YES(?,POLY)

We are left with following problem, upon which TcT provides the
certificate YES(?,POLY).

Strict Trs:
  { f(s(x1), x2, x3, x4, x5, x6, x7, x8, x9, x10) ->
    f(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
  , f(0(), s(x2), x3, x4, x5, x6, x7, x8, x9, x10) ->
    f(x2, x2, x3, x4, x5, x6, x7, x8, x9, x10)
  , f(0(), 0(), s(x3), x4, x5, x6, x7, x8, x9, x10) ->
    f(x3, x3, x3, x4, x5, x6, x7, x8, x9, x10)
  , f(0(), 0(), 0(), s(x4), x5, x6, x7, x8, x9, x10) ->
    f(x4, x4, x4, x4, x5, x6, x7, x8, x9, x10)
  , f(0(), 0(), 0(), 0(), s(x5), x6, x7, x8, x9, x10) ->
    f(x5, x5, x5, x5, x5, x6, x7, x8, x9, x10)
  , f(0(), 0(), 0(), 0(), 0(), s(x6), x7, x8, x9, x10) ->
    f(x6, x6, x6, x6, x6, x6, x7, x8, x9, x10)
  , f(0(), 0(), 0(), 0(), 0(), 0(), s(x7), x8, x9, x10) ->
    f(x7, x7, x7, x7, x7, x7, x7, x8, x9, x10)
  , f(0(), 0(), 0(), 0(), 0(), 0(), 0(), s(x8), x9, x10) ->
    f(x8, x8, x8, x8, x8, x8, x8, x8, x9, x10)
  , f(0(), 0(), 0(), 0(), 0(), 0(), 0(), 0(), s(x9), x10) ->
    f(x9, x9, x9, x9, x9, x9, x9, x9, x9, x10)
  , f(0(), 0(), 0(), 0(), 0(), 0(), 0(), 0(), 0(), s(x10)) ->
    f(x10, x10, x10, x10, x10, x10, x10, x10, x10, x10)
  , f(0(), 0(), 0(), 0(), 0(), 0(), 0(), 0(), 0(), 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,POLY)

The input was oriented with the instance of 'Polynomial Path Order
(PS)' as induced by the safe mapping

 safe(f) = {}, safe(s) = {1}, safe(0) = {}

and precedence

 empty .

Following symbols are considered recursive:

 {f}

The recursion depth is 1.

For your convenience, here are the satisfied ordering constraints:

           f(s(; x1),  x2,  x3,  x4,  x5,  x6,  x7,  x8,  x9,  x10;) > f(x1,  x2,  x3,  x4,  x5,  x6,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
          f(0(),  s(; x2),  x3,  x4,  x5,  x6,  x7,  x8,  x9,  x10;) > f(x2,  x2,  x3,  x4,  x5,  x6,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
         f(0(),  0(),  s(; x3),  x4,  x5,  x6,  x7,  x8,  x9,  x10;) > f(x3,  x3,  x3,  x4,  x5,  x6,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
        f(0(),  0(),  0(),  s(; x4),  x5,  x6,  x7,  x8,  x9,  x10;) > f(x4,  x4,  x4,  x4,  x5,  x6,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
       f(0(),  0(),  0(),  0(),  s(; x5),  x6,  x7,  x8,  x9,  x10;) > f(x5,  x5,  x5,  x5,  x5,  x6,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
      f(0(),  0(),  0(),  0(),  0(),  s(; x6),  x7,  x8,  x9,  x10;) > f(x6,  x6,  x6,  x6,  x6,  x6,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
     f(0(),  0(),  0(),  0(),  0(),  0(),  s(; x7),  x8,  x9,  x10;) > f(x7,  x7,  x7,  x7,  x7,  x7,  x7,  x8,  x9,  x10;)         
                                                                                                                                    
    f(0(),  0(),  0(),  0(),  0(),  0(),  0(),  s(; x8),  x9,  x10;) > f(x8,  x8,  x8,  x8,  x8,  x8,  x8,  x8,  x9,  x10;)         
                                                                                                                                    
   f(0(),  0(),  0(),  0(),  0(),  0(),  0(),  0(),  s(; x9),  x10;) > f(x9,  x9,  x9,  x9,  x9,  x9,  x9,  x9,  x9,  x10;)         
                                                                                                                                    
  f(0(),  0(),  0(),  0(),  0(),  0(),  0(),  0(),  0(),  s(; x10);) > f(x10,  x10,  x10,  x10,  x10,  x10,  x10,  x10,  x10,  x10;)
                                                                                                                                    
       f(0(),  0(),  0(),  0(),  0(),  0(),  0(),  0(),  0(),  0();) > 0()                                                          
                                                                                                                                    

Hurray, we answered YES(?,POLY)