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

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

Strict Trs:
  { g(X) -> u(h(X), h(X), X)
  , u(d(), c(Y), X) -> k(Y)
  , h(d()) -> c(a())
  , h(d()) -> c(b())
  , f(k(a()), k(b()), X) -> f(X, X, X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { g^#(X) -> c_1(u^#(h(X), h(X), X))
  , u^#(d(), c(Y), X) -> c_2()
  , h^#(d()) -> c_3()
  , h^#(d()) -> c_4()
  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, 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:
  { g^#(X) -> c_1(u^#(h(X), h(X), X))
  , u^#(d(), c(Y), X) -> c_2()
  , h^#(d()) -> c_3()
  , h^#(d()) -> c_4()
  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Strict Trs:
  { g(X) -> u(h(X), h(X), X)
  , u(d(), c(Y), X) -> k(Y)
  , h(d()) -> c(a())
  , h(d()) -> c(b())
  , f(k(a()), k(b()), X) -> f(X, X, X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { h(d()) -> c(a())
    , h(d()) -> c(b()) }

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

Strict DPs:
  { g^#(X) -> c_1(u^#(h(X), h(X), X))
  , u^#(d(), c(Y), X) -> c_2()
  , h^#(d()) -> c_3()
  , h^#(d()) -> c_4()
  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Strict Trs:
  { h(d()) -> c(a())
  , h(d()) -> c(b()) }
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}, Uargs(u^#) = {1, 2}

TcT has computed the following constructor-restricted matrix
interpretation.

            [h](x1) = [2]                                 
                      [0]                                 
                                                          
                [d] = [0]                                 
                      [0]                                 
                                                          
            [c](x1) = [0]                                 
                      [0]                                 
                                                          
            [k](x1) = [0]                                 
                      [0]                                 
                                                          
                [a] = [0]                                 
                      [0]                                 
                                                          
                [b] = [0]                                 
                      [0]                                 
                                                          
          [g^#](x1) = [2 1] x1 + [1]                      
                      [1 1]      [1]                      
                                                          
          [c_1](x1) = [1 0] x1 + [0]                      
                      [0 1]      [2]                      
                                                          
  [u^#](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [0 0] x3 + [1]
                      [1 1]      [0 0]      [1 1]      [2]
                                                          
              [c_2] = [1]                                 
                      [1]                                 
                                                          
          [h^#](x1) = [1 1] x1 + [2]                      
                      [1 1]      [2]                      
                                                          
              [c_3] = [1]                                 
                      [1]                                 
                                                          
              [c_4] = [1]                                 
                      [2]                                 
                                                          
  [f^#](x1, x2, x3) = [1]                                 
                      [1]                                 
                                                          
          [c_5](x1) = [2]                                 
                      [2]                                 

The following symbols are considered usable

  {h, g^#, u^#, h^#, f^#}

The order satisfies the following ordering constraints:

                  [h(d())] =  [2]                      
                              [0]                      
                           >  [0]                      
                              [0]                      
                           =  [c(a())]                 
                                                       
                  [h(d())] =  [2]                      
                              [0]                      
                           >  [0]                      
                              [0]                      
                           =  [c(b())]                 
                                                       
                  [g^#(X)] =  [2 1] X + [1]            
                              [1 1]     [1]            
                           ?  [0 0] X + [7]            
                              [1 1]     [6]            
                           =  [c_1(u^#(h(X), h(X), X))]
                                                       
       [u^#(d(), c(Y), X)] =  [0 0] X + [1]            
                              [1 1]     [2]            
                           >= [1]                      
                              [1]                      
                           =  [c_2()]                  
                                                       
                [h^#(d())] =  [2]                      
                              [2]                      
                           >  [1]                      
                              [1]                      
                           =  [c_3()]                  
                                                       
                [h^#(d())] =  [2]                      
                              [2]                      
                           >  [1]                      
                              [2]                      
                           =  [c_4()]                  
                                                       
  [f^#(k(a()), k(b()), X)] =  [1]                      
                              [1]                      
                           ?  [2]                      
                              [2]                      
                           =  [c_5(f^#(X, X, 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:
  { g^#(X) -> c_1(u^#(h(X), h(X), X))
  , u^#(d(), c(Y), X) -> c_2()
  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Weak DPs:
  { h^#(d()) -> c_3()
  , h^#(d()) -> c_4() }
Weak Trs:
  { h(d()) -> c(a())
  , h(d()) -> c(b()) }
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: g^#(X) -> c_1(u^#(h(X), h(X), X))
    , 2: u^#(d(), c(Y), X) -> c_2()
    , 3: f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X))
    , 4: h^#(d()) -> c_3()
    , 5: h^#(d()) -> c_4() }

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

Weak DPs:
  { g^#(X) -> c_1(u^#(h(X), h(X), X))
  , u^#(d(), c(Y), X) -> c_2()
  , h^#(d()) -> c_3()
  , h^#(d()) -> c_4()
  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
Weak Trs:
  { h(d()) -> c(a())
  , h(d()) -> c(b()) }
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.

{ g^#(X) -> c_1(u^#(h(X), h(X), X))
, u^#(d(), c(Y), X) -> c_2()
, h^#(d()) -> c_3()
, h^#(d()) -> c_4()
, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }

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

Weak Trs:
  { h(d()) -> c(a())
  , h(d()) -> c(b()) }
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(1))