/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES

Problem:
 app(p(),app(s(),x)) -> x
 app(fact(),0()) -> app(s(),0())
 app(fact(),app(s(),x)) -> app(app(*(),app(s(),x)),app(fact(),app(p(),app(s(),x))))
 app(app(*(),0()),y) -> 0()
 app(app(*(),app(s(),x)),y) -> app(app(+(),app(app(*(),x),y)),y)
 app(app(+(),x),0()) -> x
 app(app(+(),x),app(s(),y)) -> app(s(),app(app(+(),x),y))
 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:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   false ==> false0/0
   true ==> true0/0
   filter2 ==> filter20/0 filter21/1 filter22/2 filter23/3 filter24/4
   filter ==> filter0/0 filter1/1 filter2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   + ==> +0/0 +1/1 +2/2
   * ==> *0/0 *1/1 *2/2
   0 ==> 00/0
   fact ==> fact0/0 fact1/1
   s ==> s0/0 s1/1
   p ==> p0/0 p1/1
   
  uncurry-rules:
   app(p0(),x4) -> p1(x4)
   app(s0(),x6) -> s1(x6)
   app(fact0(),x8) -> fact1(x8)
   app(*1(x11),x12) -> *2(x11,x12)
   app(*0(),x11) -> *1(x11)
   app(+1(x14),x15) -> +2(x14,x15)
   app(+0(),x14) -> +1(x14)
   app(map1(x17),x18) -> map2(x17,x18)
   app(map0(),x17) -> map1(x17)
   app(cons1(x21),x22) -> cons2(x21,x22)
   app(cons0(),x21) -> cons1(x21)
   app(filter1(x24),x25) -> filter2(x24,x25)
   app(filter0(),x24) -> filter1(x24)
   app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
   app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
   app(filter21(x27),x28) -> filter22(x27,x28)
   app(filter20(),x27) -> filter21(x27)
  eta-rules:
   
  problem:
   p1(s1(x)) -> x
   fact1(00()) -> s1(00())
   fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
   *2(00(),y) -> 00()
   *2(s1(x),y) -> +2(*2(x,y),y)
   +2(x,00()) -> x
   +2(x,s1(y)) -> s1(+2(x,y))
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   filter2(f,nil0()) -> nil0()
   filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
   filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
   filter24(false0(),f,x,xs) -> filter2(f,xs)
   app(p0(),x4) -> p1(x4)
   app(s0(),x6) -> s1(x6)
   app(fact0(),x8) -> fact1(x8)
   app(*1(x11),x12) -> *2(x11,x12)
   app(*0(),x11) -> *1(x11)
   app(+1(x14),x15) -> +2(x14,x15)
   app(+0(),x14) -> +1(x14)
   app(map1(x17),x18) -> map2(x17,x18)
   app(map0(),x17) -> map1(x17)
   app(cons1(x21),x22) -> cons2(x21,x22)
   app(cons0(),x21) -> cons1(x21)
   app(filter1(x24),x25) -> filter2(x24,x25)
   app(filter0(),x24) -> filter1(x24)
   app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
   app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
   app(filter21(x27),x28) -> filter22(x27,x28)
   app(filter20(),x27) -> filter21(x27)
  DP Processor:
   DPs:
    fact{1,#}(s1(x)) -> p{1,#}(s1(x))
    fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x)))
    fact{1,#}(s1(x)) -> *{2,#}(s1(x),fact1(p1(s1(x))))
    *{2,#}(s1(x),y) -> *{2,#}(x,y)
    *{2,#}(s1(x),y) -> +{2,#}(*2(x,y),y)
    +{2,#}(x,s1(y)) -> +{2,#}(x,y)
    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,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
    filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
    app#(p0(),x4) -> p{1,#}(x4)
    app#(fact0(),x8) -> fact{1,#}(x8)
    app#(*1(x11),x12) -> *{2,#}(x11,x12)
    app#(+1(x14),x15) -> +{2,#}(x14,x15)
    app#(map1(x17),x18) -> map{2,#}(x17,x18)
    app#(filter1(x24),x25) -> filter{2,#}(x24,x25)
    app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30)
   TRS:
    p1(s1(x)) -> x
    fact1(00()) -> s1(00())
    fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
    *2(00(),y) -> 00()
    *2(s1(x),y) -> +2(*2(x,y),y)
    +2(x,00()) -> x
    +2(x,s1(y)) -> s1(+2(x,y))
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    filter2(f,nil0()) -> nil0()
    filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
    filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
    filter24(false0(),f,x,xs) -> filter2(f,xs)
    app(p0(),x4) -> p1(x4)
    app(s0(),x6) -> s1(x6)
    app(fact0(),x8) -> fact1(x8)
