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:
  { d(x) -> e(u(x))
  , d(u(x)) -> c(x)
  , c(u(x)) -> b(x)
  , b(u(x)) -> a(e(x))
  , v(e(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { d^#(x) -> c_1()
  , d^#(u(x)) -> c_2(c^#(x))
  , c^#(u(x)) -> c_3(b^#(x))
  , b^#(u(x)) -> c_4()
  , v^#(e(x)) -> c_5() }

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:
  { d^#(x) -> c_1()
  , d^#(u(x)) -> c_2(c^#(x))
  , c^#(u(x)) -> c_3(b^#(x))
  , b^#(u(x)) -> c_4()
  , v^#(e(x)) -> c_5() }
Strict Trs:
  { d(x) -> e(u(x))
  , d(u(x)) -> c(x)
  , c(u(x)) -> b(x)
  , b(u(x)) -> a(e(x))
  , v(e(x)) -> x }
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:
  { d^#(x) -> c_1()
  , d^#(u(x)) -> c_2(c^#(x))
  , c^#(u(x)) -> c_3(b^#(x))
  , b^#(u(x)) -> c_4()
  , v^#(e(x)) -> c_5() }
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.

    [e](x1) = [2]         
                          
    [u](x1) = [1] x1 + [1]
                          
  [d^#](x1) = [2] x1 + [2]
                          
      [c_1] = [1]         
                          
  [c_2](x1) = [1] x1 + [1]
                          
  [c^#](x1) = [1] x1 + [2]
                          
  [c_3](x1) = [1] x1 + [1]
                          
  [b^#](x1) = [1] x1 + [1]
                          
      [c_4] = [1]         
                          
  [v^#](x1) = [1] x1 + [2]
                          
      [c_5] = [1]         

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(1),O(1)).

Weak DPs:
  { d^#(x) -> c_1()
  , d^#(u(x)) -> c_2(c^#(x))
  , c^#(u(x)) -> c_3(b^#(x))
  , b^#(u(x)) -> c_4()
  , v^#(e(x)) -> c_5() }
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.

{ d^#(x) -> c_1()
, d^#(u(x)) -> c_2(c^#(x))
, c^#(u(x)) -> c_3(b^#(x))
, b^#(u(x)) -> c_4()
, v^#(e(x)) -> c_5() }

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))