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(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))

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

The following argument positions are usable:
  none

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [f](x1, x2) = [1] x1 + [1] x2 + [7]
                                     
      [s](x1) = [7]                  
                                     
      [a](x1) = [3]                  
                                     
      [c](x1) = [7]                  

The following symbols are considered usable

  {f}

The order satisfies the following ordering constraints:

     [f(X, X)] =  [2] X + [7] 
               >= [7]         
               =  [c(X)]      
                              
  [f(X, c(X))] =  [1] X + [14]
               >= [1] X + [14]
               =  [f(s(X), X)]
                              
  [f(s(X), X)] =  [1] X + [14]
               >  [1] X + [10]
               =  [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 Trs:
  { f(X, X) -> c(X)
  , f(X, c(X)) -> f(s(X), X) }
Weak Trs: { f(s(X), X) -> f(X, a(X)) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

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

The following argument positions are usable:
  none

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [f](x1, x2) = [1] x1 + [1] x2 + [7]
                                     
      [s](x1) = [4]                  
                                     
      [a](x1) = [3]                  
                                     
      [c](x1) = [7]                  

The following symbols are considered usable

  {f}

The order satisfies the following ordering constraints:

     [f(X, X)] =  [2] X + [7] 
               >= [7]         
               =  [c(X)]      
                              
  [f(X, c(X))] =  [1] X + [14]
               >  [1] X + [11]
               =  [f(s(X), X)]
                              
  [f(s(X), X)] =  [1] X + [11]
               >  [1] X + [10]
               =  [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 Trs: { f(X, X) -> c(X) }
Weak Trs:
  { 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))

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

The following argument positions are usable:
  none

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [f](x1, x2) = [1] x1 + [1] x2 + [7]
                                     
      [s](x1) = [3]                  
                                     
      [a](x1) = [3]                  
                                     
      [c](x1) = [3]                  

The following symbols are considered usable

  {f}

The order satisfies the following ordering constraints:

     [f(X, X)] =  [2] X + [7] 
               >  [3]         
               =  [c(X)]      
                              
  [f(X, c(X))] =  [1] X + [10]
               >= [1] X + [10]
               =  [f(s(X), X)]
                              
  [f(s(X), X)] =  [1] X + [10]
               >= [1] X + [10]
               =  [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(1)).

Weak 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(1))

Empty rules are trivially bounded

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