/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem:
 a__minus(0(),Y) -> 0()
 a__minus(s(X),s(Y)) -> a__minus(X,Y)
 a__geq(X,0()) -> true()
 a__geq(0(),s(Y)) -> false()
 a__geq(s(X),s(Y)) -> a__geq(X,Y)
 a__div(0(),s(Y)) -> 0()
 a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
 a__if(true(),X,Y) -> mark(X)
 a__if(false(),X,Y) -> mark(Y)
 mark(minus(X1,X2)) -> a__minus(X1,X2)
 mark(geq(X1,X2)) -> a__geq(X1,X2)
 mark(div(X1,X2)) -> a__div(mark(X1),X2)
 mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
 mark(0()) -> 0()
 mark(s(X)) -> s(mark(X))
 mark(true()) -> true()
 mark(false()) -> false()
 a__minus(X1,X2) -> minus(X1,X2)
 a__geq(X1,X2) -> geq(X1,X2)
 a__div(X1,X2) -> div(X1,X2)
 a__if(X1,X2,X3) -> if(X1,X2,X3)

Proof:
 DP Processor:
  DPs:
   a__minus#(s(X),s(Y)) -> a__minus#(X,Y)
   a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
   a__div#(s(X),s(Y)) -> a__geq#(X,Y)
   a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
   a__if#(true(),X,Y) -> mark#(X)
   a__if#(false(),X,Y) -> mark#(Y)
   mark#(minus(X1,X2)) -> a__minus#(X1,X2)
   mark#(geq(X1,X2)) -> a__geq#(X1,X2)
   mark#(div(X1,X2)) -> mark#(X1)
   mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
   mark#(if(X1,X2,X3)) -> mark#(X1)
   mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
   mark#(s(X)) -> mark#(X)
  TRS:
   a__minus(0(),Y) -> 0()
   a__minus(s(X),s(Y)) -> a__minus(X,Y)
   a__geq(X,0()) -> true()
   a__geq(0(),s(Y)) -> false()
   a__geq(s(X),s(Y)) -> a__geq(X,Y)
   a__div(0(),s(Y)) -> 0()
   a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
   a__if(true(),X,Y) -> mark(X)
   a__if(false(),X,Y) -> mark(Y)
   mark(minus(X1,X2)) -> a__minus(X1,X2)
   mark(geq(X1,X2)) -> a__geq(X1,X2)
   mark(div(X1,X2)) -> a__div(mark(X1),X2)
   mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
   mark(0()) -> 0()
   mark(s(X)) -> s(mark(X))
   mark(true()) -> true()
   mark(false()) -> false()
   a__minus(X1,X2) -> minus(X1,X2)
   a__geq(X1,X2) -> geq(X1,X2)
   a__div(X1,X2) -> div(X1,X2)
   a__if(X1,X2,X3) -> if(X1,X2,X3)
  TDG Processor:
   DPs:
    a__minus#(s(X),s(Y)) -> a__minus#(X,Y)
    a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
    a__div#(s(X),s(Y)) -> a__geq#(X,Y)
    a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
    a__if#(true(),X,Y) -> mark#(X)
    a__if#(false(),X,Y) -> mark#(Y)
    mark#(minus(X1,X2)) -> a__minus#(X1,X2)
    mark#(geq(X1,X2)) -> a__geq#(X1,X2)
    mark#(div(X1,X2)) -> mark#(X1)
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
    mark#(if(X1,X2,X3)) -> mark#(X1)
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
    mark#(s(X)) -> mark#(X)
   TRS:
    a__minus(0(),Y) -> 0()
    a__minus(s(X),s(Y)) -> a__minus(X,Y)
    a__geq(X,0()) -> true()
    a__geq(0(),s(Y)) -> false()
    a__geq(s(X),s(Y)) -> a__geq(X,Y)
    a__div(0(),s(Y)) -> 0()
    a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
    a__if(true(),X,Y) -> mark(X)
    a__if(false(),X,Y) -> mark(Y)
    mark(minus(X1,X2)) -> a__minus(X1,X2)
    mark(geq(X1,X2)) -> a__geq(X1,X2)
    mark(div(X1,X2)) -> a__div(mark(X1),X2)
    mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
    mark(0()) -> 0()
    mark(s(X)) -> s(mark(X))
    mark(true()) -> true()
    mark(false()) -> false()
    a__minus(X1,X2) -> minus(X1,X2)
    a__geq(X1,X2) -> geq(X1,X2)
    a__div(X1,X2) -> div(X1,X2)
    a__if(X1,X2,X3) -> if(X1,X2,X3)
   graph:
    mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(if(X1,X2,X3)) -> mark#(X1) ->
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
    mark#(if(X1,X2,X3)) -> mark#(X1) ->
    mark#(if(X1,X2,X3)) -> mark#(X1)
    mark#(if(X1,X2,X3)) -> mark#(X1) ->
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
    mark#(if(X1,X2,X3)) -> mark#(X1) ->
    mark#(div(X1,X2)) -> mark#(X1)
    mark#(if(X1,X2,X3)) -> mark#(X1) ->
    mark#(geq(X1,X2)) -> a__geq#(X1,X2)
    mark#(if(X1,X2,X3)) -> mark#(X1) ->
    mark#(minus(X1,X2)) -> a__minus#(X1,X2)
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) ->
    a__if#(false(),X,Y) -> mark#(Y)
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) ->
    a__if#(true(),X,Y) -> mark#(X)
