/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem:
 app(app(app(if(),true()),xs),ys) -> xs
 app(app(app(if(),false()),xs),ys) -> ys
 app(app(sub(),x),0()) -> x
 app(app(sub(),app(s(),x)),app(s(),y)) -> app(app(sub(),x),y)
 app(app(gtr(),0()),y) -> false()
 app(app(gtr(),app(s(),x)),0()) -> true()
 app(app(gtr(),app(s(),x)),app(s(),y)) -> app(app(gtr(),x),y)
 app(app(d(),x),0()) -> true()
 app(app(d(),app(s(),x)),app(s(),y)) ->
 app(app(app(if(),app(app(gtr(),x),y)),false()),app(app(d(),app(s(),x)),app(app(sub(),y),x)))
 app(len(),nil()) -> 0()
 app(len(),app(app(cons(),x),xs)) -> app(s(),app(len(),xs))
 app(app(filter(),p),nil()) -> nil()
 app(app(filter(),p),app(app(cons(),x),xs)) ->
 app(app(app(if(),app(p,x)),app(app(cons(),x),app(app(filter(),p),xs))),app(app(filter(),p),xs))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   filter ==> filter0/0 filter1/1 filter2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   len ==> len0/0 len1/1
   d ==> d0/0 d1/1 d2/2
   gtr ==> gtr0/0 gtr1/1 gtr2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   sub ==> sub0/0 sub1/1 sub2/2
   false ==> false0/0
   true ==> true0/0
   if ==> if0/0 if1/1 if2/2 if3/3
   
  uncurry-rules:
   app(if2(x5,x6),x7) -> if3(x5,x6,x7)
   app(if1(x5),x6) -> if2(x5,x6)
   app(if0(),x5) -> if1(x5)
   app(sub1(x11),x12) -> sub2(x11,x12)
   app(sub0(),x11) -> sub1(x11)
   app(s0(),x15) -> s1(x15)
   app(gtr1(x17),x18) -> gtr2(x17,x18)
   app(gtr0(),x17) -> gtr1(x17)
   app(d1(x20),x21) -> d2(x20,x21)
   app(d0(),x20) -> d1(x20)
   app(len0(),x23) -> len1(x23)
   app(cons1(x26),x27) -> cons2(x26,x27)
   app(cons0(),x26) -> cons1(x26)
   app(filter1(x29),x30) -> filter2(x29,x30)
   app(filter0(),x29) -> filter1(x29)
  eta-rules:
   
  problem:
   if3(true0(),xs,ys) -> xs
   if3(false0(),xs,ys) -> ys
   sub2(x,00()) -> x
   sub2(s1(x),s1(y)) -> sub2(x,y)
   gtr2(00(),y) -> false0()
   gtr2(s1(x),00()) -> true0()
   gtr2(s1(x),s1(y)) -> gtr2(x,y)
   d2(x,00()) -> true0()
   d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
   len1(nil0()) -> 00()
   len1(cons2(x,xs)) -> s1(len1(xs))
   filter2(p,nil0()) -> nil0()
   filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
   app(if2(x5,x6),x7) -> if3(x5,x6,x7)
   app(if1(x5),x6) -> if2(x5,x6)
   app(if0(),x5) -> if1(x5)
   app(sub1(x11),x12) -> sub2(x11,x12)
   app(sub0(),x11) -> sub1(x11)
   app(s0(),x15) -> s1(x15)
   app(gtr1(x17),x18) -> gtr2(x17,x18)
   app(gtr0(),x17) -> gtr1(x17)
   app(d1(x20),x21) -> d2(x20,x21)
   app(d0(),x20) -> d1(x20)
   app(len0(),x23) -> len1(x23)
   app(cons1(x26),x27) -> cons2(x26,x27)
   app(cons0(),x26) -> cons1(x26)
   app(filter1(x29),x30) -> filter2(x29,x30)
   app(filter0(),x29) -> filter1(x29)
  DP Processor:
   DPs:
    sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y)
    gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
    d{2,#}(s1(x),s1(y)) -> sub{2,#}(y,x)
    d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x))
    d{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
    d{2,#}(s1(x),s1(y)) -> if{3,#}(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
    len{1,#}(cons2(x,xs)) -> len{1,#}(xs)
    filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
    filter{2,#}(p,cons2(x,xs)) -> app#(p,x)
    filter{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
    app#(if2(x5,x6),x7) -> if{3,#}(x5,x6,x7)
    app#(sub1(x11),x12) -> sub{2,#}(x11,x12)
