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), s(y)) -> f(x, s(c(s(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), s(y)) -> c_2(f^#(x, s(c(s(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), s(y)) -> c_2(f^#(x, s(c(s(y))))) }
Weak Trs:
  { f(x, c(y)) -> f(x, s(f(y, y)))
  , f(s(x), s(y)) -> f(x, s(c(s(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:
  { 2: f^#(s(x), s(y)) -> c_2(f^#(x, s(c(s(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]                      
                                             
          [c](x1) = [0 0] x1 + [0]           
                    [1 1]      [0]           
                                             
          [s](x1) = [1 0] x1 + [1]           
                    [0 0]      [0]           
                                             
    [f^#](x1, x2) = [1 0] x1 + [0 4] x2 + [0]
                    [0 0]      [0 0]      [4]
                                             
    [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                    [0 0]      [0 0]      [3]
                                             
        [c_2](x1) = [1 0] x1 + [0]           
                    [0 0]      [3]           
  
  The following symbols are considered usable
  
    {f, f^#}
  
  The order satisfies the following ordering constraints:
  
         [f(x, c(y))] =  [0]                                 
                         [0]                                 
                      >= [0]                                 
                         [0]                                 
                      =  [f(x, s(f(y, y)))]                  
                                                             
      [f(s(x), s(y))] =  [0]                                 
                         [0]                                 
                      >= [0]                                 
                         [0]                                 
                      =  [f(x, s(c(s(y))))]                  
                                                             
       [f^#(x, c(y))] =  [1 0] x + [4 4] y + [0]             
                         [0 0]     [0 0]     [4]             
                      >= [1 0] x + [1 4] y + [0]             
                         [0 0]     [0 0]     [3]             
                      =  [c_1(f^#(x, s(f(y, y))), f^#(y, y))]
                                                             
    [f^#(s(x), s(y))] =  [1 0] x + [1]                       
                         [0 0]     [4]                       
                      >  [1 0] x + [0]                       
                         [0 0]     [3]                       
                      =  [c_2(f^#(x, s(c(s(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(n^1)).

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

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

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

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

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

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

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

We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.

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

Sub-proof:
----------
  The input was oriented with the instance of 'Small Polynomial Path
  Order (PS,1-bounded)' as induced by the safe mapping
  
   safe(f) = {}, safe(c) = {1}, safe(s) = {1}, safe(f^#) = {1},
   safe(c_1) = {}, safe(c_2) = {}, safe(c) = {}, safe(c_1) = {}
  
  and precedence
  
   empty .
  
  Following symbols are considered recursive:
  
   {f^#}
  
  The recursion depth is 1.
  
  Further, following argument filtering is employed:
  
   pi(f) = [], pi(c) = [1], pi(s) = [], pi(f^#) = [1, 2],
   pi(c_1) = [], pi(c_2) = [], pi(c) = [], pi(c_1) = [1]
  
  Usable defined function symbols are a subset of:
  
   {f^#}
  
  For your convenience, here are the satisfied ordering constraints:
  
    pi(f^#(x, c(y))) = f^#(c(; y); x)    
                     > c_1(f^#(y; y);)   
                     = pi(c_1(f^#(y, 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^#(y, 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^#(y, y)) }

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))