YES

Problem:
 app(app(minus(),0()),y) -> 0()
 app(app(minus(),app(s(),x)),0()) -> app(s(),x)
 app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),x),y)
 app(app(le(),0()),y) -> true()
 app(app(le(),app(s(),x)),0()) -> false()
 app(app(le(),app(s(),x)),app(s(),y)) -> app(app(le(),x),y)
 app(app(app(if(),true()),x),y) -> x
 app(app(app(if(),false()),x),y) -> y
 app(perfectp(),0()) -> false()
 app(perfectp(),app(s(),x)) -> app(app(app(app(f(),x),app(s(),0())),app(s(),x)),app(s(),x))
 app(app(app(app(f(),0()),y),0()),u) -> true()
 app(app(app(app(f(),0()),y),app(s(),z)),u) -> false()
 app(app(app(app(f(),app(s(),x)),0()),z),u) ->
 app(app(app(app(f(),x),u),app(app(minus(),z),app(s(),x))),u)
 app(app(app(app(f(),app(s(),x)),app(s(),y)),z),u) ->
 app(app(app(if(),app(app(le(),x),y)),app(app(app(app(f(),app(s(),x)),app(app(minus(),y),x)),z),u)),
     app(app(app(app(f(),x),u),z),u))
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)

Proof:
 Uncurry Processor:
  minus2(0(),y) -> 0()
  minus2(s1(x),0()) -> s1(x)
  minus2(s1(x),s1(y)) -> minus2(x,y)
  le2(0(),y) -> true()
  le2(s1(x),0()) -> false()
  le2(s1(x),s1(y)) -> le2(x,y)
  if3(true(),x,y) -> x
  if3(false(),x,y) -> y
  perfectp1(0()) -> false()
  perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
  f4(0(),y,0(),u) -> true()
  f4(0(),y,s1(z),u) -> false()
  f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
  f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
  map2(fun,nil()) -> nil()
  map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
  filter2(fun,nil()) -> nil()
  filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
  filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
  filter24(false(),fun,x,xs) -> filter2(fun,xs)
  app(minus1(x8),x9) -> minus2(x8,x9)
  app(minus(),x9) -> minus1(x9)
  app(s(),x9) -> s1(x9)
  app(le1(x8),x9) -> le2(x8,x9)
  app(le(),x9) -> le1(x9)
  app(if2(x8,x7),x9) -> if3(x8,x7,x9)
  app(if1(x8),x9) -> if2(x8,x9)
  app(if(),x9) -> if1(x9)
  app(perfectp(),x9) -> perfectp1(x9)
  app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
  app(f2(x8,x7),x9) -> f3(x8,x7,x9)
  app(f1(x8),x9) -> f2(x8,x9)
  app(f(),x9) -> f1(x9)
  app(map1(x8),x9) -> map2(x8,x9)
  app(map(),x9) -> map1(x9)
  app(cons1(x8),x9) -> cons2(x8,x9)
  app(cons(),x9) -> cons1(x9)
  app(filter1(x8),x9) -> filter2(x8,x9)
  app(filter(),x9) -> filter1(x9)
  app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
  app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
  app(filter21(x8),x9) -> filter22(x8,x9)
  app(filter2(),x9) -> filter21(x9)
  DP Processor:
   DPs:
    minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
    perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x))
    f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
    f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
    f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
    f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
    f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
    f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
    f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
    map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
    map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
    filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
    filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
    app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
    app#(le1(x8),x9) -> le{2,#}(x8,x9)
    app#(if2(x8,x7),x9) -> if{3,#}(x8,x7,x9)
    app#(perfectp(),x9) -> perfectp{1,#}(x9)
    app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9)
    app#(map1(x8),x9) -> map{2,#}(x8,x9)
    app#(filter1(x8),x9) -> filter{2,#}(x8,x9)
    app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9)
   TRS:
    minus2(0(),y) -> 0()
    minus2(s1(x),0()) -> s1(x)
    minus2(s1(x),s1(y)) -> minus2(x,y)
    le2(0(),y) -> true()
    le2(s1(x),0()) -> false()
    le2(s1(x),s1(y)) -> le2(x,y)
    if3(true(),x,y) -> x
    if3(false(),x,y) -> y
    perfectp1(0()) -> false()
    perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
    f4(0(),y,0(),u) -> true()
    f4(0(),y,s1(z),u) -> false()
    f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
    f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
    map2(fun,nil()) -> nil()
    map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
    filter2(fun,nil()) -> nil()
    filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
    filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
    filter24(false(),fun,x,xs) -> filter2(fun,xs)