    app#(gtr1(x17),x18) -> gtr{2,#}(x17,x18)
    app#(d1(x20),x21) -> d{2,#}(x20,x21)
    app#(len0(),x23) -> len{1,#}(x23)
    app#(filter1(x29),x30) -> filter{2,#}(x29,x30)
   TRS:
    if3(true0(),xs,ys) -> xs
    if3(false0(),xs,ys) -> ys
    sub2(x,00()) -> x
    sub2(s1(x),s1(y)) -> sub2(x,y)
    gtr2(00(),y) -> false0()
    gtr2(s1(x),00()) -> true0()
    gtr2(s1(x),s1(y)) -> gtr2(x,y)
    d2(x,00()) -> true0()
    d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
    len1(nil0()) -> 00()
    len1(cons2(x,xs)) -> s1(len1(xs))
    filter2(p,nil0()) -> nil0()
    filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
    app(if2(x5,x6),x7) -> if3(x5,x6,x7)
    app(if1(x5),x6) -> if2(x5,x6)
    app(if0(),x5) -> if1(x5)
    app(sub1(x11),x12) -> sub2(x11,x12)
    app(sub0(),x11) -> sub1(x11)
    app(s0(),x15) -> s1(x15)
    app(gtr1(x17),x18) -> gtr2(x17,x18)
    app(gtr0(),x17) -> gtr1(x17)
    app(d1(x20),x21) -> d2(x20,x21)
    app(d0(),x20) -> d1(x20)
    app(len0(),x23) -> len1(x23)
    app(cons1(x26),x27) -> cons2(x26,x27)
    app(cons0(),x26) -> cons1(x26)
    app(filter1(x29),x30) -> filter2(x29,x30)
    app(filter0(),x29) -> filter1(x29)
   TDG Processor:
    DPs:
     sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y)
     gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     d{2,#}(s1(x),s1(y)) -> sub{2,#}(y,x)
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x))
     d{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     d{2,#}(s1(x),s1(y)) -> if{3,#}(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
     len{1,#}(cons2(x,xs)) -> len{1,#}(xs)
     filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x)
     filter{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
     app#(if2(x5,x6),x7) -> if{3,#}(x5,x6,x7)
     app#(sub1(x11),x12) -> sub{2,#}(x11,x12)
     app#(gtr1(x17),x18) -> gtr{2,#}(x17,x18)
     app#(d1(x20),x21) -> d{2,#}(x20,x21)
     app#(len0(),x23) -> len{1,#}(x23)
     app#(filter1(x29),x30) -> filter{2,#}(x29,x30)
    TRS:
     if3(true0(),xs,ys) -> xs
     if3(false0(),xs,ys) -> ys
     sub2(x,00()) -> x
     sub2(s1(x),s1(y)) -> sub2(x,y)
     gtr2(00(),y) -> false0()
     gtr2(s1(x),00()) -> true0()
     gtr2(s1(x),s1(y)) -> gtr2(x,y)
     d2(x,00()) -> true0()
     d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
     len1(nil0()) -> 00()
     len1(cons2(x,xs)) -> s1(len1(xs))
     filter2(p,nil0()) -> nil0()
     filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
     app(if2(x5,x6),x7) -> if3(x5,x6,x7)
     app(if1(x5),x6) -> if2(x5,x6)
     app(if0(),x5) -> if1(x5)
     app(sub1(x11),x12) -> sub2(x11,x12)
     app(sub0(),x11) -> sub1(x11)
     app(s0(),x15) -> s1(x15)
     app(gtr1(x17),x18) -> gtr2(x17,x18)
     app(gtr0(),x17) -> gtr1(x17)
     app(d1(x20),x21) -> d2(x20,x21)
     app(d0(),x20) -> d1(x20)
     app(len0(),x23) -> len1(x23)
     app(cons1(x26),x27) -> cons2(x26,x27)
     app(cons0(),x26) -> cons1(x26)
     app(filter1(x29),x30) -> filter2(x29,x30)
     app(filter0(),x29) -> filter1(x29)
    graph:
     app#(filter1(x29),x30) -> filter{2,#}(x29,x30) ->
     filter{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
     app#(filter1(x29),x30) -> filter{2,#}(x29,x30) ->
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x)
     app#(filter1(x29),x30) -> filter{2,#}(x29,x30) ->
     filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
     app#(len0(),x23) -> len{1,#}(x23) ->
