Problem Transformed CSR 04 Ex2 Luc02a Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex2 Luc02a Z

stdout:

MAYBE

Problem:
terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
terms(X) -> n__terms(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__terms(X)) -> terms(X)
activate(n__first(X1,X2)) -> first(X1,X2)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex2 Luc02a Z

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex2 Luc02a Z

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
     , sqr(0()) -> 0()
     , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
     , dbl(0()) -> 0()
     , dbl(s(X)) -> s(s(dbl(X)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
     , terms(X) -> n__terms(X)
     , first(X1, X2) -> n__first(X1, X2)
     , activate(n__terms(X)) -> terms(X)
     , activate(n__first(X1, X2)) -> first(X1, X2)
     , activate(X) -> X}

Proof Output:
  None of the processors succeeded.

  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
              , 10: terms^#(X) -> c_9()
              , 11: first^#(X1, X2) -> c_10()
              , 12: activate^#(n__terms(X)) -> c_11(terms^#(X))
              , 13: activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
              , 14: activate^#(X) -> c_13()}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{9,13}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^2))    ]
                |
                |->{11}                                                  [   YES(?,O(n^2))    ]
                |
                |->{12}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^2))    ]
                |   |   |
                |   |   `->{3}                                           [     inherited      ]
                |   |       |
                |   |       |->{6}                                       [         NA         ]
                |   |       |
                |   |       `->{7}                                       [     inherited      ]
                |   |           |
                |   |           `->{6}                                   [         NA         ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^2))    ]
                |
                `->{14}                                                  [   YES(?,O(n^3))    ]

             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]



         Sub-problems:
         -------------
           * Path {5}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                dbl^#(x1) = [0 1 0] x1 + [2]
                            [6 0 0]      [0]
                            [2 3 0]      [2]
                c_4(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]

           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                dbl^#(x1) = [2 2 2] x1 + [0]
                            [0 6 0]      [0]
                            [0 0 2]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]

           * Path {9,13}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{8}: YES(?,O(n^2))
             -------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 0 2] x2 + [2]
                               [0 0 0]      [0 0 4]      [2]
                               [0 0 0]      [0 0 1]      [0]
                s(x1) = [0 0 0] x1 + [1]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [0]
                      [2]
                n__first(x1, x2) = [1 3 0] x1 + [1 2 3] x2 + [2]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [2]
                first^#(x1, x2) = [2 0 0] x1 + [3 0 2] x2 + [0]
                                  [0 2 0]      [0 2 1]      [0]
                                  [0 0 2]      [2 2 0]      [0]
                c_7() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 2 2] x1 + [3]
                          [0 0 2]      [2]
                          [0 0 2]      [7]
                activate^#(x1) = [3 0 0] x1 + [0]
                                 [0 0 2]      [0]
                                 [0 0 2]      [2]
                c_12(x1) = [1 2 0] x1 + [2]
                           [0 0 0]      [3]
                           [0 0 0]      [2]

           * Path {9,13}->{11}: YES(?,O(n^2))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(X1, X2) -> c_10()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 4 0] x2 + [0]
                               [0 0 0]      [0 1 2]      [1]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [0]
                n__first(x1, x2) = [1 0 2] x1 + [1 1 0] x2 + [2]
                                   [0 1 0]      [0 1 0]      [0]
                                   [0 0 0]      [0 0 0]      [6]
                first^#(x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [1]
                                  [0 0 0]      [0 0 0]      [2]
                                  [0 0 0]      [1 4 0]      [0]
                c_8(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [2]
                          [0 0 0]      [3]
                activate^#(x1) = [0 1 2] x1 + [0]
                                 [0 0 2]      [0]
                                 [4 0 0]      [0]
                c_10() = [0]
                         [0]
                         [0]
                c_12(x1) = [1 2 0] x1 + [5]
                           [0 2 0]      [7]
                           [0 0 0]      [7]

