WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum WORST_CASE(Omega(n^1),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)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,half,sqr,terms} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + 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)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,half,sqr,terms} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          add(x,y){x -> s(x)} =
            add(s(x),y) ->^+ s(add(x,y))
              = C[add(x,y) = add(x,y){}]

** Step 1.b: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)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,half,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()
          half#(0()) -> c_7()
          half#(dbl(X)) -> c_8()
          half#(s(0())) -> c_9()
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(0()) -> c_11()
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_13(sqr#(N))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: 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()
            half#(0()) -> c_7()
            half#(dbl(X)) -> c_8()
            half#(s(0())) -> c_9()
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(0()) -> c_11()
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(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)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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
          {1,3,5,6,7,8,9,11}
        by application of
          Pre({1,3,5,6,7,8,9,11}) = {2,4,10,12,13}.
        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: half#(0()) -> c_7()
          8: half#(dbl(X)) -> c_8()
          9: half#(s(0())) -> c_9()
          10: half#(s(s(X))) -> c_10(half#(X))
          11: sqr#(0()) -> c_11()
          12: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          13: terms#(N) -> c_13(sqr#(N))
** Step 1.b:3: 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))
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(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()
            half#(0()) -> c_7()
            half#(dbl(X)) -> c_8()
            half#(s(0())) -> c_9()
            sqr#(0()) -> c_11()
        - 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)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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():6
             -->_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():7
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          3:S:half#(s(s(X))) -> c_10(half#(X))
             -->_1 half#(s(0())) -> c_9():12
             -->_1 half#(dbl(X)) -> c_8():11
             -->_1 half#(0()) -> c_7():10
             -->_1 half#(s(s(X))) -> c_10(half#(X)):3
          
          4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(0()) -> c_11():13
             -->_3 dbl#(0()) -> c_3():7
             -->_1 add#(0(),X) -> c_1():6
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          5:S:terms#(N) -> c_13(sqr#(N))
             -->_1 sqr#(0()) -> c_11():13
             -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
          
          6:W:add#(0(),X) -> c_1()
             
          
          7:W:dbl#(0()) -> c_3()
             
          
          8:W:first#(0(),X) -> c_5()
             
          
          9:W:first#(s(X),cons(Y)) -> c_6()
             
          
          10:W:half#(0()) -> c_7()
             
          
          11:W:half#(dbl(X)) -> c_8()
             
          
          12:W:half#(s(0())) -> c_9()
             
          
          13:W:sqr#(0()) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: first#(s(X),cons(Y)) -> c_6()
          8: first#(0(),X) -> c_5()
          13: sqr#(0()) -> c_11()
          10: half#(0()) -> c_7()
          11: half#(dbl(X)) -> c_8()
          12: half#(s(0())) -> c_9()
          7: dbl#(0()) -> c_3()
          6: add#(0(),X) -> c_1()
** Step 1.b:4: 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))
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(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)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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:half#(s(s(X))) -> c_10(half#(X))
           -->_1 half#(s(s(X))) -> c_10(half#(X)):3
        
        4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
           -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
           -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
           -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
        
        5:S:terms#(N) -> c_13(sqr#(N))
           -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
        
        
        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).
        
        [(5,terms#(N) -> c_13(sqr#(N)))]
** Step 1.b:5: UsableRules 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))
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(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)))
            first(0(),X) -> nil()
            first(s(X),cons(Y)) -> cons(Y)
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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#(s(X),Y) -> c_2(add#(X,Y))
          dbl#(s(X)) -> c_4(dbl#(X))
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
** Step 1.b:6: DecomposeDG 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))
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
            ,cons,nil,recip,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
        and a lower component
          add#(s(X),Y) -> c_2(add#(X,Y))
          dbl#(s(X)) -> c_4(dbl#(X))
        Further, following extension rules are added to the lower component.
          half#(s(s(X))) -> half#(X)
          sqr#(s(X)) -> add#(sqr(X),dbl(X))
          sqr#(s(X)) -> dbl#(X)
          sqr#(s(X)) -> sqr#(X)
*** Step 1.b:6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
            ,cons,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:half#(s(s(X))) -> c_10(half#(X))
             -->_1 half#(s(s(X))) -> c_10(half#(X)):1
          