    mark#(geq(X1,X2)) -> a__geq#(X1,X2) ->
    a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
    mark#(div(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(div(X1,X2)) -> mark#(X1) ->
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
    mark#(div(X1,X2)) -> mark#(X1) ->
    mark#(if(X1,X2,X3)) -> mark#(X1)
    mark#(div(X1,X2)) -> mark#(X1) ->
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
    mark#(div(X1,X2)) -> mark#(X1) -> mark#(div(X1,X2)) -> mark#(X1)
    mark#(div(X1,X2)) -> mark#(X1) ->
    mark#(geq(X1,X2)) -> a__geq#(X1,X2)
    mark#(div(X1,X2)) -> mark#(X1) ->
    mark#(minus(X1,X2)) -> a__minus#(X1,X2)
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2) ->
    a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2) ->
    a__div#(s(X),s(Y)) -> a__geq#(X,Y)
    mark#(minus(X1,X2)) -> a__minus#(X1,X2) ->
    a__minus#(s(X),s(Y)) -> a__minus#(X,Y)
    mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
    mark#(s(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
    mark#(s(X)) -> mark#(X) -> mark#(div(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(geq(X1,X2)) -> a__geq#(X1,X2)
    mark#(s(X)) -> mark#(X) ->
    mark#(minus(X1,X2)) -> a__minus#(X1,X2)
    a__if#(false(),X,Y) -> mark#(Y) -> mark#(s(X)) -> mark#(X)
    a__if#(false(),X,Y) -> mark#(Y) ->
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
    a__if#(false(),X,Y) -> mark#(Y) ->
    mark#(if(X1,X2,X3)) -> mark#(X1)
    a__if#(false(),X,Y) -> mark#(Y) ->
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
    a__if#(false(),X,Y) -> mark#(Y) ->
    mark#(div(X1,X2)) -> mark#(X1)
    a__if#(false(),X,Y) -> mark#(Y) ->
    mark#(geq(X1,X2)) -> a__geq#(X1,X2)
    a__if#(false(),X,Y) -> mark#(Y) ->
    mark#(minus(X1,X2)) -> a__minus#(X1,X2)
    a__if#(true(),X,Y) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    a__if#(true(),X,Y) -> mark#(X) ->
    mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
    a__if#(true(),X,Y) -> mark#(X) ->
    mark#(if(X1,X2,X3)) -> mark#(X1)
    a__if#(true(),X,Y) -> mark#(X) ->
    mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
    a__if#(true(),X,Y) -> mark#(X) -> mark#(div(X1,X2)) -> mark#(X1)
    a__if#(true(),X,Y) -> mark#(X) ->
    mark#(geq(X1,X2)) -> a__geq#(X1,X2)
    a__if#(true(),X,Y) -> mark#(X) ->
    mark#(minus(X1,X2)) -> a__minus#(X1,X2)
    a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ->
    a__if#(false(),X,Y) -> mark#(Y)
    a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ->
    a__if#(true(),X,Y) -> mark#(X)
    a__div#(s(X),s(Y)) -> a__geq#(X,Y) ->
    a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
    a__geq#(s(X),s(Y)) -> a__geq#(X,Y) ->
    a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
    a__minus#(s(X),s(Y)) -> a__minus#(X,Y) -> a__minus#(s(X),s(Y)) -> a__minus#(X,Y)
   SCC Processor:
    #sccs: 3
    #rules: 10
    #arcs: 46/169
    DPs:
     mark#(if(X1,X2,X3)) -> mark#(X1)
     mark#(div(X1,X2)) -> mark#(X1)
     mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
     a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
     a__if#(true(),X,Y) -> mark#(X)
     mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3)
     a__if#(false(),X,Y) -> mark#(Y)
     mark#(s(X)) -> mark#(X)
    TRS:
     a__minus(0(),Y) -> 0()
     a__minus(s(X),s(Y)) -> a__minus(X,Y)
     a__geq(X,0()) -> true()
     a__geq(0(),s(Y)) -> false()
     a__geq(s(X),s(Y)) -> a__geq(X,Y)
     a__div(0(),s(Y)) -> 0()
     a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
     a__if(true(),X,Y) -> mark(X)
     a__if(false(),X,Y) -> mark(Y)
     mark(minus(X1,X2)) -> a__minus(X1,X2)
     mark(geq(X1,X2)) -> a__geq(X1,X2)
     mark(div(X1,X2)) -> a__div(mark(X1),X2)
     mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
     mark(0()) -> 0()
     mark(s(X)) -> s(mark(X))
     mark(true()) -> true()
     mark(false()) -> false()
     a__minus(X1,X2) -> minus(X1,X2)
     a__geq(X1,X2) -> geq(X1,X2)
     a__div(X1,X2) -> div(X1,X2)
     a__if(X1,X2,X3) -> if(X1,X2,X3)
    Arctic Interpretation Processor:
     dimension: 1
     usable rules:
      
