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

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Signature:
            {f/1,g/1} / {c/2,d/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {c,d,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x) -> c_1()
          f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
          f#(f(x)) -> c_3(f#(d(f(x))),f#(x))
          g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x) -> c_1()
            f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
            f#(f(x)) -> c_3(f#(d(f(x))),f#(x))
            g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        - Weak TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {c/2,d/1,s/1,c_1/0,c_2/1,c_3/2,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {c,d,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2}.
        Here rules are labelled as follows:
          1: f#(x) -> c_1()
          2: f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
          3: f#(f(x)) -> c_3(f#(d(f(x))),f#(x))
          4: g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
            g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        - Weak DPs:
            f#(x) -> c_1()
            f#(f(x)) -> c_3(f#(d(f(x))),f#(x))
        - Weak TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {c/2,d/1,s/1,c_1/0,c_2/1,c_3/2,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {c,d,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
             -->_1 f#(x) -> c_1():3
             -->_1 f#(c(s(x),y)) -> c_2(f#(c(x,s(y)))):1
          
          2:S:g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
             -->_1 g#(c(x,s(y))) -> c_4(g#(c(s(x),y))):2
          
          3:W:f#(x) -> c_1()
             
          
          4:W:f#(f(x)) -> c_3(f#(d(f(x))),f#(x))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: f#(f(x)) -> c_3(f#(d(f(x))),f#(x))
          3: f#(x) -> c_1()
** Step 1.b:4: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
            g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        - Weak TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {c/2,d/1,s/1,c_1/0,c_2/1,c_3/2,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {c,d,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
          g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
** Step 1.b:5: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
            g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {c/2,d/1,s/1,c_1/0,c_2/1,c_3/2,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {c,d,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(c) = [1] x2 + [8]
            p(d) = [1] x1 + [1]
            p(f) = [1]         
            p(g) = [4] x1 + [2]
            p(s) = [1] x1 + [8]
           p(f#) = [0]         
           p(g#) = [1] x1 + [2]
          p(c_1) = [1]         
          p(c_2) = [8] x1 + [0]
          p(c_3) = [1]         
          p(c_4) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        g#(c(x,s(y))) = [1] y + [18]      
                      > [1] y + [14]      
                      = c_4(g#(c(s(x),y)))
        
        
        Following rules are (at-least) weakly oriented:
        f#(c(s(x),y)) =  [0]               
                      >= [0]               
                      =  c_2(f#(c(x,s(y))))
        
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
        - Weak DPs:
            g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {c/2,d/1,s/1,c_1/0,c_2/1,c_3/2,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {c,d,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_2) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(c) = [1] x1 + [0]
              p(d) = [1] x1 + [4]
              p(f) = [8] x1 + [1]
              p(g) = [0]         
              p(s) = [1] x1 + [8]
             p(f#) = [1] x1 + [9]
             p(g#) = [0]         
            p(c_1) = [0]         
            p(c_2) = [1] x1 + [4]
            p(c_3) = [0]         
            p(c_4) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          f#(c(s(x),y)) = [1] x + [17]      
                        > [1] x + [13]      
                        = c_2(f#(c(x,s(y))))
          
          
          Following rules are (at-least) weakly oriented:
          g#(c(x,s(y))) =  [0]               
                        >= [0]               
                        =  c_4(g#(c(s(x),y)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(c(s(x),y)) -> c_2(f#(c(x,s(y))))
            g#(c(x,s(y))) -> c_4(g#(c(s(x),y)))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {c/2,d/1,s/1,c_1/0,c_2/1,c_3/2,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {c,d,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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