           * Path {9,13}->{12}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{2}: YES(?,O(n^2))
             ------------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N))
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                n__terms(x1) = [1 2 2] x1 + [0]
                               [0 1 2]      [2]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [2]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                n__first(x1, x2) = [1 2 2] x1 + [1 0 2] x2 + [0]
                                   [0 1 2]      [0 0 2]      [0]
                                   [0 0 0]      [0 0 0]      [3]
                terms^#(x1) = [4 4 4] x1 + [0]
                              [0 2 3]      [2]
                              [0 2 1]      [2]
                c_0(x1) = [2 0 0] x1 + [0]
                          [0 0 0]      [2]
                          [0 0 0]      [2]
                sqr^#(x1) = [2 2 2] x1 + [0]
                            [2 2 2]      [0]
                            [0 2 0]      [0]
                c_1() = [1]
                        [0]
                        [0]
                first^#(x1, x2) = [0 4 2] x1 + [6 0 0] x2 + [0]
                                  [1 2 3]      [0 0 2]      [2]
                                  [2 3 1]      [7 0 0]      [0]
                c_8(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [6]
                          [0 2 0]      [7]
                activate^#(x1) = [6 0 0] x1 + [4]
                                 [3 3 2]      [2]
                                 [3 3 2]      [4]
                c_11(x1) = [1 2 0] x1 + [0]
                           [0 0 2]      [3]
                           [0 2 2]      [1]
                c_12(x1) = [1 2 0] x1 + [0]
                           [0 2 0]      [2]
                           [0 2 0]      [3]

           * Path {9,13}->{12}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{6}: NA
             ------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{1}->{3}->{7}: inherited
             -------------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{7}->{6}: NA
             -----------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{10}: YES(?,O(n^2))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {terms^#(X) -> c_9()}
               Weak Rules:
                 {  activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}, Uargs(activate^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 0 2] x2 + [0]
                               [0 0 0]      [0 1 0]      [2]
                               [0 0 0]      [0 0 0]      [0]
                n__terms(x1) = [1 3 3] x1 + [2]
                               [0 1 2]      [2]
                               [0 0 0]      [2]
                s(x1) = [0 0 0] x1 + [2]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                n__first(x1, x2) = [1 4 0] x1 + [1 2 2] x2 + [2]
                                   [0 0 2]      [0 1 2]      [0]
                                   [0 0 0]      [0 0 0]      [2]
                terms^#(x1) = [0 1 2] x1 + [2]
                              [0 0 0]      [2]
                              [0 3 0]      [2]
                first^#(x1, x2) = [0 0 0] x1 + [2 3 0] x2 + [0]
                                  [0 0 0]      [6 5 0]      [0]
                                  [2 2 2]      [4 1 0]      [0]
                c_8(x1) = [1 0 0] x1 + [2]
                          [0 0 2]      [6]
                          [0 0 0]      [2]
                activate^#(x1) = [2 3 2] x1 + [0]
                                 [3 0 2]      [0]
                                 [2 2 2]      [0]
                c_9() = [1]
                        [0]
                        [0]
                c_11(x1) = [2 2 2] x1 + [2]
                           [0 0 0]      [6]
                           [0 2 2]      [3]
                c_12(x1) = [1 0 0] x1 + [6]
                           [0 0 0]      [7]
                           [0 0 0]      [6]

           * Path {9,13}->{14}: YES(?,O(n^3))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13() = [0]
                         [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_13()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 1] x2 + [0]
                               [0 0 0]      [0 1 4]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [4]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                n__first(x1, x2) = [1 0 0] x1 + [1 0 2] x2 + [2]
                                   [0 0 2]      [0 1 0]      [0]
                                   [0 0 0]      [0 0 1]      [0]
                first^#(x1, x2) = [1 0 0] x1 + [4 1 2] x2 + [0]
                                  [1 0 2]      [0 2 0]      [0]
                                  [0 0 0]      [0 0 2]      [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
                activate^#(x1) = [4 1 0] x1 + [4]
                                 [0 0 4]      [4]
                                 [2 6 0]      [4]
                c_12(x1) = [1 0 3] x1 + [6]
                           [0 0 2]      [3]
                           [0 2 0]      [7]
                c_13() = [1]
                         [0]
                         [0]

    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
              , 10: terms^#(X) -> c_9()
              , 11: first^#(X1, X2) -> c_10()
              , 12: activate^#(n__terms(X)) -> c_11(terms^#(X))
              , 13: activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
              , 14: activate^#(X) -> c_13()}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{9,13}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^2))    ]
                |
                |->{11}                                                  [   YES(?,O(n^1))    ]
                |
                |->{12}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^2))    ]
                |   |   |
                |   |   `->{3}                                           [     inherited      ]
                |   |       |
                |   |       |->{6}                                       [       MAYBE        ]
                |   |       |
                |   |       `->{7}                                       [     inherited      ]
                |   |           |
                |   |           `->{6}                                   [         NA         ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^2))    ]
                |
                `->{14}                                                  [   YES(?,O(n^1))    ]

             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {5}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                dbl^#(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 0]      [0]

           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                dbl^#(x1) = [1 2] x1 + [2]
                            [6 1]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [5]
                          [2 0]      [3]

           * Path {9,13}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{8}: YES(?,O(n^2))
             -------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [2]
                0() = [2]
                      [0]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [3]
                                   [0 1]      [0 1]      [2]
                first^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                                  [0 2]      [2 0]      [0]
                c_7() = [1]
                        [0]
                c_8(x1) = [1 0] x1 + [3]
                          [0 2]      [4]
                activate^#(x1) = [2 1] x1 + [0]
                                 [0 0]      [2]
                c_12(x1) = [1 0] x1 + [7]
                           [0 0]      [2]

           * Path {9,13}->{11}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(X1, X2) -> c_10()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                n__first(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
                                   [0 0]      [0 1]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                                  [0 0]      [4 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [1]
                                 [1 1]      [0]
                c_10() = [0]
                         [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 0]      [0]

           * Path {9,13}->{12}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{2}: YES(?,O(n^2))
             ------------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N))
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [0]
                n__terms(x1) = [1 2] x1 + [0]
                               [0 0]      [2]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [2]
                0() = [2]
                      [0]
                n__first(x1, x2) = [1 1] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 0]      [0]
                terms^#(x1) = [4 4] x1 + [0]
                              [0 2]      [0]
                c_0(x1) = [2 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [2 0] x1 + [0]
                            [2 0]      [0]
                c_1() = [1]
                        [0]
                first^#(x1, x2) = [2 2] x1 + [6 5] x2 + [0]
                                  [2 2]      [0 3]      [2]
                c_8(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
                activate^#(x1) = [6 3] x1 + [0]
                                 [4 0]      [0]
                c_11(x1) = [1 0] x1 + [3]
                           [0 2]      [0]
                c_12(x1) = [1 1] x1 + [5]
                           [0 2]      [3]

           * Path {9,13}->{12}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{6}: MAYBE
             ---------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
                  , add^#(0(), X) -> c_5()
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,13}->{12}->{1}->{3}->{7}: inherited
             -------------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{7}->{6}: NA
             -----------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{10}: YES(?,O(n^2))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {terms^#(X) -> c_9()}
               Weak Rules:
                 {  activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}, Uargs(activate^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__terms(x1) = [1 2] x1 + [2]
                               [0 1]      [2]
                s(x1) = [0 0] x1 + [4]
                        [0 0]      [0]
                n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                                   [0 0]      [0 0]      [2]
                terms^#(x1) = [0 2] x1 + [2]
                              [0 2]      [0]
                first^#(x1, x2) = [1 0] x1 + [3 0] x2 + [0]
                                  [0 0]      [0 0]      [4]
                c_8(x1) = [1 0] x1 + [2]
                          [0 0]      [3]
                activate^#(x1) = [3 2] x1 + [0]
                                 [2 5]      [0]
                c_9() = [1]
                        [0]
                c_11(x1) = [2 2] x1 + [3]
                           [2 2]      [7]
                c_12(x1) = [1 0] x1 + [7]
                           [0 0]      [7]

           * Path {9,13}->{14}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13() = [0]
                         [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_13()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [0]
                n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                   [0 1]      [0 1]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                                  [2 0]      [1 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 0]      [3]
                activate^#(x1) = [0 0] x1 + [1]
                                 [0 1]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 0]      [0]
                c_13() = [0]
                         [0]

    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
              , 10: terms^#(X) -> c_9()
              , 11: first^#(X1, X2) -> c_10()
              , 12: activate^#(n__terms(X)) -> c_11(terms^#(X))
              , 13: activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
              , 14: activate^#(X) -> c_13()}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{9,13}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [   YES(?,O(n^1))    ]
                |
                |->{12}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^1))    ]
                |   |   |
                |   |   `->{3}                                           [     inherited      ]
                |   |       |
                |   |       |->{6}                                       [       MAYBE        ]
                |   |       |
                |   |       `->{7}                                       [     inherited      ]
                |   |           |
                |   |           `->{6}                                   [         NA         ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^1))    ]
                |
                `->{14}                                                  [   YES(?,O(n^1))    ]

             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {5}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                dbl^#(x1) = [2] x1 + [0]
                c_4(x1) = [1] x1 + [7]

           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                dbl^#(x1) = [2] x1 + [0]
                c_3() = [1]
                c_4(x1) = [1] x1 + [0]

           * Path {9,13}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{8}: YES(?,O(n^1))
             -------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                s(x1) = [0] x1 + [0]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                first^#(x1, x2) = [2] x1 + [4] x2 + [0]
                c_7() = [1]
                c_8(x1) = [1] x1 + [6]
                activate^#(x1) = [4] x1 + [0]
                c_12(x1) = [1] x1 + [0]

           * Path {9,13}->{11}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(X1, X2) -> c_10()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [2] x1 + [4] x2 + [4]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [4] x1 + [0]
                c_10() = [1]
                c_12(x1) = [1] x1 + [2]

           * Path {9,13}->{12}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{2}: YES(?,O(n^1))
             ------------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N))
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [3]
                n__terms(x1) = [1] x1 + [0]
                s(x1) = [0] x1 + [5]
                0() = [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [4]
                terms^#(x1) = [0] x1 + [2]
                c_0(x1) = [1] x1 + [0]
                sqr^#(x1) = [0] x1 + [2]
                c_1() = [1]
                first^#(x1, x2) = [1] x1 + [1] x2 + [0]
                c_8(x1) = [1] x1 + [3]
                activate^#(x1) = [1] x1 + [4]
                c_11(x1) = [1] x1 + [1]
                c_12(x1) = [1] x1 + [7]

           * Path {9,13}->{12}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{6}: MAYBE
             ---------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
                  , add^#(0(), X) -> c_5()
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,13}->{12}->{1}->{3}->{7}: inherited
             -------------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{7}->{6}: NA
             -----------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{10}: YES(?,O(n^1))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {terms^#(X) -> c_9()}
               Weak Rules:
                 {  activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}, Uargs(activate^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                n__terms(x1) = [1] x1 + [0]
                s(x1) = [0] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [4]
                terms^#(x1) = [1] x1 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1) = [1] x1 + [3]
                activate^#(x1) = [2] x1 + [4]
                c_9() = [1]
                c_11(x1) = [2] x1 + [0]
                c_12(x1) = [1] x1 + [2]

           * Path {9,13}->{14}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(activate^#) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13() = [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_13()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                s(x1) = [0] x1 + [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                first^#(x1, x2) = [0] x1 + [2] x2 + [0]
                c_8(x1) = [1] x1 + [0]
                activate^#(x1) = [2] x1 + [2]
                c_12(x1) = [1] x1 + [2]
                c_13() = [1]

    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible

    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex2 Luc02a Z

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex2 Luc02a Z

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
     , sqr(0()) -> 0()
     , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
     , dbl(0()) -> 0()
     , dbl(s(X)) -> s(s(dbl(X)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
     , terms(X) -> n__terms(X)
     , first(X1, X2) -> n__first(X1, X2)
     , activate(n__terms(X)) -> terms(X)
     , activate(n__first(X1, X2)) -> first(X1, X2)
     , activate(X) -> X}

Proof Output:
  None of the processors succeeded.

