WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(x,0()) -> c_1()
          +#(x,s(y)) -> c_2(+#(x,y))
          f#(0()) -> c_3()
          f#(s(x)) -> c_4(g#(x,s(x)))
          g#(0(),y) -> c_5()
          g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
          g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1()
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(0(),y) -> c_5()
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5}
        by application of
          Pre({1,3,5}) = {2,4,6,7}.
        Here rules are labelled as follows:
          1: +#(x,0()) -> c_1()
          2: +#(x,s(y)) -> c_2(+#(x,y))
          3: f#(0()) -> c_3()
          4: f#(s(x)) -> c_4(g#(x,s(x)))
          5: g#(0(),y) -> c_5()
          6: g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
          7: g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        - Weak DPs:
            +#(x,0()) -> c_1()
            f#(0()) -> c_3()
            g#(0(),y) -> c_5()
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(x,s(y)) -> c_2(+#(x,y))
             -->_1 +#(x,0()) -> c_1():5
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          2:S:f#(s(x)) -> c_4(g#(x,s(x)))
             -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):4
             -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):3
             -->_1 g#(0(),y) -> c_5():7
          
          3:S:g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
             -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):4
             -->_1 g#(0(),y) -> c_5():7
             -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):3
             -->_2 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          4:S:g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
             -->_1 g#(0(),y) -> c_5():7
             -->_2 +#(x,0()) -> c_1():5
             -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):4
             -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):3
             -->_2 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          5:W:+#(x,0()) -> c_1()
             
          
          6:W:f#(0()) -> c_3()
             
          
          7:W:g#(0(),y) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: f#(0()) -> c_3()
          7: g#(0(),y) -> c_5()
          5: +#(x,0()) -> c_1()
* Step 4: RemoveHeads WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:+#(x,s(y)) -> c_2(+#(x,y))
           -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
        
        2:S:f#(s(x)) -> c_4(g#(x,s(x)))
           -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):4
           -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):3
        
        3:S:g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
           -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):4
           -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):3
           -->_2 +#(x,s(y)) -> c_2(+#(x,y)):1
        
        4:S:g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
           -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):4
           -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):3
           -->_2 +#(x,s(y)) -> c_2(+#(x,y)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(2,f#(s(x)) -> c_4(g#(x,s(x))))]
* Step 5: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(x,0()) -> x
          +(x,s(y)) -> s(+(x,y))
          +#(x,s(y)) -> c_2(+#(x,y))
          g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
          g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
* Step 6: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
          g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        and a lower component
          +#(x,s(y)) -> c_2(+#(x,y))
        Further, following extension rules are added to the lower component.
          g#(s(x),y) -> +#(y,x)
          g#(s(x),y) -> +#(y,s(x))
          g#(s(x),y) -> g#(x,+(y,s(x)))
          g#(s(x),y) -> g#(x,s(+(y,x)))
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x)))
             -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):2
             -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):1
          
          2:S:g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x))
             -->_1 g#(s(x),y) -> c_7(g#(x,s(+(y,x))),+#(y,x)):2
             -->_1 g#(s(x),y) -> c_6(g#(x,+(y,s(x))),+#(y,s(x))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
          g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
** Step 6.a:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,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_6) = {1},
          uargs(c_7) = {1}
        
        Following symbols are considered usable:
          {+#,f#,g#}
        TcT has computed the following interpretation:
            p(+) = [2] x1 + [0]         
            p(0) = [0]                  
            p(1) = [0]                  
            p(f) = [1]                  
            p(g) = [1] x1 + [1] x2 + [2]
            p(s) = [1] x1 + [1]         
           p(+#) = [8] x1 + [0]         
           p(f#) = [1] x1 + [8]         
           p(g#) = [3] x1 + [0]         
          p(c_1) = [0]                  
          p(c_2) = [1] x1 + [8]         
          p(c_3) = [0]                  
          p(c_4) = [1] x1 + [8]         
          p(c_5) = [1]                  
          p(c_6) = [1] x1 + [3]         
          p(c_7) = [1] x1 + [2]         
        
        Following rules are strictly oriented:
        g#(s(x),y) = [3] x + [3]         
                   > [3] x + [2]         
                   = c_7(g#(x,s(+(y,x))))
        
        
        Following rules are (at-least) weakly oriented:
        g#(s(x),y) =  [3] x + [3]         
                   >= [3] x + [3]         
                   =  c_6(g#(x,+(y,s(x))))
        
** Step 6.a:3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
        - Weak DPs:
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,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_6) = {1},
          uargs(c_7) = {1}
        
        Following symbols are considered usable:
          {+#,f#,g#}
        TcT has computed the following interpretation:
            p(+) = [8]                  
            p(0) = [2]                  
            p(1) = [2]                  
            p(f) = [2] x1 + [1]         
            p(g) = [2] x1 + [1] x2 + [0]
            p(s) = [1] x1 + [8]         
           p(+#) = [2] x2 + [2]         
           p(f#) = [1]                  
           p(g#) = [2] x1 + [1]         
          p(c_1) = [0]                  
          p(c_2) = [1]                  
          p(c_3) = [0]                  
          p(c_4) = [1] x1 + [0]         
          p(c_5) = [2]                  
          p(c_6) = [1] x1 + [14]        
          p(c_7) = [1] x1 + [15]        
        
        Following rules are strictly oriented:
        g#(s(x),y) = [2] x + [17]        
                   > [2] x + [15]        
                   = c_6(g#(x,+(y,s(x))))
        
        
        Following rules are (at-least) weakly oriented:
        g#(s(x),y) =  [2] x + [17]        
                   >= [2] x + [16]        
                   =  c_7(g#(x,s(+(y,x))))
        
** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
        - Weak DPs:
            g#(s(x),y) -> +#(y,x)
            g#(s(x),y) -> +#(y,s(x))
            g#(s(x),y) -> g#(x,+(y,s(x)))
            g#(s(x),y) -> g#(x,s(+(y,x)))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,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}
        
        Following symbols are considered usable:
          {+#,f#,g#}
        TcT has computed the following interpretation:
            p(+) = [8] x1 + [0]         
            p(0) = [0]                  
            p(1) = [4]                  
            p(f) = [1]                  
            p(g) = [2]                  
            p(s) = [1] x1 + [8]         
           p(+#) = [1] x2 + [8]         
           p(f#) = [2] x1 + [0]         
           p(g#) = [1] x1 + [9]         
          p(c_1) = [0]                  
          p(c_2) = [1] x1 + [6]         
          p(c_3) = [1]                  
          p(c_4) = [0]                  
          p(c_5) = [1]                  
          p(c_6) = [1] x1 + [1] x2 + [0]
          p(c_7) = [2] x1 + [8] x2 + [1]
        
        Following rules are strictly oriented:
        +#(x,s(y)) = [1] y + [16]
                   > [1] y + [14]
                   = c_2(+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        g#(s(x),y) =  [1] x + [17]   
                   >= [1] x + [8]    
                   =  +#(y,x)        
        
        g#(s(x),y) =  [1] x + [17]   
                   >= [1] x + [16]   
                   =  +#(y,s(x))     
        
        g#(s(x),y) =  [1] x + [17]   
                   >= [1] x + [9]    
                   =  g#(x,+(y,s(x)))
        
        g#(s(x),y) =  [1] x + [17]   
                   >= [1] x + [9]    
                   =  g#(x,s(+(y,x)))
        
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            g#(s(x),y) -> +#(y,x)
            g#(s(x),y) -> +#(y,s(x))
            g#(s(x),y) -> g#(x,+(y,s(x)))
            g#(s(x),y) -> g#(x,s(+(y,x)))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))