MAYBE

Problem:
 app(app(and(),true()),true()) -> true()
 app(app(and(),true()),false()) -> false()
 app(app(and(),false()),true()) -> false()
 app(app(and(),false()),false()) -> false()
 app(app(or(),true()),true()) -> true()
 app(app(or(),true()),false()) -> true()
 app(app(or(),false()),true()) -> true()
 app(app(or(),false()),false()) -> false()
 app(app(forall(),p),nil()) -> true()
 app(app(forall(),p),app(app(cons(),x),xs)) -> app(app(and(),app(p,x)),app(app(forall(),p),xs))
 app(app(forsome(),p),nil()) -> false()
 app(app(forsome(),p),app(app(cons(),x),xs)) -> app(app(or(),app(p,x)),app(app(forsome(),p),xs))

Proof:
 DP Processor:
  DPs:
   app#(app(forall(),p),app(app(cons(),x),xs)) -> app#(app(forall(),p),xs)
   app#(app(forall(),p),app(app(cons(),x),xs)) -> app#(p,x)
   app#(app(forall(),p),app(app(cons(),x),xs)) -> app#(and(),app(p,x))
   app#(app(forall(),p),app(app(cons(),x),xs)) -> app#(app(and(),app(p,x)),app(app(forall(),p),xs))
   app#(app(forsome(),p),app(app(cons(),x),xs)) -> app#(app(forsome(),p),xs)
   app#(app(forsome(),p),app(app(cons(),x),xs)) -> app#(p,x)
   app#(app(forsome(),p),app(app(cons(),x),xs)) -> app#(or(),app(p,x))
   app#(app(forsome(),p),app(app(cons(),x),xs)) ->
   app#(app(or(),app(p,x)),app(app(forsome(),p),xs))
  TRS:
   app(app(and(),true()),true()) -> true()
   app(app(and(),true()),false()) -> false()
   app(app(and(),false()),true()) -> false()
   app(app(and(),false()),false()) -> false()
   app(app(or(),true()),true()) -> true()
   app(app(or(),true()),false()) -> true()
   app(app(or(),false()),true()) -> true()
   app(app(or(),false()),false()) -> false()
   app(app(forall(),p),nil()) -> true()
   app(app(forall(),p),app(app(cons(),x),xs)) -> app(app(and(),app(p,x)),app(app(forall(),p),xs))
   app(app(forsome(),p),nil()) -> false()
   app(app(forsome(),p),app(app(cons(),x),xs)) -> app(app(or(),app(p,x)),app(app(forsome(),p),xs))
  Open