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:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x)))
  , g(c(x)) -> x
  , g(c(h(0()))) -> g(d(1()))
  , g(c(1())) -> g(d(h(0())))
  , g(d(x)) -> x
  , g(h(x)) -> g(x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { f^#(f(x)) -> c_1(f^#(c(f(x))))
  , f^#(f(x)) -> c_2(f^#(d(f(x))))
  , g^#(c(x)) -> c_3()
  , g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(d(x)) -> c_6()
  , g^#(h(x)) -> c_7(g^#(x)) }

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^#(f(x)) -> c_1(f^#(c(f(x))))
  , f^#(f(x)) -> c_2(f^#(d(f(x))))
  , g^#(c(x)) -> c_3()
  , g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(d(x)) -> c_6()
  , g^#(h(x)) -> c_7(g^#(x)) }
Strict Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x)))
  , g(c(x)) -> x
  , g(c(h(0()))) -> g(d(1()))
  , g(c(1())) -> g(d(h(0())))
  , g(d(x)) -> x
  , g(h(x)) -> g(x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { f(f(x)) -> f(c(f(x)))
    , f(f(x)) -> f(d(f(x))) }

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

Strict DPs:
  { f^#(f(x)) -> c_1(f^#(c(f(x))))
  , f^#(f(x)) -> c_2(f^#(d(f(x))))
  , g^#(c(x)) -> c_3()
  , g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(d(x)) -> c_6()
  , g^#(h(x)) -> c_7(g^#(x)) }
Strict Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x))) }
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_4) = {1}, Uargs(c_5) = {1}, Uargs(c_7) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

    [f](x1) = [0 2] x1 + [0]
              [0 0]      [1]
                            
    [c](x1) = [0]           
              [0]           
                            
    [d](x1) = [0]           
              [0]           
                            
    [h](x1) = [0]           
              [0]           
                            
        [0] = [0]           
              [0]           
                            
        [1] = [0]           
              [0]           
                            
  [f^#](x1) = [2]           
              [0]           
                            
  [c_1](x1) = [2]           
              [2]           
                            
  [c_2](x1) = [2]           
              [2]           
                            
  [g^#](x1) = [0]           
              [0]           
                            
      [c_3] = [1]           
              [1]           
                            
  [c_4](x1) = [1 0] x1 + [2]
              [0 1]      [2]
                            
  [c_5](x1) = [1 0] x1 + [2]
              [0 1]      [2]
                            
      [c_6] = [1]           
              [1]           
                            
  [c_7](x1) = [1 0] x1 + [2]
              [0 1]      [2]

The following symbols are considered usable

  {f, f^#, g^#}

The order satisfies the following ordering constraints:

         [f(f(x))] = [2]                  
                     [1]                  
                   > [0]                  
                     [1]                  
                   = [f(c(f(x)))]         
                                          
         [f(f(x))] = [2]                  
                     [1]                  
                   > [0]                  
                     [1]                  
                   = [f(d(f(x)))]         
                                          
       [f^#(f(x))] = [2]                  
                     [0]                  
                   ? [2]                  
                     [2]                  
                   = [c_1(f^#(c(f(x))))]  
                                          
       [f^#(f(x))] = [2]                  
                     [0]                  
                   ? [2]                  
                     [2]                  
                   = [c_2(f^#(d(f(x))))]  
                                          
       [g^#(c(x))] = [0]                  
                     [0]                  
                   ? [1]                  
                     [1]                  
                   = [c_3()]              
                                          
  [g^#(c(h(0())))] = [0]                  
                     [0]                  
                   ? [2]                  
                     [2]                  
                   = [c_4(g^#(d(1())))]   
                                          
     [g^#(c(1()))] = [0]                  
                     [0]                  
                   ? [2]                  
                     [2]                  
                   = [c_5(g^#(d(h(0()))))]
                                          
       [g^#(d(x))] = [0]                  
                     [0]                  
                   ? [1]                  
                     [1]                  
                   = [c_6()]              
                                          
       [g^#(h(x))] = [0]                  
                     [0]                  
                   ? [2]                  
                     [2]                  
                   = [c_7(g^#(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(n^1)).