     interpretation:
      [true] = 2,
      
      [a__div#](x0, x1) = 5x1 + 6,
      
      [div](x0, x1) = 4x0 + 5x1 + 0,
      
      [a__minus](x0, x1) = 2x0 + 2x1 + 0,
      
      [a__geq](x0, x1) = x0 + x1,
      
      [a__if#](x0, x1, x2) = x1 + 5x2 + 6,
      
      [mark#](x0) = x0 + 6,
      
      [false] = 0,
      
      [mark](x0) = x0,
      
      [0] = 0,
      
      [minus](x0, x1) = 0,
      
      [if](x0, x1, x2) = x0 + 1x1 + 6x2 + 7,
      
      [geq](x0, x1) = x0 + 6,
      
      [a__if](x0, x1, x2) = x0 + x1 + 8,
      
      [a__div](x0, x1) = x1,
      
      [s](x0) = x0 + 0
     orientation:
      mark#(if(X1,X2,X3)) = X1 + 1X2 + 6X3 + 7 >= X1 + 6 = mark#(X1)
      
      mark#(div(X1,X2)) = 4X1 + 5X2 + 6 >= X1 + 6 = mark#(X1)
      
      mark#(div(X1,X2)) = 4X1 + 5X2 + 6 >= 5X2 + 6 = a__div#(mark(X1),X2)
      
      a__div#(s(X),s(Y)) = 5Y + 6 >= 5Y + 6 = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
      
      a__if#(true(),X,Y) = X + 5Y + 6 >= X + 6 = mark#(X)
      
