* Step 1: ToInnermost WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
            average(s(x),y) -> average(x,s(y))
        - Signature:
            {average/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {average} and constructors {0,s}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
            average(s(x),y) -> average(x,s(y))
        - Signature:
            {average/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
          average#(0(),0()) -> c_2()
          average#(0(),s(0())) -> c_3()
          average#(0(),s(s(0()))) -> c_4()
          average#(s(x),y) -> c_5(average#(x,s(y)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(0(),0()) -> c_2()
            average#(0(),s(0())) -> c_3()
            average#(0(),s(s(0()))) -> c_4()
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Strict TRS:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
            average(s(x),y) -> average(x,s(y))
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
          average#(0(),0()) -> c_2()
          average#(0(),s(0())) -> c_3()
          average#(0(),s(s(0()))) -> c_4()
          average#(s(x),y) -> c_5(average#(x,s(y)))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(0(),0()) -> c_2()
            average#(0(),s(0())) -> c_3()
            average#(0(),s(s(0()))) -> c_4()
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4}
        by application of
          Pre({2,3,4}) = {5}.
        Here rules are labelled as follows:
          1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
          2: average#(0(),0()) -> c_2()
          3: average#(0(),s(0())) -> c_3()
          4: average#(0(),s(s(0()))) -> c_4()
          5: average#(s(x),y) -> c_5(average#(x,s(y)))
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Weak DPs:
            average#(0(),0()) -> c_2()
            average#(0(),s(0())) -> c_3()
            average#(0(),s(s(0()))) -> c_4()
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
             -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
             -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1
          
          2:S:average#(s(x),y) -> c_5(average#(x,s(y)))
             -->_1 average#(0(),s(s(0()))) -> c_4():5
             -->_1 average#(0(),s(0())) -> c_3():4
             -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
             -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1
          
          3:W:average#(0(),0()) -> c_2()
             
          
          4:W:average#(0(),s(0())) -> c_3()
             
          
          5:W:average#(0(),s(s(0()))) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: average#(0(),0()) -> c_2()
          4: average#(0(),s(0())) -> c_3()
          5: average#(0(),s(s(0()))) -> c_4()
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
          2: average#(s(x),y) -> c_5(average#(x,s(y)))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {average#}
        TcT has computed the following interpretation:
                 p(0) = [2]                  
           p(average) = [1] x2 + [1]         
                 p(s) = [1] x1 + [1]         
          p(average#) = [7] x1 + [4] x2 + [6]
               p(c_1) = [1] x1 + [0]         
               p(c_2) = [0]                  
               p(c_3) = [0]                  
               p(c_4) = [0]                  
               p(c_5) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        average#(x,s(s(s(y)))) = [7] x + [4] y + [18] 
                               > [7] x + [4] y + [13] 
                               = c_1(average#(s(x),y))
        
              average#(s(x),y) = [7] x + [4] y + [13] 
                               > [7] x + [4] y + [10] 
                               = c_5(average#(x,s(y)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
            average#(s(x),y) -> c_5(average#(x,s(y)))
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
             -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
             -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1
          
          2:W:average#(s(x),y) -> c_5(average#(x,s(y)))
             -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2
             -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y))
          2: average#(s(x),y) -> c_5(average#(x,s(y)))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))