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),y) -> f(x,s(c(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),y) -> f(x,s(c(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: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(x,c(y)) -> f(x,s(f(y,y)))
            f(s(x),y) -> f(x,s(c(y)))
        - Signature:
            {f/2} / {c/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {c,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        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))))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - 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)))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: f#(s(x),y) -> c_2(f#(x,s(c(y))))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:2.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - 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)))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_1) = {1,2},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f,f#}
        TcT has computed the following interpretation:
            p(c) = [0 0 0]      [1]             
                   [0 1 1] x1 + [1]             
                   [0 0 0]      [0]             
            p(f) = [0 0 1]      [1 0 0]      [1]
                   [0 0 0] x1 + [0 0 0] x2 + [0]
                   [0 0 0]      [0 0 0]      [0]
            p(s) = [0 0 0]      [1]             
                   [0 0 0] x1 + [0]             
                   [0 0 1]      [1]             
           p(f#) = [0 0 1]      [0 1 0]      [1]
                   [0 1 1] x1 + [0 0 0] x2 + [0]
                   [1 0 1]      [0 0 0]      [0]
          p(c_1) = [1 0 0]      [1 0 0]      [0]
                   [0 0 0] x1 + [0 0 0] x2 + [0]
                   [0 0 0]      [0 0 0]      [0]
          p(c_2) = [1 0 0]      [0]             
                   [0 0 0] x1 + [0]             
                   [0 0 0]      [0]             
        
        Following rules are strictly oriented:
        f#(s(x),y) = [0 0 1]     [0 1 0]     [2]
                     [0 0 1] x + [0 0 0] y + [1]
                     [0 0 1]     [0 0 0]     [2]
                   > [0 0 1]     [1]            
                     [0 0 0] x + [0]            
                     [0 0 0]     [0]            
                   = c_2(f#(x,s(c(y))))         
        
        
        Following rules are (at-least) weakly oriented:
        f#(x,c(y)) =  [0 0 1]     [0 1 1]     [2] 
                      [0 1 1] x + [0 0 0] y + [0] 
                      [1 0 1]     [0 0 0]     [0] 
                   >= [0 0 1]     [0 1 1]     [2] 
                      [0 0 0] x + [0 0 0] y + [0] 
                      [0 0 0]     [0 0 0]     [0] 
                   =  c_1(f#(x,s(f(y,y))),f#(y,y))
        
         f(x,c(y)) =  [0 0 1]     [2]             
                      [0 0 0] x + [0]             
                      [0 0 0]     [0]             
                   >= [0 0 1]     [2]             
                      [0 0 0] x + [0]             
                      [0 0 0]     [0]             
                   =  f(x,s(f(y,y)))              
        
         f(s(x),y) =  [0 0 1]     [1 0 0]     [2] 
                      [0 0 0] x + [0 0 0] y + [0] 
                      [0 0 0]     [0 0 0]     [0] 
                   >= [0 0 1]     [2]             
                      [0 0 0] x + [0]             
                      [0 0 0]     [0]             
                   =  f(x,s(c(y)))                
        
*** Step 1.b:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
        - Weak DPs:
            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)))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 1.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
        - Weak DPs:
            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)))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S: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)))):2
             -->_1 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2
             -->_2 f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)):1
          
          2:W:f#(s(x),y) -> c_2(f#(x,s(c(y))))
             -->_1 f#(s(x),y) -> c_2(f#(x,s(c(y)))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(s(x),y) -> c_2(f#(x,s(c(y))))
*** Step 1.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - 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),y) -> f(x,s(c(y)))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/2,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y))
             -->_2 f#(x,c(y)) -> c_1(f#(x,s(f(y,y))),f#(y,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(x,c(y)) -> c_1(f#(y,y))
*** Step 1.b:2.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - 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),y) -> f(x,s(c(y)))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,c(y)) -> c_1(f#(y,y))
*** Step 1.b:2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,c(y)) -> c_1(f#(y,y))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(x,c(y)) -> c_1(f#(y,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,c(y)) -> c_1(f#(y,y))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(c) = [1] x1 + [2]         
            p(f) = [2] x1 + [1] x2 + [1]
            p(s) = [1] x1 + [0]         
           p(f#) = [8] x2 + [8]         
          p(c_1) = [1] x1 + [8]         
          p(c_2) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
        f#(x,c(y)) = [8] y + [24]
                   > [8] y + [16]
                   = c_1(f#(y,y))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 1.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,c(y)) -> c_1(f#(y,y))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,c(y)) -> c_1(f#(y,y))
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {c,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(x,c(y)) -> c_1(f#(y,y))
             -->_1 f#(x,c(y)) -> c_1(f#(y,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(x,c(y)) -> c_1(f#(y,y))
**** Step 1.b:2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/2,f#/2} / {c/1,s/1,c_1/1,c_2/1}
        - 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))