/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(times(),x),0()) -> 0()
 app(app(times(),x),app(s(),y)) -> app(app(plus(),app(app(times(),x),y)),x)
 app(app(plus(),x),0()) -> x
 app(app(plus(),0()),x) -> x
 app(app(plus(),x),app(s(),y)) -> app(s(),app(app(plus(),x),y))
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),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
   plus ==> plus0/0 plus1/1 plus2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   times ==> times0/0 times1/1 times2/2
   
  uncurry-rules:
   app(times1(x4),x5) -> times2(x4,x5)
   app(times0(),x4) -> times1(x4)
   app(s0(),x8) -> s1(x8)
   app(plus1(x10),x11) -> plus2(x10,x11)
   app(plus0(),x10) -> plus1(x10)
   app(map1(x13),x14) -> map2(x13,x14)
   app(map0(),x13) -> map1(x13)
   app(cons1(x17),x18) -> cons2(x17,x18)
   app(cons0(),x17) -> cons1(x17)
   app(filter1(x20),x21) -> filter2(x20,x21)
   app(filter0(),x20) -> filter1(x20)
   app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
   app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
   app(filter21(x23),x24) -> filter22(x23,x24)
   app(filter20(),x23) -> filter21(x23)
  eta-rules:
   
  problem:
   times2(x,00()) -> 00()
   times2(x,s1(y)) -> plus2(times2(x,y),x)
   plus2(x,00()) -> x
   plus2(00(),x) -> x
   plus2(x,s1(y)) -> s1(plus2(x,y))
   plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
   app(times0(),x4) -> times1(x4)
   app(s0(),x8) -> s1(x8)
   app(plus1(x10),x11) -> plus2(x10,x11)
   app(plus0(),x10) -> plus1(x10)
   app(map1(x13),x14) -> map2(x13,x14)
   app(map0(),x13) -> map1(x13)
   app(cons1(x17),x18) -> cons2(x17,x18)
   app(cons0(),x17) -> cons1(x17)
   app(filter1(x20),x21) -> filter2(x20,x21)
   app(filter0(),x20) -> filter1(x20)
   app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
   app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
   app(filter21(x23),x24) -> filter22(x23,x24)
   app(filter20(),x23) -> filter21(x23)
  DP Processor:
   DPs:
    times{2,#}(x,s1(y)) -> times{2,#}(x,y)
    times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
    plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
    plus{2,#}(s1(x),y) -> plus{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#(times1(x4),x5) -> times{2,#}(x4,x5)
    app#(plus1(x10),x11) -> plus{2,#}(x10,x11)
    app#(map1(x13),x14) -> map{2,#}(x13,x14)
    app#(filter1(x20),x21) -> filter{2,#}(x20,x21)
    app#(filter23(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26)
   TRS:
    times2(x,00()) -> 00()
    times2(x,s1(y)) -> plus2(times2(x,y),x)
    plus2(x,00()) -> x
    plus2(00(),x) -> x
    plus2(x,s1(y)) -> s1(plus2(x,y))
    plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
    app(times0(),x4) -> times1(x4)
    app(s0(),x8) -> s1(x8)
    app(plus1(x10),x11) -> plus2(x10,x11)
    app(plus0(),x10) -> plus1(x10)
    app(map1(x13),x14) -> map2(x13,x14)
    app(map0(),x13) -> map1(x13)
    app(cons1(x17),x18) -> cons2(x17,x18)
    app(cons0(),x17) -> cons1(x17)
    app(filter1(x20),x21) -> filter2(x20,x21)
    app(filter0(),x20) -> filter1(x20)
    app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
    app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
    app(filter21(x23),x24) -> filter22(x23,x24)
    app(filter20(),x23) -> filter21(x23)
   TDG Processor:
    DPs:
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{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#(times1(x4),x5) -> times{2,#}(x4,x5)
     app#(plus1(x10),x11) -> plus{2,#}(x10,x11)
     app#(map1(x13),x14) -> map{2,#}(x13,x14)
     app#(filter1(x20),x21) -> filter{2,#}(x20,x21)
     app#(filter23(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26)
    TRS:
     times2(x,00()) -> 00()
     times2(x,s1(y)) -> plus2(times2(x,y),x)
     plus2(x,00()) -> x
     plus2(00(),x) -> x
     plus2(x,s1(y)) -> s1(plus2(x,y))
     plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
     app(times0(),x4) -> times1(x4)
     app(s0(),x8) -> s1(x8)
     app(plus1(x10),x11) -> plus2(x10,x11)
     app(plus0(),x10) -> plus1(x10)
     app(map1(x13),x14) -> map2(x13,x14)
     app(map0(),x13) -> map1(x13)
     app(cons1(x17),x18) -> cons2(x17,x18)
     app(cons0(),x17) -> cons1(x17)
     app(filter1(x20),x21) -> filter2(x20,x21)
     app(filter0(),x20) -> filter1(x20)
     app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
