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

Strict Trs:
  { f(c(a(), z, x)) -> b(a(), z)
  , b(y, z) -> z
  , b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Arguments of following rules are not normal-forms:

{ b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y))) }

All above mentioned rules can be savely removed.

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

Strict Trs:
  { f(c(a(), z, x)) -> b(a(), z)
  , b(y, z) -> z }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following weak dependency pairs:

Strict DPs:
  { f^#(c(a(), z, x)) -> c_1(b^#(a(), z))
  , b^#(y, z) -> c_2() }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(c(a(), z, x)) -> c_1(b^#(a(), z))
  , b^#(y, z) -> c_2() }
Strict Trs:
  { f(c(a(), z, x)) -> b(a(), z)
  , b(y, z) -> z }
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)).

Strict DPs:
  { f^#(c(a(), z, x)) -> c_1(b^#(a(), z))
  , b^#(y, z) -> c_2() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_1) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

  [c](x1, x2, x3) = [0]           
                    [0]           
                                  
              [a] = [0]           
                    [0]           
                                  
        [f^#](x1) = [0]           
                    [0]           
                                  
        [c_1](x1) = [1 0] x1 + [0]
                    [0 1]      [2]
                                  
    [b^#](x1, x2) = [1]           
                    [0]           
                                  
            [c_2] = [0]           
                    [0]           

The order satisfies the following ordering constraints:

  [f^#(c(a(), z, x))] = [0]               
                        [0]               
                      ? [1]               
                        [2]               
                      = [c_1(b^#(a(), z))]
                                          
          [b^#(y, z)] = [1]               
                        [0]               
                      > [0]               
                        [0]               
                      = [c_2()]           
                                          

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)).

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

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

  DPs:
    { 1: f^#(c(a(), z, x)) -> c_1(b^#(a(), z))
    , 2: b^#(y, z) -> c_2() }

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

Weak DPs:
  { f^#(c(a(), z, x)) -> c_1(b^#(a(), z))
  , b^#(y, z) -> c_2() }
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^#(c(a(), z, x)) -> c_1(b^#(a(), z))
, b^#(y, z) -> c_2() }

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(1))