* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          dbl#(0()) -> c_3()
          dbl#(s(X)) -> c_4(dbl#(X))
          first#(0(),X) -> c_5()
          first#(s(X),cons(Y)) -> c_6()
          sqr#(0()) -> c_7()
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_9(sqr#(N))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(0()) -> c_3()
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y)) -> c_6()
            sqr#(0()) -> c_7()
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(s(dbl(X)))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          dbl#(0()) -> c_3()
          dbl#(s(X)) -> c_4(dbl#(X))
          first#(0(),X) -> c_5()
          first#(s(X),cons(Y)) -> c_6()
          sqr#(0()) -> c_7()
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_9(sqr#(N))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(0()) -> c_3()
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y)) -> c_6()
            sqr#(0()) -> c_7()
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5,6,7}
        by application of
          Pre({1,3,5,6,7}) = {2,4,8,9}.
        Here rules are labelled as follows:
          1: add#(0(),X) -> c_1()
          2: add#(s(X),Y) -> c_2(add#(X,Y))
          3: dbl#(0()) -> c_3()
          4: dbl#(s(X)) -> c_4(dbl#(X))
          5: first#(0(),X) -> c_5()
          6: first#(s(X),cons(Y)) -> c_6()
          7: sqr#(0()) -> c_7()
          8: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          9: terms#(N) -> c_9(sqr#(N))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N))
        - Weak DPs:
            add#(0(),X) -> c_1()
            dbl#(0()) -> c_3()
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y)) -> c_6()
            sqr#(0()) -> c_7()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(0(),X) -> c_1():5
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(0()) -> c_3():6
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(0()) -> c_7():9
             -->_3 dbl#(0()) -> c_3():6
             -->_1 add#(0(),X) -> c_1():5
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          4:S:terms#(N) -> c_9(sqr#(N))
             -->_1 sqr#(0()) -> c_7():9
             -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
          
          5:W:add#(0(),X) -> c_1()
             
          
          6:W:dbl#(0()) -> c_3()
             
          
          7:W:first#(0(),X) -> c_5()
             
          
          8:W:first#(s(X),cons(Y)) -> c_6()
             
          
          9:W:sqr#(0()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: first#(s(X),cons(Y)) -> c_6()
          7: first#(0(),X) -> c_5()
          9: sqr#(0()) -> c_7()
          6: dbl#(0()) -> c_3()
          5: add#(0(),X) -> c_1()
* Step 5: RemoveHeads WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:add#(s(X),Y) -> c_2(add#(X,Y))
           -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
        
        2:S:dbl#(s(X)) -> c_4(dbl#(X))
           -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
        
        3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
           -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
           -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
           -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
        
        4:S:terms#(N) -> c_9(sqr#(N))
           -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
        
        
        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).
        
        [(4,terms#(N) -> c_9(sqr#(N)))]
* Step 6: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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:
              add#(s(X),Y) -> c_2(add#(X,Y))
          - Weak DPs:
              dbl#(s(X)) -> c_4(dbl#(X))
              sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              dbl(0()) -> 0()
              dbl(s(X)) -> s(s(dbl(X)))
              sqr(0()) -> 0()
              sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
        
        Problem (S)
          - Strict DPs:
              dbl#(s(X)) -> c_4(dbl#(X))
              sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          - Weak DPs:
              add#(s(X),Y) -> c_2(add#(X,Y))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              dbl(0()) -> 0()
              dbl(s(X)) -> s(s(dbl(X)))
              sqr(0()) -> 0()
              sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:W:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          3:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: dbl#(s(X)) -> c_4(dbl#(X))
** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          3:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
** Step 6.a:3: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
        and a lower component
          add#(s(X),Y) -> c_2(add#(X,Y))
        Further, following extension rules are added to the lower component.
          sqr#(s(X)) -> add#(sqr(X),dbl(X))
          sqr#(s(X)) -> sqr#(X)
*** Step 6.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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_8) = {1,2}
        
