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

Strict Trs:
  { f(a()) -> b()
  , f(c()) -> d()
  , f(g(x, y)) -> g(f(x), f(y))
  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
  , g(x, x) -> h(e(), x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following dependency tuples:

Strict DPs:
  { f^#(a()) -> c_1()
  , f^#(c()) -> c_2()
  , f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
  , f^#(h(x, y)) -> c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y))
  , g^#(x, 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:
  { f^#(a()) -> c_1()
  , f^#(c()) -> c_2()
  , f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
  , f^#(h(x, y)) -> c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y))
  , g^#(x, x) -> c_5() }
Weak Trs:
  { f(a()) -> b()
  , f(c()) -> d()
  , f(g(x, y)) -> g(f(x), f(y))
  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
  , g(x, x) -> h(e(), x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We estimate the number of application of {1,2,5} by applications of
Pre({1,2,5}) = {3,4}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(a()) -> c_1()
    , 2: f^#(c()) -> c_2()
    , 3: f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
    , 4: f^#(h(x, y)) ->
         c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y))
    , 5: g^#(x, x) -> c_5() }

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

Strict DPs:
  { f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
  , f^#(h(x, y)) ->
    c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y)) }
Weak DPs:
  { f^#(a()) -> c_1()
  , f^#(c()) -> c_2()
  , g^#(x, x) -> c_5() }
Weak Trs:
  { f(a()) -> b()
  , f(c()) -> d()
  , f(g(x, y)) -> g(f(x), f(y))
  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
  , g(x, x) -> h(e(), x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

{ f^#(a()) -> c_1()
, f^#(c()) -> c_2()
, g^#(x, x) -> c_5() }

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

Strict DPs:
  { f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
  , f^#(h(x, y)) ->
    c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y)) }
Weak Trs:
  { f(a()) -> b()
  , f(c()) -> d()
  , f(g(x, y)) -> g(f(x), f(y))
  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
  , g(x, x) -> h(e(), x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
  , f^#(h(x, y)) ->
    c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y)) }

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

Strict DPs:
  { f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
  , f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
Weak Trs:
  { f(a()) -> b()
  , f(c()) -> d()
  , f(g(x, y)) -> g(f(x), f(y))
  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
  , g(x, x) -> h(e(), 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:
  { f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
  , f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 1: f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
  , 2: f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_1) = {1, 2}, Uargs(c_2) = {1, 2}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
      [g](x1, x2) = [2] x1 + [2] x2 + [2]
                                         
      [h](x1, x2) = [1] x1 + [1] x2 + [2]
                                         
        [f^#](x1) = [4] x1 + [0]         
                                         
    [c_1](x1, x2) = [1] x1 + [1] x2 + [1]
                                         
    [c_2](x1, x2) = [1] x1 + [1] x2 + [0]
  
  The order satisfies the following ordering constraints:
  
    [f^#(g(x, y))] = [8] x + [8] y + [8]  
                   > [4] x + [4] y + [1]  
                   = [c_1(f^#(x), f^#(y))]
                                          
    [f^#(h(x, y))] = [4] x + [4] y + [8]  
                   > [4] x + [4] y + [0]  
                   = [c_2(f^#(x), f^#(y))]
                                          

The strictly oriented rules are moved into the weak component.

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

Weak DPs:
  { f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
  , f^#(h(x, y)) -> c_2(f^#(x), f^#(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^#(g(x, y)) -> c_1(f^#(x), f^#(y))
, f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }

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))