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


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YES

Problem:
 app(app(plus(),0()),y) -> y
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))
 app(app(times(),0()),y) -> 0()
 app(app(times(),app(s(),x)),y) -> app(app(plus(),app(app(times(),x),y)),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))
 inc() -> app(map(),app(plus(),app(s(),0())))
 double() -> app(map(),app(times(),app(s(),app(s(),0()))))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   double ==> double0/0
   inc ==> inc0/0
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   times ==> times0/0 times1/1 times2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   plus ==> plus0/0 plus1/1 plus2/2
   
  uncurry-rules:
   app(plus1(x4),x5) -> plus2(x4,x5)
   app(plus0(),x4) -> plus1(x4)
   app(s0(),x8) -> s1(x8)
   app(times1(x10),x11) -> times2(x10,x11)
   app(times0(),x10) -> times1(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)
  eta-rules:
   
  problem:
   plus2(00(),y) -> y
   plus2(s1(x),y) -> s1(plus2(x,y))
   times2(00(),y) -> 00()
   times2(s1(x),y) -> plus2(times2(x,y),y)
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   inc0() -> map1(plus1(s1(00())))
   double0() -> map1(times1(s1(s1(00()))))
   app(plus1(x4),x5) -> plus2(x4,x5)
   app(plus0(),x4) -> plus1(x4)
   app(s0(),x8) -> s1(x8)
   app(times1(x10),x11) -> times2(x10,x11)
   app(times0(),x10) -> times1(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)
  DP Processor:
   DPs:
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    times{2,#}(s1(x),y) -> times{2,#}(x,y)
    times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
    app#(times1(x10),x11) -> times{2,#}(x10,x11)
    app#(map1(x13),x14) -> map{2,#}(x13,x14)
   TRS:
    plus2(00(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    times2(00(),y) -> 00()
    times2(s1(x),y) -> plus2(times2(x,y),y)
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    inc0() -> map1(plus1(s1(00())))
    double0() -> map1(times1(s1(s1(00()))))
    app(plus1(x4),x5) -> plus2(x4,x5)
    app(plus0(),x4) -> plus1(x4)
    app(s0(),x8) -> s1(x8)
    app(times1(x10),x11) -> times2(x10,x11)
    app(times0(),x10) -> times1(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)
   TDG Processor:
    DPs:
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     app#(times1(x10),x11) -> times{2,#}(x10,x11)
     app#(map1(x13),x14) -> map{2,#}(x13,x14)
    TRS:
     plus2(00(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     times2(00(),y) -> 00()
     times2(s1(x),y) -> plus2(times2(x,y),y)
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     inc0() -> map1(plus1(s1(00())))
     double0() -> map1(times1(s1(s1(00()))))
     app(plus1(x4),x5) -> plus2(x4,x5)
     app(plus0(),x4) -> plus1(x4)
     app(s0(),x8) -> s1(x8)
     app(times1(x10),x11) -> times2(x10,x11)
     app(times0(),x10) -> times1(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)
    graph:
     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#(times1(x10),x11) -> times{2,#}(x10,x11) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     app#(times1(x10),x11) -> times{2,#}(x10,x11) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     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#(times1(x10),x11) -> times{2,#}(x10,x11)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x4),x5) -> plus{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)
     times{2,#}(s1(x),y) -> times{2,#}(x,y) ->
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y)
     times{2,#}(s1(x),y) -> times{2,#}(x,y) ->
     times{2,#}(s1(x),y) -> times{2,#}(x,y)
     times{2,#}(s1(x),y) -> plus{2,#}(times2(x,y),y) ->
     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)
    SCC Processor:
     #sccs: 3
     #rules: 5
     #arcs: 14/64
     DPs:
      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)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(00(),y) -> 00()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc0() -> map1(plus1(s1(00())))
      double0() -> map1(times1(s1(s1(00()))))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(times1(x10),x11) -> times2(x10,x11)
      app(times0(),x10) -> times1(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)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 0
       pi(app#) = 0
      problem:
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
        map{2,#}(f,cons2(x,xs)) -> app#(f,x)
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(00(),y) -> 00()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(plus1(s1(00())))
        double0() -> map1(times1(s1(s1(00()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(times1(x10),x11) -> times2(x10,x11)
        app(times0(),x10) -> times1(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)
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 5/4
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(00(),y) -> 00()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(plus1(s1(00())))
        double0() -> map1(times1(s1(s1(00()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(times1(x10),x11) -> times2(x10,x11)
        app(times0(),x10) -> times1(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)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          plus2(00(),y) -> y
          plus2(s1(x),y) -> s1(plus2(x,y))
          times2(00(),y) -> 00()
          times2(s1(x),y) -> plus2(times2(x,y),y)
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          inc0() -> map1(plus1(s1(00())))
          double0() -> map1(times1(s1(s1(00()))))
          app(plus1(x4),x5) -> plus2(x4,x5)
          app(plus0(),x4) -> plus1(x4)
          app(s0(),x8) -> s1(x8)
          app(times1(x10),x11) -> times2(x10,x11)
          app(times0(),x10) -> times1(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)
        Qed
     
     DPs:
      times{2,#}(s1(x),y) -> times{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(00(),y) -> 00()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc0() -> map1(plus1(s1(00())))
      double0() -> map1(times1(s1(s1(00()))))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(times1(x10),x11) -> times2(x10,x11)
      app(times0(),x10) -> times1(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)
     Subterm Criterion Processor:
      simple projection:
       pi(times{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(00(),y) -> 00()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(plus1(s1(00())))
        double0() -> map1(times1(s1(s1(00()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(times1(x10),x11) -> times2(x10,x11)
        app(times0(),x10) -> times1(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)
      Qed
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      times2(00(),y) -> 00()
      times2(s1(x),y) -> plus2(times2(x,y),y)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      inc0() -> map1(plus1(s1(00())))
      double0() -> map1(times1(s1(s1(00()))))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(times1(x10),x11) -> times2(x10,x11)
      app(times0(),x10) -> times1(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)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        times2(00(),y) -> 00()
        times2(s1(x),y) -> plus2(times2(x,y),y)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        inc0() -> map1(plus1(s1(00())))
        double0() -> map1(times1(s1(s1(00()))))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(times1(x10),x11) -> times2(x10,x11)
        app(times0(),x10) -> times1(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)
      Qed