    app(*1(x11),x12) -> *2(x11,x12)
    app(*0(),x11) -> *1(x11)
    app(+1(x14),x15) -> +2(x14,x15)
    app(+0(),x14) -> +1(x14)
    app(map1(x17),x18) -> map2(x17,x18)
    app(map0(),x17) -> map1(x17)
    app(cons1(x21),x22) -> cons2(x21,x22)
    app(cons0(),x21) -> cons1(x21)
    app(filter1(x24),x25) -> filter2(x24,x25)
    app(filter0(),x24) -> filter1(x24)
    app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
    app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
    app(filter21(x27),x28) -> filter22(x27,x28)
    app(filter20(),x27) -> filter21(x27)
   TDG Processor:
    DPs:
     fact{1,#}(s1(x)) -> p{1,#}(s1(x))
     fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x)))
     fact{1,#}(s1(x)) -> *{2,#}(s1(x),fact1(p1(s1(x))))
     *{2,#}(s1(x),y) -> *{2,#}(x,y)
     *{2,#}(s1(x),y) -> +{2,#}(*2(x,y),y)
     +{2,#}(x,s1(y)) -> +{2,#}(x,y)
     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,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(p0(),x4) -> p{1,#}(x4)
     app#(fact0(),x8) -> fact{1,#}(x8)
     app#(*1(x11),x12) -> *{2,#}(x11,x12)
     app#(+1(x14),x15) -> +{2,#}(x14,x15)
     app#(map1(x17),x18) -> map{2,#}(x17,x18)
     app#(filter1(x24),x25) -> filter{2,#}(x24,x25)
     app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30)
    TRS:
     p1(s1(x)) -> x
     fact1(00()) -> s1(00())
     fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
     *2(00(),y) -> 00()
     *2(s1(x),y) -> +2(*2(x,y),y)
     +2(x,00()) -> x
     +2(x,s1(y)) -> s1(+2(x,y))
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     filter2(f,nil0()) -> nil0()
     filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
     filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
     filter24(false0(),f,x,xs) -> filter2(f,xs)
     app(p0(),x4) -> p1(x4)
     app(s0(),x6) -> s1(x6)
     app(fact0(),x8) -> fact1(x8)
     app(*1(x11),x12) -> *2(x11,x12)
     app(*0(),x11) -> *1(x11)
     app(+1(x14),x15) -> +2(x14,x15)
     app(+0(),x14) -> +1(x14)
     app(map1(x17),x18) -> map2(x17,x18)
     app(map0(),x17) -> map1(x17)
     app(cons1(x21),x22) -> cons2(x21,x22)
     app(cons0(),x21) -> cons1(x21)
     app(filter1(x24),x25) -> filter2(x24,x25)
     app(filter0(),x24) -> filter1(x24)
     app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
     app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
     app(filter21(x27),x28) -> filter22(x27,x28)
     app(filter20(),x27) -> filter21(x27)
    graph:
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     filter2{4,#}(true0(),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,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x24),x25) -> filter{2,#}(x24,x25)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x17),x18) -> map{2,#}(x17,x18)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(+1(x14),x15) -> +{2,#}(x14,x15)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(*1(x11),x12) -> *{2,#}(x11,x12)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(fact0(),x8) -> fact{1,#}(x8)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(p0(),x4) -> p{1,#}(x4)
     app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30) ->
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x24),x25) -> filter{2,#}(x24,x25) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x24),x25) -> filter{2,#}(x24,x25) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x17),x18) -> map{2,#}(x17,x18) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x17),x18) -> map{2,#}(x17,x18) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(+1(x14),x15) -> +{2,#}(x14,x15) ->
     +{2,#}(x,s1(y)) -> +{2,#}(x,y)
     app#(*1(x11),x12) -> *{2,#}(x11,x12) ->
     *{2,#}(s1(x),y) -> +{2,#}(*2(x,y),y)
     app#(*1(x11),x12) -> *{2,#}(x11,x12) ->
     *{2,#}(s1(x),y) -> *{2,#}(x,y)
     app#(fact0(),x8) -> fact{1,#}(x8) ->
     fact{1,#}(s1(x)) -> *{2,#}(s1(x),fact1(p1(s1(x))))
     app#(fact0(),x8) -> fact{1,#}(x8) ->
     fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x)))
     app#(fact0(),x8) -> fact{1,#}(x8) ->
     fact{1,#}(s1(x)) -> p{1,#}(s1(x))
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x24),x25) -> filter{2,#}(x24,x25)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x17),x18) -> map{2,#}(x17,x18)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(+1(x14),x15) -> +{2,#}(x14,x15)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(*1(x11),x12) -> *{2,#}(x11,x12)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(fact0(),x8) -> fact{1,#}(x8)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(p0(),x4) -> p{1,#}(x4)
     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,#}(x,s1(y)) -> +{2,#}(x,y) ->
     +{2,#}(x,s1(y)) -> +{2,#}(x,y)
     *{2,#}(s1(x),y) -> +{2,#}(*2(x,y),y) ->
     +{2,#}(x,s1(y)) -> +{2,#}(x,y)
     *{2,#}(s1(x),y) -> *{2,#}(x,y) ->
     *{2,#}(s1(x),y) -> +{2,#}(*2(x,y),y)
     *{2,#}(s1(x),y) -> *{2,#}(x,y) ->
     *{2,#}(s1(x),y) -> *{2,#}(x,y)
     fact{1,#}(s1(x)) -> *{2,#}(s1(x),fact1(p1(s1(x)))) ->
     *{2,#}(s1(x),y) -> +{2,#}(*2(x,y),y)
     fact{1,#}(s1(x)) -> *{2,#}(s1(x),fact1(p1(s1(x)))) ->
     *{2,#}(s1(x),y) -> *{2,#}(x,y)
     fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x))) ->
     fact{1,#}(s1(x)) -> *{2,#}(s1(x),fact1(p1(s1(x))))
     fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x))) ->
     fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x)))
     fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x))) -> fact{1,#}(s1(x)) -> p{1,#}(s1(x))
    SCC Processor:
     #sccs: 4
     #rules: 12
     #arcs: 43/361
     DPs:
      filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x17),x18) -> map{2,#}(x17,x18)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x24),x25) -> filter{2,#}(x24,x25)
      filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
      filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
      app#(filter23(x27,x28,x29),x30) -> filter2{4,#}(x27,x28,x29,x30)
     TRS:
      p1(s1(x)) -> x
      fact1(00()) -> s1(00())
      fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
      *2(00(),y) -> 00()
      *2(s1(x),y) -> +2(*2(x,y),y)
      +2(x,00()) -> x
      +2(x,s1(y)) -> s1(+2(x,y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(p0(),x4) -> p1(x4)
      app(s0(),x6) -> s1(x6)
      app(fact0(),x8) -> fact1(x8)
      app(*1(x11),x12) -> *2(x11,x12)
      app(*0(),x11) -> *1(x11)
      app(+1(x14),x15) -> +2(x14,x15)
      app(+0(),x14) -> +1(x14)
      app(map1(x17),x18) -> map2(x17,x18)
      app(map0(),x17) -> map1(x17)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
      app(filter1(x24),x25) -> filter2(x24,x25)
      app(filter0(),x24) -> filter1(x24)
      app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
      app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
      app(filter21(x27),x28) -> filter22(x27,x28)
      app(filter20(),x27) -> filter21(x27)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 0
       pi(app#) = 0
       pi(filter{2,#}) = 0
       pi(filter2{4,#}) = 1
      problem:
       DPs:
        filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
        filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
        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)) -> filter2{4,#}(app(f,x),f,x,xs)
        filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
       TRS:
        p1(s1(x)) -> x
        fact1(00()) -> s1(00())
        fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
        *2(00(),y) -> 00()
        *2(s1(x),y) -> +2(*2(x,y),y)
        +2(x,00()) -> x
        +2(x,s1(y)) -> s1(+2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(p0(),x4) -> p1(x4)
        app(s0(),x6) -> s1(x6)
        app(fact0(),x8) -> fact1(x8)
        app(*1(x11),x12) -> *2(x11,x12)
        app(*0(),x11) -> *1(x11)
        app(+1(x14),x15) -> +2(x14,x15)
        app(+0(),x14) -> +1(x14)
        app(map1(x17),x18) -> map2(x17,x18)
