WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
        - Weak TRS:
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB,fib} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
        - Weak TRS:
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB,fib} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
        - Weak TRS:
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB,fib} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          FIB(s(x)){x -> s(x)} =
            FIB(s(s(x))) ->^+ c2(FIB(s(x)))
              = C[FIB(s(x)) = FIB(s(x)){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
        - Weak TRS:
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB,fib} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          FIB#(0()) -> c_1()
          FIB#(s(0())) -> c_2()
          FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
          FIB#(s(s(z0))) -> c_4(FIB#(z0))
        Weak DPs
          fib#(0()) -> c_5()
          fib#(s(0())) -> c_6()
          fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0))
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            FIB#(0()) -> c_1()
            FIB#(s(0())) -> c_2()
            FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
            FIB#(s(s(z0))) -> c_4(FIB#(z0))
        - Weak DPs:
            fib#(0()) -> c_5()
            fib#(s(0())) -> c_6()
            fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0))
        - Weak TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1,FIB#/1,fib#/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB#,fib#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3,4}.
        Here rules are labelled as follows:
          1: FIB#(0()) -> c_1()
          2: FIB#(s(0())) -> c_2()
          3: FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
          4: FIB#(s(s(z0))) -> c_4(FIB#(z0))
          5: fib#(0()) -> c_5()
          6: fib#(s(0())) -> c_6()
          7: fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
            FIB#(s(s(z0))) -> c_4(FIB#(z0))
        - Weak DPs:
            FIB#(0()) -> c_1()
            FIB#(s(0())) -> c_2()
            fib#(0()) -> c_5()
            fib#(s(0())) -> c_6()
            fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0))
        - Weak TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1,FIB#/1,fib#/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB#,fib#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
             -->_1 FIB#(s(s(z0))) -> c_4(FIB#(z0)):2
             -->_1 FIB#(s(0())) -> c_2():4
             -->_1 FIB#(s(s(z0))) -> c_3(FIB#(s(z0))):1
          
          2:S:FIB#(s(s(z0))) -> c_4(FIB#(z0))
             -->_1 FIB#(s(0())) -> c_2():4
             -->_1 FIB#(0()) -> c_1():3
             -->_1 FIB#(s(s(z0))) -> c_4(FIB#(z0)):2
             -->_1 FIB#(s(s(z0))) -> c_3(FIB#(s(z0))):1
          
          3:W:FIB#(0()) -> c_1()
             
          
          4:W:FIB#(s(0())) -> c_2()
             
          
          5:W:fib#(0()) -> c_5()
             
          
          6:W:fib#(s(0())) -> c_6()
             
          
          7:W:fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0))
             -->_2 fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0)):7
             -->_1 fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0)):7
             -->_2 fib#(s(0())) -> c_6():6
             -->_1 fib#(s(0())) -> c_6():6
             -->_2 fib#(0()) -> c_5():5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: fib#(s(s(z0))) -> c_7(fib#(s(z0)),fib#(z0))
          6: fib#(s(0())) -> c_6()
          5: fib#(0()) -> c_5()
          3: FIB#(0()) -> c_1()
          4: FIB#(s(0())) -> c_2()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
            FIB#(s(s(z0))) -> c_4(FIB#(z0))
        - Weak TRS:
            FIB(0()) -> c()
            FIB(s(0())) -> c1()
            FIB(s(s(z0))) -> c2(FIB(s(z0)))
            FIB(s(s(z0))) -> c3(FIB(z0))
            fib(0()) -> 0()
            fib(s(0())) -> s(0())
            fib(s(s(z0))) -> +(fib(s(z0)),fib(z0))
        - Signature:
            {FIB/1,fib/1,FIB#/1,fib#/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB#,fib#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
          FIB#(s(s(z0))) -> c_4(FIB#(z0))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
            FIB#(s(s(z0))) -> c_4(FIB#(z0))
        - Signature:
            {FIB/1,fib/1,FIB#/1,fib#/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB#,fib#} and constructors {+,0,c,c1,c2,c3,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},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {FIB#,fib#}
        TcT has computed the following interpretation:
             p(+) = [1] x1 + [0] 
             p(0) = [0]          
           p(FIB) = [2] x1 + [0] 
             p(c) = [1]          
            p(c1) = [4]          
            p(c2) = [1]          
            p(c3) = [2]          
           p(fib) = [1] x1 + [1] 
             p(s) = [1] x1 + [3] 
          p(FIB#) = [4] x1 + [4] 
          p(fib#) = [8]          
           p(c_1) = [1]          
           p(c_2) = [1]          
           p(c_3) = [1] x1 + [10]
           p(c_4) = [1] x1 + [6] 
           p(c_5) = [4]          
           p(c_6) = [2]          
           p(c_7) = [1] x2 + [0] 
        
        Following rules are strictly oriented:
        FIB#(s(s(z0))) = [4] z0 + [28]   
                       > [4] z0 + [26]   
                       = c_3(FIB#(s(z0)))
        
        FIB#(s(s(z0))) = [4] z0 + [28]   
                       > [4] z0 + [10]   
                       = c_4(FIB#(z0))   
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            FIB#(s(s(z0))) -> c_3(FIB#(s(z0)))
            FIB#(s(s(z0))) -> c_4(FIB#(z0))
        - Signature:
            {FIB/1,fib/1,FIB#/1,fib#/1} / {+/2,0/0,c/0,c1/0,c2/1,c3/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {FIB#,fib#} and constructors {+,0,c,c1,c2,c3,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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