  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: terms^#(N) -> c_0(sqr^#(N), N)
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
              , 10: terms^#(X) -> c_9(X)
              , 11: first^#(X1, X2) -> c_10(X1, X2)
              , 12: activate^#(n__terms(X)) -> c_11(terms^#(X))
              , 13: activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
              , 14: activate^#(X) -> c_13(X)}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{9,13}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^2))    ]
                |
                |->{11}                                                  [   YES(?,O(n^2))    ]
                |
                |->{12}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^2))    ]
                |   |   |
                |   |   `->{3}                                           [     inherited      ]
                |   |       |
                |   |       |->{6}                                       [         NA         ]
                |   |       |
                |   |       `->{7}                                       [     inherited      ]
                |   |           |
                |   |           `->{6}                                   [         NA         ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^1))    ]
                |
                `->{14}                                                  [   YES(?,O(n^2))    ]

             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]



         Sub-problems:
         -------------
           * Path {5}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                dbl^#(x1) = [0 1 0] x1 + [2]
                            [6 0 0]      [0]
                            [2 3 0]      [2]
                c_4(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]

           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                dbl^#(x1) = [2 2 2] x1 + [0]
                            [0 6 0]      [0]
                            [0 0 2]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]

           * Path {9,13}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{8}: YES(?,O(n^2))
             -------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 5 4] x2 + [0]
                               [0 0 2]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 2 2] x1 + [3]
                        [0 0 2]      [0]
                        [0 0 0]      [1]
                0() = [2]
                      [2]
                      [2]
                n__first(x1, x2) = [1 0 4] x1 + [1 2 0] x2 + [2]
                                   [0 0 0]      [0 1 0]      [2]
                                   [0 0 0]      [0 0 0]      [3]
                first^#(x1, x2) = [2 0 2] x1 + [2 0 0] x2 + [0]
                                  [2 2 2]      [3 2 0]      [0]
                                  [2 0 2]      [0 0 0]      [0]
                c_7() = [1]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 2 2] x3 + [0]
                                  [1 0 0]      [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [0 0 2]      [3]
                activate^#(x1) = [2 1 2] x1 + [0]
                                 [0 0 2]      [2]
                                 [0 0 0]      [2]
                c_12(x1) = [1 0 0] x1 + [6]
                           [0 0 0]      [6]
                           [0 0 0]      [2]

           * Path {9,13}->{11}: YES(?,O(n^2))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [3 3 3] x1 + [3 3 3] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [1 1 1] x1 + [1 1 1] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(X1, X2) -> c_10(X1, X2)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 2 2] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [2]
                n__first(x1, x2) = [1 4 0] x1 + [1 2 2] x2 + [2]
                                   [0 0 2]      [0 0 2]      [2]
                                   [0 0 0]      [0 0 0]      [0]
                first^#(x1, x2) = [2 2 0] x1 + [2 4 0] x2 + [2]
                                  [2 2 0]      [2 0 0]      [0]
                                  [2 2 2]      [0 4 0]      [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 2 0] x3 + [7]
                                  [0 0 0]      [0 0 0]      [0 0 2]      [2]
                                  [0 0 0]      [0 0 0]      [0 0 2]      [7]
                activate^#(x1) = [2 0 0] x1 + [0]
                                 [0 2 0]      [0]
                                 [0 0 0]      [2]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [1]
                           [0 0 0]      [4]
                           [0 0 0]      [2]

           * Path {9,13}->{12}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{2}: YES(?,O(n^2))
             ------------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
                              [0 1 0]      [0 0 0]      [0]
                              [0 0 1]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), N)
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(first^#) = {}, Uargs(c_8) = {3},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 2 2] x2 + [0]
                               [0 0 0]      [0 1 2]      [0]
                               [0 0 0]      [0 0 0]      [2]
                n__terms(x1) = [1 2 2] x1 + [2]
                               [0 1 1]      [0]
                               [0 0 0]      [0]
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [2]
                n__first(x1, x2) = [1 2 2] x1 + [1 2 2] x2 + [0]
                                   [0 1 0]      [0 1 0]      [0]
                                   [0 0 0]      [0 0 0]      [2]
                terms^#(x1) = [0 4 4] x1 + [0]
                              [0 0 0]      [2]
                              [4 0 4]      [0]
                c_0(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [2]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 2 2] x1 + [0]
                            [0 2 2]      [0]
                            [0 0 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                first^#(x1, x2) = [2 0 0] x1 + [2 2 4] x2 + [0]
                                  [0 0 0]      [3 3 0]      [0]
                                  [0 0 0]      [6 1 2]      [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [3]
                                  [1 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 2 0]      [2]
                activate^#(x1) = [2 5 0] x1 + [4]
                                 [2 6 0]      [0]
                                 [2 5 2]      [0]
                c_11(x1) = [2 0 0] x1 + [7]
                           [0 0 0]      [3]
                           [0 2 0]      [0]
                c_12(x1) = [1 0 0] x1 + [3]
                           [0 0 0]      [0]
                           [0 0 0]      [3]