        app(map0(),x17) -> map1(x17)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
        app(filter1(x24),x25) -> filter2(x24,x25)
        app(filter0(),x24) -> filter1(x24)
        app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
        app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
        app(filter21(x27),x28) -> filter22(x27,x28)
        app(filter20(),x27) -> filter21(x27)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        p1(s1(x)) -> x
        fact1(00()) -> s1(00())
        fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
        *2(00(),y) -> 00()
        *2(s1(x),y) -> +2(*2(x,y),y)
        +2(x,00()) -> x
        +2(x,s1(y)) -> s1(+2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(p0(),x4) -> p1(x4)
        app(s0(),x6) -> s1(x6)
        app(fact0(),x8) -> fact1(x8)
        app(*1(x11),x12) -> *2(x11,x12)
        app(*0(),x11) -> *1(x11)
        app(+1(x14),x15) -> +2(x14,x15)
        app(+0(),x14) -> +1(x14)
        app(map1(x17),x18) -> map2(x17,x18)
        app(map0(),x17) -> map1(x17)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
        app(filter1(x24),x25) -> filter2(x24,x25)
        app(filter0(),x24) -> filter1(x24)
        app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
        app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
        app(filter21(x27),x28) -> filter22(x27,x28)
        app(filter20(),x27) -> filter21(x27)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          p1(s1(x)) -> x
          fact1(00()) -> s1(00())
          fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
          *2(00(),y) -> 00()
          *2(s1(x),y) -> +2(*2(x,y),y)
          +2(x,00()) -> x
          +2(x,s1(y)) -> s1(+2(x,y))
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          filter2(f,nil0()) -> nil0()
          filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
          filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
          filter24(false0(),f,x,xs) -> filter2(f,xs)
          app(p0(),x4) -> p1(x4)
          app(s0(),x6) -> s1(x6)
          app(fact0(),x8) -> fact1(x8)
          app(*1(x11),x12) -> *2(x11,x12)
          app(*0(),x11) -> *1(x11)
          app(+1(x14),x15) -> +2(x14,x15)
          app(+0(),x14) -> +1(x14)
          app(map1(x17),x18) -> map2(x17,x18)
          app(map0(),x17) -> map1(x17)
          app(cons1(x21),x22) -> cons2(x21,x22)
          app(cons0(),x21) -> cons1(x21)
          app(filter1(x24),x25) -> filter2(x24,x25)
          app(filter0(),x24) -> filter1(x24)
          app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
          app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
          app(filter21(x27),x28) -> filter22(x27,x28)
          app(filter20(),x27) -> filter21(x27)
        Qed
       
       DPs:
        filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
        filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
        filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
       TRS:
        p1(s1(x)) -> x
        fact1(00()) -> s1(00())
        fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
        *2(00(),y) -> 00()
        *2(s1(x),y) -> +2(*2(x,y),y)
        +2(x,00()) -> x
        +2(x,s1(y)) -> s1(+2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(p0(),x4) -> p1(x4)
        app(s0(),x6) -> s1(x6)
        app(fact0(),x8) -> fact1(x8)
        app(*1(x11),x12) -> *2(x11,x12)
        app(*0(),x11) -> *1(x11)
        app(+1(x14),x15) -> +2(x14,x15)
        app(+0(),x14) -> +1(x14)
        app(map1(x17),x18) -> map2(x17,x18)
        app(map0(),x17) -> map1(x17)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
        app(filter1(x24),x25) -> filter2(x24,x25)
        app(filter0(),x24) -> filter1(x24)
        app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
        app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
        app(filter21(x27),x28) -> filter22(x27,x28)