Strict DPs:
  { f^#(f(x)) -> c_1(f^#(c(f(x))))
  , f^#(f(x)) -> c_2(f^#(d(f(x))))
  , g^#(c(x)) -> c_3()
  , g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(d(x)) -> c_6()
  , g^#(h(x)) -> c_7(g^#(x)) }
Weak Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

Consider the dependency graph:

  1: f^#(f(x)) -> c_1(f^#(c(f(x))))
  
  2: f^#(f(x)) -> c_2(f^#(d(f(x))))
  
  3: g^#(c(x)) -> c_3()
  
  4: g^#(c(h(0()))) -> c_4(g^#(d(1()))) -->_1 g^#(d(x)) -> c_6() :6
  
  5: g^#(c(1())) -> c_5(g^#(d(h(0())))) -->_1 g^#(d(x)) -> c_6() :6
  
  6: g^#(d(x)) -> c_6()
  
  7: g^#(h(x)) -> c_7(g^#(x))
     -->_1 g^#(h(x)) -> c_7(g^#(x)) :7
     -->_1 g^#(d(x)) -> c_6() :6
     -->_1 g^#(c(1())) -> c_5(g^#(d(h(0())))) :5
     -->_1 g^#(c(h(0()))) -> c_4(g^#(d(1()))) :4
     -->_1 g^#(c(x)) -> c_3() :3
  

Only the nodes {3,4,6,5,7} are reachable from nodes {3,4,5,6,7}
that start derivation from marked basic terms. The nodes not
reachable are removed from the problem.

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

Strict DPs:
  { g^#(c(x)) -> c_3()
  , g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(d(x)) -> c_6()
  , g^#(h(x)) -> c_7(g^#(x)) }
Weak Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

  DPs:
    { 1: g^#(c(x)) -> c_3()
    , 2: g^#(c(h(0()))) -> c_4(g^#(d(1())))
    , 3: g^#(c(1())) -> c_5(g^#(d(h(0()))))
    , 4: g^#(d(x)) -> c_6()
    , 5: g^#(h(x)) -> c_7(g^#(x)) }

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

Strict DPs:
  { g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(h(x)) -> c_7(g^#(x)) }
Weak DPs:
  { g^#(c(x)) -> c_3()
  , g^#(d(x)) -> c_6() }
Weak Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(x))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

  DPs:
    { 1: g^#(c(h(0()))) -> c_4(g^#(d(1())))
    , 2: g^#(c(1())) -> c_5(g^#(d(h(0()))))
    , 3: g^#(h(x)) -> c_7(g^#(x))
    , 4: g^#(c(x)) -> c_3()
    , 5: g^#(d(x)) -> c_6() }

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

Strict DPs: { g^#(h(x)) -> c_7(g^#(x)) }
Weak DPs:
  { g^#(c(x)) -> c_3()
  , g^#(c(h(0()))) -> c_4(g^#(d(1())))
  , g^#(c(1())) -> c_5(g^#(d(h(0()))))
  , g^#(d(x)) -> c_6() }
Weak Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(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.

{ g^#(c(x)) -> c_3()
, g^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(d(x)) -> c_6() }

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

Strict DPs: { g^#(h(x)) -> c_7(g^#(x)) }
Weak Trs:
  { f(f(x)) -> f(c(f(x)))
  , f(f(x)) -> f(d(f(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: { g^#(h(x)) -> c_7(g^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.

DPs:
  { 1: g^#(h(x)) -> c_7(g^#(x)) }

Sub-proof:
----------
  The input was oriented with the instance of 'Small Polynomial Path
  Order (PS,1-bounded)' as induced by the safe mapping
  
   safe(f) = {}, safe(c) = {1}, safe(d) = {1}, safe(g) = {},
   safe(h) = {1}, safe(0) = {}, safe(1) = {}, safe(f^#) = {},
   safe(c_1) = {}, safe(c_2) = {}, safe(g^#) = {}, safe(c_3) = {},
   safe(c_4) = {}, safe(c_5) = {}, safe(c_6) = {}, safe(c_7) = {}
  
  and precedence
  
   empty .
  
  Following symbols are considered recursive:
  
   {g^#}
  
  The recursion depth is 1.
  
  Further, following argument filtering is employed:
  
   pi(f) = [], pi(c) = [], pi(d) = [], pi(g) = [], pi(h) = [1],
   pi(0) = [], pi(1) = [], pi(f^#) = [], pi(c_1) = [], pi(c_2) = [],
   pi(g^#) = [1], pi(c_3) = [], pi(c_4) = [], pi(c_5) = [],
   pi(c_6) = [], pi(c_7) = [1]
  
  Usable defined function symbols are a subset of:
  
   {f^#, g^#}
  
  For your convenience, here are the satisfied ordering constraints:
  
    pi(g^#(h(x))) = g^#(h(; x);)   
                  > c_7(g^#(x;);)  
                  = pi(c_7(g^#(x)))
                                   

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: { g^#(h(x)) -> c_7(g^#(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.

{ g^#(h(x)) -> c_7(g^#(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(n^1))