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

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
            g(s(0())) -> g(f(s(0())))
        - Signature:
            {f/1,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#(f(x)) -> c_1(f#(x))
          f#(s(x)) -> c_2(f#(x))
          g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(x))
            f#(s(x)) -> c_2(f#(x))
            g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
            g(s(0())) -> g(f(s(0())))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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(f(x)) -> f(x)
          f(s(x)) -> f(x)
          f#(f(x)) -> c_1(f#(x))
          f#(s(x)) -> c_2(f#(x))
          g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
** Step 1.b:3: DecomposeDG WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(x))
            f#(s(x)) -> c_2(f#(x))
            g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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 = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
        and a lower component
          f#(f(x)) -> c_1(f#(x))
          f#(s(x)) -> c_2(f#(x))
        Further, following extension rules are added to the lower component.
          g#(s(0())) -> f#(s(0()))
          g#(s(0())) -> g#(f(s(0())))
*** Step 1.b:3.a:1: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0())))
             -->_1 g#(s(0())) -> c_3(g#(f(s(0()))),f#(s(0()))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g#(s(0())) -> c_3(g#(f(s(0()))))
*** Step 1.b:3.a:2: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(s(0())) -> c_3(g#(f(s(0()))))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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 = 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_3) = {1}
        
        Following symbols are considered usable:
          {f,f#,g#}
        TcT has computed the following interpretation:
            p(0) = [0]         
            p(f) = [0]         
            p(g) = [1] x1 + [1]
            p(s) = [12]        
           p(f#) = [1] x1 + [0]
           p(g#) = [2] x1 + [3]
          p(c_1) = [1]         
          p(c_2) = [2] x1 + [1]
          p(c_3) = [2] x1 + [2]
        
        Following rules are strictly oriented:
        g#(s(0())) = [27]              
                   > [8]               
                   = c_3(g#(f(s(0()))))
        
        
        Following rules are (at-least) weakly oriented:
        f(f(x)) =  [0] 
                >= [0] 
                =  f(x)
        
        f(s(x)) =  [0] 
                >= [0] 
                =  f(x)
        
*** Step 1.b:3.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(s(0())) -> c_3(g#(f(s(0()))))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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).

*** Step 1.b:3.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(x))
            f#(s(x)) -> c_2(f#(x))
        - Weak DPs:
            g#(s(0())) -> f#(s(0()))
            g#(s(0())) -> g#(f(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,c_2/1,c_3/2}
        - 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(g#) = {1},
            uargs(c_1) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [12]                 
              p(f) = [1] x1 + [0]         
              p(g) = [2] x1 + [1]         
              p(s) = [1] x1 + [4]         
             p(f#) = [1] x1 + [0]         
             p(g#) = [1] x1 + [2]         
            p(c_1) = [1] x1 + [5]         
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
          f#(s(x)) = [1] x + [4]
                   > [1] x + [0]
                   = c_2(f#(x)) 
          
          
          Following rules are (at-least) weakly oriented:
            f#(f(x)) =  [1] x + [0]  
                     >= [1] x + [5]  
                     =  c_1(f#(x))   
          
          g#(s(0())) =  [18]         
                     >= [16]         
                     =  f#(s(0()))   
          
          g#(s(0())) =  [18]         
                     >= [18]         
                     =  g#(f(s(0())))
          
             f(f(x)) =  [1] x + [0]  
                     >= [1] x + [0]  
                     =  f(x)         
          
             f(s(x)) =  [1] x + [4]  
                     >= [1] x + [0]  
                     =  f(x)         
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:3.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(x))
        - Weak DPs:
            f#(s(x)) -> c_2(f#(x))
            g#(s(0())) -> f#(s(0()))
            g#(s(0())) -> g#(f(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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 any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [0]         
            p(f) = [2] x1 + [1]
            p(g) = [1] x1 + [0]
            p(s) = [1] x1 + [0]
           p(f#) = [4] x1 + [0]
           p(g#) = [9]         
          p(c_1) = [2] x1 + [3]
          p(c_2) = [1] x1 + [0]
          p(c_3) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        f#(f(x)) = [8] x + [4]
                 > [8] x + [3]
                 = c_1(f#(x)) 
        
        
        Following rules are (at-least) weakly oriented:
          f#(s(x)) =  [4] x + [0]  
                   >= [4] x + [0]  
                   =  c_2(f#(x))   
        
        g#(s(0())) =  [9]          
                   >= [0]          
                   =  f#(s(0()))   
        
        g#(s(0())) =  [9]          
                   >= [9]          
                   =  g#(f(s(0())))
        
*** Step 1.b:3.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(f(x)) -> c_1(f#(x))
            f#(s(x)) -> c_2(f#(x))
            g#(s(0())) -> f#(s(0()))
            g#(s(0())) -> g#(f(s(0())))
        - Weak TRS:
            f(f(x)) -> f(x)
            f(s(x)) -> f(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,s/1,c_1/1,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).

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