WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} 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(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          g(x,y){y -> s(y)} =
            g(x,s(y)) ->^+ g(x,y)
              = C[g(x,y) = g(x,y){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
          g#(X,s(Y)) -> c_2(g#(X,Y))
          g#(0(),Y) -> c_3()
          h#(X,Z) -> c_4(f#(X,s(X),Z))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/2,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4}
        by application of
          Pre({3,4}) = {1,2}.
        Here rules are labelled as follows:
          1: f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
          2: g#(X,s(Y)) -> c_2(g#(X,Y))
          3: g#(0(),Y) -> c_3()
          4: h#(X,Z) -> c_4(f#(X,s(X),Z))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Weak DPs:
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/2,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
             -->_2 g#(X,s(Y)) -> c_2(g#(X,Y)):2
             -->_2 g#(0(),Y) -> c_3():3
          
          2:S:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(0(),Y) -> c_3():3
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
          
          3:W:g#(0(),Y) -> c_3()
             
          
          4:W:h#(X,Z) -> c_4(f#(X,s(X),Z))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: h#(X,Z) -> c_4(f#(X,s(X),Z))
          3: g#(0(),Y) -> c_3()
** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Weak TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/2,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
             -->_2 g#(X,s(Y)) -> c_2(g#(X,Y)):2
          
          2:S:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(X,Y,g(X,Y)) -> c_1(g#(X,Y))
** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Weak TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g#(X,s(Y)) -> c_2(g#(X,Y))
** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} 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_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#,h#}
        TcT has computed the following interpretation:
            p(0) = [0]                  
            p(f) = [1] x2 + [1] x3 + [0]
            p(g) = [1] x1 + [2]         
            p(h) = [1] x1 + [1] x2 + [1]
            p(s) = [1] x1 + [2]         
           p(f#) = [1] x1 + [1] x3 + [0]
           p(g#) = [8] x2 + [0]         
           p(h#) = [8] x2 + [0]         
          p(c_1) = [0]                  
          p(c_2) = [1] x1 + [12]        
          p(c_3) = [0]                  
          p(c_4) = [1] x1 + [1]         
        
        Following rules are strictly oriented:
        g#(X,s(Y)) = [8] Y + [16]
                   > [8] Y + [12]
                   = c_2(g#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} 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))