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

Strict Trs:
  { f(X, X) -> c(X)
  , f(X, c(X)) -> f(s(X), X)
  , f(s(X), X) -> f(X, a(X)) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following weak dependency pairs:

Strict DPs:
  { f^#(X, X) -> c_1(X)
  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
  , f^#(s(X), X) -> c_3(f^#(X, a(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^#(X, X) -> c_1(X)
  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
Strict Trs:
  { f(X, X) -> c(X)
  , f(X, c(X)) -> f(s(X), X)
  , f(s(X), X) -> f(X, a(X)) }
Obligation:
  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^#(X, X) -> c_1(X)
  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
Obligation:
  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_2) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

        [s](x1) = [0]           
                  [0]           
                                
        [a](x1) = [0]           
                  [0]           
                                
        [c](x1) = [0]           
                  [0]           
                                
  [f^#](x1, x2) = [1]           
                  [0]           
                                
      [c_1](x1) = [0]           
                  [0]           
                                
      [c_2](x1) = [1 0] x1 + [0]
                  [0 1]      [2]
                                
      [c_3](x1) = [0]           
                  [0]           

The order satisfies the following ordering constraints:

     [f^#(X, X)] = [1]                
                   [0]                
                 > [0]                
                   [0]                
                 = [c_1(X)]           
                                      
  [f^#(X, c(X))] = [1]                
                   [0]                
                 ? [1]                
                   [2]                
                 = [c_2(f^#(s(X), X))]
                                      
  [f^#(s(X), X)] = [1]                
                   [0]                
                 > [0]                
                   [0]                
                 = [c_3(f^#(X, a(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^#(X, c(X)) -> c_2(f^#(s(X), X)) }
Weak DPs:
  { f^#(X, X) -> c_1(X)
  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
Obligation:
  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^#(s(X), X) -> c_3(f^#(X, a(X))) }

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

Strict DPs: { f^#(X, c(X)) -> c_2(f^#(s(X), X)) }
Weak DPs: { f^#(X, X) -> c_1(X) }
Obligation:
  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^#(X, c(X)) -> c_2(f^#(s(X), X)) }

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

Strict DPs: { f^#(X, c(X)) -> c_1() }
Weak DPs: { f^#(X, X) -> c_2(X) }
Obligation:
  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^#(X, c(X)) -> c_1()
  , 2: f^#(X, X) -> c_2(X) }

Sub-proof:
----------
  The following argument positions are usable:
    none
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [c](x1) = [3]                  
                                         
    [f^#](x1, x2) = [2] x1 + [2] x2 + [1]
                                         
            [c_1] = [0]                  
                                         
        [c_2](x1) = [0]                  
  
  The order satisfies the following ordering constraints:
  
       [f^#(X, X)] = [4] X + [1]
                   > [0]        
                   = [c_2(X)]   
                                
    [f^#(X, c(X))] = [2] X + [7]
                   > [0]        
                   = [c_1()]    
                                

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^#(X, X) -> c_2(X)
  , f^#(X, c(X)) -> c_1() }
Obligation:
  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^#(X, X) -> c_2(X)
, f^#(X, c(X)) -> c_1() }

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

Rules: Empty
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^1))