/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(inc(),xs) -> app(app(map(),app(plus(),app(s(),0()))),xs)
 app(double(),xs) -> app(app(map(),app(times(),app(s(),app(s(),0())))),xs)
 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))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   double ==> double0/0 double1/1
   map ==> map0/0 map1/1 map2/2
   inc ==> inc0/0 inc1/1
   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(inc0(),x13) -> inc1(x13)
   app(map1(x15),x16) -> map2(x15,x16)
   app(map0(),x15) -> map1(x15)
   app(double0(),x18) -> double1(x18)
   app(cons1(x21),x22) -> cons2(x21,x22)
   app(cons0(),x21) -> cons1(x21)
  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)
   inc1(xs) -> map2(plus1(s1(00())),xs)
   double1(xs) -> map2(times1(s1(s1(00()))),xs)
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   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(inc0(),x13) -> inc1(x13)
   app(map1(x15),x16) -> map2(x15,x16)
   app(map0(),x15) -> map1(x15)
   app(double0(),x18) -> double1(x18)
   app(cons1(x21),x22) -> cons2(x21,x22)
   app(cons0(),x21) -> cons1(x21)
  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)
    inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
    double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs)
    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#(inc0(),x13) -> inc{1,#}(x13)
    app#(map1(x15),x16) -> map{2,#}(x15,x16)
    app#(double0(),x18) -> double{1,#}(x18)
   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)
    inc1(xs) -> map2(plus1(s1(00())),xs)
    double1(xs) -> map2(times1(s1(s1(00()))),xs)
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    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(inc0(),x13) -> inc1(x13)
    app(map1(x15),x16) -> map2(x15,x16)
    app(map0(),x15) -> map1(x15)
    app(double0(),x18) -> double1(x18)
    app(cons1(x21),x22) -> cons2(x21,x22)
    app(cons0(),x21) -> cons1(x21)
   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)
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
     double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs)
     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#(inc0(),x13) -> inc{1,#}(x13)
     app#(map1(x15),x16) -> map{2,#}(x15,x16)
     app#(double0(),x18) -> double{1,#}(x18)
    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)
     inc1(xs) -> map2(plus1(s1(00())),xs)
     double1(xs) -> map2(times1(s1(s1(00()))),xs)
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     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(inc0(),x13) -> inc1(x13)
     app(map1(x15),x16) -> map2(x15,x16)
     app(map0(),x15) -> map1(x15)
     app(double0(),x18) -> double1(x18)
     app(cons1(x21),x22) -> cons2(x21,x22)
     app(cons0(),x21) -> cons1(x21)
    graph:
     app#(double0(),x18) -> double{1,#}(x18) ->
     double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs)
     app#(map1(x15),x16) -> map{2,#}(x15,x16) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x15),x16) -> map{2,#}(x15,x16) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(inc0(),x13) -> inc{1,#}(x13) ->
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),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)
     double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(double0(),x18) -> double{1,#}(x18)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x15),x16) -> map{2,#}(x15,x16)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(inc0(),x13) -> inc{1,#}(x13)
     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)
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),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: 9
     #arcs: 22/144
     DPs:
      app#(double0(),x18) -> double{1,#}(x18)
      double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(inc0(),x13) -> inc{1,#}(x13)
      inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
      app#(map1(x15),x16) -> map{2,#}(x15,x16)
     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)
      inc1(xs) -> map2(plus1(s1(00())),xs)
      double1(xs) -> map2(times1(s1(s1(00()))),xs)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      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(inc0(),x13) -> inc1(x13)
      app(map1(x15),x16) -> map2(x15,x16)
      app(map0(),x15) -> map1(x15)
      app(double0(),x18) -> double1(x18)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
     Subterm Criterion Processor:
      simple projection:
       pi(inc{1,#}) = 0
       pi(map{2,#}) = 1
       pi(double{1,#}) = 0
       pi(app#) = 1
      problem:
       DPs:
        app#(double0(),x18) -> double{1,#}(x18)
        double{1,#}(xs) -> map{2,#}(times1(s1(s1(00()))),xs)
        app#(inc0(),x13) -> inc{1,#}(x13)
        inc{1,#}(xs) -> map{2,#}(plus1(s1(00())),xs)
        app#(map1(x15),x16) -> map{2,#}(x15,x16)
       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)
        inc1(xs) -> map2(plus1(s1(00())),xs)
        double1(xs) -> map2(times1(s1(s1(00()))),xs)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        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(inc0(),x13) -> inc1(x13)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(double0(),x18) -> double1(x18)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 13/25
       
     
     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)
      inc1(xs) -> map2(plus1(s1(00())),xs)
      double1(xs) -> map2(times1(s1(s1(00()))),xs)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      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(inc0(),x13) -> inc1(x13)
      app(map1(x15),x16) -> map2(x15,x16)
      app(map0(),x15) -> map1(x15)
      app(double0(),x18) -> double1(x18)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
     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)
        inc1(xs) -> map2(plus1(s1(00())),xs)
        double1(xs) -> map2(times1(s1(s1(00()))),xs)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        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(inc0(),x13) -> inc1(x13)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(double0(),x18) -> double1(x18)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
      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)
      inc1(xs) -> map2(plus1(s1(00())),xs)
      double1(xs) -> map2(times1(s1(s1(00()))),xs)
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      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(inc0(),x13) -> inc1(x13)
      app(map1(x15),x16) -> map2(x15,x16)
      app(map0(),x15) -> map1(x15)
      app(double0(),x18) -> double1(x18)
      app(cons1(x21),x22) -> cons2(x21,x22)
      app(cons0(),x21) -> cons1(x21)
     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)
        inc1(xs) -> map2(plus1(s1(00())),xs)
        double1(xs) -> map2(times1(s1(s1(00()))),xs)
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        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(inc0(),x13) -> inc1(x13)
        app(map1(x15),x16) -> map2(x15,x16)
        app(map0(),x15) -> map1(x15)
        app(double0(),x18) -> double1(x18)
        app(cons1(x21),x22) -> cons2(x21,x22)
        app(cons0(),x21) -> cons1(x21)
      Qed