      mark#(if(X1,X2,X3)) = X1 + 1X2 + 6X3 + 7 >= X2 + 5X3 + 6 = a__if#(mark(X1),X2,X3)
      
      a__if#(false(),X,Y) = X + 5Y + 6 >= Y + 6 = mark#(Y)
      
      mark#(s(X)) = X + 6 >= X + 6 = mark#(X)
      
      a__minus(0(),Y) = 2Y + 2 >= 0 = 0()
      
      a__minus(s(X),s(Y)) = 2X + 2Y + 2 >= 2X + 2Y + 0 = a__minus(X,Y)
      
      a__geq(X,0()) = X + 0 >= 2 = true()
      
      a__geq(0(),s(Y)) = Y + 0 >= 0 = false()
      
      a__geq(s(X),s(Y)) = X + Y + 0 >= X + Y = a__geq(X,Y)
      
      a__div(0(),s(Y)) = Y + 0 >= 0 = 0()
      
      a__div(s(X),s(Y)) = Y + 0 >= X + 5Y + 8 = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
      
      a__if(true(),X,Y) = X + 8 >= X = mark(X)
      
      a__if(false(),X,Y) = X + 8 >= Y = mark(Y)
      
      mark(minus(X1,X2)) = 0 >= 2X1 + 2X2 + 0 = a__minus(X1,X2)
      
      mark(geq(X1,X2)) = X1 + 6 >= X1 + X2 = a__geq(X1,X2)
      
      mark(div(X1,X2)) = 4X1 + 5X2 + 0 >= X2 = a__div(mark(X1),X2)
      
      mark(if(X1,X2,X3)) = X1 + 1X2 + 6X3 + 7 >= X1 + X2 + 8 = a__if(mark(X1),X2,X3)
      
      mark(0()) = 0 >= 0 = 0()
      
      mark(s(X)) = X + 0 >= X + 0 = s(mark(X))
      
      mark(true()) = 2 >= 2 = true()
      
      mark(false()) = 0 >= 0 = false()
      
      a__minus(X1,X2) = 2X1 + 2X2 + 0 >= 0 = minus(X1,X2)
      
      a__geq(X1,X2) = X1 + X2 >= X1 + 6 = geq(X1,X2)
      
      a__div(X1,X2) = X2 >= 4X1 + 5X2 + 0 = div(X1,X2)
      
      a__if(X1,X2,X3) = X1 + X2 + 8 >= X1 + 1X2 + 6X3 + 7 = if(X1,X2,X3)
     problem:
      DPs:
       mark#(if(X1,X2,X3)) -> mark#(X1)
       mark#(div(X1,X2)) -> mark#(X1)
       mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
       a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
       a__if#(true(),X,Y) -> mark#(X)
       a__if#(false(),X,Y) -> mark#(Y)
       mark#(s(X)) -> mark#(X)
      TRS:
       a__minus(0(),Y) -> 0()
       a__minus(s(X),s(Y)) -> a__minus(X,Y)
       a__geq(X,0()) -> true()
       a__geq(0(),s(Y)) -> false()
       a__geq(s(X),s(Y)) -> a__geq(X,Y)
       a__div(0(),s(Y)) -> 0()
       a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
       a__if(true(),X,Y) -> mark(X)
       a__if(false(),X,Y) -> mark(Y)
       mark(minus(X1,X2)) -> a__minus(X1,X2)
       mark(geq(X1,X2)) -> a__geq(X1,X2)
       mark(div(X1,X2)) -> a__div(mark(X1),X2)
       mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
       mark(0()) -> 0()
       mark(s(X)) -> s(mark(X))
       mark(true()) -> true()
       mark(false()) -> false()
       a__minus(X1,X2) -> minus(X1,X2)
       a__geq(X1,X2) -> geq(X1,X2)
       a__div(X1,X2) -> div(X1,X2)
       a__if(X1,X2,X3) -> if(X1,X2,X3)
     Restore Modifier:
      DPs:
       mark#(if(X1,X2,X3)) -> mark#(X1)
       mark#(div(X1,X2)) -> mark#(X1)
       mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
       a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
       a__if#(true(),X,Y) -> mark#(X)
       a__if#(false(),X,Y) -> mark#(Y)
       mark#(s(X)) -> mark#(X)
      TRS:
       a__minus(0(),Y) -> 0()
       a__minus(s(X),s(Y)) -> a__minus(X,Y)
       a__geq(X,0()) -> true()
       a__geq(0(),s(Y)) -> false()
       a__geq(s(X),s(Y)) -> a__geq(X,Y)
       a__div(0(),s(Y)) -> 0()
       a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
       a__if(true(),X,Y) -> mark(X)
       a__if(false(),X,Y) -> mark(Y)
       mark(minus(X1,X2)) -> a__minus(X1,X2)
       mark(geq(X1,X2)) -> a__geq(X1,X2)
       mark(div(X1,X2)) -> a__div(mark(X1),X2)
       mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
       mark(0()) -> 0()
       mark(s(X)) -> s(mark(X))
       mark(true()) -> true()
       mark(false()) -> false()
       a__minus(X1,X2) -> minus(X1,X2)
       a__geq(X1,X2) -> geq(X1,X2)
       a__div(X1,X2) -> div(X1,X2)
       a__if(X1,X2,X3) -> if(X1,X2,X3)
      Matrix Interpretation Processor: dim=1
       
