* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(x),s(0())) -> f(g(x),g(x))
            g(0()) -> 0()
            g(s(x)) -> s(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(g(x),s(0())) -> f(g(x),g(x))
            g(0()) -> 0()
            g(s(x)) -> s(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)) ->^+ 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())) -> f(g(x),g(x))
            g(0()) -> 0()
            g(s(x)) -> s(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_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          f#(g(x),s(0())) -> c_1(f#(g(x),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: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x),s(0())) -> c_1(f#(g(x),g(x)))
            g#(0()) -> c_2()
            g#(s(x)) -> c_3(g#(x))
        - Strict TRS:
            f(g(x),s(0())) -> f(g(x),g(x))
            g(0()) -> 0()
            g(s(x)) -> s(g(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,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#(0()) -> c_2()
          g#(s(x)) -> c_3(g#(x))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(0()) -> c_2()
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,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
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: g#(0()) -> c_2()
          2: g#(s(x)) -> c_3(g#(x))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x)) -> c_3(g#(x))
        - Weak DPs:
            g#(0()) -> c_2()
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,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:g#(s(x)) -> c_3(g#(x))
             -->_1 g#(0()) -> c_2():2
             -->_1 g#(s(x)) -> c_3(g#(x)):1
          
          2: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.
          2: g#(0()) -> c_2()
** Step 1.b:5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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: g#(s(x)) -> c_3(g#(x))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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_3) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [1]          
            p(f) = [1] x2 + [2] 
            p(g) = [0]          
            p(s) = [1] x1 + [4] 
           p(f#) = [1] x2 + [1] 
           p(g#) = [4] x1 + [0] 
          p(c_1) = [2]          
          p(c_2) = [0]          
          p(c_3) = [1] x1 + [15]
        
        Following rules are strictly oriented:
        g#(s(x)) = [4] x + [16]
                 > [4] x + [15]
                 = c_3(g#(x))  
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 1.b:5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(s(x)) -> c_3(g#(x))
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,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:W:g#(s(x)) -> c_3(g#(x))
             -->_1 g#(s(x)) -> c_3(g#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(s(x)) -> c_3(g#(x))
*** Step 1.b:5.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/2,g/1,f#/2,g#/1} / {0/0,s/1,c_1/1,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))