     app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
     app(filter21(x23),x24) -> filter22(x23,x24)
     app(filter20(),x23) -> filter21(x23)
    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(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x20),x21) -> filter{2,#}(x20,x21)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x13),x14) -> map{2,#}(x13,x14)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x10),x11) -> plus{2,#}(x10,x11)
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(times1(x4),x5) -> times{2,#}(x4,x5)
     app#(filter23(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26) ->
     filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter23(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26) ->
     filter2{4,#}(true0(),f,x,xs) -> filter{2,#}(f,xs)
     app#(filter1(x20),x21) -> filter{2,#}(x20,x21) ->
     filter{2,#}(f,cons2(x,xs)) -> filter2{4,#}(app(f,x),f,x,xs)
     app#(filter1(x20),x21) -> filter{2,#}(x20,x21) ->
     filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x13),x14) -> map{2,#}(x13,x14) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x13),x14) -> map{2,#}(x13,x14) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(plus1(x10),x11) -> plus{2,#}(x10,x11) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     app#(plus1(x10),x11) -> plus{2,#}(x10,x11) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     app#(times1(x4),x5) -> times{2,#}(x4,x5) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     app#(times1(x4),x5) -> times{2,#}(x4,x5) ->
     times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter23(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(filter1(x20),x21) -> filter{2,#}(x20,x21)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x13),x14) -> map{2,#}(x13,x14)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x10),x11) -> plus{2,#}(x10,x11)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(times1(x4),x5) -> times{2,#}(x4,x5)
     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)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x) ->
     plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) ->
     times{2,#}(x,s1(y)) -> plus{2,#}(times2(x,y),x)
     times{2,#}(x,s1(y)) -> times{2,#}(x,y) -> times{2,#}(x,s1(y)) -> times{2,#}(x,y)
    SCC Processor:
     #sccs: 3
     #rules: 12
     #arcs: 36/225
     DPs:
      filter2{4,#}(false0(),f,x,xs) -> filter{2,#}(f,xs)
      filter{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x13),x14) -> map{2,#}(x13,x14)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(filter1(x20),x21) -> filter{2,#}(x20,x21)
      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(x23,x24,x25),x26) -> filter2{4,#}(x23,x24,x25,x26)
     TRS:
      times2(x,00()) -> 00()
      times2(x,s1(y)) -> plus2(times2(x,y),x)
      plus2(x,00()) -> x
      plus2(00(),x) -> x
      plus2(x,s1(y)) -> s1(plus2(x,y))
      plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
      app(times0(),x4) -> times1(x4)
      app(s0(),x8) -> s1(x8)
      app(plus1(x10),x11) -> plus2(x10,x11)
      app(plus0(),x10) -> plus1(x10)
      app(map1(x13),x14) -> map2(x13,x14)
      app(map0(),x13) -> map1(x13)
      app(cons1(x17),x18) -> cons2(x17,x18)
      app(cons0(),x17) -> cons1(x17)
      app(filter1(x20),x21) -> filter2(x20,x21)
      app(filter0(),x20) -> filter1(x20)
      app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
      app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
      app(filter21(x23),x24) -> filter22(x23,x24)
      app(filter20(),x23) -> filter21(x23)
     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:
        times2(x,00()) -> 00()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,00()) -> x
        plus2(00(),x) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
        app(times0(),x4) -> times1(x4)
        app(s0(),x8) -> s1(x8)
        app(plus1(x10),x11) -> plus2(x10,x11)
        app(plus0(),x10) -> plus1(x10)
        app(map1(x13),x14) -> map2(x13,x14)
        app(map0(),x13) -> map1(x13)
        app(cons1(x17),x18) -> cons2(x17,x18)
        app(cons0(),x17) -> cons1(x17)
        app(filter1(x20),x21) -> filter2(x20,x21)
        app(filter0(),x20) -> filter1(x20)
        app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
        app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
