/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(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(minus(),x),0()) -> x
 app(app(minus(),app(s(),x)),app(s(),y)) -> app(app(minus(),x),y)
 app(app(div(),0()),app(s(),y)) -> 0()
 app(app(div(),app(s(),x)),app(s(),y)) -> app(s(),app(app(div(),app(app(minus(),x),y)),app(s(),y)))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   div ==> div0/0 div1/1 div2/2
   s ==> s0/0 s1/1
   0 ==> 00/0
   minus ==> minus0/0 minus1/1 minus2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   
  uncurry-rules:
   app(map1(x4),x5) -> map2(x4,x5)
   app(map0(),x4) -> map1(x4)
   app(cons1(x8),x9) -> cons2(x8,x9)
   app(cons0(),x8) -> cons1(x8)
   app(minus1(x11),x12) -> minus2(x11,x12)
   app(minus0(),x11) -> minus1(x11)
   app(s0(),x15) -> s1(x15)
   app(div1(x17),x18) -> div2(x17,x18)
   app(div0(),x17) -> div1(x17)
  eta-rules:
   
  problem:
   map2(f,nil0()) -> nil0()
   map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
   minus2(x,00()) -> x
   minus2(s1(x),s1(y)) -> minus2(x,y)
   div2(00(),s1(y)) -> 00()
   div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
   app(map1(x4),x5) -> map2(x4,x5)
   app(map0(),x4) -> map1(x4)
   app(cons1(x8),x9) -> cons2(x8,x9)
   app(cons0(),x8) -> cons1(x8)
   app(minus1(x11),x12) -> minus2(x11,x12)
   app(minus0(),x11) -> minus1(x11)
   app(s0(),x15) -> s1(x15)
   app(div1(x17),x18) -> div2(x17,x18)
   app(div0(),x17) -> div1(x17)
  DP Processor:
   DPs:
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    div{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
    div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y))
    app#(map1(x4),x5) -> map{2,#}(x4,x5)
    app#(minus1(x11),x12) -> minus{2,#}(x11,x12)
    app#(div1(x17),x18) -> div{2,#}(x17,x18)
   TRS:
    map2(f,nil0()) -> nil0()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    minus2(x,00()) -> x
    minus2(s1(x),s1(y)) -> minus2(x,y)
    div2(00(),s1(y)) -> 00()
    div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
    app(map1(x4),x5) -> map2(x4,x5)
    app(map0(),x4) -> map1(x4)
    app(cons1(x8),x9) -> cons2(x8,x9)
    app(cons0(),x8) -> cons1(x8)
    app(minus1(x11),x12) -> minus2(x11,x12)
    app(minus0(),x11) -> minus1(x11)
    app(s0(),x15) -> s1(x15)
    app(div1(x17),x18) -> div2(x17,x18)
    app(div0(),x17) -> div1(x17)
   TDG Processor:
    DPs:
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     div{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y))
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     app#(minus1(x11),x12) -> minus{2,#}(x11,x12)
     app#(div1(x17),x18) -> div{2,#}(x17,x18)
    TRS:
     map2(f,nil0()) -> nil0()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     minus2(x,00()) -> x
     minus2(s1(x),s1(y)) -> minus2(x,y)
     div2(00(),s1(y)) -> 00()
     div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
     app(map1(x4),x5) -> map2(x4,x5)
     app(map0(),x4) -> map1(x4)
     app(cons1(x8),x9) -> cons2(x8,x9)
     app(cons0(),x8) -> cons1(x8)
     app(minus1(x11),x12) -> minus2(x11,x12)
     app(minus0(),x11) -> minus1(x11)
     app(s0(),x15) -> s1(x15)
     app(div1(x17),x18) -> div2(x17,x18)
     app(div0(),x17) -> div1(x17)
    graph:
     div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y)) ->
     div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y))
     div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y)) ->
     div{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     div{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(div1(x17),x18) -> div{2,#}(x17,x18) ->
     div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y))
     app#(div1(x17),x18) -> div{2,#}(x17,x18) ->
     div{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(minus1(x11),x12) -> minus{2,#}(x11,x12) ->
     minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(div1(x17),x18) -> div{2,#}(x17,x18)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(minus1(x11),x12) -> minus{2,#}(x11,x12)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x4),x5) -> map{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)
    SCC Processor:
     #sccs: 3
     #rules: 5
     #arcs: 14/64
     DPs:
      app#(map1(x4),x5) -> map{2,#}(x4,x5)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     TRS:
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      minus2(x,00()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      div2(00(),s1(y)) -> 00()
      div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
      app(map1(x4),x5) -> map2(x4,x5)
      app(map0(),x4) -> map1(x4)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons0(),x8) -> cons1(x8)
      app(minus1(x11),x12) -> minus2(x11,x12)
      app(minus0(),x11) -> minus1(x11)
      app(s0(),x15) -> s1(x15)
      app(div1(x17),x18) -> div2(x17,x18)
      app(div0(),x17) -> div1(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:
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        div2(00(),s1(y)) -> 00()
        div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map0(),x4) -> map1(x4)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons0(),x8) -> cons1(x8)
        app(minus1(x11),x12) -> minus2(x11,x12)
        app(minus0(),x11) -> minus1(x11)
        app(s0(),x15) -> s1(x15)
        app(div1(x17),x18) -> div2(x17,x18)
        app(div0(),x17) -> div1(x17)
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 5/4
       DPs:
        map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
       TRS:
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        div2(00(),s1(y)) -> 00()
        div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map0(),x4) -> map1(x4)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons0(),x8) -> cons1(x8)
        app(minus1(x11),x12) -> minus2(x11,x12)
        app(minus0(),x11) -> minus1(x11)
        app(s0(),x15) -> s1(x15)
        app(div1(x17),x18) -> div2(x17,x18)
        app(div0(),x17) -> div1(x17)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          map2(f,nil0()) -> nil0()
          map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
          minus2(x,00()) -> x
          minus2(s1(x),s1(y)) -> minus2(x,y)
          div2(00(),s1(y)) -> 00()
          div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
          app(map1(x4),x5) -> map2(x4,x5)
          app(map0(),x4) -> map1(x4)
          app(cons1(x8),x9) -> cons2(x8,x9)
          app(cons0(),x8) -> cons1(x8)
          app(minus1(x11),x12) -> minus2(x11,x12)
          app(minus0(),x11) -> minus1(x11)
          app(s0(),x15) -> s1(x15)
          app(div1(x17),x18) -> div2(x17,x18)
          app(div0(),x17) -> div1(x17)
        Qed
     