        Following symbols are considered usable:
          {add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = [5]                   
             p(add) = [4] x1 + [0]          
            p(cons) = [8]                   
             p(dbl) = [3] x1 + [0]          
           p(first) = [1]                   
             p(nil) = [0]                   
           p(recip) = [2]                   
               p(s) = [1] x1 + [4]          
             p(sqr) = [1] x1 + [3]          
           p(terms) = [1]                   
            p(add#) = [0]                   
            p(dbl#) = [0]                   
          p(first#) = [2]                   
            p(sqr#) = [4] x1 + [8]          
          p(terms#) = [2]                   
             p(c_1) = [1]                   
             p(c_2) = [1]                   
             p(c_3) = [1]                   
             p(c_4) = [1]                   
             p(c_5) = [1]                   
             p(c_6) = [1]                   
             p(c_7) = [0]                   
             p(c_8) = [8] x1 + [1] x2 + [12]
             p(c_9) = [1] x1 + [0]          
        
        Following rules are strictly oriented:
        sqr#(s(X)) = [4] X + [24]                    
                   > [4] X + [20]                    
                   = c_8(add#(sqr(X),dbl(X)),sqr#(X))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

*** Step 6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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: add#(s(X),Y) -> c_2(add#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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_2) = {1}
        
        Following symbols are considered usable:
          {add,dbl,sqr,add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = 0                 
             p(add) = x1 + x2           
            p(cons) = 0                 
             p(dbl) = 2*x1              
           p(first) = x1^2 + x2 + 2*x2^2
             p(nil) = 1                 
           p(recip) = 1 + x1            
               p(s) = 1 + x1            
             p(sqr) = x1^2              
           p(terms) = 1 + x1            
            p(add#) = 6 + x1            
            p(dbl#) = 1 + x1^2          
          p(first#) = 0                 
            p(sqr#) = 5 + 2*x1^2        
          p(terms#) = 1 + x1 + x1^2     
             p(c_1) = 0                 
             p(c_2) = x1                
             p(c_3) = 0                 
             p(c_4) = 1 + x1            
             p(c_5) = 0                 
             p(c_6) = 0                 
             p(c_7) = 1                 
             p(c_8) = 1                 
             p(c_9) = 1                 
        
        Following rules are strictly oriented:
        add#(s(X),Y) = 7 + X         
                     > 6 + X         
                     = c_2(add#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
         sqr#(s(X)) =  7 + 4*X + 2*X^2      
                    >= 6 + X^2              
                    =  add#(sqr(X),dbl(X))  
        
         sqr#(s(X)) =  7 + 4*X + 2*X^2      
                    >= 5 + 2*X^2            
                    =  sqr#(X)              
        
         add(0(),X) =  X                    
                    >= X                    
                    =  X                    
        
        add(s(X),Y) =  1 + X + Y            
                    >= 1 + X + Y            
                    =  s(add(X,Y))          
        
           dbl(0()) =  0                    
                    >= 0                    
                    =  0()                  
        
          dbl(s(X)) =  2 + 2*X              
                    >= 2 + 2*X              
                    =  s(s(dbl(X)))         
        
           sqr(0()) =  0                    
                    >= 0                    
                    =  0()                  
        
          sqr(s(X)) =  1 + 2*X + X^2        
                    >= 1 + 2*X + X^2        
                    =  s(add(sqr(X),dbl(X)))
        
**** Step 6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(X)) -> sqr#(X)
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:W:sqr#(s(X)) -> add#(sqr(X),dbl(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          3:W:sqr#(s(X)) -> sqr#(X)
             -->_1 sqr#(s(X)) -> sqr#(X):3
             -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sqr#(s(X)) -> sqr#(X)
          2: sqr#(s(X)) -> add#(sqr(X),dbl(X))
          1: add#(s(X),Y) -> c_2(add#(X,Y))
**** Step 6.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          2:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):3
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          3:W:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: add#(s(X),Y) -> c_2(add#(X,Y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          2:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
** Step 6.b:3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          dbl#(s(X)) -> c_4(dbl#(X))
          sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
** Step 6.b:4: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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:
              dbl#(s(X)) -> c_4(dbl#(X))
          - Weak DPs:
              sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
        
