YES(O(1),O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { f(0()) -> cons(0())
  , f(s(0())) -> f(p(s(0())))
  , p(s(X)) -> X }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We add following weak dependency pairs:

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(X)) -> 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(n^2)).

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(X)) -> c_3() }
Strict Trs:
  { f(0()) -> cons(0())
  , f(s(0())) -> f(p(s(0())))
  , p(s(X)) -> X }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We replace rewrite rules by usable rules:

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

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(s(0())) -> c_2(f^#(p(s(0()))))
  , p^#(s(X)) -> c_3() }
Strict Trs: { p(s(X)) -> X }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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 following constructor-restricted matrix
interpretation.

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

This order satisfies following ordering constraints:

  [p(s(X))] = [2] X + [4]
            > [1] X + [0]
            = [X]        
                         

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^2)).

Strict DPs: { f^#(s(0())) -> c_2(f^#(p(s(0())))) }
Weak DPs:
  { f^#(0()) -> c_1()
  , p^#(s(X)) -> c_3() }
Weak Trs: { p(s(X)) -> X }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

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

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

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict DPs: { f^#(s(0())) -> c_2(f^#(p(s(0())))) }
Weak Trs: { p(s(X)) -> X }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

The following argument positions are usable:
  Uargs(c_2) = {1}

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

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

This order satisfies following ordering constraints:

      [p(s(X))] =  [1 0] X + [0]        
                   [0 2]     [2]        
                >= [1 0] X + [0]        
                   [0 1]     [0]        
                =  [X]                  
                                        
  [f^#(s(0()))] =  [1]                  
                   [0]                  
                >  [0]                  
                   [0]                  
                =  [c_2(f^#(p(s(0()))))]
                                        

Hurray, we answered YES(O(1),O(n^2))