     len{1,#}(cons2(x,xs)) -> len{1,#}(xs)
     app#(d1(x20),x21) -> d{2,#}(x20,x21) ->
     d{2,#}(s1(x),s1(y)) -> if{3,#}(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
     app#(d1(x20),x21) -> d{2,#}(x20,x21) ->
     d{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     app#(d1(x20),x21) -> d{2,#}(x20,x21) ->
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x))
     app#(d1(x20),x21) -> d{2,#}(x20,x21) ->
     d{2,#}(s1(x),s1(y)) -> sub{2,#}(y,x)
     app#(gtr1(x17),x18) -> gtr{2,#}(x17,x18) ->
     gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     app#(sub1(x11),x12) -> sub{2,#}(x11,x12) ->
     sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(filter1(x29),x30) -> filter{2,#}(x29,x30)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(len0(),x23) -> len{1,#}(x23)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(d1(x20),x21) -> d{2,#}(x20,x21)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(gtr1(x17),x18) -> gtr{2,#}(x17,x18)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(sub1(x11),x12) -> sub{2,#}(x11,x12)
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(if2(x5,x6),x7) -> if{3,#}(x5,x6,x7)
     filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs) ->
     filter{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
     filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs) ->
     filter{2,#}(p,cons2(x,xs)) -> app#(p,x)
     filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs) ->
     filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
     len{1,#}(cons2(x,xs)) -> len{1,#}(xs) ->
     len{1,#}(cons2(x,xs)) -> len{1,#}(xs)
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x)) ->
     d{2,#}(s1(x),s1(y)) -> if{3,#}(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x)) ->
     d{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x)) ->
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x))
     d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x)) ->
     d{2,#}(s1(x),s1(y)) -> sub{2,#}(y,x)
     d{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y) ->
     gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     d{2,#}(s1(x),s1(y)) -> sub{2,#}(y,x) ->
     sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y)
     gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y) ->
     gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y) -> sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y)
    SCC Processor:
     #sccs: 5
     #rules: 7
     #arcs: 28/256
     DPs:
      app#(filter1(x29),x30) -> filter{2,#}(x29,x30)
      filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
      filter{2,#}(p,cons2(x,xs)) -> app#(p,x)
     TRS:
      if3(true0(),xs,ys) -> xs
      if3(false0(),xs,ys) -> ys
      sub2(x,00()) -> x
      sub2(s1(x),s1(y)) -> sub2(x,y)
      gtr2(00(),y) -> false0()
      gtr2(s1(x),00()) -> true0()
      gtr2(s1(x),s1(y)) -> gtr2(x,y)
      d2(x,00()) -> true0()
      d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
      len1(nil0()) -> 00()
      len1(cons2(x,xs)) -> s1(len1(xs))
      filter2(p,nil0()) -> nil0()
      filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
      app(if2(x5,x6),x7) -> if3(x5,x6,x7)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if0(),x5) -> if1(x5)
      app(sub1(x11),x12) -> sub2(x11,x12)
      app(sub0(),x11) -> sub1(x11)
      app(s0(),x15) -> s1(x15)
