/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:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   plus#(s(x),y) -> plus#(x,y)
   plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
  TRS:
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
  TDG Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    plus#(s(x),y) -> plus#(x,y)
    plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
   TRS:
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
   graph:
    plus#(s(x),y) -> plus#(x,y) ->
    plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
    plus#(s(x),y) -> plus#(x,y) ->
    plus#(s(x),y) -> plus#(x,y)
    plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
    plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
    plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
    plus#(s(x),y) -> plus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 8/25
    DPs:
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
    Subterm Criterion Processor:
     simple projection:
      pi(minus) = 0
      pi(quot#) = 0
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
     Qed
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
     problem:
      DPs:
       
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
    TRS:
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = [0,0,1,1]
     problem:
      DPs:
       plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
     Usable Rule Processor:
      DPs:
       plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
      TRS:
       minus(s(x),s(y)) -> minus(x,y)
       minus(x,0()) -> x
      Semantic Labeling Processor:
       dimension: 2
       usable rules:
        
       interpretation:
                       [1]
        [s](x0) = x0 + [1],
        
                          [1 1]     [0 1]  
        [minus](x0, x1) = [1 1]x0 + [0 0]x1,
        
              [0]
        [0] = [0]
        labeled:
         plus#
         minus
        usable (for model):
         plus#
         minus
         s
         0
        argument filtering:
         pi(0) = []
         pi(minus) = 1
         pi(s) = []
         pi(plus#) = []
       precedence:
        plus# ~ s ~ minus ~ 0
       problem:
        DPs:
         
        TRS:
         minus(s(x),s(y)) -> minus(x,y)
         minus(x,0()) -> x
       Qed