MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            fib(x) -> fibiter(x,0(),0(),s(0()))
            fibiter(b,c,x,y) -> if(lt(c,b),b,c,x,y)
            if(false(),b,c,x,y) -> x
            if(true(),b,c,x,y) -> fibiter(b,s(c),y,plus(x,y))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {fib/1,fibiter/4,if/5,lt/2,plus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib,fibiter,if,lt,plus} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
          fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
          if#(false(),b,c,x,y) -> c_3()
          if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
          lt#(x,0()) -> c_5()
          lt#(0(),s(y)) -> c_6()
          lt#(s(x),s(y)) -> c_7(lt#(x,y))
          plus#(0(),y) -> c_8()
          plus#(s(x),y) -> c_9(plus#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
            fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
            if#(false(),b,c,x,y) -> c_3()
            if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
            lt#(x,0()) -> c_5()
            lt#(0(),s(y)) -> c_6()
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
            plus#(0(),y) -> c_8()
            plus#(s(x),y) -> c_9(plus#(x,y))
        - Weak TRS:
            fib(x) -> fibiter(x,0(),0(),s(0()))
            fibiter(b,c,x,y) -> if(lt(c,b),b,c,x,y)
            if(false(),b,c,x,y) -> x
            if(true(),b,c,x,y) -> fibiter(b,s(c),y,plus(x,y))
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {fib/1,fibiter/4,if/5,lt/2,plus/2,fib#/1,fibiter#/4,if#/5,lt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/1
            ,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibiter#,if#,lt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          lt(x,0()) -> false()
          lt(0(),s(y)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
          fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
          if#(false(),b,c,x,y) -> c_3()
          if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
          lt#(x,0()) -> c_5()
          lt#(0(),s(y)) -> c_6()
          lt#(s(x),s(y)) -> c_7(lt#(x,y))
          plus#(0(),y) -> c_8()
          plus#(s(x),y) -> c_9(plus#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
            fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
            if#(false(),b,c,x,y) -> c_3()
            if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
            lt#(x,0()) -> c_5()
            lt#(0(),s(y)) -> c_6()
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
            plus#(0(),y) -> c_8()
            plus#(s(x),y) -> c_9(plus#(x,y))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {fib/1,fibiter/4,if/5,lt/2,plus/2,fib#/1,fibiter#/4,if#/5,lt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/1
            ,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibiter#,if#,lt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,5,6,8}
        by application of
          Pre({3,5,6,8}) = {2,4,7,9}.
        Here rules are labelled as follows:
          1: fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
          2: fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
          3: if#(false(),b,c,x,y) -> c_3()
          4: if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
          5: lt#(x,0()) -> c_5()
          6: lt#(0(),s(y)) -> c_6()
          7: lt#(s(x),s(y)) -> c_7(lt#(x,y))
          8: plus#(0(),y) -> c_8()
          9: plus#(s(x),y) -> c_9(plus#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
            fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
            if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
            plus#(s(x),y) -> c_9(plus#(x,y))
        - Weak DPs:
            if#(false(),b,c,x,y) -> c_3()
            lt#(x,0()) -> c_5()
            lt#(0(),s(y)) -> c_6()
            plus#(0(),y) -> c_8()
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {fib/1,fibiter/4,if/5,lt/2,plus/2,fib#/1,fibiter#/4,if#/5,lt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/1
            ,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibiter#,if#,lt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
             -->_1 fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b)):2
          
          2:S:fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
             -->_2 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
             -->_1 if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y)):3
             -->_2 lt#(0(),s(y)) -> c_6():8
             -->_2 lt#(x,0()) -> c_5():7
             -->_1 if#(false(),b,c,x,y) -> c_3():6
          
          3:S:if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
             -->_2 plus#(s(x),y) -> c_9(plus#(x,y)):5
             -->_2 plus#(0(),y) -> c_8():9
             -->_1 fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b)):2
          
          4:S:lt#(s(x),s(y)) -> c_7(lt#(x,y))
             -->_1 lt#(0(),s(y)) -> c_6():8
             -->_1 lt#(x,0()) -> c_5():7
             -->_1 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
          
          5:S:plus#(s(x),y) -> c_9(plus#(x,y))
             -->_1 plus#(0(),y) -> c_8():9
             -->_1 plus#(s(x),y) -> c_9(plus#(x,y)):5
          
          6:W:if#(false(),b,c,x,y) -> c_3()
             
          
          7:W:lt#(x,0()) -> c_5()
             
          
          8:W:lt#(0(),s(y)) -> c_6()
             
          
          9:W:plus#(0(),y) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: if#(false(),b,c,x,y) -> c_3()
          9: plus#(0(),y) -> c_8()
          7: lt#(x,0()) -> c_5()
          8: lt#(0(),s(y)) -> c_6()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
            fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
            if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
            lt#(s(x),s(y)) -> c_7(lt#(x,y))
            plus#(s(x),y) -> c_9(plus#(x,y))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(y)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {fib/1,fibiter/4,if/5,lt/2,plus/2,fib#/1,fibiter#/4,if#/5,lt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/1
            ,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fib#,fibiter#,if#,lt#,plus#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:fib#(x) -> c_1(fibiter#(x,0(),0(),s(0())))
           -->_1 fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b)):2
        
        2:S:fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
           -->_2 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
           -->_1 if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y)):3
        
        3:S:if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
           -->_2 plus#(s(x),y) -> c_9(plus#(x,y)):5
           -->_1 fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b)):2
        
        4:S:lt#(s(x),s(y)) -> c_7(lt#(x,y))
           -->_1 lt#(s(x),s(y)) -> c_7(lt#(x,y)):4
        
        5:S:plus#(s(x),y) -> c_9(plus#(x,y))
           -->_1 plus#(s(x),y) -> c_9(plus#(x,y)):5
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(1,fib#(x) -> c_1(fibiter#(x,0(),0(),s(0()))))]
* Step 6: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          fibiter#(b,c,x,y) -> c_2(if#(lt(c,b),b,c,x,y),lt#(c,b))
          if#(true(),b,c,x,y) -> c_4(fibiter#(b,s(c),y,plus(x,y)),plus#(x,y))
          lt#(s(x),s(y)) -> c_7(lt#(x,y))
          plus#(s(x),y) -> c_9(plus#(x,y))
      - Weak TRS:
          lt(x,0()) -> false()
          lt(0(),s(y)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
      - Signature:
          {fib/1,fibiter/4,if/5,lt/2,plus/2,fib#/1,fibiter#/4,if#/5,lt#/2,plus#/2} / {0/0,false/0,s/1,true/0,c_1/1
          ,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {fib#,fibiter#,if#,lt#,plus#} and constructors {0,false,s
          ,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE