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:
  { p(m, n, s(r)) -> p(m, r, n)
  , p(m, s(n), 0()) -> p(0(), n, m)
  , p(m, 0(), 0()) -> m }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { p^#(m, n, s(r)) -> c_1(p^#(m, r, n))
  , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m))
  , p^#(m, 0(), 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(n^1)).

Strict DPs:
  { p^#(m, n, s(r)) -> c_1(p^#(m, r, n))
  , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m))
  , p^#(m, 0(), 0()) -> c_3() }
Strict Trs:
  { p(m, n, s(r)) -> p(m, r, n)
  , p(m, s(n), 0()) -> p(0(), n, m)
  , p(m, 0(), 0()) -> m }
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:
  { p^#(m, n, s(r)) -> c_1(p^#(m, r, n))
  , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m))
  , p^#(m, 0(), 0()) -> c_3() }
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_1) = {1}, Uargs(c_2) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

            [s](x1) = [1] x1 + [0]
                                  
                [0] = [0]         
                                  
  [p^#](x1, x2, x3) = [1]         
                                  
          [c_1](x1) = [1] x1 + [1]
                                  
          [c_2](x1) = [1] x1 + [0]
                                  
              [c_3] = [0]         

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:
  { p^#(m, n, s(r)) -> c_1(p^#(m, r, n))
  , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) }
Weak DPs: { p^#(m, 0(), 0()) -> c_3() }
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.

{ p^#(m, 0(), 0()) -> c_3() }

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

Strict DPs:
  { p^#(m, n, s(r)) -> c_1(p^#(m, r, n))
  , p^#(m, s(n), 0()) -> c_2(p^#(0(), n, m)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

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

 safe(s) = {1}, safe(0) = {}, safe(p^#) = {}, safe(c_1) = {},
 safe(c_2) = {}

and precedence

 empty .

Following symbols are considered recursive:

 {p^#}

The recursion depth is 1.

Further, following argument filtering is employed:

 pi(s) = [1], pi(0) = [], pi(p^#) = [1, 2, 3], pi(c_1) = [1],
 pi(c_2) = [1]

Usable defined function symbols are a subset of:

 {p^#}

For your convenience, here are the satisfied ordering constraints:

    pi(p^#(m, n, s(r))) = p^#(m,  n,  s(; r);)   
                        > c_1(p^#(m,  r,  n;);)  
                        = pi(c_1(p^#(m, r, n)))  
                                                 
  pi(p^#(m, s(n), 0())) = p^#(m,  s(; n),  0();) 
                        > c_2(p^#(0(),  n,  m;);)
                        = pi(c_2(p^#(0(), n, m)))
                                                 

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