          2:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_12(sqr#(X))
*** Step 1.b:6.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
            ,cons,nil,recip,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(s(X)) -> c_12(sqr#(X))
*** Step 1.b:6.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(sqr#(X))
        - Signature:
            {add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
            ,cons,nil,recip,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_10) = {1},
            uargs(c_12) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                  
               p(add) = [2] x1 + [1] x2 + [0]
              p(cons) = [1] x1 + [2]         
               p(dbl) = [1] x1 + [0]         
             p(first) = [8] x2 + [1]         
              p(half) = [1] x1 + [0]         
               p(nil) = [0]                  
             p(recip) = [0]                  
                 p(s) = [1] x1 + [1]         
               p(sqr) = [2] x1 + [0]         
             p(terms) = [1] x1 + [2]         
              p(add#) = [8] x1 + [2] x2 + [2]
              p(dbl#) = [8] x1 + [1]         
            p(first#) = [1] x1 + [1] x2 + [0]
             p(half#) = [1] x1 + [5]         
              p(sqr#) = [8] x1 + [0]         
            p(terms#) = [1]                  
               p(c_1) = [0]                  
               p(c_2) = [0]                  
               p(c_3) = [2]                  
               p(c_4) = [1] x1 + [0]         
               p(c_5) = [0]                  
               p(c_6) = [8]                  
               p(c_7) = [2]                  
               p(c_8) = [0]                  
               p(c_9) = [1]                  
              p(c_10) = [1] x1 + [1]         
              p(c_11) = [2]                  
              p(c_12) = [1] x1 + [0]         
              p(c_13) = [1]                  
          
          Following rules are strictly oriented:
          half#(s(s(X))) = [1] X + [7]   
                         > [1] X + [6]   
                         = c_10(half#(X))
          
              sqr#(s(X)) = [8] X + [8]   
                         > [8] X + [0]   
                         = c_12(sqr#(X)) 
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(sqr#(X))
        - Signature:
            {add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
        - Weak DPs:
            half#(s(s(X))) -> half#(X)
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          3:W:half#(s(s(X))) -> half#(X)
             -->_1 half#(s(s(X))) -> half#(X):3
          
          4:W:sqr#(s(X)) -> add#(sqr(X),dbl(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          5:W:sqr#(s(X)) -> dbl#(X)
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          6:W:sqr#(s(X)) -> sqr#(X)
             -->_1 sqr#(s(X)) -> sqr#(X):6
             -->_1 sqr#(s(X)) -> dbl#(X):5
             -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: half#(s(s(X))) -> half#(X)
*** Step 1.b:6.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
        - Weak DPs:
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
            ,cons,nil,recip,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {add#,dbl#,first#,half#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = 0        
             p(add) = 8        
            p(cons) = 1 + x1   
             p(dbl) = 4        
           p(first) = 1 + 2*x1 
            p(half) = 2*x1     
             p(nil) = 2        
           p(recip) = 1        
               p(s) = 4 + x1   
             p(sqr) = 14 + x1  
           p(terms) = 1 + x1   
            p(add#) = 2        
            p(dbl#) = 12 + 4*x1
          p(first#) = 0        
           p(half#) = 8        
            p(sqr#) = 4*x1     
          p(terms#) = 2 + 8*x1 
             p(c_1) = 0        
             p(c_2) = x1       
             p(c_3) = 8        
             p(c_4) = 12 + x1  
             p(c_5) = 4        
             p(c_6) = 1        
             p(c_7) = 0        
             p(c_8) = 0        
             p(c_9) = 4        
            p(c_10) = 0        
            p(c_11) = 0        
            p(c_12) = 0        
            p(c_13) = 1 + x1   
        
