WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(0()),g(x)) -> f(x,g(x))
            g(s(x)) -> g(x)
        - Signature:
            {f/2,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(s(0()),g(x)) -> f(x,g(x))
            g(s(x)) -> g(x)
        - Signature:
            {f/2,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)) ->^+ g(x)
              = C[g(x) = g(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(s(0()),g(x)) -> f(x,g(x))
            g(s(x)) -> g(x)
        - Signature:
            {f/2,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#(s(0()),g(x)) -> c_1(f#(x,g(x)),g#(x))
          g#(s(x)) -> c_2(g#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: RemoveInapplicable WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(0()),g(x)) -> c_1(f#(x,g(x)),g#(x))
            g#(s(x)) -> c_2(g#(x))
        - Weak TRS:
            f(s(0()),g(x)) -> f(x,g(x))
            g(s(x)) -> g(x)
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/2,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveInapplicable
    + Details:
        Only the nodes
          {2}
        are reachable from nodes
          {2}
        that start derivation from marked basic terms.
        The nodes not reachable are removed from the problem.
** Step 1.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x)) -> c_2(g#(x))
        - Weak TRS:
            f(s(0()),g(x)) -> f(x,g(x))
            g(s(x)) -> g(x)
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/2,c_2/1}
        - 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_2(g#(x))
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x)) -> c_2(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/2,c_2/1}
        - 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_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [0]         
              p(f) = [0]         
              p(g) = [0]         
              p(s) = [1] x1 + [9]
             p(f#) = [0]         
             p(g#) = [3] x1 + [0]
            p(c_1) = [0]         
            p(c_2) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          g#(s(x)) = [3] x + [27]
                   > [3] x + [0] 
                   = c_2(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:5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(s(x)) -> c_2(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/2,c_2/1}
        - 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))