/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:
 from(X) -> cons(X,n__from(s(X)))
 2ndspos(0(),Z) -> rnil()
 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
 2ndsneg(0(),Z) -> rnil()
 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
 pi(X) -> 2ndspos(X,from(0()))
 plus(0(),Y) -> Y
 plus(s(X),Y) -> s(plus(X,Y))
 times(0(),Y) -> 0()
 times(s(X),Y) -> plus(Y,times(X,Y))
 square(X) -> times(X,X)
 from(X) -> n__from(X)
 activate(n__from(X)) -> from(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   2ndspos#(s(N),cons(X,Z)) -> activate#(Z)
   2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
   2ndspos#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
   2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z))
   2ndsneg#(s(N),cons(X,Z)) -> activate#(Z)
   2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
   2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
   2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z))
   pi#(X) -> from#(0())
   pi#(X) -> 2ndspos#(X,from(0()))
   plus#(s(X),Y) -> plus#(X,Y)
   times#(s(X),Y) -> times#(X,Y)
   times#(s(X),Y) -> plus#(Y,times(X,Y))
   square#(X) -> times#(X,X)
   activate#(n__from(X)) -> from#(X)
  TRS:
   from(X) -> cons(X,n__from(s(X)))
   2ndspos(0(),Z) -> rnil()
   2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
   2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
   2ndsneg(0(),Z) -> rnil()
   2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
   2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
   pi(X) -> 2ndspos(X,from(0()))
   plus(0(),Y) -> Y
   plus(s(X),Y) -> s(plus(X,Y))
   times(0(),Y) -> 0()
   times(s(X),Y) -> plus(Y,times(X,Y))
   square(X) -> times(X,X)
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X
  TDG Processor:
   DPs:
    2ndspos#(s(N),cons(X,Z)) -> activate#(Z)
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z))
    2ndsneg#(s(N),cons(X,Z)) -> activate#(Z)
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z))
    pi#(X) -> from#(0())
    pi#(X) -> 2ndspos#(X,from(0()))
    plus#(s(X),Y) -> plus#(X,Y)
    times#(s(X),Y) -> times#(X,Y)
    times#(s(X),Y) -> plus#(Y,times(X,Y))
    square#(X) -> times#(X,X)
    activate#(n__from(X)) -> from#(X)
   TRS:
    from(X) -> cons(X,n__from(s(X)))
    2ndspos(0(),Z) -> rnil()
    2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
    2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
    2ndsneg(0(),Z) -> rnil()
    2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
    2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
    pi(X) -> 2ndspos(X,from(0()))
    plus(0(),Y) -> Y
    plus(s(X),Y) -> s(plus(X,Y))
    times(0(),Y) -> 0()
    times(s(X),Y) -> plus(Y,times(X,Y))
    square(X) -> times(X,X)
    from(X) -> n__from(X)
    activate(n__from(X)) -> from(X)
    activate(X) -> X
   graph:
    square#(X) -> times#(X,X) -> times#(s(X),Y) -> plus#(Y,times(X,Y))
    square#(X) -> times#(X,X) -> times#(s(X),Y) -> times#(X,Y)
    times#(s(X),Y) -> times#(X,Y) ->
    times#(s(X),Y) -> plus#(Y,times(X,Y))
    times#(s(X),Y) -> times#(X,Y) ->
    times#(s(X),Y) -> times#(X,Y)
    times#(s(X),Y) -> plus#(Y,times(X,Y)) -> plus#(s(X),Y) -> plus#(X,Y)
    plus#(s(X),Y) -> plus#(X,Y) -> plus#(s(X),Y) -> plus#(X,Y)
    pi#(X) -> 2ndspos#(X,from(0())) ->
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z))
    pi#(X) -> 2ndspos#(X,from(0())) ->
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    pi#(X) -> 2ndspos#(X,from(0())) ->
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
    pi#(X) -> 2ndspos#(X,from(0())) ->
    2ndspos#(s(N),cons(X,Z)) -> activate#(Z)
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(X)
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) ->
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z))
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) ->
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) ->
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z)) ->
    2ndspos#(s(N),cons(X,Z)) -> activate#(Z)
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) ->
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z))
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) ->
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) ->
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z))) ->
    2ndsneg#(s(N),cons(X,Z)) -> activate#(Z)