       usable rules:
        a__minus(0(),Y) -> 0()
        a__minus(s(X),s(Y)) -> a__minus(X,Y)
        a__geq(X,0()) -> true()
        a__geq(0(),s(Y)) -> false()
        a__geq(s(X),s(Y)) -> a__geq(X,Y)
        a__div(0(),s(Y)) -> 0()
        a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
        a__if(true(),X,Y) -> mark(X)
        a__if(false(),X,Y) -> mark(Y)
        mark(minus(X1,X2)) -> a__minus(X1,X2)
        mark(geq(X1,X2)) -> a__geq(X1,X2)
        mark(div(X1,X2)) -> a__div(mark(X1),X2)
        mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
        mark(0()) -> 0()
        mark(s(X)) -> s(mark(X))
        mark(true()) -> true()
        mark(false()) -> false()
        a__minus(X1,X2) -> minus(X1,X2)
        a__geq(X1,X2) -> geq(X1,X2)
        a__div(X1,X2) -> div(X1,X2)
        a__if(X1,X2,X3) -> if(X1,X2,X3)
       interpretation:
        [true] = 0,
        
        [a__div#](x0, x1) = 6x0 + 7,
        
        [div](x0, x1) = 3x0 + 3,
        
        [a__minus](x0, x1) = 0,
        
        [a__geq](x0, x1) = 1,
        
        [a__if#](x0, x1, x2) = x0 + 2x1 + 4x2 + 1,
        
        [mark#](x0) = 2x0 + 1,
        
        [false] = 1,
        
        [mark](x0) = x0,
        
        [0] = 0,
        
        [minus](x0, x1) = 0,
        
        [if](x0, x1, x2) = 2x0 + x1 + x2,
        
        [geq](x0, x1) = 1,
        
        [a__if](x0, x1, x2) = 2x0 + x1 + x2,
        
        [a__div](x0, x1) = 3x0 + 3,
        
        [s](x0) = 2x0 + 3
       orientation:
        mark#(if(X1,X2,X3)) = 4X1 + 2X2 + 2X3 + 1 >= 2X1 + 1 = mark#(X1)
        
        mark#(div(X1,X2)) = 6X1 + 7 >= 2X1 + 1 = mark#(X1)
        
        mark#(div(X1,X2)) = 6X1 + 7 >= 6X1 + 7 = a__div#(mark(X1),X2)
        
        a__div#(s(X),s(Y)) = 12X + 25 >= 20 = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
        
        a__if#(true(),X,Y) = 2X + 4Y + 1 >= 2X + 1 = mark#(X)
        
