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

We add following weak dependency pairs:

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

TcT has computed following constructor-restricted matrix
interpretation.

              [0] = [2]         
                                
          [s](x1) = [1] x1 + [1]
                                
  [add^#](x1, x2) = [2] x2 + [1]
                                
            [c_1] = [0]         
                                
        [c_2](x1) = [1] x1 + [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: { add^#(s(x), y) -> c_2(add^#(x, y)) }
Weak DPs: { add^#(0(), x) -> c_1() }
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.

{ add^#(0(), x) -> c_1() }

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

Strict DPs: { add^#(s(x), y) -> c_2(add^#(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

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

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

          [s](x1) = [1] x1 + [3]         
                                         
  [add^#](x1, x2) = [1] x1 + [3] x2 + [3]
                                         
        [c_2](x1) = [1] x1 + [2]         

This order satisfies following ordering constraints:

  [add^#(s(x), y)] = [1] x + [3] y + [6]
                   > [1] x + [3] y + [5]
                   = [c_2(add^#(x, y))] 
                                        

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