      app(gtr1(x17),x18) -> gtr2(x17,x18)
      app(gtr0(),x17) -> gtr1(x17)
      app(d1(x20),x21) -> d2(x20,x21)
      app(d0(),x20) -> d1(x20)
      app(len0(),x23) -> len1(x23)
      app(cons1(x26),x27) -> cons2(x26,x27)
      app(cons0(),x26) -> cons1(x26)
      app(filter1(x29),x30) -> filter2(x29,x30)
      app(filter0(),x29) -> filter1(x29)
     Subterm Criterion Processor:
      simple projection:
       pi(filter{2,#}) = 0
       pi(app#) = 0
      problem:
       DPs:
        filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
        filter{2,#}(p,cons2(x,xs)) -> app#(p,x)
       TRS:
        if3(true0(),xs,ys) -> xs
        if3(false0(),xs,ys) -> ys
        sub2(x,00()) -> x
        sub2(s1(x),s1(y)) -> sub2(x,y)
        gtr2(00(),y) -> false0()
        gtr2(s1(x),00()) -> true0()
        gtr2(s1(x),s1(y)) -> gtr2(x,y)
        d2(x,00()) -> true0()
        d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
        len1(nil0()) -> 00()
        len1(cons2(x,xs)) -> s1(len1(xs))
        filter2(p,nil0()) -> nil0()
        filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
        app(if2(x5,x6),x7) -> if3(x5,x6,x7)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if0(),x5) -> if1(x5)
        app(sub1(x11),x12) -> sub2(x11,x12)
        app(sub0(),x11) -> sub1(x11)
        app(s0(),x15) -> s1(x15)
        app(gtr1(x17),x18) -> gtr2(x17,x18)
        app(gtr0(),x17) -> gtr1(x17)
        app(d1(x20),x21) -> d2(x20,x21)
        app(d0(),x20) -> d1(x20)
        app(len0(),x23) -> len1(x23)
        app(cons1(x26),x27) -> cons2(x26,x27)
        app(cons0(),x26) -> cons1(x26)
        app(filter1(x29),x30) -> filter2(x29,x30)
        app(filter0(),x29) -> filter1(x29)
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 5/4
       DPs:
        filter{2,#}(p,cons2(x,xs)) -> filter{2,#}(p,xs)
       TRS:
        if3(true0(),xs,ys) -> xs
        if3(false0(),xs,ys) -> ys
        sub2(x,00()) -> x
        sub2(s1(x),s1(y)) -> sub2(x,y)
        gtr2(00(),y) -> false0()
        gtr2(s1(x),00()) -> true0()
        gtr2(s1(x),s1(y)) -> gtr2(x,y)
        d2(x,00()) -> true0()
        d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
        len1(nil0()) -> 00()
        len1(cons2(x,xs)) -> s1(len1(xs))
        filter2(p,nil0()) -> nil0()
        filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
        app(if2(x5,x6),x7) -> if3(x5,x6,x7)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if0(),x5) -> if1(x5)
        app(sub1(x11),x12) -> sub2(x11,x12)
        app(sub0(),x11) -> sub1(x11)
        app(s0(),x15) -> s1(x15)
        app(gtr1(x17),x18) -> gtr2(x17,x18)
        app(gtr0(),x17) -> gtr1(x17)
        app(d1(x20),x21) -> d2(x20,x21)
        app(d0(),x20) -> d1(x20)
        app(len0(),x23) -> len1(x23)
        app(cons1(x26),x27) -> cons2(x26,x27)
        app(cons0(),x26) -> cons1(x26)
        app(filter1(x29),x30) -> filter2(x29,x30)
        app(filter0(),x29) -> filter1(x29)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          if3(true0(),xs,ys) -> xs
          if3(false0(),xs,ys) -> ys
          sub2(x,00()) -> x
          sub2(s1(x),s1(y)) -> sub2(x,y)
          gtr2(00(),y) -> false0()
          gtr2(s1(x),00()) -> true0()
          gtr2(s1(x),s1(y)) -> gtr2(x,y)
          d2(x,00()) -> true0()
          d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
          len1(nil0()) -> 00()
          len1(cons2(x,xs)) -> s1(len1(xs))
