WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          a(x){x -> b(x)} =
            a(b(x)) ->^+ b(a(x))
              = C[a(x) = a(x){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#(b(x)) -> c_1(a#(x))
          a#(c(x)) -> c_2()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(b(x)) -> c_1(a#(x))
            a#(c(x)) -> c_2()
        - Weak TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1}.
        Here rules are labelled as follows:
          1: a#(b(x)) -> c_1(a#(x))
          2: a#(c(x)) -> c_2()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(b(x)) -> c_1(a#(x))
        - Weak DPs:
            a#(c(x)) -> c_2()
        - Weak TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:a#(b(x)) -> c_1(a#(x))
             -->_1 a#(c(x)) -> c_2():2
             -->_1 a#(b(x)) -> c_1(a#(x)):1
          
          2:W:a#(c(x)) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: a#(c(x)) -> c_2()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(b(x)) -> c_1(a#(x))
        - Weak TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          a#(b(x)) -> c_1(a#(x))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(b(x)) -> c_1(a#(x))
        - Signature:
            {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c}
    + 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}
        
        Following symbols are considered usable:
          {a#}
        TcT has computed the following interpretation:
            p(a) = [2] x1 + [8]
            p(b) = [1] x1 + [5]
            p(c) = [1]         
           p(a#) = [4] x1 + [2]
          p(c_1) = [1] x1 + [0]
          p(c_2) = [1]         
        
        Following rules are strictly oriented:
        a#(b(x)) = [4] x + [22]
                 > [4] x + [2] 
                 = c_1(a#(x))  
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            a#(b(x)) -> c_1(a#(x))
        - Signature:
            {a/1,a#/1} / {b/1,c/1,c_1/1,c_2/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#} and constructors {b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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