YES(O(1),O(1))

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

Strict Trs:
  { p(f(f(x))) -> q(f(g(x)))
  , p(g(g(x))) -> q(g(f(x)))
  , q(f(f(x))) -> p(f(g(x)))
  , q(g(g(x))) -> p(g(f(x))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }

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:
  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
Strict Trs:
  { p(f(f(x))) -> q(f(g(x)))
  , p(g(g(x))) -> q(g(f(x)))
  , q(f(f(x))) -> p(f(g(x)))
  , q(g(g(x))) -> p(g(f(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)).

Strict DPs:
  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
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:
  none

TcT has computed the following constructor-restricted matrix
interpretation.

    [f](x1) = [2]           
              [2]           
                            
    [g](x1) = [0]           
              [0]           
                            
  [p^#](x1) = [0 2] x1 + [0]
              [2 0]      [0]
                            
  [c_1](x1) = [1]           
              [1]           
                            
  [q^#](x1) = [0]           
              [0]           
                            
  [c_2](x1) = [1]           
              [1]           
                            
  [c_3](x1) = [1]           
              [0]           
                            
  [c_4](x1) = [1]           
              [0]           

The following symbols are considered usable

  {p^#, q^#}

The order satisfies the following ordering constraints:

  [p^#(f(f(x)))] = [4]                
                   [4]                
                 > [1]                
                   [1]                
                 = [c_1(q^#(f(g(x))))]
                                      
  [p^#(g(g(x)))] = [0]                
                   [0]                
                 ? [1]                
                   [1]                
                 = [c_2(q^#(g(f(x))))]
                                      
  [q^#(f(f(x)))] = [0]                
                   [0]                
                 ? [1]                
                   [0]                
                 = [c_3(p^#(f(g(x))))]
                                      
  [q^#(g(g(x)))] = [0]                
                   [0]                
                 ? [1]                
                   [0]                
                 = [c_4(p^#(g(f(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)).

Strict DPs:
  { p^#(g(g(x))) -> c_2(q^#(g(f(x))))
  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
Weak DPs: { p^#(f(f(x))) -> c_1(q^#(f(g(x)))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: p^#(g(g(x))) -> c_2(q^#(g(f(x))))
    , 2: q^#(f(f(x))) -> c_3(p^#(f(g(x))))
    , 3: q^#(g(g(x))) -> c_4(p^#(g(f(x))))
    , 4: p^#(f(f(x))) -> c_1(q^#(f(g(x)))) }

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

Weak DPs:
  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
  , q^#(g(g(x))) -> c_4(p^#(g(f(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.

{ p^#(f(f(x))) -> c_1(q^#(f(g(x))))
, p^#(g(g(x))) -> c_2(q^#(g(f(x))))
, q^#(f(f(x))) -> c_3(p^#(f(g(x))))
, q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }

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))