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:
  { a__c() -> a__f(g(c()))
  , a__c() -> c()
  , a__f(X) -> f(X)
  , a__f(g(X)) -> g(X)
  , mark(g(X)) -> g(X)
  , mark(c()) -> a__c()
  , mark(f(X)) -> a__f(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { a__c^#() -> c_1(a__f^#(g(c())))
  , a__c^#() -> c_2()
  , a__f^#(X) -> c_3()
  , a__f^#(g(X)) -> c_4()
  , mark^#(g(X)) -> c_5()
  , mark^#(c()) -> c_6(a__c^#())
  , mark^#(f(X)) -> c_7(a__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(n^1)).

Strict DPs:
  { a__c^#() -> c_1(a__f^#(g(c())))
  , a__c^#() -> c_2()
  , a__f^#(X) -> c_3()
  , a__f^#(g(X)) -> c_4()
  , mark^#(g(X)) -> c_5()
  , mark^#(c()) -> c_6(a__c^#())
  , mark^#(f(X)) -> c_7(a__f^#(X)) }
Strict Trs:
  { a__c() -> a__f(g(c()))
  , a__c() -> c()
  , a__f(X) -> f(X)
  , a__f(g(X)) -> g(X)
  , mark(g(X)) -> g(X)
  , mark(c()) -> a__c()
  , mark(f(X)) -> a__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:
  { a__c^#() -> c_1(a__f^#(g(c())))
  , a__c^#() -> c_2()
  , a__f^#(X) -> c_3()
  , a__f^#(g(X)) -> c_4()
  , mark^#(g(X)) -> c_5()
  , mark^#(c()) -> c_6(a__c^#())
  , mark^#(f(X)) -> c_7(a__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_1) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

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

This order satisfies following ordering constraints:


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)).

Strict DPs: { a__c^#() -> c_1(a__f^#(g(c()))) }
Weak DPs:
  { a__c^#() -> c_2()
  , a__f^#(X) -> c_3()
  , a__f^#(g(X)) -> c_4()
  , mark^#(g(X)) -> c_5()
  , mark^#(c()) -> c_6(a__c^#())
  , mark^#(f(X)) -> c_7(a__f^#(X)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ a__c^#() -> c_2()
, a__f^#(X) -> c_3()
, a__f^#(g(X)) -> c_4()
, mark^#(g(X)) -> c_5()
, mark^#(f(X)) -> c_7(a__f^#(X)) }

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

Strict DPs: { a__c^#() -> c_1(a__f^#(g(c()))) }
Weak DPs: { mark^#(c()) -> c_6(a__c^#()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { a__c^#() -> c_1(a__f^#(g(c()))) }

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

Strict DPs: { a__c^#() -> c_1() }
Weak DPs: { mark^#(c()) -> c_2(a__c^#()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

Consider the dependency graph

  1: a__c^#() -> c_1()
  
  2: mark^#(c()) -> c_2(a__c^#()) -->_1 a__c^#() -> c_1() :1
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { mark^#(c()) -> c_2(a__c^#()) }


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

Strict DPs: { a__c^#() -> c_1() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

Consider the dependency graph

  1: a__c^#() -> c_1()
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { a__c^#() -> c_1() }


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

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

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


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