    app(minus1(x8),x9) -> minus2(x8,x9)
    app(minus(),x9) -> minus1(x9)
    app(s(),x9) -> s1(x9)
    app(le1(x8),x9) -> le2(x8,x9)
    app(le(),x9) -> le1(x9)
    app(if2(x8,x7),x9) -> if3(x8,x7,x9)
    app(if1(x8),x9) -> if2(x8,x9)
    app(if(),x9) -> if1(x9)
    app(perfectp(),x9) -> perfectp1(x9)
    app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
    app(f2(x8,x7),x9) -> f3(x8,x7,x9)
    app(f1(x8),x9) -> f2(x8,x9)
    app(f(),x9) -> f1(x9)
    app(map1(x8),x9) -> map2(x8,x9)
    app(map(),x9) -> map1(x9)
    app(cons1(x8),x9) -> cons2(x8,x9)
    app(cons(),x9) -> cons1(x9)
    app(filter1(x8),x9) -> filter2(x8,x9)
    app(filter(),x9) -> filter1(x9)
    app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
    app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
    app(filter21(x8),x9) -> filter22(x8,x9)
    app(filter2(),x9) -> filter21(x9)
   TDG Processor:
    DPs:
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x))
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
     app#(le1(x8),x9) -> le{2,#}(x8,x9)
     app#(if2(x8,x7),x9) -> if{3,#}(x8,x7,x9)
     app#(perfectp(),x9) -> perfectp{1,#}(x9)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9)
     app#(map1(x8),x9) -> map{2,#}(x8,x9)
     app#(filter1(x8),x9) -> filter{2,#}(x8,x9)
     app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9)
    TRS:
     minus2(0(),y) -> 0()
     minus2(s1(x),0()) -> s1(x)
     minus2(s1(x),s1(y)) -> minus2(x,y)
     le2(0(),y) -> true()
     le2(s1(x),0()) -> false()
     le2(s1(x),s1(y)) -> le2(x,y)
     if3(true(),x,y) -> x
     if3(false(),x,y) -> y
     perfectp1(0()) -> false()
     perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
     f4(0(),y,0(),u) -> true()
     f4(0(),y,s1(z),u) -> false()
     f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
     f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     map2(fun,nil()) -> nil()
     map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
     filter2(fun,nil()) -> nil()
     filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
     filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
     filter24(false(),fun,x,xs) -> filter2(fun,xs)
     app(minus1(x8),x9) -> minus2(x8,x9)
     app(minus(),x9) -> minus1(x9)
     app(s(),x9) -> s1(x9)
     app(le1(x8),x9) -> le2(x8,x9)
     app(le(),x9) -> le1(x9)
     app(if2(x8,x7),x9) -> if3(x8,x7,x9)
     app(if1(x8),x9) -> if2(x8,x9)
     app(if(),x9) -> if1(x9)
     app(perfectp(),x9) -> perfectp1(x9)
     app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
     app(f2(x8,x7),x9) -> f3(x8,x7,x9)
     app(f1(x8),x9) -> f2(x8,x9)
     app(f(),x9) -> f1(x9)
     app(map1(x8),x9) -> map2(x8,x9)
     app(map(),x9) -> map1(x9)
     app(cons1(x8),x9) -> cons2(x8,x9)
     app(cons(),x9) -> cons1(x9)
     app(filter1(x8),x9) -> filter2(x8,x9)
     app(filter(),x9) -> filter1(x9)
     app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
     app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
     app(filter21(x8),x9) -> filter22(x8,x9)
     app(filter2(),x9) -> filter21(x9)
    graph:
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x8),x9) -> filter{2,#}(x8,x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x8),x9) -> map{2,#}(x8,x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(perfectp(),x9) -> perfectp{1,#}(x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(if2(x8,x7),x9) -> if{3,#}(x8,x7,x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(le1(x8),x9) -> le{2,#}(x8,x9)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
     app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9) ->
     filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9) ->
     filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter1(x8),x9) -> filter{2,#}(x8,x9) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     app#(filter1(x8),x9) -> filter{2,#}(x8,x9) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x8),x9) -> map{2,#}(x8,x9) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x8),x9) -> map{2,#}(x8,x9) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9) ->
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
     app#(le1(x8),x9) -> le{2,#}(x8,x9) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(perfectp(),x9) -> perfectp{1,#}(x9) ->
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x))