        app(filter21(x23),x24) -> filter22(x23,x24)
        app(filter20(),x23) -> filter21(x23)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        times2(x,00()) -> 00()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,00()) -> x
        plus2(00(),x) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
        app(times0(),x4) -> times1(x4)
        app(s0(),x8) -> s1(x8)
        app(plus1(x10),x11) -> plus2(x10,x11)
        app(plus0(),x10) -> plus1(x10)
        app(map1(x13),x14) -> map2(x13,x14)
        app(map0(),x13) -> map1(x13)
        app(cons1(x17),x18) -> cons2(x17,x18)
        app(cons0(),x17) -> cons1(x17)
        app(filter1(x20),x21) -> filter2(x20,x21)
        app(filter0(),x20) -> filter1(x20)
        app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
        app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
        app(filter21(x23),x24) -> filter22(x23,x24)
        app(filter20(),x23) -> filter21(x23)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          times2(x,00()) -> 00()
          times2(x,s1(y)) -> plus2(times2(x,y),x)
          plus2(x,00()) -> x
          plus2(00(),x) -> x
          plus2(x,s1(y)) -> s1(plus2(x,y))
          plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
          app(times0(),x4) -> times1(x4)
          app(s0(),x8) -> s1(x8)
          app(plus1(x10),x11) -> plus2(x10,x11)
          app(plus0(),x10) -> plus1(x10)
          app(map1(x13),x14) -> map2(x13,x14)
          app(map0(),x13) -> map1(x13)
          app(cons1(x17),x18) -> cons2(x17,x18)
          app(cons0(),x17) -> cons1(x17)
          app(filter1(x20),x21) -> filter2(x20,x21)
          app(filter0(),x20) -> filter1(x20)
          app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
          app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
          app(filter21(x23),x24) -> filter22(x23,x24)
          app(filter20(),x23) -> filter21(x23)
        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:
        times2(x,00()) -> 00()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,00()) -> x
        plus2(00(),x) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
        app(times0(),x4) -> times1(x4)
        app(s0(),x8) -> s1(x8)
        app(plus1(x10),x11) -> plus2(x10,x11)
        app(plus0(),x10) -> plus1(x10)
        app(map1(x13),x14) -> map2(x13,x14)
        app(map0(),x13) -> map1(x13)
        app(cons1(x17),x18) -> cons2(x17,x18)
        app(cons0(),x17) -> cons1(x17)
        app(filter1(x20),x21) -> filter2(x20,x21)
        app(filter0(),x20) -> filter1(x20)
        app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
        app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
        app(filter21(x23),x24) -> filter22(x23,x24)
        app(filter20(),x23) -> filter21(x23)
       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:
          times2(x,00()) -> 00()
          times2(x,s1(y)) -> plus2(times2(x,y),x)
          plus2(x,00()) -> x
          plus2(00(),x) -> x
          plus2(x,s1(y)) -> s1(plus2(x,y))
          plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
          app(times0(),x4) -> times1(x4)
          app(s0(),x8) -> s1(x8)
          app(plus1(x10),x11) -> plus2(x10,x11)
          app(plus0(),x10) -> plus1(x10)
          app(map1(x13),x14) -> map2(x13,x14)
          app(map0(),x13) -> map1(x13)
          app(cons1(x17),x18) -> cons2(x17,x18)
          app(cons0(),x17) -> cons1(x17)
          app(filter1(x20),x21) -> filter2(x20,x21)
          app(filter0(),x20) -> filter1(x20)
          app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
          app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
          app(filter21(x23),x24) -> filter22(x23,x24)
          app(filter20(),x23) -> filter21(x23)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      times{2,#}(x,s1(y)) -> times{2,#}(x,y)
     TRS:
      times2(x,00()) -> 00()
      times2(x,s1(y)) -> plus2(times2(x,y),x)
      plus2(x,00()) -> x
      plus2(00(),x) -> x
      plus2(x,s1(y)) -> s1(plus2(x,y))
      plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
      app(times0(),x4) -> times1(x4)
      app(s0(),x8) -> s1(x8)
      app(plus1(x10),x11) -> plus2(x10,x11)
      app(plus0(),x10) -> plus1(x10)
      app(map1(x13),x14) -> map2(x13,x14)
      app(map0(),x13) -> map1(x13)
