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

Strict Trs:
  { fst(0(), Z) -> nil()
  , fst(s(), cons(Y)) -> cons(Y)
  , from(X) -> cons(X)
  , add(0(), X) -> X
  , add(s(), Y) -> s()
  , len(nil()) -> 0()
  , len(cons(X)) -> s() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following weak dependency pairs:

Strict DPs:
  { fst^#(0(), Z) -> c_1()
  , fst^#(s(), cons(Y)) -> c_2()
  , from^#(X) -> c_3()
  , add^#(0(), X) -> c_4()
  , add^#(s(), Y) -> c_5()
  , len^#(nil()) -> c_6()
  , len^#(cons(X)) -> c_7() }

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:
  { fst^#(0(), Z) -> c_1()
  , fst^#(s(), cons(Y)) -> c_2()
  , from^#(X) -> c_3()
  , add^#(0(), X) -> c_4()
  , add^#(s(), Y) -> c_5()
  , len^#(nil()) -> c_6()
  , len^#(cons(X)) -> c_7() }
Strict Trs:
  { fst(0(), Z) -> nil()
  , fst(s(), cons(Y)) -> cons(Y)
  , from(X) -> cons(X)
  , add(0(), X) -> X
  , add(s(), Y) -> s()
  , len(nil()) -> 0()
  , len(cons(X)) -> s() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),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),O(1)).

Strict DPs:
  { fst^#(0(), Z) -> c_1()
  , fst^#(s(), cons(Y)) -> c_2()
  , from^#(X) -> c_3()
  , add^#(0(), X) -> c_4()
  , add^#(s(), Y) -> c_5()
  , len^#(nil()) -> c_6()
  , len^#(cons(X)) -> c_7() }
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:
  none

TcT has computed the following constructor-restricted matrix
interpretation.

              [0] = [1]           
                    [0]           
                                  
            [nil] = [0]           
                    [1]           
                                  
              [s] = [0]           
                    [0]           
                                  
       [cons](x1) = [0]           
                    [0]           
                                  
  [fst^#](x1, x2) = [0 0] x1 + [0]
                    [2 0]      [0]
                                  
            [c_1] = [0]           
                    [0]           
                                  
            [c_2] = [0]           
                    [0]           
                                  
     [from^#](x1) = [1]           
                    [0]           
                                  
            [c_3] = [0]           
                    [0]           
                                  
  [add^#](x1, x2) = [0 1] x1 + [0]
                    [0 0]      [0]
                                  
            [c_4] = [0]           
                    [0]           
                                  
            [c_5] = [0]           
                    [0]           
                                  
      [len^#](x1) = [0]           
                    [0]           
                                  
            [c_6] = [0]           
                    [0]           
                                  
            [c_7] = [0]           
                    [0]           

The order satisfies the following ordering constraints:

        [fst^#(0(), Z)] =  [0]    
                           [2]    
                        >= [0]    
                           [0]    
                        =  [c_1()]
                                  
  [fst^#(s(), cons(Y))] =  [0]    
                           [0]    
                        >= [0]    
                           [0]    
                        =  [c_2()]
                                  
            [from^#(X)] =  [1]    
                           [0]    
                        >  [0]    
                           [0]    
                        =  [c_3()]
                                  
        [add^#(0(), X)] =  [0]    
                           [0]    
                        >= [0]    
                           [0]    
                        =  [c_4()]
                                  
        [add^#(s(), Y)] =  [0]    
                           [0]    
                        >= [0]    
                           [0]    
                        =  [c_5()]
                                  
         [len^#(nil())] =  [0]    
                           [0]    
                        >= [0]    
                           [0]    
                        =  [c_6()]
                                  
       [len^#(cons(X))] =  [0]    
                           [0]    
                        >= [0]    
                           [0]    
                        =  [c_7()]
                                  

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)).

Strict DPs:
  { fst^#(0(), Z) -> c_1()
  , fst^#(s(), cons(Y)) -> c_2()
  , add^#(0(), X) -> c_4()
  , add^#(s(), Y) -> c_5()
  , len^#(nil()) -> c_6()
  , len^#(cons(X)) -> c_7() }
Weak DPs: { from^#(X) -> c_3() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We estimate the number of application of {1,2,3,4,5,6} by
applications of Pre({1,2,3,4,5,6}) = {}. Here rules are labeled as
follows:

  DPs:
    { 1: fst^#(0(), Z) -> c_1()
    , 2: fst^#(s(), cons(Y)) -> c_2()
    , 3: add^#(0(), X) -> c_4()
    , 4: add^#(s(), Y) -> c_5()
    , 5: len^#(nil()) -> c_6()
    , 6: len^#(cons(X)) -> c_7()
    , 7: from^#(X) -> c_3() }

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

Weak DPs:
  { fst^#(0(), Z) -> c_1()
  , fst^#(s(), cons(Y)) -> c_2()
  , from^#(X) -> c_3()
  , add^#(0(), X) -> c_4()
  , add^#(s(), Y) -> c_5()
  , len^#(nil()) -> c_6()
  , len^#(cons(X)) -> c_7() }
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.

{ fst^#(0(), Z) -> c_1()
, fst^#(s(), cons(Y)) -> c_2()
, from^#(X) -> c_3()
, add^#(0(), X) -> c_4()
, add^#(s(), Y) -> c_5()
, len^#(nil()) -> c_6()
, len^#(cons(X)) -> c_7() }

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(1))