          filter2(p,nil0()) -> nil0()
          filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
          app(if2(x5,x6),x7) -> if3(x5,x6,x7)
          app(if1(x5),x6) -> if2(x5,x6)
          app(if0(),x5) -> if1(x5)
          app(sub1(x11),x12) -> sub2(x11,x12)
          app(sub0(),x11) -> sub1(x11)
          app(s0(),x15) -> s1(x15)
          app(gtr1(x17),x18) -> gtr2(x17,x18)
          app(gtr0(),x17) -> gtr1(x17)
          app(d1(x20),x21) -> d2(x20,x21)
          app(d0(),x20) -> d1(x20)
          app(len0(),x23) -> len1(x23)
          app(cons1(x26),x27) -> cons2(x26,x27)
          app(cons0(),x26) -> cons1(x26)
          app(filter1(x29),x30) -> filter2(x29,x30)
          app(filter0(),x29) -> filter1(x29)
        Qed
     
     DPs:
      len{1,#}(cons2(x,xs)) -> len{1,#}(xs)
     TRS:
      if3(true0(),xs,ys) -> xs
      if3(false0(),xs,ys) -> ys
      sub2(x,00()) -> x
      sub2(s1(x),s1(y)) -> sub2(x,y)
      gtr2(00(),y) -> false0()
      gtr2(s1(x),00()) -> true0()
      gtr2(s1(x),s1(y)) -> gtr2(x,y)
      d2(x,00()) -> true0()
      d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
      len1(nil0()) -> 00()
      len1(cons2(x,xs)) -> s1(len1(xs))
      filter2(p,nil0()) -> nil0()
      filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
      app(if2(x5,x6),x7) -> if3(x5,x6,x7)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if0(),x5) -> if1(x5)
      app(sub1(x11),x12) -> sub2(x11,x12)
      app(sub0(),x11) -> sub1(x11)
      app(s0(),x15) -> s1(x15)
      app(gtr1(x17),x18) -> gtr2(x17,x18)
      app(gtr0(),x17) -> gtr1(x17)
      app(d1(x20),x21) -> d2(x20,x21)
      app(d0(),x20) -> d1(x20)
      app(len0(),x23) -> len1(x23)
      app(cons1(x26),x27) -> cons2(x26,x27)
      app(cons0(),x26) -> cons1(x26)
      app(filter1(x29),x30) -> filter2(x29,x30)
      app(filter0(),x29) -> filter1(x29)
     Subterm Criterion Processor:
      simple projection:
       pi(len{1,#}) = 0
      problem:
       DPs:
        
       TRS:
        if3(true0(),xs,ys) -> xs
        if3(false0(),xs,ys) -> ys
        sub2(x,00()) -> x
        sub2(s1(x),s1(y)) -> sub2(x,y)
        gtr2(00(),y) -> false0()
        gtr2(s1(x),00()) -> true0()
        gtr2(s1(x),s1(y)) -> gtr2(x,y)
        d2(x,00()) -> true0()
        d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
        len1(nil0()) -> 00()
        len1(cons2(x,xs)) -> s1(len1(xs))
        filter2(p,nil0()) -> nil0()
        filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
        app(if2(x5,x6),x7) -> if3(x5,x6,x7)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if0(),x5) -> if1(x5)
        app(sub1(x11),x12) -> sub2(x11,x12)
        app(sub0(),x11) -> sub1(x11)
        app(s0(),x15) -> s1(x15)
        app(gtr1(x17),x18) -> gtr2(x17,x18)
        app(gtr0(),x17) -> gtr1(x17)
        app(d1(x20),x21) -> d2(x20,x21)
        app(d0(),x20) -> d1(x20)
        app(len0(),x23) -> len1(x23)
        app(cons1(x26),x27) -> cons2(x26,x27)
        app(cons0(),x26) -> cons1(x26)
        app(filter1(x29),x30) -> filter2(x29,x30)
        app(filter0(),x29) -> filter1(x29)
      Qed
     
     DPs:
      d{2,#}(s1(x),s1(y)) -> d{2,#}(s1(x),sub2(y,x))
     TRS:
      if3(true0(),xs,ys) -> xs
      if3(false0(),xs,ys) -> ys
      sub2(x,00()) -> x
      sub2(s1(x),s1(y)) -> sub2(x,y)
      gtr2(00(),y) -> false0()
      gtr2(s1(x),00()) -> true0()
      gtr2(s1(x),s1(y)) -> gtr2(x,y)
