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

We add following weak dependency pairs:

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

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:
  { +^#(0(), y) -> c_1()
  , +^#(s(x), y) -> c_2(+^#(x, s(y)))
  , +^#(s(x), y) -> c_3(+^#(x, y)) }
Strict Trs:
  { +(0(), y) -> y
  , +(s(x), y) -> +(x, s(y))
  , +(s(x), y) -> s(+(x, y)) }
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:
  { +^#(0(), y) -> c_1()
  , +^#(s(x), y) -> c_2(+^#(x, s(y)))
  , +^#(s(x), y) -> c_3(+^#(x, y)) }
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_2) = {1}, Uargs(c_3) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

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

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: { +^#(s(x), y) -> c_2(+^#(x, s(y))) }
Weak DPs:
  { +^#(0(), y) -> c_1()
  , +^#(s(x), y) -> c_3(+^#(x, y)) }
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.

{ +^#(0(), y) -> c_1() }

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

Strict DPs: { +^#(s(x), y) -> c_2(+^#(x, s(y))) }
Weak DPs: { +^#(s(x), y) -> c_3(+^#(x, y)) }
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(+^#) = {2}, safe(c_2) = {}, safe(c_3) = {}

and precedence

 empty .

Following symbols are considered recursive:

 {+^#}

The recursion depth is 1.

Further, following argument filtering is employed:

 pi(s) = [1], pi(+^#) = [1, 2], pi(c_2) = [1], pi(c_3) = [1]

Usable defined function symbols are a subset of:

 {+^#}

For your convenience, here are the satisfied ordering constraints:

  pi(+^#(s(x), y)) = +^#(s(; x); y)       
                   > c_2(+^#(x; s(; y));) 
                   = pi(c_2(+^#(x, s(y))))
                                          
  pi(+^#(s(x), y)) = +^#(s(; x); y)       
                   > c_3(+^#(x; y);)      
                   = pi(c_3(+^#(x, y)))   
                                          

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