WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e,f} and constructors {a,b,c,d,g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e,f} and constructors {a,b,c,d,g}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          e(x){x -> g(x)} =
            e(g(x)) ->^+ e(x)
              = C[e(x) = e(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e,f} and constructors {a,b,c,d,g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          e#(g(X)) -> c_1(e#(X))
          f#(a()) -> c_2(f#(c(a())))
          f#(a()) -> c_3(f#(d(a())))
          f#(c(X)) -> c_4()
          f#(c(a())) -> c_5(f#(d(b())))
          f#(c(b())) -> c_6(f#(d(a())))
          f#(d(X)) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            e#(g(X)) -> c_1(e#(X))
            f#(a()) -> c_2(f#(c(a())))
            f#(a()) -> c_3(f#(d(a())))
            f#(c(X)) -> c_4()
            f#(c(a())) -> c_5(f#(d(b())))
            f#(c(b())) -> c_6(f#(d(a())))
            f#(d(X)) -> c_7()
        - Weak TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,7}
        by application of
          Pre({4,7}) = {2,3,5,6}.
        Here rules are labelled as follows:
          1: e#(g(X)) -> c_1(e#(X))
          2: f#(a()) -> c_2(f#(c(a())))
          3: f#(a()) -> c_3(f#(d(a())))
          4: f#(c(X)) -> c_4()
          5: f#(c(a())) -> c_5(f#(d(b())))
          6: f#(c(b())) -> c_6(f#(d(a())))
          7: f#(d(X)) -> c_7()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            e#(g(X)) -> c_1(e#(X))
            f#(a()) -> c_2(f#(c(a())))
            f#(a()) -> c_3(f#(d(a())))
            f#(c(a())) -> c_5(f#(d(b())))
            f#(c(b())) -> c_6(f#(d(a())))
        - Weak DPs:
            f#(c(X)) -> c_4()
            f#(d(X)) -> c_7()
        - Weak TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4,5}
        by application of
          Pre({3,4,5}) = {2}.
        Here rules are labelled as follows:
          1: e#(g(X)) -> c_1(e#(X))
          2: f#(a()) -> c_2(f#(c(a())))
          3: f#(a()) -> c_3(f#(d(a())))
          4: f#(c(a())) -> c_5(f#(d(b())))
          5: f#(c(b())) -> c_6(f#(d(a())))
          6: f#(c(X)) -> c_4()
          7: f#(d(X)) -> c_7()
** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            e#(g(X)) -> c_1(e#(X))
            f#(a()) -> c_2(f#(c(a())))
        - Weak DPs:
            f#(a()) -> c_3(f#(d(a())))
            f#(c(X)) -> c_4()
            f#(c(a())) -> c_5(f#(d(b())))
            f#(c(b())) -> c_6(f#(d(a())))
            f#(d(X)) -> c_7()
        - Weak TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {}.
        Here rules are labelled as follows:
          1: e#(g(X)) -> c_1(e#(X))
          2: f#(a()) -> c_2(f#(c(a())))
          3: f#(a()) -> c_3(f#(d(a())))
          4: f#(c(X)) -> c_4()
          5: f#(c(a())) -> c_5(f#(d(b())))
          6: f#(c(b())) -> c_6(f#(d(a())))
          7: f#(d(X)) -> c_7()
** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            e#(g(X)) -> c_1(e#(X))
        - Weak DPs:
            f#(a()) -> c_2(f#(c(a())))
            f#(a()) -> c_3(f#(d(a())))
            f#(c(X)) -> c_4()
            f#(c(a())) -> c_5(f#(d(b())))
            f#(c(b())) -> c_6(f#(d(a())))
            f#(d(X)) -> c_7()
        - Weak TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:e#(g(X)) -> c_1(e#(X))
             -->_1 e#(g(X)) -> c_1(e#(X)):1
          
          2:W:f#(a()) -> c_2(f#(c(a())))
             -->_1 f#(c(a())) -> c_5(f#(d(b()))):5
             -->_1 f#(c(X)) -> c_4():4
          
          3:W:f#(a()) -> c_3(f#(d(a())))
             -->_1 f#(d(X)) -> c_7():7
          
          4:W:f#(c(X)) -> c_4()
             
          
          5:W:f#(c(a())) -> c_5(f#(d(b())))
             -->_1 f#(d(X)) -> c_7():7
          
          6:W:f#(c(b())) -> c_6(f#(d(a())))
             -->_1 f#(d(X)) -> c_7():7
          
          7:W:f#(d(X)) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: f#(c(b())) -> c_6(f#(d(a())))
          3: f#(a()) -> c_3(f#(d(a())))
          2: f#(a()) -> c_2(f#(c(a())))
          4: f#(c(X)) -> c_4()
          5: f#(c(a())) -> c_5(f#(d(b())))
          7: f#(d(X)) -> c_7()
** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            e#(g(X)) -> c_1(e#(X))
        - Weak TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          e#(g(X)) -> c_1(e#(X))
** Step 1.b:7: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            e#(g(X)) -> c_1(e#(X))
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(c_1) = {1}
        
        Following symbols are considered usable:
          {e#,f#}
        TcT has computed the following interpretation:
            p(a) = 0        
            p(b) = 0        
            p(c) = x1       
            p(d) = x1       
            p(e) = 0        
            p(f) = 0        
            p(g) = 1 + x1   
           p(e#) = 15 + 8*x1
           p(f#) = 0        
          p(c_1) = x1       
          p(c_2) = x1       
          p(c_3) = x1       
          p(c_4) = 0        
          p(c_5) = x1       
          p(c_6) = x1       
          p(c_7) = 4        
        
        Following rules are strictly oriented:
        e#(g(X)) = 23 + 8*X  
                 > 15 + 8*X  
                 = c_1(e#(X))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            e#(g(X)) -> c_1(e#(X))
        - Signature:
            {e/1,f/1,e#/1,f#/1} / {a/0,b/0,c/1,d/1,g/1,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {e#,f#} and constructors {a,b,c,d,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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