MAYBE

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

Proof:
 DP Processor:
  DPs:
   min#(s(x),s(y)) -> min#(x,y)
   max#(s(x),s(y)) -> max#(x,y)
   +#(s(x),y) -> +#(x,y)
   -#(s(x),s(y)) -> -#(x,y)
   *#(x,s(y)) -> *#(x,y)
   *#(x,s(y)) -> +#(x,*(x,y))
   f#(s(x)) -> +#(s(x),s(s(s(s(0())))))
   f#(s(x)) -> max#(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))
   f#(s(x)) -> +#(s(x),s(s(s(0()))))
   f#(s(x)) -> *#(s(x),s(x))
   f#(s(x)) -> max#(*(s(x),s(x)),+(s(x),s(s(s(0())))))
   f#(s(x)) ->
   -#(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))
   f#(s(x)) ->
   f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
  TRS:
   min(0(),y) -> 0()
   min(x,0()) -> 0()
   min(s(x),s(y)) -> s(min(x,y))
   max(0(),y) -> y
   max(x,0()) -> x
   max(s(x),s(y)) -> s(max(x,y))
   +(0(),y) -> y
   +(s(x),y) -> s(+(x,y))
   -(x,0()) -> x
   -(s(x),s(y)) -> -(x,y)
   *(x,0()) -> 0()
   *(x,s(y)) -> +(x,*(x,y))
   f(s(x)) ->
   f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
  TDG Processor:
   DPs:
    min#(s(x),s(y)) -> min#(x,y)
    max#(s(x),s(y)) -> max#(x,y)
    +#(s(x),y) -> +#(x,y)
    -#(s(x),s(y)) -> -#(x,y)
    *#(x,s(y)) -> *#(x,y)
    *#(x,s(y)) -> +#(x,*(x,y))
    f#(s(x)) -> +#(s(x),s(s(s(s(0())))))
    f#(s(x)) -> max#(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))
    f#(s(x)) -> +#(s(x),s(s(s(0()))))
    f#(s(x)) -> *#(s(x),s(x))
    f#(s(x)) -> max#(*(s(x),s(x)),+(s(x),s(s(s(0())))))
    f#(s(x)) ->
    -#(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
   TRS:
    min(0(),y) -> 0()
    min(x,0()) -> 0()
    min(s(x),s(y)) -> s(min(x,y))
    max(0(),y) -> y
    max(x,0()) -> x
    max(s(x),s(y)) -> s(max(x,y))
    +(0(),y) -> y
    +(s(x),y) -> s(+(x,y))
    -(x,0()) -> x
    -(s(x),s(y)) -> -(x,y)
    *(x,0()) -> 0()
    *(x,s(y)) -> +(x,*(x,y))
    f(s(x)) ->
    f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
   graph:
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) ->
    -#(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) -> max#(*(s(x),s(x)),+(s(x),s(s(s(0())))))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) -> *#(s(x),s(x))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) -> +#(s(x),s(s(s(0()))))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) -> max#(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))
    f#(s(x)) ->
    f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))))) ->
    f#(s(x)) -> +#(s(x),s(s(s(s(0())))))
    f#(s(x)) -> *#(s(x),s(x)) -> *#(x,s(y)) -> +#(x,*(x,y))
    f#(s(x)) -> *#(s(x),s(x)) ->
    *#(x,s(y)) -> *#(x,y)
    f#(s(x)) ->
    -#(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))) ->
    -#(s(x),s(y)) -> -#(x,y)
    f#(s(x)) -> +#(s(x),s(s(s(s(0()))))) -> +#(s(x),y) -> +#(x,y)
    f#(s(x)) -> +#(s(x),s(s(s(0())))) ->
    +#(s(x),y) -> +#(x,y)
    f#(s(x)) -> max#(*(s(x),s(x)),+(s(x),s(s(s(0()))))) ->
    max#(s(x),s(y)) -> max#(x,y)
    f#(s(x)) -> max#(s(*(s(x),s(x))),+(s(x),s(s(s(s(0())))))) ->
    max#(s(x),s(y)) -> max#(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),s(y)) -> -#(x,y) -> -#(s(x),s(y)) -> -#(x,y)
    +#(s(x),y) -> +#(x,y) -> +#(s(x),y) -> +#(x,y)
    max#(s(x),s(y)) -> max#(x,y) -> max#(s(x),s(y)) -> max#(x,y)
    min#(s(x),s(y)) -> min#(x,y) -> min#(s(x),s(y)) -> min#(x,y)
   SCC Processor:
    #sccs: 6
    #rules: 6
    #arcs: 21/169
    DPs:
     min#(s(x),s(y)) -> min#(x,y)
    TRS:
     min(0(),y) -> 0()
     min(x,0()) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(0(),y) -> y
     max(x,0()) -> x
     max(s(x),s(y)) -> s(max(x,y))
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) ->
     f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    Subterm Criterion Processor:
     simple projection:
      pi(min#) = 1
     problem:
      DPs:
       