        Problem (S)
          - Strict DPs:
              sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
          - Weak DPs:
              dbl#(s(X)) -> c_4(dbl#(X))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
*** Step 6.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
        - Weak DPs:
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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: dbl#(s(X)) -> c_4(dbl#(X))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.b:4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
        - Weak DPs:
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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_4) = {1},
          uargs(c_8) = {1,2}
        
        Following symbols are considered usable:
          {add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = 0                           
             p(add) = 1 + x1 + x1*x2 + x1^2 + x2^2
            p(cons) = 0                           
             p(dbl) = 1 + 8*x1                    
           p(first) = 2*x1 + x1*x2 + x1^2 + x2    
             p(nil) = 0                           
           p(recip) = 0                           
               p(s) = 1 + x1                      
             p(sqr) = 1                           
           p(terms) = 1 + x1 + 8*x1^2             
            p(add#) = x1 + 2*x1^2 + x2 + x2^2     
            p(dbl#) = 8 + 8*x1                    
          p(first#) = 4*x1 + x1^2 + x2            
            p(sqr#) = 2 + 10*x1 + 9*x1^2          
          p(terms#) = 0                           
             p(c_1) = 0                           
             p(c_2) = 1 + x1                      
             p(c_3) = 1                           
             p(c_4) = x1                          
             p(c_5) = 0                           
             p(c_6) = 0                           
             p(c_7) = 0                           
             p(c_8) = x1 + x2                     
             p(c_9) = 0                           
        
        Following rules are strictly oriented:
        dbl#(s(X)) = 16 + 8*X    
                   > 8 + 8*X     
                   = c_4(dbl#(X))
        
        
        Following rules are (at-least) weakly oriented:
        sqr#(s(X)) =  21 + 28*X + 9*X^2   
                   >= 10 + 18*X + 9*X^2   
                   =  c_8(sqr#(X),dbl#(X))
        
**** Step 6.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 6.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          2:W:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2
             -->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
          1: dbl#(s(X)) -> c_4(dbl#(X))
**** Step 6.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Weak DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
             -->_2 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):1
          
          2:W:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: dbl#(s(X)) -> c_4(dbl#(X))
*** Step 6.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_8(sqr#(X))
*** Step 6.b:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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: sqr#(s(X)) -> c_8(sqr#(X))
          
        The strictly oriented rules are moved into the weak component.
**** Step 6.b:4.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,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_8) = {1}
        
        Following symbols are considered usable:
          {add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = [0]                  
             p(add) = [0]                  
            p(cons) = [2]                  
             p(dbl) = [8]                  
           p(first) = [1]                  
             p(nil) = [4]                  
           p(recip) = [8]                  
               p(s) = [1] x1 + [2]         
             p(sqr) = [8]                  
           p(terms) = [8] x1 + [0]         
            p(add#) = [1] x1 + [1] x2 + [2]
            p(dbl#) = [1]                  
          p(first#) = [1] x1 + [2] x2 + [2]
            p(sqr#) = [1] x1 + [1]         
          p(terms#) = [2]                  
             p(c_1) = [1]                  
             p(c_2) = [1]                  
             p(c_3) = [1]                  
             p(c_4) = [1] x1 + [1]         
             p(c_5) = [0]                  
             p(c_6) = [1]                  
             p(c_7) = [4]                  
             p(c_8) = [1] x1 + [0]         
             p(c_9) = [8]                  
        
        Following rules are strictly oriented:
        sqr#(s(X)) = [1] X + [3] 
                   > [1] X + [1] 
                   = c_8(sqr#(X))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 6.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

WORST_CASE(?,O(n^3))