WORST_CASE(?,O(n^2))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            div(x,y) -> quot(x,y,y)
            div(0(),y) -> 0()
            div(div(x,y),z) -> div(x,times(y,z))
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            quot(x,0(),s(z)) -> s(div(x,s(z)))
            quot(0(),s(y),z) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,plus,quot,times} and constructors {0,s}
    + Applied Processor:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          div(div(x,y),z) -> div(x,times(y,z))
        All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            div(x,y) -> quot(x,y,y)
            div(0(),y) -> 0()
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            quot(x,0(),s(z)) -> s(div(x,s(z)))
            quot(0(),s(y),z) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,plus,quot,times} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          div#(x,y) -> c_1(quot#(x,y,y))
          div#(0(),y) -> c_2()
          plus#(x,0()) -> c_3()
          plus#(0(),y) -> c_4()
          plus#(s(x),y) -> c_5(plus#(x,y))
          quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          quot#(0(),s(y),z) -> c_7()
          quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
          times#(0(),y) -> c_9()
          times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          times#(s(0()),y) -> c_11()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            div#(0(),y) -> c_2()
            plus#(x,0()) -> c_3()
            plus#(0(),y) -> c_4()
            plus#(s(x),y) -> c_5(plus#(x,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(0(),s(y),z) -> c_7()
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
            times#(0(),y) -> c_9()
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
            times#(s(0()),y) -> c_11()
        - Weak TRS:
            div(x,y) -> quot(x,y,y)
            div(0(),y) -> 0()
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            quot(x,0(),s(z)) -> s(div(x,s(z)))
            quot(0(),s(y),z) -> 0()
            quot(s(x),s(y),z) -> quot(x,y,z)
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          plus(x,0()) -> x
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          times(0(),y) -> 0()
          times(s(x),y) -> plus(y,times(x,y))
          times(s(0()),y) -> y
          div#(x,y) -> c_1(quot#(x,y,y))
          div#(0(),y) -> c_2()
          plus#(x,0()) -> c_3()
          plus#(0(),y) -> c_4()
          plus#(s(x),y) -> c_5(plus#(x,y))
          quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          quot#(0(),s(y),z) -> c_7()
          quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
          times#(0(),y) -> c_9()
          times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          times#(s(0()),y) -> c_11()
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            div#(0(),y) -> c_2()
            plus#(x,0()) -> c_3()
            plus#(0(),y) -> c_4()
            plus#(s(x),y) -> c_5(plus#(x,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(0(),s(y),z) -> c_7()
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
            times#(0(),y) -> c_9()
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
            times#(s(0()),y) -> c_11()
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,4,7,9,11}
        by application of
          Pre({2,3,4,7,9,11}) = {1,5,6,8,10}.
        Here rules are labelled as follows:
          1: div#(x,y) -> c_1(quot#(x,y,y))
          2: div#(0(),y) -> c_2()
          3: plus#(x,0()) -> c_3()
          4: plus#(0(),y) -> c_4()
          5: plus#(s(x),y) -> c_5(plus#(x,y))
          6: quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          7: quot#(0(),s(y),z) -> c_7()
          8: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
          9: times#(0(),y) -> c_9()
          10: times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          11: times#(s(0()),y) -> c_11()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            plus#(s(x),y) -> c_5(plus#(x,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak DPs:
            div#(0(),y) -> c_2()
            plus#(x,0()) -> c_3()
            plus#(0(),y) -> c_4()
            quot#(0(),s(y),z) -> c_7()
            times#(0(),y) -> c_9()
            times#(s(0()),y) -> c_11()
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(x,y) -> c_1(quot#(x,y,y))
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):4
             -->_1 quot#(0(),s(y),z) -> c_7():9
          
          2:S:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(0(),y) -> c_4():8
             -->_1 plus#(x,0()) -> c_3():7
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
          3:S:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
             -->_1 div#(0(),y) -> c_2():6
             -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1
          
          4:S:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
             -->_1 quot#(0(),s(y),z) -> c_7():9
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):4
             -->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):3
          
          5:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
             -->_2 times#(s(0()),y) -> c_11():11
             -->_2 times#(0(),y) -> c_9():10
             -->_1 plus#(0(),y) -> c_4():8
             -->_1 plus#(x,0()) -> c_3():7
             -->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):5
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
          6:W:div#(0(),y) -> c_2()
             
          
          7:W:plus#(x,0()) -> c_3()
             
          
          8:W:plus#(0(),y) -> c_4()
             
          
          9:W:quot#(0(),s(y),z) -> c_7()
             
          
          10:W:times#(0(),y) -> c_9()
             