     DPs:
      div{2,#}(s1(x),s1(y)) -> div{2,#}(minus2(x,y),s1(y))
     TRS:
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      minus2(x,00()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      div2(00(),s1(y)) -> 00()
      div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
      app(map1(x4),x5) -> map2(x4,x5)
      app(map0(),x4) -> map1(x4)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons0(),x8) -> cons1(x8)
      app(minus1(x11),x12) -> minus2(x11,x12)
      app(minus0(),x11) -> minus1(x11)
      app(s0(),x15) -> s1(x15)
      app(div1(x17),x18) -> div2(x17,x18)
      app(div0(),x17) -> div1(x17)
     Subterm Criterion Processor:
      simple projection:
       pi(cons2) = [0,0,0,1,1,1]
       pi(minus2) = [0,0]
       pi(s1) = [0,0,0]
       pi(div{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        div2(00(),s1(y)) -> 00()
        div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map0(),x4) -> map1(x4)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons0(),x8) -> cons1(x8)
        app(minus1(x11),x12) -> minus2(x11,x12)
        app(minus0(),x11) -> minus1(x11)
        app(s0(),x15) -> s1(x15)
        app(div1(x17),x18) -> div2(x17,x18)
        app(div0(),x17) -> div1(x17)
      Qed
     
     DPs:
      minus{2,#}(s1(x),s1(y)) -> minus{2,#}(x,y)
     TRS:
      map2(f,nil0()) -> nil0()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      minus2(x,00()) -> x
      minus2(s1(x),s1(y)) -> minus2(x,y)
      div2(00(),s1(y)) -> 00()
      div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
      app(map1(x4),x5) -> map2(x4,x5)
      app(map0(),x4) -> map1(x4)
      app(cons1(x8),x9) -> cons2(x8,x9)
      app(cons0(),x8) -> cons1(x8)
      app(minus1(x11),x12) -> minus2(x11,x12)
      app(minus0(),x11) -> minus1(x11)
      app(s0(),x15) -> s1(x15)
      app(div1(x17),x18) -> div2(x17,x18)
      app(div0(),x17) -> div1(x17)
     Subterm Criterion Processor:
      simple projection:
       pi(minus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        map2(f,nil0()) -> nil0()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        minus2(x,00()) -> x
        minus2(s1(x),s1(y)) -> minus2(x,y)
        div2(00(),s1(y)) -> 00()
        div2(s1(x),s1(y)) -> s1(div2(minus2(x,y),s1(y)))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map0(),x4) -> map1(x4)
        app(cons1(x8),x9) -> cons2(x8,x9)
        app(cons0(),x8) -> cons1(x8)
        app(minus1(x11),x12) -> minus2(x11,x12)
        app(minus0(),x11) -> minus1(x11)
        app(s0(),x15) -> s1(x15)
        app(div1(x17),x18) -> div2(x17,x18)
        app(div0(),x17) -> div1(x17)
      Qed