YES(O(1),O(1))

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

Strict Trs:
  { f(0()) -> cons(0())
  , f(s(0())) -> f(p(s(0())))
  , p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(0())) -> c_3() }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(0())) -> c_3() }
Strict Trs:
  { f(0()) -> cons(0())
  , f(s(0())) -> f(p(s(0())))
  , p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We replace rewrite rules by usable rules:

  Strict Usable Rules: { p(s(0())) -> 0() }

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

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(0())) -> c_3() }
Strict Trs: { p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

The following argument positions are usable:
  Uargs(f^#) = {1}, Uargs(c_2) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

        [0] = [0]           
              [0]           
                            
    [s](x1) = [0]           
              [2]           
                            
    [p](x1) = [0 2] x1 + [0]
              [0 0]      [0]
                            
  [f^#](x1) = [1 0] x1 + [0]
              [0 0]      [0]
                            
      [c_1] = [1]           
              [1]           
                            
  [c_2](x1) = [1 0] x1 + [2]
              [0 1]      [2]
                            
  [p^#](x1) = [2]           
              [2]           
                            
      [c_3] = [1]           
              [1]           

The following symbols are considered usable

  {p, f^#, p^#}

The order satisfies the following ordering constraints:

    [p(s(0()))] = [4]                  
                  [0]                  
                > [0]                  
                  [0]                  
                = [0()]                
                                       
     [f^#(0())] = [0]                  
                  [0]                  
                ? [1]                  
                  [1]                  
                = [c_1()]              
                                       
  [f^#(s(0()))] = [0]                  
                  [0]                  
                ? [6]                  
                  [2]                  
                = [c_2(f^#(p(s(0()))))]
                                       
  [p^#(s(0()))] = [2]                  
                  [2]                  
                > [1]                  
                  [1]                  
                = [c_3()]              
                                       

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(1),O(1)).

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0())))) }
Weak DPs: { p^#(s(0())) -> c_3() }
Weak Trs: { p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: f^#(0()) -> c_1()
    , 2: f^#(s(0())) -> c_2(f^#(p(s(0()))))
    , 3: p^#(s(0())) -> c_3() }

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

Strict DPs: { f^#(s(0())) -> c_2(f^#(p(s(0())))) }
Weak DPs:
  { f^#(0()) -> c_1()
  , p^#(s(0())) -> c_3() }
Weak Trs: { p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: f^#(s(0())) -> c_2(f^#(p(s(0()))))
    , 2: f^#(0()) -> c_1()
    , 3: p^#(s(0())) -> c_3() }

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

Weak DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(0())) -> c_3() }
Weak Trs: { p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(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^#(p(s(0()))))
, p^#(s(0())) -> c_3() }

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

Weak Trs: { p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(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(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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