WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2} / {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#(x,0()) -> c_1()
          f#(s(x),s(y)) -> c_2(f#(x,y))
          g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,0()) -> c_1()
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f(x,0()) -> s(0())
          f(s(x),s(y)) -> s(f(x,y))
          f#(x,0()) -> c_1()
          f#(s(x),s(y)) -> c_2(f#(x,y))
          g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,0()) -> c_1()
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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}) = {2,3}.
        Here rules are labelled as follows:
          1: f#(x,0()) -> c_1()
          2: f#(s(x),s(y)) -> c_2(f#(x,y))
          3: g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak DPs:
            f#(x,0()) -> c_1()
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x),s(y)) -> c_2(f#(x,y))
             -->_1 f#(x,0()) -> c_1():3
             -->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
          
          2:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
             -->_2 f#(x,0()) -> c_1():3
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):2
             -->_2 f#(s(x),s(y)) -> c_2(f#(x,y)):1
          
          3:W:f#(x,0()) -> c_1()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(x,0()) -> c_1()
* Step 5: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              f#(s(x),s(y)) -> c_2(f#(x,y))
          - Weak DPs:
              g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
          - Weak TRS:
              f(x,0()) -> s(0())
              f(s(x),s(y)) -> s(f(x,y))
          - Signature:
              {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
          - Weak DPs:
              f#(s(x),s(y)) -> c_2(f#(x,y))
          - Weak TRS:
              f(x,0()) -> s(0())
              f(s(x),s(y)) -> s(f(x,y))
          - Signature:
              {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
** Step 5.a:1: DecomposeDG WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
        - Weak DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        and a lower component
          f#(s(x),s(y)) -> c_2(f#(x,y))
        Further, following extension rules are added to the lower component.
          g#(0(),x) -> f#(x,x)
          g#(0(),x) -> g#(f(x,x),x)
*** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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,f#,g#}
        TcT has computed the following interpretation:
            p(0) = [15]                 
            p(f) = [10]                 
            p(g) = [1] x2 + [1]         
            p(s) = [0]                  
           p(f#) = [6] x1 + [2] x2 + [8]
           p(g#) = [1] x1 + [2] x2 + [6]
          p(c_1) = [1]                  
          p(c_2) = [2] x1 + [0]         
          p(c_3) = [1] x1 + [1]         
        
        Following rules are strictly oriented:
        g#(0(),x) = [2] x + [21]             
                  > [2] x + [17]             
                  = c_3(g#(f(x,x),x),f#(x,x))
        
        
        Following rules are (at-least) weakly oriented:
            f(x,0()) =  [10]     
                     >= [0]      
                     =  s(0())   
        
        f(s(x),s(y)) =  [10]     
                     >= [0]      
                     =  s(f(x,y))
        
**** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
**** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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).

*** Step 5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
        - Weak DPs:
            g#(0(),x) -> f#(x,x)
            g#(0(),x) -> g#(f(x,x),x)
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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: f#(s(x),s(y)) -> c_2(f#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
        - Weak DPs:
            g#(0(),x) -> f#(x,x)
            g#(0(),x) -> g#(f(x,x),x)
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [3]                  
            p(f) = [0]                  
            p(g) = [1] x1 + [2] x2 + [2]
            p(s) = [1] x1 + [1]         
           p(f#) = [2] x1 + [0]         
           p(g#) = [6] x2 + [0]         
          p(c_1) = [4]                  
          p(c_2) = [1] x1 + [0]         
          p(c_3) = [1] x2 + [1]         
        
        Following rules are strictly oriented:
        f#(s(x),s(y)) = [2] x + [2] 
                      > [2] x + [0] 
                      = c_2(f#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        g#(0(),x) =  [6] x + [0] 
                  >= [2] x + [0] 
                  =  f#(x,x)     
        
        g#(0(),x) =  [6] x + [0] 
                  >= [6] x + [0] 
                  =  g#(f(x,x),x)
        
**** Step 5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> f#(x,x)
            g#(0(),x) -> g#(f(x,x),x)
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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 5.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> f#(x,x)
            g#(0(),x) -> g#(f(x,x),x)
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(s(x),s(y)) -> c_2(f#(x,y))
             -->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
          
          2:W:g#(0(),x) -> f#(x,x)
             -->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
          
          3:W:g#(0(),x) -> g#(f(x,x),x)
             -->_1 g#(0(),x) -> g#(f(x,x),x):3
             -->_1 g#(0(),x) -> f#(x,x):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: g#(0(),x) -> g#(f(x,x),x)
          2: g#(0(),x) -> f#(x,x)
          1: f#(s(x),s(y)) -> c_2(f#(x,y))
**** Step 5.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - 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).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
             -->_2 f#(s(x),s(y)) -> c_2(f#(x,y)):2
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
          
          2:W:f#(s(x),s(y)) -> c_2(f#(x,y))
             -->_1 f#(s(x),s(y)) -> c_2(f#(x,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),s(y)) -> c_2(f#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g#(0(),x) -> c_3(g#(f(x,x),x))
** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,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 = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g#(0(),x) -> c_3(g#(f(x,x),x))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, 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 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {f,f#,g#}
        TcT has computed the following interpretation:
            p(0) = [2]                  
            p(f) = [0]                  
            p(g) = [1] x1 + [2] x2 + [8]
            p(s) = [0]                  
           p(f#) = [1] x1 + [2] x2 + [2]
           p(g#) = [8] x1 + [0]         
          p(c_1) = [4]                  
          p(c_2) = [8] x1 + [1]         
          p(c_3) = [4] x1 + [12]        
        
        Following rules are strictly oriented:
        g#(0(),x) = [16]             
                  > [12]             
                  = c_3(g#(f(x,x),x))
        
        
        Following rules are (at-least) weakly oriented:
            f(x,0()) =  [0]      
                     >= [0]      
                     =  s(0())   
        
        f(s(x),s(y)) =  [0]      
                     >= [0]      
                     =  s(f(x,y))
        
*** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,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 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,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#(0(),x) -> c_3(g#(f(x,x),x))
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(0(),x) -> c_3(g#(f(x,x),x))
*** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,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(?,O(n^1))