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 following 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 following constructor-restricted matrix
interpretation.

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

This order satisfies following ordering constraints:

  [p(s(0()))] = [4]  
              > [0]  
              = [0()]
                     

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)).

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))

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)).

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))

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)).

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

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

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


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


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