* 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: WeightGap 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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
            p(c) = [1] x1 + [14]
            p(f) = [1] x2 + [0] 
            p(s) = [1] x1 + [2] 
          
          Following rules are strictly oriented:
          f(x,c(y)) = [1] y + [14]  
                    > [1] y + [2]   
                    = f(x,s(f(y,y)))
          
          
          Following rules are (at-least) weakly oriented:
          f(s(x),s(y)) =  [1] y + [2]    
                       >= [1] y + [18]   
                       =  f(x,s(c(s(y))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: MI 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:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))):
        
        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 2 0]       [0]              
                 [0 0 2] x_1 + [1]              
                 [0 0 0]       [4]              
          p(f) = [0 0 3]       [1 2 0]       [0]
                 [4 0 0] x_1 + [0 0 0] x_2 + [0]
                 [0 0 0]       [0 0 0]       [0]
          p(s) = [1 0 0]       [2]              
                 [0 0 1] x_1 + [0]              
                 [0 0 1]       [4]              
        
        Following rules are strictly oriented:
        f(s(x),s(y)) = [0 0 3]     [1 0 2]     [14]
                       [4 0 0] x + [0 0 0] y + [8] 
                       [0 0 0]     [0 0 0]     [0] 
                     > [0 0 3]     [1 0 2]     [12]
                       [4 0 0] x + [0 0 0] y + [0] 
                       [0 0 0]     [0 0 0]     [0] 
                     = f(x,s(c(s(y))))             
        
        
        Following rules are (at-least) weakly oriented:
        f(x,c(y)) =  [0 0 3]     [1 2 4]     [2]
                     [4 0 0] x + [0 0 0] y + [0]
                     [0 0 0]     [0 0 0]     [0]
                  >= [0 0 3]     [1 2 3]     [2]
                     [4 0 0] x + [0 0 0] y + [0]
                     [0 0 0]     [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))