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x8),x9) -> filter{2,#}(x8,x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x8),x9) -> map{2,#}(x8,x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f3(x8,x7,x6),x9) -> f{4,#}(x8,x7,x6,x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(perfectp(),x9) -> perfectp{1,#}(x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(if2(x8,x7),x9) -> if{3,#}(x8,x7,x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(le1(x8),x9) -> le{2,#}(x8,x9)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(minus1(x8),x9) -> minus{2,#}(x8,x9)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u) ->
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u) ->
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x)) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> if{3,#}(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> le{2,#}(x,y)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> minus{2,#}(y,x)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
     perfectp{1,#}(s1(x)) -> f{4,#}(x,s1(0()),s1(x),s1(x)) ->
     f{4,#}(s1(x),0(),z,u) -> minus{2,#}(z,s1(x))
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y) ->
     le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) -> minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    SCC Processor:
     #sccs: 4
     #rules: 14
     #arcs: 73/576
     DPs:
      filter2{4,#}(false(),fun,x,xs) -> filter{2,#}(fun,xs)
      filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(map1(x8),x9) -> map{2,#}(x8,x9)
      map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
      map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(filter1(x8),x9) -> filter{2,#}(x8,x9)
      filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
      filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
      app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9)
     TRS:
      minus2(0(),y) -> 0()
      minus2(s1(x),0()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true(),x,y) -> x
      if3(false(),x,y) -> y
      perfectp1(0()) -> false()
      perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
      f4(0(),y,0(),u) -> true()
      f4(0(),y,s1(z),u) -> false()
      f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus(),x9) -> minus1(x9)
      app(s(),x9) -> s1(x9)
      app(le1(x8),x9) -> le2(x8,x9)
      app(le(),x9) -> le1(x9)
      app(if2(x8,x7),x9) -> if3(x8,x7,x9)
      app(if1(x8),x9) -> if2(x8,x9)
      app(if(),x9) -> if1(x9)
      app(perfectp(),x9) -> perfectp1(x9)
      app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
      app(f2(x8,x7),x9) -> f3(x8,x7,x9)
      app(f1(x8),x9) -> f2(x8,x9)
      app(f(),x9) -> f1(x9)
      app(map1(x8),x9) -> map2(x8,x9)
      app(map(),x9) -> map1(x9)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons(),x9) -> cons1(x9)
      app(filter1(x8),x9) -> filter2(x8,x9)
      app(filter(),x9) -> filter1(x9)
      app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
      app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
      app(filter21(x8),x9) -> filter22(x8,x9)
      app(filter2(),x9) -> filter21(x9)
     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(),fun,x,xs) -> filter{2,#}(fun,xs)
        app#(map1(x8),x9) -> map{2,#}(x8,x9)
        app#(filter1(x8),x9) -> filter{2,#}(x8,x9)
        filter2{4,#}(true(),fun,x,xs) -> filter{2,#}(fun,xs)
        app#(filter23(x8,x7,x6),x9) -> filter2{4,#}(x8,x7,x6,x9)
       TRS:
        minus2(0(),y) -> 0()
        minus2(s1(x),0()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true(),x,y) -> x
        if3(false(),x,y) -> y
        perfectp1(0()) -> false()
        perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
        f4(0(),y,0(),u) -> true()
        f4(0(),y,s1(z),u) -> false()
        f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus(),x9) -> minus1(x9)
        app(s(),x9) -> s1(x9)
        app(le1(x8),x9) -> le2(x8,x9)
        app(le(),x9) -> le1(x9)
        app(if2(x8,x7),x9) -> if3(x8,x7,x9)
        app(if1(x8),x9) -> if2(x8,x9)
        app(if(),x9) -> if1(x9)
        app(perfectp(),x9) -> perfectp1(x9)
        app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
        app(f2(x8,x7),x9) -> f3(x8,x7,x9)
        app(f1(x8),x9) -> f2(x8,x9)
        app(f(),x9) -> f1(x9)
        app(map1(x8),x9) -> map2(x8,x9)
        app(map(),x9) -> map1(x9)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons(),x9) -> cons1(x9)
        app(filter1(x8),x9) -> filter2(x8,x9)
        app(filter(),x9) -> filter1(x9)
        app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
        app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
        app(filter21(x8),x9) -> filter22(x8,x9)
        app(filter2(),x9) -> filter21(x9)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 20/25
       