        a__if#(false(),X,Y) = 2X + 4Y + 2 >= 2Y + 1 = mark#(Y)
        
        mark#(s(X)) = 4X + 7 >= 2X + 1 = mark#(X)
        
        a__minus(0(),Y) = 0 >= 0 = 0()
        
        a__minus(s(X),s(Y)) = 0 >= 0 = a__minus(X,Y)
        
        a__geq(X,0()) = 1 >= 0 = true()
        
        a__geq(0(),s(Y)) = 1 >= 1 = false()
        
        a__geq(s(X),s(Y)) = 1 >= 1 = a__geq(X,Y)
        
        a__div(0(),s(Y)) = 3 >= 0 = 0()
        
        a__div(s(X),s(Y)) = 6X + 12 >= 11 = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
        
        a__if(true(),X,Y) = X + Y >= X = mark(X)
        
        a__if(false(),X,Y) = X + Y + 2 >= Y = mark(Y)
        
        mark(minus(X1,X2)) = 0 >= 0 = a__minus(X1,X2)
        
        mark(geq(X1,X2)) = 1 >= 1 = a__geq(X1,X2)
        
        mark(div(X1,X2)) = 3X1 + 3 >= 3X1 + 3 = a__div(mark(X1),X2)
        
        mark(if(X1,X2,X3)) = 2X1 + X2 + X3 >= 2X1 + X2 + X3 = a__if(mark(X1),X2,X3)
        
        mark(0()) = 0 >= 0 = 0()
        
        mark(s(X)) = 2X + 3 >= 2X + 3 = s(mark(X))
        
        mark(true()) = 0 >= 0 = true()
        
        mark(false()) = 1 >= 1 = false()
        
        a__minus(X1,X2) = 0 >= 0 = minus(X1,X2)
        
        a__geq(X1,X2) = 1 >= 1 = geq(X1,X2)
        
        a__div(X1,X2) = 3X1 + 3 >= 3X1 + 3 = div(X1,X2)
        
        a__if(X1,X2,X3) = 2X1 + X2 + X3 >= 2X1 + X2 + X3 = if(X1,X2,X3)
       problem:
        DPs:
         mark#(if(X1,X2,X3)) -> mark#(X1)
         mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
         a__if#(true(),X,Y) -> mark#(X)
        TRS:
         a__minus(0(),Y) -> 0()
         a__minus(s(X),s(Y)) -> a__minus(X,Y)
         a__geq(X,0()) -> true()
         a__geq(0(),s(Y)) -> false()
         a__geq(s(X),s(Y)) -> a__geq(X,Y)
         a__div(0(),s(Y)) -> 0()
         a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
         a__if(true(),X,Y) -> mark(X)
         a__if(false(),X,Y) -> mark(Y)
         mark(minus(X1,X2)) -> a__minus(X1,X2)
         mark(geq(X1,X2)) -> a__geq(X1,X2)
         mark(div(X1,X2)) -> a__div(mark(X1),X2)
         mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
         mark(0()) -> 0()
         mark(s(X)) -> s(mark(X))
         mark(true()) -> true()
         mark(false()) -> false()
         a__minus(X1,X2) -> minus(X1,X2)
         a__geq(X1,X2) -> geq(X1,X2)
         a__div(X1,X2) -> div(X1,X2)
         a__if(X1,X2,X3) -> if(X1,X2,X3)
       Restore Modifier:
        DPs:
         mark#(if(X1,X2,X3)) -> mark#(X1)
         mark#(div(X1,X2)) -> a__div#(mark(X1),X2)
         a__if#(true(),X,Y) -> mark#(X)
        TRS:
         a__minus(0(),Y) -> 0()
         a__minus(s(X),s(Y)) -> a__minus(X,Y)
         a__geq(X,0()) -> true()
         a__geq(0(),s(Y)) -> false()
         a__geq(s(X),s(Y)) -> a__geq(X,Y)
         a__div(0(),s(Y)) -> 0()
         a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
         a__if(true(),X,Y) -> mark(X)
         a__if(false(),X,Y) -> mark(Y)
         mark(minus(X1,X2)) -> a__minus(X1,X2)
         mark(geq(X1,X2)) -> a__geq(X1,X2)
         mark(div(X1,X2)) -> a__div(mark(X1),X2)
         mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
         mark(0()) -> 0()
         mark(s(X)) -> s(mark(X))
         mark(true()) -> true()
         mark(false()) -> false()
         a__minus(X1,X2) -> minus(X1,X2)
         a__geq(X1,X2) -> geq(X1,X2)
         a__div(X1,X2) -> div(X1,X2)
         a__if(X1,X2,X3) -> if(X1,X2,X3)
        SCC Processor:
         #sccs: 1
         #rules: 1
         #arcs: 30/9
         DPs:
          mark#(if(X1,X2,X3)) -> mark#(X1)
         TRS:
          a__minus(0(),Y) -> 0()
          a__minus(s(X),s(Y)) -> a__minus(X,Y)
          a__geq(X,0()) -> true()
          a__geq(0(),s(Y)) -> false()
          a__geq(s(X),s(Y)) -> a__geq(X,Y)
          a__div(0(),s(Y)) -> 0()
          a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
          a__if(true(),X,Y) -> mark(X)
          a__if(false(),X,Y) -> mark(Y)
          mark(minus(X1,X2)) -> a__minus(X1,X2)
          mark(geq(X1,X2)) -> a__geq(X1,X2)
          mark(div(X1,X2)) -> a__div(mark(X1),X2)
          mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
          mark(0()) -> 0()
          mark(s(X)) -> s(mark(X))
          mark(true()) -> true()
          mark(false()) -> false()
          a__minus(X1,X2) -> minus(X1,X2)
          a__geq(X1,X2) -> geq(X1,X2)
          a__div(X1,X2) -> div(X1,X2)
          a__if(X1,X2,X3) -> if(X1,X2,X3)
         Size-Change Termination Processor:
          DPs:
           