        Following rules are strictly oriented:
        dbl#(s(X)) = 28 + 4*X    
                   > 24 + 4*X    
                   = c_4(dbl#(X))
        
        
        Following rules are (at-least) weakly oriented:
        add#(s(X),Y) =  2                  
                     >= 2                  
                     =  c_2(add#(X,Y))     
        
          sqr#(s(X)) =  16 + 4*X           
                     >= 2                  
                     =  add#(sqr(X),dbl(X))
        
          sqr#(s(X)) =  16 + 4*X           
                     >= 12 + 4*X           
                     =  dbl#(X)            
        
          sqr#(s(X)) =  16 + 4*X           
                     >= 4*X                
                     =  sqr#(X)            
        
*** Step 1.b:6.b:3: NaturalPI WORST_CASE(?,O(n^2))
    + 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)) -> add#(sqr(X),dbl(X))
            sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
            ,cons,nil,recip,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any 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},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {add,dbl,sqr,add#,dbl#,first#,half#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = 0                
             p(add) = x1 + 4*x2        
            p(cons) = 1                
             p(dbl) = 2*x1             
           p(first) = 1 + x2 + x2^2    
            p(half) = 1 + 2*x1 + 2*x1^2
             p(nil) = 1                
           p(recip) = 1 + x1           
               p(s) = 2 + x1           
             p(sqr) = 2*x1 + 2*x1^2    
           p(terms) = 1                
            p(add#) = 5 + x1 + x2      
            p(dbl#) = x1^2             
          p(first#) = 4 + 2*x2         
           p(half#) = 1 + 2*x1^2       
            p(sqr#) = 6 + 2*x1^2       
          p(terms#) = 1 + x1 + x1^2    
             p(c_1) = 4                
             p(c_2) = x1               
             p(c_3) = 1                
             p(c_4) = x1               
             p(c_5) = 4                
             p(c_6) = 2                
             p(c_7) = 1                
             p(c_8) = 2                
             p(c_9) = 2                
            p(c_10) = 1 + x1           
            p(c_11) = 1                
            p(c_12) = x3               
            p(c_13) = 4                
        
        Following rules are strictly oriented:
        add#(s(X),Y) = 7 + X + Y     
                     > 5 + X + Y     
                     = c_2(add#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
         dbl#(s(X)) =  4 + 4*X + X^2        
                    >= X^2                  
                    =  c_4(dbl#(X))         
        
         sqr#(s(X)) =  14 + 8*X + 2*X^2     
                    >= 5 + 4*X + 2*X^2      
                    =  add#(sqr(X),dbl(X))  
        
         sqr#(s(X)) =  14 + 8*X + 2*X^2     
                    >= X^2                  
                    =  dbl#(X)              
        
         sqr#(s(X)) =  14 + 8*X + 2*X^2     
                    >= 6 + 2*X^2            
                    =  sqr#(X)              
        
         add(0(),X) =  4*X                  
                    >= X                    
                    =  X                    
        
        add(s(X),Y) =  2 + X + 4*Y          
                    >= 2 + X + 4*Y          
                    =  s(add(X,Y))          
        
           dbl(0()) =  0                    
                    >= 0                    
                    =  0()                  
        
          dbl(s(X)) =  4 + 2*X              
                    >= 4 + 2*X              
                    =  s(s(dbl(X)))         
        
           sqr(0()) =  0                    
                    >= 0                    
                    =  0()                  
        
          sqr(s(X)) =  12 + 10*X + 2*X^2    
                    >= 2 + 10*X + 2*X^2     
                    =  s(add(sqr(X),dbl(X)))
        
*** Step 1.b:6.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
            sqr#(s(X)) -> add#(sqr(X),dbl(X))
            sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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(Omega(n^1),O(n^3))