     
     DPs:
      f{4,#}(s1(x),0(),z,u) -> f{4,#}(x,u,minus2(z,s1(x)),u)
      f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(x,u,z,u)
      f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
     TRS:
      minus2(0(),y) -> 0()
      minus2(s1(x),0()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true(),x,y) -> x
      if3(false(),x,y) -> y
      perfectp1(0()) -> false()
      perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
      f4(0(),y,0(),u) -> true()
      f4(0(),y,s1(z),u) -> false()
      f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus(),x9) -> minus1(x9)
      app(s(),x9) -> s1(x9)
      app(le1(x8),x9) -> le2(x8,x9)
      app(le(),x9) -> le1(x9)
      app(if2(x8,x7),x9) -> if3(x8,x7,x9)
      app(if1(x8),x9) -> if2(x8,x9)
      app(if(),x9) -> if1(x9)
      app(perfectp(),x9) -> perfectp1(x9)
      app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
      app(f2(x8,x7),x9) -> f3(x8,x7,x9)
      app(f1(x8),x9) -> f2(x8,x9)
      app(f(),x9) -> f1(x9)
      app(map1(x8),x9) -> map2(x8,x9)
      app(map(),x9) -> map1(x9)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons(),x9) -> cons1(x9)
      app(filter1(x8),x9) -> filter2(x8,x9)
      app(filter(),x9) -> filter1(x9)
      app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
      app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
      app(filter21(x8),x9) -> filter22(x8,x9)
      app(filter2(),x9) -> filter21(x9)
     Subterm Criterion Processor:
      simple projection:
       pi(f{4,#}) = 0
      problem:
       DPs:
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
       TRS:
        minus2(0(),y) -> 0()
        minus2(s1(x),0()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true(),x,y) -> x
        if3(false(),x,y) -> y
        perfectp1(0()) -> false()
        perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
        f4(0(),y,0(),u) -> true()
        f4(0(),y,s1(z),u) -> false()
        f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus(),x9) -> minus1(x9)
        app(s(),x9) -> s1(x9)
        app(le1(x8),x9) -> le2(x8,x9)
        app(le(),x9) -> le1(x9)
        app(if2(x8,x7),x9) -> if3(x8,x7,x9)
        app(if1(x8),x9) -> if2(x8,x9)
        app(if(),x9) -> if1(x9)
        app(perfectp(),x9) -> perfectp1(x9)
        app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
        app(f2(x8,x7),x9) -> f3(x8,x7,x9)
        app(f1(x8),x9) -> f2(x8,x9)
        app(f(),x9) -> f1(x9)
        app(map1(x8),x9) -> map2(x8,x9)
        app(map(),x9) -> map1(x9)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons(),x9) -> cons1(x9)
        app(filter1(x8),x9) -> filter2(x8,x9)
        app(filter(),x9) -> filter1(x9)
        app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
        app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
        app(filter21(x8),x9) -> filter22(x8,x9)
        app(filter2(),x9) -> filter21(x9)
      EDG Processor:
       DPs:
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
       TRS:
        minus2(0(),y) -> 0()
        minus2(s1(x),0()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true(),x,y) -> x
        if3(false(),x,y) -> y
        perfectp1(0()) -> false()
        perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
        f4(0(),y,0(),u) -> true()
        f4(0(),y,s1(z),u) -> false()
        f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus(),x9) -> minus1(x9)
        app(s(),x9) -> s1(x9)
        app(le1(x8),x9) -> le2(x8,x9)
        app(le(),x9) -> le1(x9)
        app(if2(x8,x7),x9) -> if3(x8,x7,x9)
        app(if1(x8),x9) -> if2(x8,x9)
        app(if(),x9) -> if1(x9)
        app(perfectp(),x9) -> perfectp1(x9)
        app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
        app(f2(x8,x7),x9) -> f3(x8,x7,x9)
        app(f1(x8),x9) -> f2(x8,x9)
        app(f(),x9) -> f1(x9)
        app(map1(x8),x9) -> map2(x8,x9)
        app(map(),x9) -> map1(x9)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons(),x9) -> cons1(x9)
        app(filter1(x8),x9) -> filter2(x8,x9)
        app(filter(),x9) -> filter1(x9)
        app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
        app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
        app(filter21(x8),x9) -> filter22(x8,x9)
        app(filter2(),x9) -> filter21(x9)
       graph:
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u) ->
        f{4,#}(s1(x),s1(y),z,u) -> f{4,#}(s1(x),minus2(y,x),z,u)
       Matrix Interpretation Processor: dim=1
        