      d2(x,00()) -> true0()
      d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
      len1(nil0()) -> 00()
      len1(cons2(x,xs)) -> s1(len1(xs))
      filter2(p,nil0()) -> nil0()
      filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
      app(if2(x5,x6),x7) -> if3(x5,x6,x7)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if0(),x5) -> if1(x5)
      app(sub1(x11),x12) -> sub2(x11,x12)
      app(sub0(),x11) -> sub1(x11)
      app(s0(),x15) -> s1(x15)
      app(gtr1(x17),x18) -> gtr2(x17,x18)
      app(gtr0(),x17) -> gtr1(x17)
      app(d1(x20),x21) -> d2(x20,x21)
      app(d0(),x20) -> d1(x20)
      app(len0(),x23) -> len1(x23)
      app(cons1(x26),x27) -> cons2(x26,x27)
      app(cons0(),x26) -> cons1(x26)
      app(filter1(x29),x30) -> filter2(x29,x30)
      app(filter0(),x29) -> filter1(x29)
     Subterm Criterion Processor:
      simple projection:
       pi(sub2) = 0
       pi(d{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        if3(true0(),xs,ys) -> xs
        if3(false0(),xs,ys) -> ys
        sub2(x,00()) -> x
        sub2(s1(x),s1(y)) -> sub2(x,y)
        gtr2(00(),y) -> false0()
        gtr2(s1(x),00()) -> true0()
        gtr2(s1(x),s1(y)) -> gtr2(x,y)
        d2(x,00()) -> true0()
        d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
        len1(nil0()) -> 00()
        len1(cons2(x,xs)) -> s1(len1(xs))
        filter2(p,nil0()) -> nil0()
        filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
        app(if2(x5,x6),x7) -> if3(x5,x6,x7)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if0(),x5) -> if1(x5)
        app(sub1(x11),x12) -> sub2(x11,x12)
        app(sub0(),x11) -> sub1(x11)
        app(s0(),x15) -> s1(x15)
        app(gtr1(x17),x18) -> gtr2(x17,x18)
        app(gtr0(),x17) -> gtr1(x17)
        app(d1(x20),x21) -> d2(x20,x21)
        app(d0(),x20) -> d1(x20)
        app(len0(),x23) -> len1(x23)
        app(cons1(x26),x27) -> cons2(x26,x27)
        app(cons0(),x26) -> cons1(x26)
        app(filter1(x29),x30) -> filter2(x29,x30)
        app(filter0(),x29) -> filter1(x29)
      Qed
     
     DPs:
      gtr{2,#}(s1(x),s1(y)) -> gtr{2,#}(x,y)
     TRS:
      if3(true0(),xs,ys) -> xs
      if3(false0(),xs,ys) -> ys
      sub2(x,00()) -> x
      sub2(s1(x),s1(y)) -> sub2(x,y)
      gtr2(00(),y) -> false0()
      gtr2(s1(x),00()) -> true0()
      gtr2(s1(x),s1(y)) -> gtr2(x,y)
      d2(x,00()) -> true0()
      d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
      len1(nil0()) -> 00()
      len1(cons2(x,xs)) -> s1(len1(xs))
      filter2(p,nil0()) -> nil0()
      filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
      app(if2(x5,x6),x7) -> if3(x5,x6,x7)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if0(),x5) -> if1(x5)
      app(sub1(x11),x12) -> sub2(x11,x12)
      app(sub0(),x11) -> sub1(x11)
      app(s0(),x15) -> s1(x15)
      app(gtr1(x17),x18) -> gtr2(x17,x18)
      app(gtr0(),x17) -> gtr1(x17)
      app(d1(x20),x21) -> d2(x20,x21)
      app(d0(),x20) -> d1(x20)
      app(len0(),x23) -> len1(x23)
      app(cons1(x26),x27) -> cons2(x26,x27)
      app(cons0(),x26) -> cons1(x26)
      app(filter1(x29),x30) -> filter2(x29,x30)
      app(filter0(),x29) -> filter1(x29)
     Subterm Criterion Processor:
      simple projection:
       pi(gtr{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        if3(true0(),xs,ys) -> xs
        if3(false0(),xs,ys) -> ys
        sub2(x,00()) -> x