          
          11:W:times#(s(0()),y) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: times#(0(),y) -> c_9()
          11: times#(s(0()),y) -> c_11()
          7: plus#(x,0()) -> c_3()
          8: plus#(0(),y) -> c_4()
          6: div#(0(),y) -> c_2()
          9: quot#(0(),s(y),z) -> c_7()
* Step 6: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            plus#(s(x),y) -> c_5(plus#(x,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              div#(x,y) -> c_1(quot#(x,y,y))
              quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
              quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
          - Weak DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
              times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          - Weak TRS:
              plus(x,0()) -> x
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
              times(0(),y) -> 0()
              times(s(x),y) -> plus(y,times(x,y))
              times(s(0()),y) -> y
          - Signature:
              {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
              ,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
              times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          - Weak DPs:
              div#(x,y) -> c_1(quot#(x,y,y))
              quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
              quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
          - Weak TRS:
              plus(x,0()) -> x
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
              times(0(),y) -> 0()
              times(s(x),y) -> plus(y,times(x,y))
              times(s(0()),y) -> y
          - Signature:
              {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
              ,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(x,y) -> c_1(quot#(x,y,y))
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):4
          
          2:W:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
          3:S:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
             -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1
          
          4:S:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
             -->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):3
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):4
          
          5:W:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
             -->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          2: plus#(s(x),y) -> c_5(plus#(x,y))
** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          div#(x,y) -> c_1(quot#(x,y,y))
          quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
** Step 6.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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:
          3: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
          
        Consider the set of all dependency pairs
          1: div#(x,y) -> c_1(quot#(x,y,y))
          2: quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          3: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {3}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
*** Step 6.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_6) = {1},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {div#,plus#,quot#,times#}
        TcT has computed the following interpretation:
               p(0) = [0]                   
             p(div) = [2] x1 + [8] x2 + [0] 
            p(plus) = [2] x2 + [2]          
            p(quot) = [8] x1 + [1] x2 + [1] 
               p(s) = [1] x1 + [3]          
           p(times) = [1]                   
            p(div#) = [4] x1 + [10] x2 + [0]
           p(plus#) = [1] x1 + [0]          
           p(quot#) = [4] x1 + [10] x3 + [0]
          p(times#) = [1] x2 + [1]          
             p(c_1) = [1] x1 + [0]          
             p(c_2) = [4]                   
             p(c_3) = [1]                   
             p(c_4) = [1]                   
             p(c_5) = [1]                   
             p(c_6) = [1] x1 + [0]          
             p(c_7) = [0]                   
             p(c_8) = [1] x1 + [10]         
             p(c_9) = [8]                   
            p(c_10) = [4] x1 + [0]          
            p(c_11) = [0]                   
        
        Following rules are strictly oriented:
        quot#(s(x),s(y),z) = [4] x + [10] z + [12]
                           > [4] x + [10] z + [10]
                           = c_8(quot#(x,y,z))    
        
        
        Following rules are (at-least) weakly oriented:
                div#(x,y) =  [4] x + [10] y + [0] 
                          >= [4] x + [10] y + [0] 
                          =  c_1(quot#(x,y,y))    
        
        quot#(x,0(),s(z)) =  [4] x + [10] z + [30]
                          >= [4] x + [10] z + [30]
                          =  c_6(div#(x,s(z)))    
        
*** Step 6.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
        - Weak DPs:
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:div#(x,y) -> c_1(quot#(x,y,y))
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):3
          
          2:W:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
             -->_1 div#(x,y) -> c_1(quot#(x,y,y)):1
          
          3:W:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):3
             -->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: div#(x,y) -> c_1(quot#(x,y,y))
          2: quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          3: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
*** Step 6.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak DPs:
            div#(x,y) -> c_1(quot#(x,y,y))
            quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
            quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
          2:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
             -->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):2
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
          3:W:div#(x,y) -> c_1(quot#(x,y,y))
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):5
          
          4:W:quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
             -->_1 div#(x,y) -> c_1(quot#(x,y,y)):3
          
          5:W:quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
             -->_1 quot#(s(x),s(y),z) -> c_8(quot#(x,y,z)):5
             -->_1 quot#(x,0(),s(z)) -> c_6(div#(x,s(z))):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: div#(x,y) -> c_1(quot#(x,y,y))
          4: quot#(x,0(),s(z)) -> c_6(div#(x,s(z)))
          5: quot#(s(x),s(y),z) -> c_8(quot#(x,y,z))
** Step 6.b:2: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
          - Weak DPs:
              times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          - Weak TRS:
              plus(x,0()) -> x
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
              times(0(),y) -> 0()
              times(s(x),y) -> plus(y,times(x,y))
              times(s(0()),y) -> y
          - Signature:
              {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
              ,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
        
