/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(sumwith(),f),nil()) -> nil()
 app(app(sumwith(),f),app(app(cons(),x),xs)) -> app(app(plus(),app(f,x)),app(app(sumwith(),f),xs))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   sumwith ==> sumwith0/0 sumwith1/1 sumwith2/2
   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(sumwith1(x10),x11) -> sumwith2(x10,x11)
   app(sumwith0(),x10) -> sumwith1(x10)
   app(cons1(x14),x15) -> cons2(x14,x15)
   app(cons0(),x14) -> cons1(x14)
  eta-rules:
   
  problem:
   plus2(00(),y) -> y
   plus2(s1(x),y) -> s1(plus2(x,y))
   sumwith2(f,nil0()) -> nil0()
   sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
   app(plus1(x4),x5) -> plus2(x4,x5)
   app(plus0(),x4) -> plus1(x4)
   app(s0(),x8) -> s1(x8)
   app(sumwith1(x10),x11) -> sumwith2(x10,x11)
   app(sumwith0(),x10) -> sumwith1(x10)
   app(cons1(x14),x15) -> cons2(x14,x15)
   app(cons0(),x14) -> cons1(x14)
  DP Processor:
   DPs:
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
    sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x)
    sumwith{2,#}(f,cons2(x,xs)) -> plus{2,#}(app(f,x),sumwith2(f,xs))
    app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
    app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11)
   TRS:
    plus2(00(),y) -> y
    plus2(s1(x),y) -> s1(plus2(x,y))
    sumwith2(f,nil0()) -> nil0()
    sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
    app(plus1(x4),x5) -> plus2(x4,x5)
    app(plus0(),x4) -> plus1(x4)
    app(s0(),x8) -> s1(x8)
    app(sumwith1(x10),x11) -> sumwith2(x10,x11)
    app(sumwith0(),x10) -> sumwith1(x10)
    app(cons1(x14),x15) -> cons2(x14,x15)
    app(cons0(),x14) -> cons1(x14)
   TDG Processor:
    DPs:
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
     sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x)
     sumwith{2,#}(f,cons2(x,xs)) -> plus{2,#}(app(f,x),sumwith2(f,xs))
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11)
    TRS:
     plus2(00(),y) -> y
     plus2(s1(x),y) -> s1(plus2(x,y))
     sumwith2(f,nil0()) -> nil0()
     sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
     app(plus1(x4),x5) -> plus2(x4,x5)
     app(plus0(),x4) -> plus1(x4)
     app(s0(),x8) -> s1(x8)
     app(sumwith1(x10),x11) -> sumwith2(x10,x11)
     app(sumwith0(),x10) -> sumwith1(x10)
     app(cons1(x14),x15) -> cons2(x14,x15)
     app(cons0(),x14) -> cons1(x14)
    graph:
     app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11) ->
     sumwith{2,#}(f,cons2(x,xs)) -> plus{2,#}(app(f,x),sumwith2(f,xs))
     app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11) ->
     sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11) ->
     sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11)
     sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs) ->
     sumwith{2,#}(f,cons2(x,xs)) -> plus{2,#}(app(f,x),sumwith2(f,xs))
     sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs) ->
     sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x)
     sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs) ->
     sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
     sumwith{2,#}(f,cons2(x,xs)) -> plus{2,#}(app(f,x),sumwith2(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)
    SCC Processor:
     #sccs: 2
     #rules: 4
     #arcs: 11/36
     DPs:
      app#(sumwith1(x10),x11) -> sumwith{2,#}(x10,x11)
      sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
      sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      sumwith2(f,nil0()) -> nil0()
      sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(sumwith1(x10),x11) -> sumwith2(x10,x11)
      app(sumwith0(),x10) -> sumwith1(x10)
      app(cons1(x14),x15) -> cons2(x14,x15)
      app(cons0(),x14) -> cons1(x14)
     Subterm Criterion Processor:
      simple projection:
       pi(sumwith{2,#}) = 0
       pi(app#) = 0
      problem:
       DPs:
        sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
        sumwith{2,#}(f,cons2(x,xs)) -> app#(f,x)
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        sumwith2(f,nil0()) -> nil0()
        sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(sumwith1(x10),x11) -> sumwith2(x10,x11)
        app(sumwith0(),x10) -> sumwith1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 5/4
       DPs:
        sumwith{2,#}(f,cons2(x,xs)) -> sumwith{2,#}(f,xs)
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        sumwith2(f,nil0()) -> nil0()
        sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(sumwith1(x10),x11) -> sumwith2(x10,x11)
        app(sumwith0(),x10) -> sumwith1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
       Subterm Criterion Processor:
        simple projection:
         pi(sumwith{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          plus2(00(),y) -> y
          plus2(s1(x),y) -> s1(plus2(x,y))
          sumwith2(f,nil0()) -> nil0()
          sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
          app(plus1(x4),x5) -> plus2(x4,x5)
          app(plus0(),x4) -> plus1(x4)
          app(s0(),x8) -> s1(x8)
          app(sumwith1(x10),x11) -> sumwith2(x10,x11)
          app(sumwith0(),x10) -> sumwith1(x10)
          app(cons1(x14),x15) -> cons2(x14,x15)
          app(cons0(),x14) -> cons1(x14)
        Qed
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      plus2(00(),y) -> y
      plus2(s1(x),y) -> s1(plus2(x,y))
      sumwith2(f,nil0()) -> nil0()
      sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus0(),x4) -> plus1(x4)
      app(s0(),x8) -> s1(x8)
      app(sumwith1(x10),x11) -> sumwith2(x10,x11)
      app(sumwith0(),x10) -> sumwith1(x10)
      app(cons1(x14),x15) -> cons2(x14,x15)
      app(cons0(),x14) -> cons1(x14)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        plus2(00(),y) -> y
        plus2(s1(x),y) -> s1(plus2(x,y))
        sumwith2(f,nil0()) -> nil0()
        sumwith2(f,cons2(x,xs)) -> plus2(app(f,x),sumwith2(f,xs))
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus0(),x4) -> plus1(x4)
        app(s0(),x8) -> s1(x8)
        app(sumwith1(x10),x11) -> sumwith2(x10,x11)
        app(sumwith0(),x10) -> sumwith1(x10)
        app(cons1(x14),x15) -> cons2(x14,x15)
        app(cons0(),x14) -> cons1(x14)
      Qed