        sub2(s1(x),s1(y)) -> sub2(x,y)
        gtr2(00(),y) -> false0()
        gtr2(s1(x),00()) -> true0()
        gtr2(s1(x),s1(y)) -> gtr2(x,y)
        d2(x,00()) -> true0()
        d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
        len1(nil0()) -> 00()
        len1(cons2(x,xs)) -> s1(len1(xs))
        filter2(p,nil0()) -> nil0()
        filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
        app(if2(x5,x6),x7) -> if3(x5,x6,x7)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if0(),x5) -> if1(x5)
        app(sub1(x11),x12) -> sub2(x11,x12)
        app(sub0(),x11) -> sub1(x11)
        app(s0(),x15) -> s1(x15)
        app(gtr1(x17),x18) -> gtr2(x17,x18)
        app(gtr0(),x17) -> gtr1(x17)
        app(d1(x20),x21) -> d2(x20,x21)
        app(d0(),x20) -> d1(x20)
        app(len0(),x23) -> len1(x23)
        app(cons1(x26),x27) -> cons2(x26,x27)
        app(cons0(),x26) -> cons1(x26)
        app(filter1(x29),x30) -> filter2(x29,x30)
        app(filter0(),x29) -> filter1(x29)
      Qed
     
     DPs:
      sub{2,#}(s1(x),s1(y)) -> sub{2,#}(x,y)
     TRS:
      if3(true0(),xs,ys) -> xs
      if3(false0(),xs,ys) -> ys
      sub2(x,00()) -> x
      sub2(s1(x),s1(y)) -> sub2(x,y)
      gtr2(00(),y) -> false0()
      gtr2(s1(x),00()) -> true0()
      gtr2(s1(x),s1(y)) -> gtr2(x,y)
      d2(x,00()) -> true0()
      d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
      len1(nil0()) -> 00()
      len1(cons2(x,xs)) -> s1(len1(xs))
      filter2(p,nil0()) -> nil0()
      filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
      app(if2(x5,x6),x7) -> if3(x5,x6,x7)
      app(if1(x5),x6) -> if2(x5,x6)
      app(if0(),x5) -> if1(x5)
      app(sub1(x11),x12) -> sub2(x11,x12)
      app(sub0(),x11) -> sub1(x11)
      app(s0(),x15) -> s1(x15)
      app(gtr1(x17),x18) -> gtr2(x17,x18)
      app(gtr0(),x17) -> gtr1(x17)
      app(d1(x20),x21) -> d2(x20,x21)
      app(d0(),x20) -> d1(x20)
      app(len0(),x23) -> len1(x23)
      app(cons1(x26),x27) -> cons2(x26,x27)
      app(cons0(),x26) -> cons1(x26)
      app(filter1(x29),x30) -> filter2(x29,x30)
      app(filter0(),x29) -> filter1(x29)
     Subterm Criterion Processor:
      simple projection:
       pi(sub{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        if3(true0(),xs,ys) -> xs
        if3(false0(),xs,ys) -> ys
        sub2(x,00()) -> x
        sub2(s1(x),s1(y)) -> sub2(x,y)
        gtr2(00(),y) -> false0()
        gtr2(s1(x),00()) -> true0()
        gtr2(s1(x),s1(y)) -> gtr2(x,y)
        d2(x,00()) -> true0()
        d2(s1(x),s1(y)) -> if3(gtr2(x,y),false0(),d2(s1(x),sub2(y,x)))
        len1(nil0()) -> 00()
        len1(cons2(x,xs)) -> s1(len1(xs))
        filter2(p,nil0()) -> nil0()
        filter2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,filter2(p,xs)),filter2(p,xs))
        app(if2(x5,x6),x7) -> if3(x5,x6,x7)
        app(if1(x5),x6) -> if2(x5,x6)
        app(if0(),x5) -> if1(x5)
        app(sub1(x11),x12) -> sub2(x11,x12)
        app(sub0(),x11) -> sub1(x11)
        app(s0(),x15) -> s1(x15)
        app(gtr1(x17),x18) -> gtr2(x17,x18)
        app(gtr0(),x17) -> gtr1(x17)
        app(d1(x20),x21) -> d2(x20,x21)
        app(d0(),x20) -> d1(x20)
        app(len0(),x23) -> len1(x23)
        app(cons1(x26),x27) -> cons2(x26,x27)
        app(cons0(),x26) -> cons1(x26)
        app(filter1(x29),x30) -> filter2(x29,x30)
        app(filter0(),x29) -> filter1(x29)
      Qed