/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:
 plus(0(),x) -> x
 plus(s(x),y) -> s(plus(p(s(x)),y))
 times(0(),y) -> 0()
 times(s(x),y) -> plus(y,times(p(s(x)),y))
 p(s(0())) -> 0()
 p(s(s(x))) -> s(p(s(x)))
 fac(0(),x) -> x
 fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
 factorial(x) -> fac(x,s(0()))

Proof:
 DP Processor:
  DPs:
   plus#(s(x),y) -> p#(s(x))
   plus#(s(x),y) -> plus#(p(s(x)),y)
   times#(s(x),y) -> p#(s(x))
   times#(s(x),y) -> times#(p(s(x)),y)
   times#(s(x),y) -> plus#(y,times(p(s(x)),y))
   p#(s(s(x))) -> p#(s(x))
   fac#(s(x),y) -> times#(s(x),y)
   fac#(s(x),y) -> p#(s(x))
   fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
   factorial#(x) -> fac#(x,s(0()))
  TRS:
   plus(0(),x) -> x
   plus(s(x),y) -> s(plus(p(s(x)),y))
   times(0(),y) -> 0()
   times(s(x),y) -> plus(y,times(p(s(x)),y))
   p(s(0())) -> 0()
   p(s(s(x))) -> s(p(s(x)))
   fac(0(),x) -> x
   fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
   factorial(x) -> fac(x,s(0()))
  TDG Processor:
   DPs:
    plus#(s(x),y) -> p#(s(x))
    plus#(s(x),y) -> plus#(p(s(x)),y)
    times#(s(x),y) -> p#(s(x))
    times#(s(x),y) -> times#(p(s(x)),y)
    times#(s(x),y) -> plus#(y,times(p(s(x)),y))
    p#(s(s(x))) -> p#(s(x))
    fac#(s(x),y) -> times#(s(x),y)
    fac#(s(x),y) -> p#(s(x))
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
    factorial#(x) -> fac#(x,s(0()))
   TRS:
    plus(0(),x) -> x
    plus(s(x),y) -> s(plus(p(s(x)),y))
    times(0(),y) -> 0()
    times(s(x),y) -> plus(y,times(p(s(x)),y))
    p(s(0())) -> 0()
    p(s(s(x))) -> s(p(s(x)))
    fac(0(),x) -> x
    fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
    factorial(x) -> fac(x,s(0()))
   graph:
    factorial#(x) -> fac#(x,s(0())) ->
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
    factorial#(x) -> fac#(x,s(0())) -> fac#(s(x),y) -> p#(s(x))
    factorial#(x) -> fac#(x,s(0())) ->
    fac#(s(x),y) -> times#(s(x),y)
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y)) ->
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y)) ->
    fac#(s(x),y) -> p#(s(x))
    fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y)) ->
    fac#(s(x),y) -> times#(s(x),y)
    fac#(s(x),y) -> times#(s(x),y) ->
    times#(s(x),y) -> plus#(y,times(p(s(x)),y))
    fac#(s(x),y) -> times#(s(x),y) ->
    times#(s(x),y) -> times#(p(s(x)),y)
    fac#(s(x),y) -> times#(s(x),y) -> times#(s(x),y) -> p#(s(x))
    fac#(s(x),y) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
    times#(s(x),y) -> times#(p(s(x)),y) ->
    times#(s(x),y) -> plus#(y,times(p(s(x)),y))
    times#(s(x),y) -> times#(p(s(x)),y) ->
    times#(s(x),y) -> times#(p(s(x)),y)
    times#(s(x),y) -> times#(p(s(x)),y) -> times#(s(x),y) -> p#(s(x))
    times#(s(x),y) -> p#(s(x)) ->
    p#(s(s(x))) -> p#(s(x))
    times#(s(x),y) -> plus#(y,times(p(s(x)),y)) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    times#(s(x),y) -> plus#(y,times(p(s(x)),y)) -> plus#(s(x),y) -> p#(s(x))
    p#(s(s(x))) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
    plus#(s(x),y) -> p#(s(x)) -> p#(s(s(x))) -> p#(s(x))
    plus#(s(x),y) -> plus#(p(s(x)),y) ->
    plus#(s(x),y) -> plus#(p(s(x)),y)
    plus#(s(x),y) -> plus#(p(s(x)),y) -> plus#(s(x),y) -> p#(s(x))
   SCC Processor:
    #sccs: 4
    #rules: 4
    #arcs: 20/100
    DPs:
     fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
    TRS:
     plus(0(),x) -> x
     plus(s(x),y) -> s(plus(p(s(x)),y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(p(s(x)),y))
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     fac(0(),x) -> x
     fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
     factorial(x) -> fac(x,s(0()))
    Usable Rule Processor:
     DPs:
      fac#(s(x),y) -> fac#(p(s(x)),times(s(x),y))
     TRS:
      times(s(x),y) -> plus(y,times(p(s(x)),y))
      plus(0(),x) -> x
      plus(s(x),y) -> s(plus(p(s(x)),y))
      p(s(0())) -> 0()
      p(s(s(x))) -> s(p(s(x)))
      times(0(),y) -> 0()
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
      interpretation:
       [fac#](x0, x1) = 3x0 + 0,
       
