WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(x,c(y)) -> f(x,s(f(y,y)))
            f(s(x),s(y)) -> f(x,s(c(s(y))))
        - Signature:
            {f/2} / {c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x,c(y)) -> f(x,s(f(y,y)))
            f(s(x),s(y)) -> f(x,s(c(s(y))))
        - Signature:
            {f/2} / {c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y){y -> c(y)} =
            f(x,c(y)) ->^+ f(x,s(f(y,y)))
              = C[f(y,y) = f(x,y){x -> y}]

** Step 1.b:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(x,c(y)) -> f(x,s(f(y,y)))
            f(s(x),s(y)) -> f(x,s(c(s(y))))
        - Signature:
            {f/2} / {c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(f) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(c) = [1 0 0]      [0]             
                 [0 1 0] x1 + [2]             
                 [0 0 0]      [1]             
          p(f) = [0 0 0]      [1 2 0]      [0]
                 [2 0 0] x1 + [0 0 0] x2 + [0]
                 [0 0 0]      [1 2 0]      [0]
          p(s) = [1 0 0]      [0]             
                 [0 0 0] x1 + [0]             
                 [0 0 0]      [2]             
        
        Following rules are strictly oriented:
        f(x,c(y)) = [0 0 0]     [1 2 0]     [4]
                    [2 0 0] x + [0 0 0] y + [0]
                    [0 0 0]     [1 2 0]     [4]
                  > [0 0 0]     [1 2 0]     [0]
                    [2 0 0] x + [0 0 0] y + [0]
                    [0 0 0]     [1 2 0]     [0]
                  = f(x,s(f(y,y)))             
        
        
        Following rules are (at-least) weakly oriented:
        f(s(x),s(y)) =  [0 0 0]     [1 0 0]     [0]
                        [2 0 0] x + [0 0 0] y + [0]
                        [0 0 0]     [1 0 0]     [0]
                     >= [0 0 0]     [1 0 0]     [0]
                        [2 0 0] x + [0 0 0] y + [0]
                        [0 0 0]     [1 0 0]     [0]
                     =  f(x,s(c(s(y))))            
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(s(x),s(y)) -> f(x,s(c(s(y))))
        - Weak TRS:
            f(x,c(y)) -> f(x,s(f(y,y)))
        - Signature:
            {f/2} / {c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(f) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(c) = [1 1 0]      [0]             
                 [0 0 3] x1 + [1]             
                 [0 0 0]      [0]             
          p(f) = [0 0 2]      [1 1 0]      [0]
                 [2 0 0] x1 + [0 0 0] x2 + [2]
                 [0 0 1]      [0 0 0]      [0]
          p(s) = [1 0 0]      [0]             
                 [0 0 1] x1 + [0]             
                 [0 0 1]      [1]             
        
        Following rules are strictly oriented:
        f(s(x),s(y)) = [0 0 2]     [1 0 1]     [2]
                       [2 0 0] x + [0 0 0] y + [2]
                       [0 0 1]     [0 0 0]     [1]
                     > [0 0 2]     [1 0 1]     [0]
                       [2 0 0] x + [0 0 0] y + [2]
                       [0 0 1]     [0 0 0]     [0]
                     = f(x,s(c(s(y))))            
        
        
        Following rules are (at-least) weakly oriented:
        f(x,c(y)) =  [0 0 2]     [1 1 3]     [1]
                     [2 0 0] x + [0 0 0] y + [2]
                     [0 0 1]     [0 0 0]     [0]
                  >= [0 0 2]     [1 1 3]     [0]
                     [2 0 0] x + [0 0 0] y + [2]
                     [0 0 1]     [0 0 0]     [0]
                  =  f(x,s(f(y,y)))             
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,c(y)) -> f(x,s(f(y,y)))
            f(s(x),s(y)) -> f(x,s(c(s(y))))
        - Signature:
            {f/2} / {c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))