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(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h}
    + 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(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          g(x){x -> h(x)} =
            g(h(x)) ->^+ g(x)
              = C[g(x) = g(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0,c/1,d/1,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,c,d,h}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
          f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
          g#(c(x)) -> c_3()
          g#(c(1())) -> c_4(g#(d(h(0()))))
          g#(c(h(0()))) -> c_5(g#(d(1())))
          g#(d(x)) -> c_6()
          g#(h(x)) -> c_7(g#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
            g#(c(x)) -> c_3()
            g#(c(1())) -> c_4(g#(d(h(0()))))
            g#(c(h(0()))) -> c_5(g#(d(1())))
            g#(d(x)) -> c_6()
            g#(h(x)) -> c_7(g#(x))
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,6}
        by application of
          Pre({3,6}) = {4,5,7}.
        Here rules are labelled as follows:
          1: f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
          2: f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
          3: g#(c(x)) -> c_3()
          4: g#(c(1())) -> c_4(g#(d(h(0()))))
          5: g#(c(h(0()))) -> c_5(g#(d(1())))
          6: g#(d(x)) -> c_6()
          7: g#(h(x)) -> c_7(g#(x))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
            g#(c(1())) -> c_4(g#(d(h(0()))))
            g#(c(h(0()))) -> c_5(g#(d(1())))
            g#(h(x)) -> c_7(g#(x))
        - Weak DPs:
            g#(c(x)) -> c_3()
            g#(d(x)) -> c_6()
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + 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}) = {5}.
        Here rules are labelled as follows:
          1: f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
          2: f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
          3: g#(c(1())) -> c_4(g#(d(h(0()))))
          4: g#(c(h(0()))) -> c_5(g#(d(1())))
          5: g#(h(x)) -> c_7(g#(x))
          6: g#(c(x)) -> c_3()
          7: g#(d(x)) -> c_6()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
            g#(h(x)) -> c_7(g#(x))
        - Weak DPs:
            g#(c(x)) -> c_3()
            g#(c(1())) -> c_4(g#(d(h(0()))))
            g#(c(h(0()))) -> c_5(g#(d(1())))
            g#(d(x)) -> c_6()
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          2:S:f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          3:S:g#(h(x)) -> c_7(g#(x))
             -->_1 g#(c(h(0()))) -> c_5(g#(d(1()))):6
             -->_1 g#(c(1())) -> c_4(g#(d(h(0())))):5
             -->_1 g#(d(x)) -> c_6():7
             -->_1 g#(c(x)) -> c_3():4
             -->_1 g#(h(x)) -> c_7(g#(x)):3
          
          4:W:g#(c(x)) -> c_3()
             
          
          5:W:g#(c(1())) -> c_4(g#(d(h(0()))))
             -->_1 g#(d(x)) -> c_6():7
          
          6:W:g#(c(h(0()))) -> c_5(g#(d(1())))
             -->_1 g#(d(x)) -> c_6():7
          
          7:W:g#(d(x)) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: g#(c(x)) -> c_3()
          5: g#(c(1())) -> c_4(g#(d(h(0()))))
          6: g#(c(h(0()))) -> c_5(g#(d(1())))
          7: g#(d(x)) -> c_6()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
            f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
            g#(h(x)) -> c_7(g#(x))
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/2,c_2/2,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(f(x)) -> c_1(f#(c(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          2:S:f#(f(x)) -> c_2(f#(d(f(x))),f#(x))
             -->_2 f#(f(x)) -> c_2(f#(d(f(x))),f#(x)):2
             -->_2 f#(f(x)) -> c_1(f#(c(f(x))),f#(x)):1
          
          3:S:g#(h(x)) -> c_7(g#(x))
             -->_1 g#(h(x)) -> c_7(g#(x)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(f(x)) -> c_1(f#(x))
          f#(f(x)) -> c_2(f#(x))
** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x)) -> c_1(f#(x))
            f#(f(x)) -> c_2(f#(x))
            g#(h(x)) -> c_7(g#(x))
        - Weak TRS:
            f(f(x)) -> f(c(f(x)))
            f(f(x)) -> f(d(f(x)))
            g(c(x)) -> x
            g(c(1())) -> g(d(h(0())))
            g(c(h(0()))) -> g(d(1()))
            g(d(x)) -> x
            g(h(x)) -> g(x)
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g#(h(x)) -> c_7(g#(x))
** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(h(x)) -> c_7(g#(x))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + 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_7) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [0]          
            p(1) = [0]          
            p(c) = [1] x1 + [0] 
            p(d) = [1] x1 + [1] 
            p(f) = [2] x1 + [2] 
            p(g) = [8] x1 + [1] 
            p(h) = [1] x1 + [3] 
           p(f#) = [0]          
           p(g#) = [6] x1 + [0] 
          p(c_1) = [0]          
          p(c_2) = [0]          
          p(c_3) = [8]          
          p(c_4) = [1] x1 + [0] 
          p(c_5) = [2]          
          p(c_6) = [2]          
          p(c_7) = [1] x1 + [10]
        
        Following rules are strictly oriented:
        g#(h(x)) = [6] x + [18]
                 > [6] x + [10]
                 = c_7(g#(x))  
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(h(x)) -> c_7(g#(x))
        - Signature:
            {f/1,g/1,f#/1,g#/1} / {0/0,1/0,c/1,d/1,h/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/0,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,c,d,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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