      TRS:
       min(0(),y) -> 0()
       min(x,0()) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(0(),y) -> y
       max(x,0()) -> x
       max(s(x),s(y)) -> s(max(x,y))
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) ->
       f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
     Qed
    
    DPs:
     f#(s(x)) ->
     f#(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    TRS:
     min(0(),y) -> 0()
     min(x,0()) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(0(),y) -> y
     max(x,0()) -> x
     max(s(x),s(y)) -> s(max(x,y))
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) ->
     f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    Open
    
    DPs:
     -#(s(x),s(y)) -> -#(x,y)
    TRS:
     min(0(),y) -> 0()
     min(x,0()) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(0(),y) -> y
     max(x,0()) -> x
     max(s(x),s(y)) -> s(max(x,y))
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) ->
     f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    Subterm Criterion Processor:
     simple projection:
      pi(-#) = 1
     problem:
      DPs:
       
      TRS:
       min(0(),y) -> 0()
       min(x,0()) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(0(),y) -> y
       max(x,0()) -> x
       max(s(x),s(y)) -> s(max(x,y))
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) ->
       f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
     Qed
    
    DPs:
     *#(x,s(y)) -> *#(x,y)
    TRS:
     min(0(),y) -> 0()
     min(x,0()) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(0(),y) -> y
     max(x,0()) -> x
     max(s(x),s(y)) -> s(max(x,y))
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) ->
     f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    Subterm Criterion Processor:
     simple projection:
      pi(*#) = 1
     problem:
      DPs:
       
      TRS:
       min(0(),y) -> 0()
       min(x,0()) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(0(),y) -> y
       max(x,0()) -> x
       max(s(x),s(y)) -> s(max(x,y))
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) ->
       f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
     Qed
    
    DPs:
     max#(s(x),s(y)) -> max#(x,y)
    TRS:
     min(0(),y) -> 0()
     min(x,0()) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(0(),y) -> y
     max(x,0()) -> x
     max(s(x),s(y)) -> s(max(x,y))
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) ->
     f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    Subterm Criterion Processor:
     simple projection:
      pi(max#) = 1
     problem:
      DPs:
       
      TRS:
       min(0(),y) -> 0()
       min(x,0()) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(0(),y) -> y
       max(x,0()) -> x
       max(s(x),s(y)) -> s(max(x,y))
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) ->
       f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
     Qed
    
    DPs:
     +#(s(x),y) -> +#(x,y)
    TRS:
     min(0(),y) -> 0()
     min(x,0()) -> 0()
     min(s(x),s(y)) -> s(min(x,y))
     max(0(),y) -> y
     max(x,0()) -> x
     max(s(x),s(y)) -> s(max(x,y))
     +(0(),y) -> y
     +(s(x),y) -> s(+(x,y))
     -(x,0()) -> x
     -(s(x),s(y)) -> -(x,y)
     *(x,0()) -> 0()
     *(x,s(y)) -> +(x,*(x,y))
     f(s(x)) ->
     f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
    Subterm Criterion Processor:
     simple projection:
      pi(+#) = 0
     problem:
      DPs:
       
      TRS:
       min(0(),y) -> 0()
       min(x,0()) -> 0()
       min(s(x),s(y)) -> s(min(x,y))
       max(0(),y) -> y
       max(x,0()) -> x
       max(s(x),s(y)) -> s(max(x,y))
       +(0(),y) -> y
       +(s(x),y) -> s(+(x,y))
       -(x,0()) -> x
       -(s(x),s(y)) -> -(x,y)
       *(x,0()) -> 0()
       *(x,s(y)) -> +(x,*(x,y))
       f(s(x)) ->
       f(-(max(*(s(x),s(x)),+(s(x),s(s(s(0()))))),max(s(*(s(x),s(x))),+(s(x),s(s(s(s(0()))))))))
     Qed