           * Path {9,13}->{12}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{6}: NA
             ------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{1}->{3}->{7}: inherited
             -------------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{7}->{6}: NA
             -----------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{10}: YES(?,O(n^1))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [3 3 3] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                activate^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {terms^#(X) -> c_9(X)}
               Weak Rules:
                 {  activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0 0] x1 + [1 3 2] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                n__terms(x1) = [1 2 0] x1 + [3]
                               [0 0 0]      [4]
                               [0 0 0]      [6]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [4]
                        [0 0 0]      [0]
                n__first(x1, x2) = [1 2 0] x1 + [1 0 2] x2 + [0]
                                   [0 0 0]      [0 0 4]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [4]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [2 0 0] x2 + [0]
                                  [0 0 0]      [0 0 2]      [0]
                                  [0 1 0]      [3 0 2]      [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 2] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [1 0 0]      [0 0 0]      [0 0 0]      [3]
                activate^#(x1) = [2 0 1] x1 + [0]
                                 [4 0 0]      [0]
                                 [0 2 0]      [0]
                c_9(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_11(x1) = [2 0 0] x1 + [3]
                           [0 0 0]      [7]
                           [0 0 0]      [7]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 3 1]      [0]
                           [0 0 0]      [0]

           * Path {9,13}->{14}: YES(?,O(n^2))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                n__terms(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                n__first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                activate^#(x1) = [3 3 3] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_13(x1) = [1 1 1] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_13(X)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 4 4] x2 + [0]
                               [0 0 2]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 2 2] x1 + [2]
                        [0 0 2]      [4]
                        [0 0 0]      [2]
                n__first(x1, x2) = [1 0 2] x1 + [1 2 2] x2 + [2]
                                   [0 1 0]      [0 1 1]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                first^#(x1, x2) = [2 2 0] x1 + [2 2 0] x2 + [0]
                                  [2 2 0]      [2 2 0]      [0]
                                  [2 0 2]      [2 0 0]      [0]
                c_8(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [6]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [6]
                activate^#(x1) = [2 5 0] x1 + [4]
                                 [4 0 0]      [0]
                                 [4 0 0]      [0]
                c_12(x1) = [1 0 0] x1 + [7]
                           [0 0 0]      [7]
                           [0 0 0]      [0]
                c_13(x1) = [0 1 0] x1 + [1]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: terms^#(N) -> c_0(sqr^#(N), N)
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
              , 10: terms^#(X) -> c_9(X)
              , 11: first^#(X1, X2) -> c_10(X1, X2)
              , 12: activate^#(n__terms(X)) -> c_11(terms^#(X))
              , 13: activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
              , 14: activate^#(X) -> c_13(X)}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{9,13}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^2))    ]
                |
                |->{11}                                                  [   YES(?,O(n^1))    ]
                |
                |->{12}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^2))    ]
                |   |   |
                |   |   `->{3}                                           [     inherited      ]
                |   |       |
                |   |       |->{6}                                       [       MAYBE        ]
                |   |       |
                |   |       `->{7}                                       [     inherited      ]
                |   |           |
                |   |           `->{6}                                   [         NA         ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^2))    ]
                |
                `->{14}                                                  [   YES(?,O(n^2))    ]

             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {5}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                dbl^#(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 0]      [0]

           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                dbl^#(x1) = [1 2] x1 + [2]
                            [6 1]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [5]
                          [2 0]      [3]

           * Path {9,13}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{8}: YES(?,O(n^2))
             -------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 3] x1 + [1 2] x2 + [0]
                               [0 1]      [0 0]      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 0]      [2]
                0() = [0]
                      [2]
                n__first(x1, x2) = [1 2] x1 + [1 0] x2 + [4]
                                   [0 1]      [0 1]      [0]
                first^#(x1, x2) = [2 0] x1 + [2 0] x2 + [4]
                                  [2 2]      [4 2]      [0]
                c_7() = [1]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 2] x3 + [4]
                                  [0 0]      [0 0]      [0 2]      [7]
                activate^#(x1) = [2 1] x1 + [0]
                                 [0 1]      [2]
                c_12(x1) = [1 0] x1 + [3]
                           [0 0]      [2]

