WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          g(x){x -> s(x)} =
            g(s(x)) ->^+ s(g(x))
              = C[g(x) = g(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
          g#(0()) -> c_2()
          g#(s(x)) -> c_3(g#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
            g#(0()) -> c_2()
            g#(s(x)) -> c_3(g#(x))
        - Weak TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/3,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
          2: g#(0()) -> c_2()
          3: g#(s(x)) -> c_3(g#(x))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
            g#(s(x)) -> c_3(g#(x))
        - Weak DPs:
            g#(0()) -> c_2()
        - Weak TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/3,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
             -->_3 g#(s(x)) -> c_3(g#(x)):2
             -->_3 g#(0()) -> c_2():3
             -->_1 f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x)):1
          
          2:S:g#(s(x)) -> c_3(g#(x))
             -->_1 g#(0()) -> c_2():3
             -->_1 g#(s(x)) -> c_3(g#(x)):2
          
          3:W:g#(0()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: g#(0()) -> c_2()
** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
            g#(s(x)) -> c_3(g#(x))
        - Weak TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/3,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x))
             -->_3 g#(s(x)) -> c_3(g#(x)):2
             -->_1 f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(s(0())),g#(x)):1
          
          2:S:g#(s(x)) -> c_3(g#(x))
             -->_1 g#(s(x)) -> c_3(g#(x)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(x))
** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x),s(0()),y) -> c_1(f#(g(s(0())),y,g(x)),g#(x))
            g#(s(x)) -> c_3(g#(x))
        - Weak TRS:
            f(g(x),s(0()),y) -> f(g(s(0())),y,g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g#(s(x)) -> c_3(g#(x))
** Step 1.b:6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [0]                  
              p(f) = [2]                  
              p(g) = [0]                  
              p(s) = [1] x1 + [3]         
             p(f#) = [0]                  
             p(g#) = [6] x1 + [2]         
            p(c_1) = [8] x1 + [1] x2 + [1]
            p(c_2) = [0]                  
            p(c_3) = [1] x1 + [5]         
          
          Following rules are strictly oriented:
          g#(s(x)) = [6] x + [20]
                   > [6] x + [7] 
                   = c_3(g#(x))  
          
          
          Following rules are (at-least) weakly oriented:
          
        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:
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/2,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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