    2ndsneg#(s(N),cons(X,Z)) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(X)
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) ->
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z))
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) ->
    2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) ->
    2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z)) ->
    2ndsneg#(s(N),cons(X,Z)) -> activate#(Z)
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(X)
    2ndspos#(s(N),cons(X,Z)) -> activate#(Z) ->
    activate#(n__from(X)) -> from#(X)
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) ->
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z))
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) ->
    2ndspos#(s(N),cons2(X,cons(Y,Z))) -> activate#(Z)
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) ->
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
    2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z))) ->
    2ndspos#(s(N),cons(X,Z)) -> activate#(Z)
   SCC Processor:
    #sccs: 3
    #rules: 6
    #arcs: 30/225
    DPs:
     2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
     2ndspos#(s(N),cons2(X,cons(Y,Z))) -> 2ndsneg#(N,activate(Z))
     2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
     2ndsneg#(s(N),cons2(X,cons(Y,Z))) -> 2ndspos#(N,activate(Z))
    TRS:
     from(X) -> cons(X,n__from(s(X)))
     2ndspos(0(),Z) -> rnil()
     2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
     2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
     2ndsneg(0(),Z) -> rnil()
     2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
     2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
     pi(X) -> 2ndspos(X,from(0()))
     plus(0(),Y) -> Y
     plus(s(X),Y) -> s(plus(X,Y))
     times(0(),Y) -> 0()
     times(s(X),Y) -> plus(Y,times(X,Y))
     square(X) -> times(X,X)
     from(X) -> n__from(X)
     activate(n__from(X)) -> from(X)
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(2ndspos#) = 0
      pi(2ndsneg#) = 0
     problem:
      DPs:
       2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
       2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
      TRS:
       from(X) -> cons(X,n__from(s(X)))
       2ndspos(0(),Z) -> rnil()
       2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
       2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
       2ndsneg(0(),Z) -> rnil()
       2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
       2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
       pi(X) -> 2ndspos(X,from(0()))
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       times(0(),Y) -> 0()
       times(s(X),Y) -> plus(Y,times(X,Y))
       square(X) -> times(X,X)
       from(X) -> n__from(X)
       activate(n__from(X)) -> from(X)
       activate(X) -> X
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 8/4
      DPs:
       2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
      TRS:
       from(X) -> cons(X,n__from(s(X)))
       2ndspos(0(),Z) -> rnil()
       2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
       2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
       2ndsneg(0(),Z) -> rnil()
       2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
       2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
       pi(X) -> 2ndspos(X,from(0()))
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       times(0(),Y) -> 0()
       times(s(X),Y) -> plus(Y,times(X,Y))
       square(X) -> times(X,X)
       from(X) -> n__from(X)
       activate(n__from(X)) -> from(X)
       activate(X) -> X
      EDG Processor:
       DPs:
        2ndspos#(s(N),cons(X,Z)) -> 2ndspos#(s(N),cons2(X,activate(Z)))
       TRS:
        from(X) -> cons(X,n__from(s(X)))
        2ndspos(0(),Z) -> rnil()
        2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
        2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
        2ndsneg(0(),Z) -> rnil()
        2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
        2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
        pi(X) -> 2ndspos(X,from(0()))
        plus(0(),Y) -> Y
        plus(s(X),Y) -> s(plus(X,Y))
        times(0(),Y) -> 0()
        times(s(X),Y) -> plus(Y,times(X,Y))
        square(X) -> times(X,X)
        from(X) -> n__from(X)
        activate(n__from(X)) -> from(X)
        activate(X) -> X
       graph:
        