           * Path {9,13}->{11}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(X1, X2) -> c_10(X1, X2)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [3]
                        [0 0]      [0]
                n__first(x1, x2) = [1 4] x1 + [1 0] x2 + [2]
                                   [0 0]      [0 0]      [2]
                first^#(x1, x2) = [2 0] x1 + [2 0] x2 + [4]
                                  [3 0]      [6 0]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [2]
                                  [0 0]      [1 0]      [2 2]      [3]
                activate^#(x1) = [2 2] x1 + [0]
                                 [1 2]      [2]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                               [1 0]      [0 0]      [0]
                c_12(x1) = [1 0] x1 + [3]
                           [0 0]      [6]

           * Path {9,13}->{12}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{2}: YES(?,O(n^2))
             ------------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                              [0 1]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), N)
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(first^#) = {}, Uargs(c_8) = {3},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                               [0 0]      [0 1]      [1]
                n__terms(x1) = [1 2] x1 + [2]
                               [0 1]      [2]
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [2]
                      [2]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [2]
                terms^#(x1) = [0 2] x1 + [6]
                              [0 2]      [0]
                c_0(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 2] x1 + [4]
                            [2 2]      [0]
                c_1() = [1]
                        [0]
                first^#(x1, x2) = [3 0] x1 + [3 5] x2 + [0]
                                  [0 0]      [6 0]      [0]
                c_8(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [2]
                                  [0 0]      [0 0]      [2 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [3 3]      [0]
                c_11(x1) = [2 2] x1 + [0]
                           [2 0]      [0]
                c_12(x1) = [1 0] x1 + [6]
                           [0 0]      [7]

           * Path {9,13}->{12}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{6}: MAYBE
             ---------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N), N)
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
                  , add^#(0(), X) -> c_5(X)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,13}->{12}->{1}->{3}->{7}: inherited
             -------------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{7}->{6}: NA
             -----------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{10}: YES(?,O(n^2))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [3 3] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {terms^#(X) -> c_9(X)}
               Weak Rules:
                 {  activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 0]      [2]
                n__terms(x1) = [1 2] x1 + [2]
                               [0 1]      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                n__first(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                                   [0 0]      [0 1]      [2]
                terms^#(x1) = [0 1] x1 + [2]
                              [0 2]      [2]
                first^#(x1, x2) = [2 0] x1 + [4 4] x2 + [0]
                                  [1 0]      [0 2]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [6]
                                  [0 0]      [1 0]      [0 0]      [3]
                activate^#(x1) = [4 3] x1 + [0]
                                 [3 3]      [0]
                c_9(x1) = [0 1] x1 + [1]
                          [0 0]      [0]
                c_11(x1) = [4 2] x1 + [2]
                           [2 2]      [4]
                c_12(x1) = [1 2] x1 + [7]
                           [0 0]      [7]

           * Path {9,13}->{14}: YES(?,O(n^2))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__terms(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                n__first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_13(x1) = [1 1] x1 + [0]
                           [0 0]      [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_13(X)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 4] x2 + [0]
                               [0 0]      [0 0]      [2]
                s(x1) = [1 7] x1 + [2]
                        [0 0]      [2]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                                   [0 1]      [0 1]      [0]
                first^#(x1, x2) = [0 2] x1 + [2 2] x2 + [0]
                                  [2 2]      [0 0]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [5]
                                  [0 0]      [0 0]      [0 0]      [0]
                activate^#(x1) = [2 5] x1 + [2]
                                 [4 1]      [4]
                c_12(x1) = [1 0] x1 + [2]
                           [2 2]      [2]
                c_13(x1) = [0 1] x1 + [1]
                           [0 1]      [0]

    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:

             {  1: terms^#(N) -> c_0(sqr^#(N), N)
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
              , 10: terms^#(X) -> c_9(X)
              , 11: first^#(X1, X2) -> c_10(X1, X2)
              , 12: activate^#(n__terms(X)) -> c_11(terms^#(X))
              , 13: activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
              , 14: activate^#(X) -> c_13(X)}

           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)

