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

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

Strict Trs:
  { f(n__b(), X, n__c()) -> f(X, c(), X)
  , c() -> n__c()
  , c() -> b()
  , b() -> n__b()
  , activate(X) -> X
  , activate(n__b()) -> b()
  , activate(n__c()) -> c() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#())
  , activate^#(n__c()) -> c_7(c^#()) }

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:
  { f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#())
  , activate^#(n__c()) -> c_7(c^#()) }
Strict Trs:
  { f(n__b(), X, n__c()) -> f(X, c(), X)
  , c() -> n__c()
  , c() -> b()
  , b() -> n__b()
  , activate(X) -> X
  , activate(n__b()) -> b()
  , activate(n__c()) -> c() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { c() -> n__c()
    , c() -> b()
    , b() -> n__b() }

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

Strict DPs:
  { f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
  , c^#() -> c_2()
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#())
  , activate^#(n__c()) -> c_7(c^#()) }
Strict Trs:
  { c() -> n__c()
  , c() -> b()
  , b() -> n__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(f^#) = {2}, Uargs(c_1) = {1}, Uargs(c_3) = {1},
  Uargs(c_6) = {1}, Uargs(c_7) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

             [n__b] = [0]                           
                                                    
             [n__c] = [1]                           
                                                    
                [c] = [2]                           
                                                    
                [b] = [1]                           
                                                    
  [f^#](x1, x2, x3) = [1] x1 + [2] x2 + [1] x3 + [2]
                                                    
          [c_1](x1) = [1] x1 + [1]                  
                                                    
              [c^#] = [2]                           
                                                    
              [c_2] = [1]                           
                                                    
          [c_3](x1) = [1] x1 + [2]                  
                                                    
              [b^#] = [1]                           
                                                    
              [c_4] = [1]                           
                                                    
   [activate^#](x1) = [2] x1 + [2]                  
                                                    
              [c_5] = [1]                           
                                                    
          [c_6](x1) = [1] x1 + [1]                  
                                                    
          [c_7](x1) = [1] x1 + [1]                  

This order satisfies following ordering constraints:

  [c()] = [2]     
        > [1]     
        = [n__c()]
                  
  [c()] = [2]     
        > [1]     
        = [b()]   
                  
  [b()] = [1]     
        > [0]     
        = [n__b()]
                  

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:
  { f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
  , c^#() -> c_3(b^#())
  , b^#() -> c_4()
  , activate^#(n__b()) -> c_6(b^#()) }
Weak DPs:
  { c^#() -> c_2()
  , activate^#(X) -> c_5()
  , activate^#(n__c()) -> c_7(c^#()) }
Weak Trs:
  { c() -> n__c()
  , c() -> b()
  , b() -> n__b() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

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

  DPs:
    { 1: f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
    , 2: c^#() -> c_3(b^#())
    , 3: b^#() -> c_4()
    , 4: activate^#(n__b()) -> c_6(b^#())
    , 5: c^#() -> c_2()
    , 6: activate^#(X) -> c_5()
    , 7: activate^#(n__c()) -> c_7(c^#()) }

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

Strict DPs:
  { c^#() -> c_3(b^#())
  , activate^#(n__b()) -> c_6(b^#()) }
Weak DPs:
  { f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
  , c^#() -> c_2()
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__c()) -> c_7(c^#()) }
Weak Trs:
  { c() -> n__c()
  , c() -> b()
  , b() -> n__b() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

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

  DPs:
    { 1: c^#() -> c_3(b^#())
    , 2: activate^#(n__b()) -> c_6(b^#())
    , 3: f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
    , 4: c^#() -> c_2()
    , 5: b^#() -> c_4()
    , 6: activate^#(X) -> c_5()
    , 7: activate^#(n__c()) -> c_7(c^#()) }

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

Strict DPs: { c^#() -> c_3(b^#()) }
Weak DPs:
  { f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
  , c^#() -> c_2()
  , b^#() -> c_4()
  , activate^#(X) -> c_5()
  , activate^#(n__b()) -> c_6(b^#())
  , activate^#(n__c()) -> c_7(c^#()) }
Weak Trs:
  { c() -> n__c()
  , c() -> b()
  , b() -> n__b() }
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.

{ f^#(n__b(), X, n__c()) -> c_1(f^#(X, c(), X))
, c^#() -> c_2()
, b^#() -> c_4()
, activate^#(X) -> c_5()
, activate^#(n__b()) -> c_6(b^#()) }

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

Strict DPs: { c^#() -> c_3(b^#()) }
Weak DPs: { activate^#(n__c()) -> c_7(c^#()) }
Weak Trs:
  { c() -> n__c()
  , c() -> b()
  , b() -> n__b() }
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:

  { c^#() -> c_3(b^#()) }

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

Strict DPs: { c^#() -> c_1() }
Weak DPs: { activate^#(n__c()) -> c_2(c^#()) }
Weak Trs:
  { c() -> n__c()
  , c() -> b()
  , b() -> n__b() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,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)).

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

Consider the dependency graph

  1: c^#() -> c_1()
  
  2: activate^#(n__c()) -> c_2(c^#()) -->_1 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).

  { activate^#(n__c()) -> c_2(c^#()) }


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

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

Consider the dependency graph

  1: 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).

  { 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(1))