      app(cons1(x17),x18) -> cons2(x17,x18)
      app(cons0(),x17) -> cons1(x17)
      app(filter1(x20),x21) -> filter2(x20,x21)
      app(filter0(),x20) -> filter1(x20)
      app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
      app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
      app(filter21(x23),x24) -> filter22(x23,x24)
      app(filter20(),x23) -> filter21(x23)
     Subterm Criterion Processor:
      simple projection:
       pi(times{2,#}) = 1
      problem:
       DPs:
        
       TRS:
        times2(x,00()) -> 00()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,00()) -> x
        plus2(00(),x) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
        app(times0(),x4) -> times1(x4)
        app(s0(),x8) -> s1(x8)
        app(plus1(x10),x11) -> plus2(x10,x11)
        app(plus0(),x10) -> plus1(x10)
        app(map1(x13),x14) -> map2(x13,x14)
        app(map0(),x13) -> map1(x13)
        app(cons1(x17),x18) -> cons2(x17,x18)
        app(cons0(),x17) -> cons1(x17)
        app(filter1(x20),x21) -> filter2(x20,x21)
        app(filter0(),x20) -> filter1(x20)
        app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
        app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
        app(filter21(x23),x24) -> filter22(x23,x24)
        app(filter20(),x23) -> filter21(x23)
      Qed
     
     DPs:
      plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      times2(x,00()) -> 00()
      times2(x,s1(y)) -> plus2(times2(x,y),x)
      plus2(x,00()) -> x
      plus2(00(),x) -> x
      plus2(x,s1(y)) -> s1(plus2(x,y))
      plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
      app(times0(),x4) -> times1(x4)
      app(s0(),x8) -> s1(x8)
      app(plus1(x10),x11) -> plus2(x10,x11)
      app(plus0(),x10) -> plus1(x10)
      app(map1(x13),x14) -> map2(x13,x14)
      app(map0(),x13) -> map1(x13)
      app(cons1(x17),x18) -> cons2(x17,x18)
      app(cons0(),x17) -> cons1(x17)
      app(filter1(x20),x21) -> filter2(x20,x21)
      app(filter0(),x20) -> filter1(x20)
      app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
      app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
      app(filter21(x23),x24) -> filter22(x23,x24)
      app(filter20(),x23) -> filter21(x23)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        plus{2,#}(x,s1(y)) -> plus{2,#}(x,y)
       TRS:
        times2(x,00()) -> 00()
        times2(x,s1(y)) -> plus2(times2(x,y),x)
        plus2(x,00()) -> x
        plus2(00(),x) -> x
        plus2(x,s1(y)) -> s1(plus2(x,y))
        plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
        app(times0(),x4) -> times1(x4)
        app(s0(),x8) -> s1(x8)
        app(plus1(x10),x11) -> plus2(x10,x11)
        app(plus0(),x10) -> plus1(x10)
        app(map1(x13),x14) -> map2(x13,x14)
        app(map0(),x13) -> map1(x13)
        app(cons1(x17),x18) -> cons2(x17,x18)
        app(cons0(),x17) -> cons1(x17)
        app(filter1(x20),x21) -> filter2(x20,x21)
        app(filter0(),x20) -> filter1(x20)
        app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
        app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
        app(filter21(x23),x24) -> filter22(x23,x24)
        app(filter20(),x23) -> filter21(x23)
      Subterm Criterion Processor:
       simple projection:
        pi(plus{2,#}) = 1
       problem:
        DPs:
         
        TRS:
         times2(x,00()) -> 00()
         times2(x,s1(y)) -> plus2(times2(x,y),x)
         plus2(x,00()) -> x
         plus2(00(),x) -> x
         plus2(x,s1(y)) -> s1(plus2(x,y))
         plus2(s1(x),y) -> s1(plus2(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(times1(x4),x5) -> times2(x4,x5)
         app(times0(),x4) -> times1(x4)
         app(s0(),x8) -> s1(x8)
         app(plus1(x10),x11) -> plus2(x10,x11)
         app(plus0(),x10) -> plus1(x10)
         app(map1(x13),x14) -> map2(x13,x14)
         app(map0(),x13) -> map1(x13)
         app(cons1(x17),x18) -> cons2(x17,x18)
         app(cons0(),x17) -> cons1(x17)
         app(filter1(x20),x21) -> filter2(x20,x21)
         app(filter0(),x20) -> filter1(x20)
         app(filter23(x23,x24,x25),x26) -> filter24(x23,x24,x25,x26)
         app(filter22(x23,x24),x25) -> filter23(x23,x24,x25)
         app(filter21(x23),x24) -> filter22(x23,x24)
         app(filter20(),x23) -> filter21(x23)
       Qed