        interpretation:
         [f{4,#}](x0, x1, x2, x3) = 1/2x1 + 1/2,
         
         [filter21](x0) = x0 + 3,
         
         [filter22](x0, x1) = 0,
         
         [filter24](x0, x1, x2, x3) = 2,
         
         [filter23](x0, x1, x2) = 2x2 + 3/2,
         
         [filter2](x0, x1) = 2,
         
         [filter1](x0) = 1,
         
         [cons2](x0, x1) = 2,
         
         [cons1](x0) = 5/2,
         
         [map2](x0, x1) = 2x0 + 2x1 + 5/2,
         
         [map1](x0) = 2x0 + 1,
         
         [f1](x0) = 1/2x0 + 1/2,
         
         [f2](x0, x1) = x0 + 2x1 + 3,
         
         [f4](x0, x1, x2, x3) = 0,
         
         [f3](x0, x1, x2) = 0,
         
         [perfectp1](x0) = x0 + 5/2,
         
         [if1](x0) = 1/2x0,
         
         [if3](x0, x1, x2) = x1 + 3/2x2,
         
         [if2](x0, x1) = x0 + 1/2x1,
         
         [le2](x0, x1) = 2x0 + x1,
         
         [le1](x0) = x0,
         
         [s1](x0) = 2x0 + 2,
         
         [minus2](x0, x1) = x0 + 1,
         
         [minus1](x0) = x0,
         
         [filter2] = 2,
         
         [filter] = 1/2,
         
         [cons] = 2,
         
         [nil] = 3/2,
         
         [map] = 0,
         
         [f] = 3,
         
         [perfectp] = 3/2,
         
         [if] = 7/2,
         
         [false] = 0,
         
         [true] = 0,
         
         [le] = 0,
         
         [s] = 0,
         
         [app](x0, x1) = 2x0 + 2x1 + 2,
         
         [0] = 3/2,
         
         [minus] = 0
        orientation:
         f{4,#}(s1(x),s1(y),z,u) = y + 3/2 >= 1/2y + 1 = f{4,#}(s1(x),minus2(y,x),z,u)
         
         minus2(0(),y) = 5/2 >= 3/2 = 0()
         
         minus2(s1(x),0()) = 2x + 3 >= 2x + 2 = s1(x)
         
         minus2(s1(x),s1(y)) = 2x + 3 >= x + 1 = minus2(x,y)
         
         le2(0(),y) = y + 3 >= 0 = true()
         
         le2(s1(x),0()) = 4x + 11/2 >= 0 = false()
         
         le2(s1(x),s1(y)) = 4x + 2y + 6 >= 2x + y = le2(x,y)
         
         if3(true(),x,y) = x + 3/2y >= x = x
         
         if3(false(),x,y) = x + 3/2y >= y = y
         
         perfectp1(0()) = 4 >= 0 = false()
         
         perfectp1(s1(x)) = 2x + 9/2 >= 0 = f4(x,s1(0()),s1(x),s1(x))
         
         f4(0(),y,0(),u) = 0 >= 0 = true()
         
