(+(x,y))
 +(1(x),0(y)) -> 1(+(x,y))
 +(1(x),1(y)) -> 0(+(+(x,y),1(#())))
 +(+(x,y),z) -> +(x,+(y,z))
 -(#(),x) -> #()
 -(x,#()) -> x
 -(0(x),0(y)) -> 0(-(x,y))
 -(0(x),1(y)) -> 1(-(-(x,y),1(#())))
 -(1(x),0(y)) -> 1(-(x,y))
 -(1(x),1(y)) -> 0(-(x,y))
 not(true()) -> false()
 not(false()) -> true()
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 eq(#(),#()) -> true()
 eq(#(),1(y)) -> false()
 eq(1(x),#()) -> false()
 eq(#(),0(y)) -> eq(#(),y)
 eq(0(x),#()) -> eq(x,#())
 eq(1(x),1(y)) -> eq(x,y)
 eq(0(x),1(y)) -> false()
 eq(1(x),0(y)) -> false()
 eq(0(x),0(y)) -> eq(x,y)
 ge(0(x),0(y)) -> ge(x,y)
 ge(0(x),1(y)) -> not(ge(y,x))
 ge(1(x),0(y)) -> ge(x,y)
 ge(1(x),1(y)) -> ge(x,y)
 ge(x,#()) -> true()
 ge(#(),0(x)) -> ge(#(),x)
 ge(#(),1(x)) -> false()
 log(x) -> -(log'(x),1(#()))
 log'(#()) -> #()
 log'(1(x)) -> +(log'(x),1(#()))
 log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#())
 *(#(),x) -> #()
 *(0(x),y) -> 0(*(x,y))
 *(1(x),y) -> +(0(*(x,y)),y)
 *(*(x,y),z) -> *(x,*(y,z))
 *(x,+(y,z)) -> +(*(x,y),*(x,z))
 app(nil(),l) -> l
 app(cons(x,l1),l2) -> cons(x,app(l1,l2))
 sum(nil()) -> 0(#())
 sum(cons(x,l)) -> +(x,sum(l))
 sum(app(l1,l2)) -> +(sum(l1),sum(l2))
 prod(nil()) -> 1(#())
 prod(cons(x,l)) -> *(x,prod(l))
 prod(app(l1,l2)) -> *(prod(l1),prod(l2))
 mem(x,nil()) -> false()
 mem(x,cons(y,l)) -> if(eq(x,y),true(),mem(x,l))
 inter(x,nil()) -> nil()
 inter(nil(),x) -> nil()
 inter(app(l1,l2),l3) -> app(inter(l1,l3),inter(l2,l3))
 inter(l1,app(l2,l3)) -> app(inter(l1,l2),inter(l1,l3))
 inter(cons(x,l1),l2) -> ifinter(mem(x,l2),x,l1,l2)
 inter(l1,cons(x,l2)) -> ifinter(mem(x,l1),x,l2,l1)
 ifinter(true(),x,l1,l2) -> cons(x,inter(l1,l2))
 ifinter(false(),x,l1,l2) -> inter(l1,l2)

Proof:
 Open