       [times](x0, x1) = x1 + 0,
       
       [p](x0) = -1x0 + 0,
       
       [s](x0) = 1x0 + 1,
       
       [plus](x0, x1) = -1x0 + x1 + 0,
       
       [0] = 3
      orientation:
       fac#(s(x),y) = 4x + 4 >= 3x + 3 = fac#(p(s(x)),times(s(x),y))
       
       times(s(x),y) = y + 0 >= y + 0 = plus(y,times(p(s(x)),y))
       
       plus(0(),x) = x + 2 >= x = x
       
       plus(s(x),y) = x + y + 0 >= x + 1y + 1 = s(plus(p(s(x)),y))
       
       p(s(0())) = 3 >= 3 = 0()
       
       p(s(s(x))) = 1x + 1 >= 1x + 1 = s(p(s(x)))
       
       times(0(),y) = y + 0 >= 3 = 0()
      problem:
       DPs:
        
       TRS:
        times(s(x),y) -> plus(y,times(p(s(x)),y))
        plus(0(),x) -> x
        plus(s(x),y) -> s(plus(p(s(x)),y))
        p(s(0())) -> 0()
        p(s(s(x))) -> s(p(s(x)))
        times(0(),y) -> 0()
      Qed
    
    DPs:
     times#(s(x),y) -> times#(p(s(x)),y)
    TRS:
     plus(0(),x) -> x
     plus(s(x),y) -> s(plus(p(s(x)),y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(p(s(x)),y))
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     fac(0(),x) -> x
     fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
     factorial(x) -> fac(x,s(0()))
    Usable Rule Processor:
     DPs:
      times#(s(x),y) -> times#(p(s(x)),y)
     TRS:
      p(s(0())) -> 0()
      p(s(s(x))) -> s(p(s(x)))
     Semantic Labeling Processor:
      dimension: 2
      usable rules:
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
      interpretation:
                 [1 0]  
       [p](x0) = [1 0]x0,
       
                 [0 1]     [0]
       [s](x0) = [0 1]x0 + [1],
       
             [0]
       [0] = [1]
       labeled:
        times#
        s
        p
        0
       usable (for model):
        times#
        s
        p
        0
       argument filtering:
        pi(0) = []
        pi(s) = []
        pi(p) = []
        pi(times#) = 0
      precedence:
       s > times# ~ p ~ 0
      problem:
       DPs:
        
       TRS:
        p(s(0())) -> 0()
        p(s(s(x))) -> s(p(s(x)))
      Qed
    
    DPs:
     plus#(s(x),y) -> plus#(p(s(x)),y)
    TRS:
     plus(0(),x) -> x
     plus(s(x),y) -> s(plus(p(s(x)),y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(p(s(x)),y))
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     fac(0(),x) -> x
     fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
     factorial(x) -> fac(x,s(0()))
    Usable Rule Processor:
     DPs:
      plus#(s(x),y) -> plus#(p(s(x)),y)
     TRS:
      p(s(0())) -> 0()
      p(s(s(x))) -> s(p(s(x)))
     Semantic Labeling Processor:
      dimension: 2
      usable rules:
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
      interpretation:
                 [1 0]  
       [p](x0) = [1 0]x0,
       
                 [0 1]     [0]
       [s](x0) = [0 1]x0 + [1],
       
             [0]
       [0] = [1]
       labeled:
        plus#
        s
        p
        0
       usable (for model):
        plus#
        s
        p
        0
       argument filtering:
        pi(0) = []
        pi(s) = []
        pi(p) = []
        pi(plus#) = 0
      precedence:
       s > plus# ~ p ~ 0
      problem:
       DPs:
        
       TRS:
        p(s(0())) -> 0()
        p(s(s(x))) -> s(p(s(x)))
      Qed
    
    DPs:
     p#(s(s(x))) -> p#(s(x))
    TRS:
     plus(0(),x) -> x
     plus(s(x),y) -> s(plus(p(s(x)),y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(p(s(x)),y))
     p(s(0())) -> 0()
     p(s(s(x))) -> s(p(s(x)))
     fac(0(),x) -> x
     fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
     factorial(x) -> fac(x,s(0()))
    Subterm Criterion Processor:
     simple projection:
      pi(p#) = 0
     problem:
      DPs:
       
      TRS:
       plus(0(),x) -> x
       plus(s(x),y) -> s(plus(p(s(x)),y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(p(s(x)),y))
       p(s(0())) -> 0()
       p(s(s(x))) -> s(p(s(x)))
       fac(0(),x) -> x
       fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
       factorial(x) -> fac(x,s(0()))
     Qed