             ->{9,13}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [   YES(?,O(n^1))    ]
                |
                |->{12}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^1))    ]
                |   |   |
                |   |   `->{3}                                           [     inherited      ]
                |   |       |
                |   |       |->{6}                                       [       MAYBE        ]
                |   |       |
                |   |       `->{7}                                       [     inherited      ]
                |   |           |
                |   |           `->{6}                                   [         NA         ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^1))    ]
                |
                `->{14}                                                  [   YES(?,O(n^1))    ]

             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {5}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                dbl^#(x1) = [2] x1 + [0]
                c_4(x1) = [1] x1 + [7]

           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                dbl^#(x1) = [2] x1 + [0]
                c_3() = [1]
                c_4(x1) = [1] x1 + [0]

           * Path {9,13}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{8}: YES(?,O(n^1))
             -------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                s(x1) = [1] x1 + [0]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                first^#(x1, x2) = [2] x1 + [4] x2 + [0]
                c_7() = [1]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [6]
                activate^#(x1) = [4] x1 + [0]
                c_12(x1) = [1] x1 + [0]

           * Path {9,13}->{11}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1, x2) = [1] x1 + [1] x2 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(X1, X2) -> c_10(X1, X2)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [2] x1 + [4] x2 + [2]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [3]
                activate^#(x1) = [4] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [1]
                c_12(x1) = [1] x1 + [5]

           * Path {9,13}->{12}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{2}: YES(?,O(n^1))
             ------------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [1] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_11(x1) = [1] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), N)
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(first^#) = {}, Uargs(c_8) = {3},
                 Uargs(activate^#) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                n__terms(x1) = [1] x1 + [0]
                s(x1) = [1] x1 + [2]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                terms^#(x1) = [2] x1 + [4]
                c_0(x1, x2) = [1] x1 + [0] x2 + [0]
                sqr^#(x1) = [2] x1 + [4]
                c_1() = [1]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [4]
                activate^#(x1) = [2] x1 + [4]
                c_11(x1) = [1] x1 + [0]
                c_12(x1) = [1] x1 + [3]

           * Path {9,13}->{12}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{6}: MAYBE
             ---------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N), N)
                  , activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))
                  , add^#(0(), X) -> c_5(X)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,13}->{12}->{1}->{3}->{7}: inherited
             -------------------------------------------

             This path is subsumed by the proof of path {9,13}->{12}->{1}->{3}->{7}->{6}.

           * Path {9,13}->{12}->{1}->{3}->{7}->{6}: NA
             -----------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(s(dbl(X)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(add(X, Y))}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

             We have not generated a proof for the resulting sub-problem.

           * Path {9,13}->{12}->{10}: YES(?,O(n^1))
             --------------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {1}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9(x1) = [1] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_11(x1) = [1] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13(x1) = [0] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {terms^#(X) -> c_9(X)}
               Weak Rules:
                 {  activate^#(n__terms(X)) -> c_11(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(n__first) = {}, Uargs(terms^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_11) = {1}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                n__terms(x1) = [1] x1 + [2]
                s(x1) = [1] x1 + [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [4]
                terms^#(x1) = [1] x1 + [2]
                first^#(x1, x2) = [0] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [3]
                activate^#(x1) = [2] x1 + [0]
                c_9(x1) = [1] x1 + [1]
                c_11(x1) = [2] x1 + [0]
                c_12(x1) = [1] x1 + [7]

           * Path {9,13}->{14}: YES(?,O(n^1))
             --------------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(n__terms) = {}, Uargs(s) = {},
                 Uargs(add) = {}, Uargs(dbl) = {}, Uargs(first) = {},
                 Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}, Uargs(c_11) = {}, Uargs(c_12) = {1},
                 Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                n__terms(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [1] x1 + [0]
                c_13(x1) = [1] x1 + [0]

             We apply the sub-processor on the resulting sub-problem:

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_13(X)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(Y, X, activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_12(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {3}, Uargs(activate^#) = {},
                 Uargs(c_12) = {1}, Uargs(c_13) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [0] x1 + [2] x2 + [2]
                c_8(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [1]
                activate^#(x1) = [2] x1 + [1]
                c_12(x1) = [1] x1 + [3]
                c_13(x1) = [0] x1 + [0]

    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible

    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.

    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.