MAYBE

Problem:
 app(app(*(),x),app(app(+(),y),z)) -> app(app(+(),app(app(*(),x),y)),app(app(*(),x),z))
 app(app(*(),app(app(+(),y),z)),x) -> app(app(+(),app(app(*(),x),y)),app(app(*(),x),z))
 app(app(*(),app(app(*(),x),y)),z) -> app(app(*(),x),app(app(*(),y),z))
 app(app(+(),app(app(+(),x),y)),z) -> app(app(+(),x),app(app(+(),y),z))
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 app(app(filter(),f),nil()) -> nil()
 app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
 app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
 app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

Proof:
 Uncurry Processor:
  *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
  *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
  *2(*2(x,y),z) -> *2(x,*2(y,z))
  +2(+2(x,y),z) -> +2(x,+2(y,z))
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  filter2(f,nil()) -> nil()
  filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
  filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
  filter24(false(),f,x,xs) -> filter2(f,xs)
  app(*1(x7),x8) -> *2(x7,x8)
  app(*(),x8) -> *1(x8)
  app(+1(x7),x8) -> +2(x7,x8)
  app(+(),x8) -> +1(x8)
  app(map1(x7),x8) -> map2(x7,x8)
  app(map(),x8) -> map1(x8)
  app(cons1(x7),x8) -> cons2(x7,x8)
  app(cons(),x8) -> cons1(x8)
  app(filter1(x7),x8) -> filter2(x7,x8)
  app(filter(),x8) -> filter1(x8)
  app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
  app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
  app(filter21(x7),x8) -> filter22(x7,x8)
  app(filter2(),x8) -> filter21(x8)
  DP Processor:
   DPs:
    *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
    *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
    *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
    *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
    *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
    *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
    *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
    *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
    +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
    +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
    filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
    filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
    app#(*1(x7),x8) -> *{2,#}(x7,x8)
    app#(+1(x7),x8) -> +{2,#}(x7,x8)
    app#(map1(x7),x8) -> map{2,#}(x7,x8)
    app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
    app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
   TRS:
    *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
    *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
    *2(*2(x,y),z) -> *2(x,*2(y,z))
    +2(+2(x,y),z) -> +2(x,+2(y,z))
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    filter2(f,nil()) -> nil()
    filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
    filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
    filter24(false(),f,x,xs) -> filter2(f,xs)
    app(*1(x7),x8) -> *2(x7,x8)
    app(*(),x8) -> *1(x8)
    app(+1(x7),x8) -> +2(x7,x8)
    app(+(),x8) -> +1(x8)
    app(map1(x7),x8) -> map2(x7,x8)
    app(map(),x8) -> map1(x8)
    app(cons1(x7),x8) -> cons2(x7,x8)
    app(cons(),x8) -> cons1(x8)
    app(filter1(x7),x8) -> filter2(x7,x8)
    app(filter(),x8) -> filter1(x8)
    app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
    app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
    app(filter21(x7),x8) -> filter22(x7,x8)
    app(filter2(),x8) -> filter21(x8)
   TDG Processor:
    DPs:
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(*1(x7),x8) -> *{2,#}(x7,x8)
     app#(+1(x7),x8) -> +{2,#}(x7,x8)
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
    TRS:
     *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
     *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
     *2(*2(x,y),z) -> *2(x,*2(y,z))
     +2(+2(x,y),z) -> +2(x,+2(y,z))
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     filter2(f,nil()) -> nil()
     filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
     filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
     filter24(false(),f,x,xs) -> filter2(f,xs)
     app(*1(x7),x8) -> *2(x7,x8)
     app(*(),x8) -> *1(x8)
     app(+1(x7),x8) -> +2(x7,x8)
     app(+(),x8) -> +1(x8)
     app(map1(x7),x8) -> map2(x7,x8)
     app(map(),x8) -> map1(x8)
     app(cons1(x7),x8) -> cons2(x7,x8)
     app(cons(),x8) -> cons1(x8)
     app(filter1(x7),x8) -> filter2(x7,x8)
     app(filter(),x8) -> filter1(x8)
     app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
     app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
     app(filter21(x7),x8) -> filter22(x7,x8)
     app(filter2(),x8) -> filter21(x8)
    graph:
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(+1(x7),x8) -> +{2,#}(x7,x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(*1(x7),x8) -> *{2,#}(x7,x8)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8) ->
     filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8) ->
     filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x7),x8) -> map{2,#}(x7,x8) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x7),x8) -> map{2,#}(x7,x8) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(+1(x7),x8) -> +{2,#}(x7,x8) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     app#(+1(x7),x8) -> +{2,#}(x7,x8) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     app#(*1(x7),x8) -> *{2,#}(x7,x8) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x7),x8) -> map{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(+1(x7),x8) -> +{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(*1(x7),x8) -> *{2,#}(x7,x8)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z)) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z)) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z)) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z)) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z)) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z)) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z)) ->
     +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(+2(y,z),x) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(x,+2(y,z)) -> +{2,#}(*2(x,y),*2(x,z))
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) ->
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
     *{2,#}(x,+2(y,z)) -> *{2,#}(x,y) -> *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
    SCC Processor:
     #sccs: 3
     #rules: 17
     #arcs: 90/441
     DPs:
      filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x7),x8) -> map{2,#}(x7,x8)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
      filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
      filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
      app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
     TRS:
      *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
      *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
      *2(*2(x,y),z) -> *2(x,*2(y,z))
      +2(+2(x,y),z) -> +2(x,+2(y,z))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(*1(x7),x8) -> *2(x7,x8)
      app(*(),x8) -> *1(x8)
      app(+1(x7),x8) -> +2(x7,x8)
      app(+(),x8) -> +1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
       pi(filter{2,#}) = 1
       pi(filter2{4,#}) = 3
      problem:
       DPs:
        filter2{4,#}(false(),f,x,xs) -> filter{2,#}(f,xs)
        app#(map1(x7),x8) -> map{2,#}(x7,x8)
        app#(filter1(x7),x8) -> filter{2,#}(x7,x8)
        filter2{4,#}(true(),f,x,xs) -> filter{2,#}(f,xs)
        app#(filter23(x7,x6,x5),x8) -> filter2{4,#}(x7,x6,x5,x8)
       TRS:
        *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
        *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
        *2(*2(x,y),z) -> *2(x,*2(y,z))
        +2(+2(x,y),z) -> +2(x,+2(y,z))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(*1(x7),x8) -> *2(x7,x8)
        app(*(),x8) -> *1(x8)
        app(+1(x7),x8) -> +2(x7,x8)
        app(+(),x8) -> +1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 20/25
       
