MAYBE

Problem:
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 +(0(),y) -> y
 +(s(x),y) -> s(+(x,y))
 *(x,0()) -> 0()
 *(x,s(y)) -> +(x,*(x,y))
 f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))

Proof:
 DP Processor:
  DPs:
   -#(s(x),s(y)) -> -#(x,y)
   +#(s(x),y) -> +#(x,y)
   *#(x,s(y)) -> *#(x,y)
   *#(x,s(y)) -> +#(x,*(x,y))
   f#(s(x)) -> *#(s(s(0())),s(x))
   f#(s(x)) -> -#(*(s(s(0())),s(x)),s(s(x)))
   f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x))))
  TRS:
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   +(0(),y) -> y
   +(s(x),y) -> s(+(x,y))
   *(x,0()) -> 0()
   *(x,s(y)) -> +(x,*(x,y))
   f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
  TDG Processor:
   DPs:
    -#(s(x),s(y)) -> -#(x,y)
    +#(s(x),y) -> +#(x,y)
    *#(x,s(y)) -> *#(x,y)
    *#(x,s(y)) -> +#(x,*(x,y))
    f#(s(x)) -> *#(s(s(0())),s(x))
    f#(s(x)) -> -#(*(s(s(0())),s(x)),s(s(x)))
    f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x))))
   TRS:
    -(x,0()) -> x
    -(s(x),s(y)) -> -(x,y)
    +(0(),y) -> y
    +(s(x),y) -> s(+(x,y))
    *(x,0()) -> 0()
    *(x,s(y)) -> +(x,*(x,y))
    f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
   graph:
    f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x)))) ->
    f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x))))
    f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x)))) ->
    f#(s(x)) -> -#(*(s(s(0())),s(x)),s(s(x)))
    f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x)))) ->
    f#(s(x)) -> *#(s(s(0())),s(x))
    f#(s(x)) -> *#(s(s(0())),s(x)) -> *#(x,s(y)) -> +#(x,*(x,y))
    f#(s(x)) -> *#(s(s(0())),s(x)) ->
    *#(x,s(y)) -> *#(x,y)
    f#(s(x)) -> -#(*(s(s(0())),s(x)),s(s(x))) -> -#(s(x),s(y)) -> -#(x,y)
    *#(x,s(y)) -> *#(x,y) -> *#(x,s(y)) -> +#(x,*(x,y))
    *#(x,s(y)) -> *#(x,y) -> *#(x,s(y)) -> *#(x,y)
    *#(x,s(y)) -> +#(x,*(x,y)) -> +#(s(x),y) -> +#(x,y)
    +#(s(x),y) -> +#(x,y) -> +#(s(x),y) -> +#(x,y)
    -#(s(x),s(y)) -> -#(x,y) -> -#(s(x),s(y)) -> -#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 4
    #arcs: 11/49
    DPs:
     f#(s(x)) -> f#(-(*(s(s(0())),s(x)),s(s(x))))
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
    Open
    
    DPs:
     -#(s(x),s(y)) -> -#(x,y)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
    LPO Processor:
     argument filtering:
      pi(0) = []
      pi(-) = [0]
      pi(s) = [0]
      pi(+) = [0,1]
      pi(*) = [0,1]
      pi(f) = []
      pi(-#) = [0]
     precedence:
      * > + > -# ~ f ~ s ~ - ~ 0
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
     Qed
    
    DPs:
     *#(x,s(y)) -> *#(x,y)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
    LPO Processor:
     argument filtering:
      pi(0) = []
      pi(-) = [0]
      pi(s) = [0]
      pi(+) = [0,1]
      pi(*) = [0,1]
      pi(f) = []
      pi(*#) = [1]
     precedence:
      * > + > *# ~ f ~ s ~ - ~ 0
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
     Qed
    
    DPs:
     +#(s(x),y) -> +#(x,y)
    TRS:
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
    LPO Processor:
     argument filtering:
      pi(0) = []
      pi(-) = [0]
      pi(s) = [0]
      pi(+) = [0,1]
      pi(*) = [0,1]
      pi(f) = []
      pi(+#) = [0]
     precedence:
      * > + > +# ~ f ~ s ~ - ~ 0
     problem:
      DPs:
       
      TRS:
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) -> f(-(*(s(s(0())),s(x)),s(s(x))))
     Qed