YES(O(1),O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
  { f(0()) -> s(0())
  , f(s(0())) -> s(s(0()))
  , f(s(0())) -> *(s(s(0())), f(0()))
  , f(+(x, y)) -> *(f(x), f(y))
  , f(+(x, s(0()))) -> +(s(s(0())), f(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2()
  , f^#(s(0())) -> c_3(f^#(0()))
  , f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
  , f^#(+(x, s(0()))) -> c_5(f^#(x)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2()
  , f^#(s(0())) -> c_3(f^#(0()))
  , f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
  , f^#(+(x, s(0()))) -> c_5(f^#(x)) }
Strict Trs:
  { f(0()) -> s(0())
  , f(s(0())) -> s(s(0()))
  , f(s(0())) -> *(s(s(0())), f(0()))
  , f(+(x, y)) -> *(f(x), f(y))
  , f(+(x, s(0()))) -> +(s(s(0())), f(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2()
  , f^#(s(0())) -> c_3(f^#(0()))
  , f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
  , f^#(+(x, s(0()))) -> c_5(f^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_3) = {1}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

            [0] = [0]                  
                                       
        [s](x1) = [0]                  
                                       
    [+](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
      [f^#](x1) = [1]                  
                                       
          [c_1] = [0]                  
                                       
          [c_2] = [0]                  
                                       
      [c_3](x1) = [1] x1 + [1]         
                                       
  [c_4](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
      [c_5](x1) = [1] x1 + [1]         

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(s(0())) -> c_3(f^#(0()))
  , f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
  , f^#(+(x, s(0()))) -> c_5(f^#(x)) }
Weak DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

We estimate the number of application of {1} by applications of
Pre({1}) = {2,3}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(s(0())) -> c_3(f^#(0()))
    , 2: f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
    , 3: f^#(+(x, s(0()))) -> c_5(f^#(x))
    , 4: f^#(0()) -> c_1()
    , 5: f^#(s(0())) -> c_2() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
  , f^#(+(x, s(0()))) -> c_5(f^#(x)) }
Weak DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2()
  , f^#(s(0())) -> c_3(f^#(0())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ f^#(0()) -> c_1()
, f^#(s(0())) -> c_2()
, f^#(s(0())) -> c_3(f^#(0())) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
  , f^#(+(x, s(0()))) -> c_5(f^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following argument positions are usable:
  Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}

TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).

            [0] = [0]                  
                                       
        [s](x1) = [1] x1 + [0]         
                                       
    [+](x1, x2) = [1] x1 + [1] x2 + [1]
                                       
      [f^#](x1) = [1] x1 + [0]         
                                       
  [c_4](x1, x2) = [1] x1 + [1] x2 + [0]
                                       
      [c_5](x1) = [1] x1 + [0]         

This order satisfies following ordering constraints:

       [f^#(+(x, y))] = [1] x + [1] y + [1]  
                      > [1] x + [1] y + [0]  
                      = [c_4(f^#(x), f^#(y))]
                                             
  [f^#(+(x, s(0())))] = [1] x + [1]          
                      > [1] x + [0]          
                      = [c_5(f^#(x))]        
                                             

Hurray, we answered YES(O(1),O(n^1))