        app(filter20(),x27) -> filter21(x27)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
         pi(filter2{4,#}) = 3
        problem:
         DPs:
          filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
          filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
         TRS:
          p1(s1(x)) -> x
          fact1(00()) -> s1(00())
          fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
          *2(00(),y) -> 00()
          *2(s1(x),y) -> +2(*2(x,y),y)
          +2(x,00()) -> x
          +2(x,s1(y)) -> s1(+2(x,y))
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          filter2(f,nil0()) -> nil0()
          filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
          filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
          filter24(false0(),f,x,xs) -> filter2(f,xs)
          app(p0(),x4) -> p1(x4)
          app(s0(),x6) -> s1(x6)
          app(fact0(),x8) -> fact1(x8)
          app(*1(x11),x12) -> *2(x11,x12)
          app(*0(),x11) -> *1(x11)
          app(+1(x14),x15) -> +2(x14,x15)
          app(+0(),x14) -> +1(x14)
          app(map1(x17),x18) -> map2(x17,x18)
          app(map0(),x17) -> map1(x17)
          app(cons1(x21),x22) -> cons2(x21,x22)
          app(cons0(),x21) -> cons1(x21)
          app(filter1(x24),x25) -> filter2(x24,x25)
          app(filter0(),x24) -> filter1(x24)
          app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
          app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
          app(filter21(x27),x28) -> filter22(x27,x28)
          app(filter20(),x27) -> filter21(x27)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x)))
     TRS:
      p1(s1(x)) -> x
      fact1(00()) -> s1(00())
      fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
      *2(00(),y) -> 00()
      *2(s1(x),y) -> +2(*2(x,y),y)
      +2(x,00()) -> x
      +2(x,s1(y)) -> s1(+2(x,y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(p0(),x4) -> p1(x4)
      app(s0(),x6) -> s1(x6)
      app(fact0(),x8) -> fact1(x8)
      app(*1(x11),x12) -> *2(x11,x12)
      app(*0(),x11) -> *1(x11)
      app(+1(x14),x15) -> +2(x14,x15)
      app(+0(),x14) -> +1(x14)
      app(map1(x17),x18) -> map2(x17,x18)
      app(map0(),x17) -> map1(x17)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
      app(filter1(x24),x25) -> filter2(x24,x25)
      app(filter0(),x24) -> filter1(x24)
      app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
      app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
      app(filter21(x27),x28) -> filter22(x27,x28)
      app(filter20(),x27) -> filter21(x27)
     Usable Rule Processor:
      DPs:
       fact{1,#}(s1(x)) -> fact{1,#}(p1(s1(x)))
      TRS:
       p1(s1(x)) -> x
      Semantic Labeling Processor:
       dimension: 2
       usable rules:
        
       interpretation:
                   [0 1]  
        [p1](x0) = [0 1]x0,
        
                   [1 1]     [1]
        [s1](x0) = [1 1]x0 + [0]
        labeled:
         fact{1,#}
        usable (for model):
         fact{1,#}
         s1
         p1
        argument filtering:
         pi(p1) = 0
         pi(s1) = 0
         pi(fact{1,#}) = []
       precedence:
        fact{1,#} ~ s1 ~ p1
       problem:
        DPs:
         
        TRS:
         p1(s1(x)) -> x
       Qed
     
     DPs:
      *{2,#}(s1(x),y) -> *{2,#}(x,y)
     TRS:
      p1(s1(x)) -> x
      fact1(00()) -> s1(00())
      fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
      *2(00(),y) -> 00()
      *2(s1(x),y) -> +2(*2(x,y),y)
      +2(x,00()) -> x
      +2(x,s1(y)) -> s1(+2(x,y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(p0(),x4) -> p1(x4)
      app(s0(),x6) -> s1(x6)
      app(fact0(),x8) -> fact1(x8)
      app(*1(x11),x12) -> *2(x11,x12)
      app(*0(),x11) -> *1(x11)
      app(+1(x14),x15) -> +2(x14,x15)
      app(+0(),x14) -> +1(x14)
      app(map1(x17),x18) -> map2(x17,x18)
      app(map0(),x17) -> map1(x17)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