         f4(0(),y,s1(z),u) = 0 >= 0 = false()
         
         f4(s1(x),0(),z,u) = 0 >= 0 = f4(x,u,minus2(z,s1(x)),u)
         
         f4(s1(x),s1(y),z,u) = 0 >= 0 = if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
         
         map2(fun,nil()) = 2fun + 11/2 >= 3/2 = nil()
         
         map2(fun,cons2(x,xs)) = 2fun + 13/2 >= 2 = cons2(app(fun,x),map2(fun,xs))
         
         filter2(fun,nil()) = 2 >= 3/2 = nil()
         
         filter2(fun,cons2(x,xs)) = 2 >= 2 = filter24(app(fun,x),fun,x,xs)
         
         filter24(true(),fun,x,xs) = 2 >= 2 = cons2(x,filter2(fun,xs))
         
         filter24(false(),fun,x,xs) = 2 >= 2 = filter2(fun,xs)
         
         app(minus1(x8),x9) = 2x8 + 2x9 + 2 >= x8 + 1 = minus2(x8,x9)
         
         app(minus(),x9) = 2x9 + 2 >= x9 = minus1(x9)
         
         app(s(),x9) = 2x9 + 2 >= 2x9 + 2 = s1(x9)
         
         app(le1(x8),x9) = 2x8 + 2x9 + 2 >= 2x8 + x9 = le2(x8,x9)
         
         app(le(),x9) = 2x9 + 2 >= x9 = le1(x9)
         
         app(if2(x8,x7),x9) = x7 + 2x8 + 2x9 + 2 >= x7 + 3/2x9 = if3(x8,x7,x9)
         
         app(if1(x8),x9) = x8 + 2x9 + 2 >= x8 + 1/2x9 = if2(x8,x9)
         
         app(if(),x9) = 2x9 + 9 >= 1/2x9 = if1(x9)
         
         app(perfectp(),x9) = 2x9 + 5 >= x9 + 5/2 = perfectp1(x9)
         
         app(f3(x8,x7,x6),x9) = 2x9 + 2 >= 0 = f4(x8,x7,x6,x9)
         
         app(f2(x8,x7),x9) = 4x7 + 2x8 + 2x9 + 8 >= 0 = f3(x8,x7,x9)
         
         app(f1(x8),x9) = x8 + 2x9 + 3 >= x8 + 2x9 + 3 = f2(x8,x9)
         
         app(f(),x9) = 2x9 + 8 >= 1/2x9 + 1/2 = f1(x9)
         
         app(map1(x8),x9) = 4x8 + 2x9 + 4 >= 2x8 + 2x9 + 5/2 = map2(x8,x9)
         
         app(map(),x9) = 2x9 + 2 >= 2x9 + 1 = map1(x9)
         
         app(cons1(x8),x9) = 2x9 + 7 >= 2 = cons2(x8,x9)
         
         app(cons(),x9) = 2x9 + 6 >= 5/2 = cons1(x9)
         
         app(filter1(x8),x9) = 2x9 + 4 >= 2 = filter2(x8,x9)
         
         app(filter(),x9) = 2x9 + 3 >= 1 = filter1(x9)
         
         app(filter23(x8,x7,x6),x9) = 4x6 + 2x9 + 5 >= 2 = filter24(x8,x7,x6,x9)
         
         app(filter22(x8,x7),x9) = 2x9 + 2 >= 2x9 + 3/2 = filter23(x8,x7,x9)
         
         app(filter21(x8),x9) = 2x8 + 2x9 + 8 >= 0 = filter22(x8,x9)
         
         app(filter2(),x9) = 2x9 + 6 >= x9 + 3 = filter21(x9)
        problem:
         DPs:
          