        Problem (S)
          - Strict DPs:
              times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          - Weak DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
          - Weak TRS:
              plus(x,0()) -> x
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
              times(0(),y) -> 0()
              times(s(x),y) -> plus(y,times(x,y))
              times(s(0()),y) -> y
          - Signature:
              {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1
              ,c_6/1,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
*** Step 6.b:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
        - Weak DPs:
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: plus#(s(x),y) -> c_5(plus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.b:2.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
        - Weak DPs:
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_10) = {1,2}
        
        Following symbols are considered usable:
          {div#,plus#,quot#,times#}
        TcT has computed the following interpretation:
               p(0) = 0                                      
             p(div) = x1^2 + x2 + x2^2                       
            p(plus) = 4 + x1                                 
            p(quot) = x1*x2 + x1^2 + x2 + 2*x2*x3 + x2^2 + x3
               p(s) = 3 + x1                                 
           p(times) = 0                                      
            p(div#) = 2 + 2*x1 + 4*x1^2 + x2^2               
           p(plus#) = x1                                     
           p(quot#) = 2 + x1*x3 + 4*x2*x3 + 4*x3             
          p(times#) = 1 + 5*x1*x2 + 2*x2^2                   
             p(c_1) = 4                                      
             p(c_2) = 1                                      
             p(c_3) = 0                                      
             p(c_4) = 0                                      
             p(c_5) = 2 + x1                                 
             p(c_6) = 2                                      
             p(c_7) = 0                                      
             p(c_8) = 1                                      
             p(c_9) = 1                                      
            p(c_10) = x1 + x2                                
            p(c_11) = 1                                      
        
        Following rules are strictly oriented:
        plus#(s(x),y) = 3 + x          
                      > 2 + x          
                      = c_5(plus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        times#(s(x),y) =  1 + 5*x*y + 15*y + 2*y^2             
                       >= 1 + 5*x*y + y + 2*y^2                
                       =  c_10(plus#(y,times(x,y)),times#(x,y))
        
**** Step 6.b:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.b:2.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
          2:W:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
             -->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):2
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
          1: plus#(s(x),y) -> c_5(plus#(x,y))
**** Step 6.b:2.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 6.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
             -->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):1
          
          2:W:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: plus#(s(x),y) -> c_5(plus#(x,y))
*** Step 6.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y))
             -->_2 times#(s(x),y) -> c_10(plus#(y,times(x,y)),times#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          times#(s(x),y) -> c_10(times#(x,y))
*** Step 6.b:2.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(x),y) -> c_10(times#(x,y))
        - Weak TRS:
            plus(x,0()) -> x
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            times(0(),y) -> 0()
            times(s(x),y) -> plus(y,times(x,y))
            times(s(0()),y) -> y
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          times#(s(x),y) -> c_10(times#(x,y))
*** Step 6.b:2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(x),y) -> c_10(times#(x,y))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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: times#(s(x),y) -> c_10(times#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.b:2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(s(x),y) -> c_10(times#(x,y))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} 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_10) = {1}
        
        Following symbols are considered usable:
          {div#,plus#,quot#,times#}
        TcT has computed the following interpretation:
               p(0) = [1]                           
             p(div) = [2]                           
            p(plus) = [2] x2 + [0]                  
            p(quot) = [1] x1 + [4] x2 + [8] x3 + [1]
               p(s) = [1] x1 + [3]                  
           p(times) = [1]                           
            p(div#) = [1] x1 + [1]                  
           p(plus#) = [1] x2 + [0]                  
           p(quot#) = [1] x1 + [4] x2 + [1]         
          p(times#) = [2] x1 + [2]                  
             p(c_1) = [1] x1 + [0]                  
             p(c_2) = [1]                           
             p(c_3) = [0]                           
             p(c_4) = [1]                           
             p(c_5) = [2]                           
             p(c_6) = [8] x1 + [1]                  
             p(c_7) = [1]                           
             p(c_8) = [1] x1 + [0]                  
             p(c_9) = [4]                           
            p(c_10) = [1] x1 + [3]                  
            p(c_11) = [0]                           
        
        Following rules are strictly oriented:
        times#(s(x),y) = [2] x + [8]      
                       > [2] x + [5]      
                       = c_10(times#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 6.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            times#(s(x),y) -> c_10(times#(x,y))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.b:2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            times#(s(x),y) -> c_10(times#(x,y))
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:times#(s(x),y) -> c_10(times#(x,y))
             -->_1 times#(s(x),y) -> c_10(times#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: times#(s(x),y) -> c_10(times#(x,y))
**** Step 6.b:2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {div/2,plus/2,quot/3,times/2,div#/2,plus#/2,quot#/3,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1
            ,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,plus#,quot#,times#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))