      app(filter1(x24),x25) -> filter2(x24,x25)
      app(filter0(),x24) -> filter1(x24)
      app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
      app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
      app(filter21(x27),x28) -> filter22(x27,x28)
      app(filter20(),x27) -> filter21(x27)
     Subterm Criterion Processor:
      simple projection:
       pi(*{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        p1(s1(x)) -> x
        fact1(00()) -> s1(00())
        fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
        *2(00(),y) -> 00()
        *2(s1(x),y) -> +2(*2(x,y),y)
        +2(x,00()) -> x
        +2(x,s1(y)) -> s1(+2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(p0(),x4) -> p1(x4)
        app(s0(),x6) -> s1(x6)
        app(fact0(),x8) -> fact1(x8)
        app(*1(x11),x12) -> *2(x11,x12)
        app(*0(),x11) -> *1(x11)
        app(+1(x14),x15) -> +2(x14,x15)
        app(+0(),x14) -> +1(x14)
        app(map1(x17),x18) -> map2(x17,x18)
        app(map0(),x17) -> map1(x17)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
        app(filter1(x24),x25) -> filter2(x24,x25)
        app(filter0(),x24) -> filter1(x24)
        app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
        app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
        app(filter21(x27),x28) -> filter22(x27,x28)
        app(filter20(),x27) -> filter21(x27)
      Qed
     
     DPs:
      +{2,#}(x,s1(y)) -> +{2,#}(x,y)
     TRS:
      p1(s1(x)) -> x
      fact1(00()) -> s1(00())
      fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
      *2(00(),y) -> 00()
      *2(s1(x),y) -> +2(*2(x,y),y)
      +2(x,00()) -> x
      +2(x,s1(y)) -> s1(+2(x,y))
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      filter2(f,nil0()) -> nil0()
      filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
      filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
      filter24(false0(),f,x,xs) -> filter2(f,xs)
      app(p0(),x4) -> p1(x4)
      app(s0(),x6) -> s1(x6)
      app(fact0(),x8) -> fact1(x8)
      app(*1(x11),x12) -> *2(x11,x12)
      app(*0(),x11) -> *1(x11)
      app(+1(x14),x15) -> +2(x14,x15)
      app(+0(),x14) -> +1(x14)
      app(map1(x17),x18) -> map2(x17,x18)
      app(map0(),x17) -> map1(x17)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
      app(filter1(x24),x25) -> filter2(x24,x25)
      app(filter0(),x24) -> filter1(x24)
      app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
      app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
      app(filter21(x27),x28) -> filter22(x27,x28)
      app(filter20(),x27) -> filter21(x27)
     Subterm Criterion Processor:
      simple projection:
       pi(+{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        p1(s1(x)) -> x
        fact1(00()) -> s1(00())
        fact1(s1(x)) -> *2(s1(x),fact1(p1(s1(x))))
        *2(00(),y) -> 00()
        *2(s1(x),y) -> +2(*2(x,y),y)
        +2(x,00()) -> x
        +2(x,s1(y)) -> s1(+2(x,y))
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        filter2(f,nil0()) -> nil0()
        filter2(f,cons2(x,xs)) -> filter24(app(f,x),f,x,xs)
        filter24(true0(),f,x,xs) -> cons2(x,filter2(f,xs))
        filter24(false0(),f,x,xs) -> filter2(f,xs)
        app(p0(),x4) -> p1(x4)
        app(s0(),x6) -> s1(x6)
        app(fact0(),x8) -> fact1(x8)
        app(*1(x11),x12) -> *2(x11,x12)
        app(*0(),x11) -> *1(x11)
        app(+1(x14),x15) -> +2(x14,x15)
        app(+0(),x14) -> +1(x14)
        app(map1(x17),x18) -> map2(x17,x18)
        app(map0(),x17) -> map1(x17)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
        app(filter1(x24),x25) -> filter2(x24,x25)
        app(filter0(),x24) -> filter1(x24)
        app(filter23(x27,x28,x29),x30) -> filter24(x27,x28,x29,x30)
        app(filter22(x27,x28),x29) -> filter23(x27,x28,x29)
        app(filter21(x27),x28) -> filter22(x27,x28)
        app(filter20(),x27) -> filter21(x27)
      Qed