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

We add following weak dependency pairs:

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

We replace rewrite rules by usable rules:

  Strict Usable Rules: { g(x, y) -> y }

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

Strict DPs:
  { f^#(s(x), y) -> c_1(f^#(x, g(x, y)))
  , f^#(0(), y) -> c_2()
  , g^#(x, y) -> c_3() }
Strict Trs: { g(x, y) -> 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(f^#) = {2}, Uargs(c_1) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

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

This order satisfies following ordering constraints:

  [g(x, y)] = [1] y + [1]
            > [1] y + [0]
            = [y]        
                         

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),O(1)).

Weak DPs:
  { f^#(s(x), y) -> c_1(f^#(x, g(x, y)))
  , f^#(0(), y) -> c_2()
  , g^#(x, y) -> c_3() }
Weak Trs: { g(x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ f^#(s(x), y) -> c_1(f^#(x, g(x, y)))
, f^#(0(), y) -> c_2()
, g^#(x, y) -> c_3() }

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

Weak Trs: { g(x, y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),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),O(1)).

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

Empty rules are trivially bounded

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