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 0/1
        
      
      DPs:
       2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
      TRS:
       from(X) -> cons(X,n__from(s(X)))
       2ndspos(0(),Z) -> rnil()
       2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
       2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
       2ndsneg(0(),Z) -> rnil()
       2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
       2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
       pi(X) -> 2ndspos(X,from(0()))
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       times(0(),Y) -> 0()
       times(s(X),Y) -> plus(Y,times(X,Y))
       square(X) -> times(X,X)
       from(X) -> n__from(X)
       activate(n__from(X)) -> from(X)
       activate(X) -> X
      EDG Processor:
       DPs:
        2ndsneg#(s(N),cons(X,Z)) -> 2ndsneg#(s(N),cons2(X,activate(Z)))
       TRS:
        from(X) -> cons(X,n__from(s(X)))
        2ndspos(0(),Z) -> rnil()
        2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
        2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
        2ndsneg(0(),Z) -> rnil()
        2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
        2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
        pi(X) -> 2ndspos(X,from(0()))
        plus(0(),Y) -> Y
        plus(s(X),Y) -> s(plus(X,Y))
        times(0(),Y) -> 0()
        times(s(X),Y) -> plus(Y,times(X,Y))
        square(X) -> times(X,X)
        from(X) -> n__from(X)
        activate(n__from(X)) -> from(X)
        activate(X) -> X
       graph:
        
       SCC Processor:
        #sccs: 0
        #rules: 0
        #arcs: 0/1
        
    
    DPs:
     times#(s(X),Y) -> times#(X,Y)
    TRS:
     from(X) -> cons(X,n__from(s(X)))
     2ndspos(0(),Z) -> rnil()
     2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
     2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
     2ndsneg(0(),Z) -> rnil()
     2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
     2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
     pi(X) -> 2ndspos(X,from(0()))
     plus(0(),Y) -> Y
     plus(s(X),Y) -> s(plus(X,Y))
     times(0(),Y) -> 0()
     times(s(X),Y) -> plus(Y,times(X,Y))
     square(X) -> times(X,X)
     from(X) -> n__from(X)
     activate(n__from(X)) -> from(X)
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      TRS:
       from(X) -> cons(X,n__from(s(X)))
       2ndspos(0(),Z) -> rnil()
       2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
       2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
       2ndsneg(0(),Z) -> rnil()
       2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
       2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
       pi(X) -> 2ndspos(X,from(0()))
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       times(0(),Y) -> 0()
       times(s(X),Y) -> plus(Y,times(X,Y))
       square(X) -> times(X,X)
       from(X) -> n__from(X)
       activate(n__from(X)) -> from(X)
       activate(X) -> X
     Qed
    
    DPs:
     plus#(s(X),Y) -> plus#(X,Y)
    TRS:
     from(X) -> cons(X,n__from(s(X)))
     2ndspos(0(),Z) -> rnil()
     2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
     2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
     2ndsneg(0(),Z) -> rnil()
     2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
     2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
     pi(X) -> 2ndspos(X,from(0()))
     plus(0(),Y) -> Y
     plus(s(X),Y) -> s(plus(X,Y))
     times(0(),Y) -> 0()
     times(s(X),Y) -> plus(Y,times(X,Y))
     square(X) -> times(X,X)
     from(X) -> n__from(X)
     activate(n__from(X)) -> from(X)
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       
      TRS:
       from(X) -> cons(X,n__from(s(X)))
       2ndspos(0(),Z) -> rnil()
       2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
       2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
       2ndsneg(0(),Z) -> rnil()
       2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
       2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
       pi(X) -> 2ndspos(X,from(0()))
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       times(0(),Y) -> 0()
       times(s(X),Y) -> plus(Y,times(X,Y))
       square(X) -> times(X,X)
       from(X) -> n__from(X)
       activate(n__from(X)) -> from(X)
       activate(X) -> X
     Qed