         TRS:
          minus2(0(),y) -> 0()
          minus2(s1(x),0()) -> s1(x)
          minus2(s1(x),s1(y)) -> minus2(x,y)
          le2(0(),y) -> true()
          le2(s1(x),0()) -> false()
          le2(s1(x),s1(y)) -> le2(x,y)
          if3(true(),x,y) -> x
          if3(false(),x,y) -> y
          perfectp1(0()) -> false()
          perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
          f4(0(),y,0(),u) -> true()
          f4(0(),y,s1(z),u) -> false()
          f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
          f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
          map2(fun,nil()) -> nil()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil()) -> nil()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false(),fun,x,xs) -> filter2(fun,xs)
          app(minus1(x8),x9) -> minus2(x8,x9)
          app(minus(),x9) -> minus1(x9)
          app(s(),x9) -> s1(x9)
          app(le1(x8),x9) -> le2(x8,x9)
          app(le(),x9) -> le1(x9)
          app(if2(x8,x7),x9) -> if3(x8,x7,x9)
          app(if1(x8),x9) -> if2(x8,x9)
          app(if(),x9) -> if1(x9)
          app(perfectp(),x9) -> perfectp1(x9)
          app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
          app(f2(x8,x7),x9) -> f3(x8,x7,x9)
          app(f1(x8),x9) -> f2(x8,x9)
          app(f(),x9) -> f1(x9)
          app(map1(x8),x9) -> map2(x8,x9)
          app(map(),x9) -> map1(x9)
          app(cons1(x8),x9) -> cons2(x8,x9)
          app(cons(),x9) -> cons1(x9)
          app(filter1(x8),x9) -> filter2(x8,x9)
          app(filter(),x9) -> filter1(x9)
          app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
          app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
          app(filter21(x8),x9) -> filter22(x8,x9)
          app(filter2(),x9) -> filter21(x9)
        Qed
     
