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, c(y)) -> f(x, s(f(y, y)))
  , f(s(x), y) -> f(x, s(c(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following dependency tuples:

Strict DPs:
  { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
  , f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) }

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, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
  , f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) }
Weak Trs:
  { f(x, c(y)) -> f(x, s(f(y, y)))
  , f(s(x), y) -> f(x, s(c(y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.

DPs:
  { 1: f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
  , 2: f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_1) = {1, 2}, Uargs(c_2) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA) and not(IDA(1)).
  
      [f](x1, x2) = [0 0] x1 + [0]           
                    [0 1]      [0]           
                                             
          [c](x1) = [1 1] x1 + [4]           
                    [0 0]      [0]           
                                             
          [s](x1) = [0 0] x1 + [0]           
                    [0 1]      [4]           
                                             
    [f^#](x1, x2) = [0 1] x1 + [2 0] x2 + [0]
                    [0 0]      [0 1]      [0]
                                             
    [c_1](x1, x2) = [1 1] x1 + [1 0] x2 + [3]
                    [0 0]      [0 0]      [0]
                                             
        [c_2](x1) = [1 0] x1 + [1]           
                    [0 0]      [0]           
  
  The following symbols are considered usable
  
    {f, f^#}
  
  The order satisfies the following ordering constraints:
  
      [f(x, c(y))] =  [0 0] x + [0]                       
                      [0 1]     [0]                       
                   >= [0 0] x + [0]                       
                      [0 1]     [0]                       
                   =  [f(x, s(f(y, y)))]                  
                                                          
      [f(s(x), y)] =  [0 0] x + [0]                       
                      [0 1]     [4]                       
                   >= [0 0] x + [0]                       
                      [0 1]     [0]                       
                   =  [f(x, s(c(y)))]                     
                                                          
    [f^#(x, c(y))] =  [0 1] x + [2 2] y + [8]             
                      [0 0]     [0 0]     [0]             
                   >  [0 1] x + [2 2] y + [7]             
                      [0 0]     [0 0]     [0]             
                   =  [c_1(f^#(x, s(f(y, y))), f^#(y, y))]
                                                          
    [f^#(s(x), y)] =  [0 1] x + [2 0] y + [4]             
                      [0 0]     [0 1]     [0]             
                   >  [0 1] x + [1]                       
                      [0 0]     [0]                       
                   =  [c_2(f^#(x, s(c(y))))]              
                                                          

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, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
  , f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) }
Weak Trs:
  { f(x, c(y)) -> f(x, s(f(y, y)))
  , f(s(x), y) -> f(x, s(c(y))) }
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.

{ f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
, f^#(s(x), y) -> c_2(f^#(x, s(c(y)))) }

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

Weak Trs:
  { f(x, c(y)) -> f(x, s(f(y, y)))
  , f(s(x), y) -> f(x, s(c(y))) }
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)).

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

Empty rules are trivially bounded

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