     
     DPs:
      *{2,#}(x,+2(y,z)) -> *{2,#}(x,z)
      *{2,#}(x,+2(y,z)) -> *{2,#}(x,y)
      *{2,#}(+2(y,z),x) -> *{2,#}(x,z)
      *{2,#}(+2(y,z),x) -> *{2,#}(x,y)
      *{2,#}(*2(x,y),z) -> *{2,#}(y,z)
      *{2,#}(*2(x,y),z) -> *{2,#}(x,*2(y,z))
     TRS:
      *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
      *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
      *2(*2(x,y),z) -> *2(x,*2(y,z))
      +2(+2(x,y),z) -> +2(x,+2(y,z))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(*1(x7),x8) -> *2(x7,x8)
      app(*(),x8) -> *1(x8)
      app(+1(x7),x8) -> +2(x7,x8)
      app(+(),x8) -> +1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Open
     
     DPs:
      +{2,#}(+2(x,y),z) -> +{2,#}(y,z)
      +{2,#}(+2(x,y),z) -> +{2,#}(x,+2(y,z))
     TRS:
      *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
      *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
      *2(*2(x,y),z) -> *2(x,*2(y,z))
      +2(+2(x,y),z) -> +2(x,+2(y,z))
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil()) -> nil()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false(),f,x,xs) -> filter2(f,xs)
      app(*1(x7),x8) -> *2(x7,x8)
      app(*(),x8) -> *1(x8)
      app(+1(x7),x8) -> +2(x7,x8)
      app(+(),x8) -> +1(x8)
      app(map1(x7),x8) -> map2(x7,x8)
      app(map(),x8) -> map1(x8)
      app(cons1(x7),x8) -> cons2(x7,x8)
      app(cons(),x8) -> cons1(x8)
      app(filter1(x7),x8) -> filter2(x7,x8)
      app(filter(),x8) -> filter1(x8)
      app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
      app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
      app(filter21(x7),x8) -> filter22(x7,x8)
      app(filter2(),x8) -> filter21(x8)
     Subterm Criterion Processor:
      simple projection:
       pi(+{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        *2(x,+2(y,z)) -> +2(*2(x,y),*2(x,z))
        *2(+2(y,z),x) -> +2(*2(x,y),*2(x,z))
        *2(*2(x,y),z) -> *2(x,*2(y,z))
        +2(+2(x,y),z) -> +2(x,+2(y,z))
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil()) -> nil()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false(),f,x,xs) -> filter2(f,xs)
        app(*1(x7),x8) -> *2(x7,x8)
        app(*(),x8) -> *1(x8)
        app(+1(x7),x8) -> +2(x7,x8)
        app(+(),x8) -> +1(x8)
        app(map1(x7),x8) -> map2(x7,x8)
        app(map(),x8) -> map1(x8)
        app(cons1(x7),x8) -> cons2(x7,x8)
        app(cons(),x8) -> cons1(x8)
        app(filter1(x7),x8) -> filter2(x7,x8)
        app(filter(),x8) -> filter1(x8)
        app(filter23(x7,x6,x5),x8) -> filter24(x7,x6,x5,x8)
        app(filter22(x7,x6),x8) -> filter23(x7,x6,x8)
        app(filter21(x7),x8) -> filter22(x7,x8)
        app(filter2(),x8) -> filter21(x8)
      Qed