     DPs:
      le{2,#}(s1(x),s1(y)) -> le{2,#}(x,y)
     TRS:
      minus2(0(),y) -> 0()
      minus2(s1(x),0()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true(),x,y) -> x
      if3(false(),x,y) -> y
      perfectp1(0()) -> false()
      perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
      f4(0(),y,0(),u) -> true()
      f4(0(),y,s1(z),u) -> false()
      f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus(),x9) -> minus1(x9)
      app(s(),x9) -> s1(x9)
      app(le1(x8),x9) -> le2(x8,x9)
      app(le(),x9) -> le1(x9)
      app(if2(x8,x7),x9) -> if3(x8,x7,x9)
      app(if1(x8),x9) -> if2(x8,x9)
      app(if(),x9) -> if1(x9)
      app(perfectp(),x9) -> perfectp1(x9)
      app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
      app(f2(x8,x7),x9) -> f3(x8,x7,x9)
      app(f1(x8),x9) -> f2(x8,x9)
      app(f(),x9) -> f1(x9)
      app(map1(x8),x9) -> map2(x8,x9)
      app(map(),x9) -> map1(x9)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons(),x9) -> cons1(x9)
      app(filter1(x8),x9) -> filter2(x8,x9)
      app(filter(),x9) -> filter1(x9)
      app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
      app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
      app(filter21(x8),x9) -> filter22(x8,x9)
      app(filter2(),x9) -> filter21(x9)
     Subterm Criterion Processor:
      simple projection:
       pi(le{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        minus2(0(),y) -> 0()
        minus2(s1(x),0()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true(),x,y) -> x
        if3(false(),x,y) -> y
        perfectp1(0()) -> false()
        perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
        f4(0(),y,0(),u) -> true()
        f4(0(),y,s1(z),u) -> false()
        f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus(),x9) -> minus1(x9)
        app(s(),x9) -> s1(x9)
        app(le1(x8),x9) -> le2(x8,x9)
        app(le(),x9) -> le1(x9)
        app(if2(x8,x7),x9) -> if3(x8,x7,x9)
        app(if1(x8),x9) -> if2(x8,x9)
        app(if(),x9) -> if1(x9)
        app(perfectp(),x9) -> perfectp1(x9)
        app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
        app(f2(x8,x7),x9) -> f3(x8,x7,x9)
        app(f1(x8),x9) -> f2(x8,x9)
        app(f(),x9) -> f1(x9)
        app(map1(x8),x9) -> map2(x8,x9)
        app(map(),x9) -> map1(x9)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons(),x9) -> cons1(x9)
        app(filter1(x8),x9) -> filter2(x8,x9)
        app(filter(),x9) -> filter1(x9)
        app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
        app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
        app(filter21(x8),x9) -> filter22(x8,x9)
        app(filter2(),x9) -> filter21(x9)
      Qed
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      minus2(0(),y) -> 0()
      minus2(s1(x),0()) -> s1(x)
      minus2(s1(x),s1(y)) -> minus2(x,y)
      le2(0(),y) -> true()
      le2(s1(x),0()) -> false()
      le2(s1(x),s1(y)) -> le2(x,y)
      if3(true(),x,y) -> x
      if3(false(),x,y) -> y
      perfectp1(0()) -> false()
      perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
      f4(0(),y,0(),u) -> true()
      f4(0(),y,s1(z),u) -> false()
      f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
      f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
      map2(fun,nil()) -> nil()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil()) -> nil()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false(),fun,x,xs) -> filter2(fun,xs)
      app(minus1(x8),x9) -> minus2(x8,x9)
      app(minus(),x9) -> minus1(x9)
      app(s(),x9) -> s1(x9)
      app(le1(x8),x9) -> le2(x8,x9)
      app(le(),x9) -> le1(x9)
      app(if2(x8,x7),x9) -> if3(x8,x7,x9)
      app(if1(x8),x9) -> if2(x8,x9)
      app(if(),x9) -> if1(x9)
      app(perfectp(),x9) -> perfectp1(x9)
      app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
      app(f2(x8,x7),x9) -> f3(x8,x7,x9)
      app(f1(x8),x9) -> f2(x8,x9)
      app(f(),x9) -> f1(x9)
      app(map1(x8),x9) -> map2(x8,x9)
      app(map(),x9) -> map1(x9)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons(),x9) -> cons1(x9)
      app(filter1(x8),x9) -> filter2(x8,x9)
      app(filter(),x9) -> filter1(x9)
      app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
      app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
      app(filter21(x8),x9) -> filter22(x8,x9)
      app(filter2(),x9) -> filter21(x9)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        minus2(0(),y) -> 0()
        minus2(s1(x),0()) -> s1(x)
        minus2(s1(x),s1(y)) -> minus2(x,y)
        le2(0(),y) -> true()
        le2(s1(x),0()) -> false()
        le2(s1(x),s1(y)) -> le2(x,y)
        if3(true(),x,y) -> x
        if3(false(),x,y) -> y
        perfectp1(0()) -> false()
        perfectp1(s1(x)) -> f4(x,s1(0()),s1(x),s1(x))
        f4(0(),y,0(),u) -> true()
        f4(0(),y,s1(z),u) -> false()
        f4(s1(x),0(),z,u) -> f4(x,u,minus2(z,s1(x)),u)
        f4(s1(x),s1(y),z,u) -> if3(le2(x,y),f4(s1(x),minus2(y,x),z,u),f4(x,u,z,u))
        map2(fun,nil()) -> nil()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil()) -> nil()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false(),fun,x,xs) -> filter2(fun,xs)
        app(minus1(x8),x9) -> minus2(x8,x9)
        app(minus(),x9) -> minus1(x9)
        app(s(),x9) -> s1(x9)
        app(le1(x8),x9) -> le2(x8,x9)
        app(le(),x9) -> le1(x9)
        app(if2(x8,x7),x9) -> if3(x8,x7,x9)
        app(if1(x8),x9) -> if2(x8,x9)
        app(if(),x9) -> if1(x9)
        app(perfectp(),x9) -> perfectp1(x9)
        app(f3(x8,x7,x6),x9) -> f4(x8,x7,x6,x9)
        app(f2(x8,x7),x9) -> f3(x8,x7,x9)
        app(f1(x8),x9) -> f2(x8,x9)
        app(f(),x9) -> f1(x9)
        app(map1(x8),x9) -> map2(x8,x9)
        app(map(),x9) -> map1(x9)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons(),x9) -> cons1(x9)
        app(filter1(x8),x9) -> filter2(x8,x9)
        app(filter(),x9) -> filter1(x9)
        app(filter23(x8,x7,x6),x9) -> filter24(x8,x7,x6,x9)
        app(filter22(x8,x7),x9) -> filter23(x8,x7,x9)
        app(filter21(x8),x9) -> filter22(x8,x9)
        app(filter2(),x9) -> filter21(x9)
      Qed