          TRS:
           a__minus(0(),Y) -> 0()
           a__minus(s(X),s(Y)) -> a__minus(X,Y)
           a__geq(X,0()) -> true()
           a__geq(0(),s(Y)) -> false()
           a__geq(s(X),s(Y)) -> a__geq(X,Y)
           a__div(0(),s(Y)) -> 0()
           a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
           a__if(true(),X,Y) -> mark(X)
           a__if(false(),X,Y) -> mark(Y)
           mark(minus(X1,X2)) -> a__minus(X1,X2)
           mark(geq(X1,X2)) -> a__geq(X1,X2)
           mark(div(X1,X2)) -> a__div(mark(X1),X2)
           mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
           mark(0()) -> 0()
           mark(s(X)) -> s(mark(X))
           mark(true()) -> true()
           mark(false()) -> false()
           a__minus(X1,X2) -> minus(X1,X2)
           a__geq(X1,X2) -> geq(X1,X2)
           a__div(X1,X2) -> div(X1,X2)
           a__if(X1,X2,X3) -> if(X1,X2,X3)
          The DP: mark#(if(X1,X2,X3)) -> mark#(X1) has the edges:
            0 >   0
          Qed
    
    DPs:
     a__geq#(s(X),s(Y)) -> a__geq#(X,Y)
    TRS:
     a__minus(0(),Y) -> 0()
     a__minus(s(X),s(Y)) -> a__minus(X,Y)
     a__geq(X,0()) -> true()
     a__geq(0(),s(Y)) -> false()
     a__geq(s(X),s(Y)) -> a__geq(X,Y)
     a__div(0(),s(Y)) -> 0()
     a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
     a__if(true(),X,Y) -> mark(X)
     a__if(false(),X,Y) -> mark(Y)
     mark(minus(X1,X2)) -> a__minus(X1,X2)
     mark(geq(X1,X2)) -> a__geq(X1,X2)
     mark(div(X1,X2)) -> a__div(mark(X1),X2)
     mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
     mark(0()) -> 0()
     mark(s(X)) -> s(mark(X))
     mark(true()) -> true()
     mark(false()) -> false()
     a__minus(X1,X2) -> minus(X1,X2)
     a__geq(X1,X2) -> geq(X1,X2)
     a__div(X1,X2) -> div(X1,X2)
     a__if(X1,X2,X3) -> if(X1,X2,X3)
    Subterm Criterion Processor:
     simple projection:
      pi(a__geq#) = 0
     problem:
      DPs:
       
