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

Strict Trs:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))

We add the following weak dependency pairs:

Strict DPs:
{ plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z)))
, times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) }

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:
{ plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z)))
, times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) }
Strict Trs:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X)) }
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(plus) = {2}, Uargs(plus^#) = {2}, Uargs(c_1) = {1},
Uargs(c_2) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

[plus](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 1]      [0 1]      [1]

[times](x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[2 1]      [2 0]      [0]

[s](x1) = [1 2] x1 + [2]
[0 0]      [0]

[plus^#](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 0]      [0 0]      [0]

[c_1](x1) = [1 0] x1 + [0]
[0 1]      [0]

[times^#](x1, x2) = [2 1] x1 + [1 0] x2 + [0]
[0 0]      [0 0]      [0]

[c_2](x1) = [1 0] x1 + [0]
[0 1]      [0]

The order satisfies the following ordering constraints:

[plus(plus(X, Y), Z)] = [0 1] X + [0 1] Y + [1 0] Z + [1]
[0 1]     [0 1]     [0 1]     [2]
> [0 1] X + [0 1] Y + [1 0] Z + [0]
[0 1]     [0 1]     [0 1]     [2]
= [plus(X, plus(Y, Z))]

[times(X, s(Y))] = [1 2] X + [1 2] Y + [2]
[2 1]     [2 4]     [4]
> [1 1] X + [1 2] Y + [0]
[2 1]     [2 1]     [1]
= [plus(X, times(Y, X))]

[plus^#(plus(X, Y), Z)] = [0 1] X + [0 1] Y + [1 0] Z + [1]
[0 0]     [0 0]     [0 0]     [0]
> [0 1] X + [0 1] Y + [1 0] Z + [0]
[0 0]     [0 0]     [0 0]     [0]
= [c_1(plus^#(X, plus(Y, Z)))]

[times^#(X, s(Y))] = [2 1] X + [1 2] Y + [2]
[0 0]     [0 0]     [0]
> [1 1] X + [1 2] Y + [0]
[0 0]     [0 0]     [0]
= [c_2(plus^#(X, times(Y, X)))]

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:
{ plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z)))
, times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) }
Weak Trs:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X)) }
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.

{ plus^#(plus(X, Y), Z) -> c_1(plus^#(X, plus(Y, Z)))
, times^#(X, s(Y)) -> c_2(plus^#(X, times(Y, X))) }

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

Weak Trs:
{ plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
, times(X, s(Y)) -> plus(X, times(Y, X)) }
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))
```