      TRS:
       a__minus(0(),Y) -> 0()
       a__minus(s(X),s(Y)) -> a__minus(X,Y)
       a__geq(X,0()) -> true()
       a__geq(0(),s(Y)) -> false()
       a__geq(s(X),s(Y)) -> a__geq(X,Y)
       a__div(0(),s(Y)) -> 0()
       a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
       a__if(true(),X,Y) -> mark(X)
       a__if(false(),X,Y) -> mark(Y)
       mark(minus(X1,X2)) -> a__minus(X1,X2)
       mark(geq(X1,X2)) -> a__geq(X1,X2)
       mark(div(X1,X2)) -> a__div(mark(X1),X2)
       mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
       mark(0()) -> 0()
       mark(s(X)) -> s(mark(X))
       mark(true()) -> true()
       mark(false()) -> false()
       a__minus(X1,X2) -> minus(X1,X2)
       a__geq(X1,X2) -> geq(X1,X2)
       a__div(X1,X2) -> div(X1,X2)
       a__if(X1,X2,X3) -> if(X1,X2,X3)
     Qed
    
    DPs:
     a__minus#(s(X),s(Y)) -> a__minus#(X,Y)
    TRS:
     a__minus(0(),Y) -> 0()
     a__minus(s(X),s(Y)) -> a__minus(X,Y)
     a__geq(X,0()) -> true()
     a__geq(0(),s(Y)) -> false()
     a__geq(s(X),s(Y)) -> a__geq(X,Y)
     a__div(0(),s(Y)) -> 0()
     a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
     a__if(true(),X,Y) -> mark(X)
     a__if(false(),X,Y) -> mark(Y)
     mark(minus(X1,X2)) -> a__minus(X1,X2)
     mark(geq(X1,X2)) -> a__geq(X1,X2)
     mark(div(X1,X2)) -> a__div(mark(X1),X2)
     mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
     mark(0()) -> 0()
     mark(s(X)) -> s(mark(X))
     mark(true()) -> true()
     mark(false()) -> false()
     a__minus(X1,X2) -> minus(X1,X2)
     a__geq(X1,X2) -> geq(X1,X2)
     a__div(X1,X2) -> div(X1,X2)
     a__if(X1,X2,X3) -> if(X1,X2,X3)
    Subterm Criterion Processor:
     simple projection:
      pi(a__minus#) = 0
     problem:
      DPs:
       
      TRS:
       a__minus(0(),Y) -> 0()
       a__minus(s(X),s(Y)) -> a__minus(X,Y)
       a__geq(X,0()) -> true()
       a__geq(0(),s(Y)) -> false()
       a__geq(s(X),s(Y)) -> a__geq(X,Y)
       a__div(0(),s(Y)) -> 0()
       a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())
       a__if(true(),X,Y) -> mark(X)
       a__if(false(),X,Y) -> mark(Y)
       mark(minus(X1,X2)) -> a__minus(X1,X2)
       mark(geq(X1,X2)) -> a__geq(X1,X2)
       mark(div(X1,X2)) -> a__div(mark(X1),X2)
       mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
       mark(0()) -> 0()
       mark(s(X)) -> s(mark(X))
       mark(true()) -> true()
       mark(false()) -> false()
       a__minus(X1,X2) -> minus(X1,X2)
       a__geq(X1,X2) -> geq(X1,X2)
       a__div(X1,X2) -> div(X1,X2)
       a__if(X1,X2,X3) -> if(X1,X2,X3)
     Qed