Problem Transformed CSR 04 Ex26 Luc03b Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex26 Luc03b Z

stdout:

MAYBE

Problem:
terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
sqr(0()) -> 0()
sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
dbl(0()) -> 0()
dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
terms(X) -> n__terms(X)
add(X1,X2) -> n__add(X1,X2)
s(X) -> n__s(X)
dbl(X) -> n__dbl(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__terms(X)) -> terms(X)
activate(n__add(X1,X2)) -> add(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__dbl(X)) -> dbl(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 Ex26 Luc03b Z

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex26 Luc03b 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(n__add(sqr(activate(X)), dbl(activate(X))))
     , dbl(0()) -> 0()
     , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(n__add(activate(X), Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) ->
       cons(Y, n__first(activate(X), activate(Z)))
     , terms(X) -> n__terms(X)
     , add(X1, X2) -> n__add(X1, X2)
     , s(X) -> n__s(X)
     , dbl(X) -> n__dbl(X)
     , first(X1, X2) -> n__first(X1, X2)
     , activate(n__terms(X)) -> terms(X)
     , activate(n__add(X1, X2)) -> add(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__dbl(X)) -> dbl(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), s^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) ->
                   c_2(s^#(n__add(sqr(activate(X)), dbl(activate(X)))))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(s^#(n__s(n__dbl(activate(X)))))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
              , 10: terms^#(X) -> c_9()
              , 11: add^#(X1, X2) -> c_10()
              , 12: s^#(X) -> c_11()
              , 13: dbl^#(X) -> c_12()
              , 14: first^#(X1, X2) -> c_13()
              , 15: activate^#(n__terms(X)) -> c_14(terms^#(X))
              , 16: activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
              , 17: activate^#(n__s(X)) -> c_16(s^#(X))
              , 18: activate^#(n__dbl(X)) -> c_17(dbl^#(X))
              , 19: activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
              , 20: activate^#(X) -> c_19()}

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

             ->{9,19}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^1))    ]
                |
                |->{14}                                                  [   YES(?,O(n^1))    ]
                |
                |->{15}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^2))    ]
                |   |   |
                |   |   |->{3}                                           [     inherited      ]
                |   |   |   |
                |   |   |   `->{12}                                      [         NA         ]
                |   |   |
                |   |   `->{12}                                          [   YES(?,O(n^2))    ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^2))    ]
                |
                |->{16}                                                  [     inherited      ]
                |   |
                |   |->{6}                                               [   YES(?,O(n^2))    ]
                |   |
                |   |->{7}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [       MAYBE        ]
                |   |
                |   `->{11}                                              [   YES(?,O(n^1))    ]
                |
                |->{17}                                                  [   YES(?,O(n^2))    ]
                |   |
                |   `->{12}                                              [   YES(?,O(n^2))    ]
                |
                |->{18}                                                  [     inherited      ]
                |   |
                |   |->{4}                                               [   YES(?,O(n^2))    ]
                |   |
                |   |->{5}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [         NA         ]
                |   |
                |   `->{13}                                              [   YES(?,O(n^2))    ]
                |
                `->{20}                                                  [   YES(?,O(n^2))    ]



         Sub-problems:
         -------------
           * Path {9,19}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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^#(0(), X) -> c_7()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1, 2}, Uargs(activate^#) = {},
                 Uargs(c_18) = {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) = [2 2] x1 + [4]
                        [2 2]      [0]
                0() = [2]
                      [2]
                n__first(x1, x2) = [1 4] x1 + [1 2] x2 + [2]
                                   [0 0]      [0 0]      [2]
                first^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                                  [2 0]      [4 0]      [0]
                c_7() = [1]
                        [0]
                c_8(x1, x2) = [2 2] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [7]
                activate^#(x1) = [2 2] x1 + [0]
                                 [0 2]      [0]
                c_18(x1) = [1 0] x1 + [7]
                           [0 0]      [3]

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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_13()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1, 2}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [2 4] x1 + [0]
                        [2 2]      [2]
                n__first(x1, x2) = [1 5] x1 + [1 0] x2 + [2]
                                   [0 0]      [0 1]      [2]
                first^#(x1, x2) = [2 2] x1 + [2 0] x2 + [4]
                                  [2 0]      [4 0]      [0]
                c_8(x1, x2) = [2 2] x1 + [1 0] x2 + [6]
                              [0 0]      [0 0]      [0]
                activate^#(x1) = [2 2] x1 + [0]
                                 [2 3]      [0]
                c_13() = [1]
                         [0]
                c_18(x1) = [1 0] x1 + [3]
                           [0 0]      [7]

           * Path {9,19}->{15}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 1]      [0]
                n__terms(x1) = [1 1] x1 + [1]
                               [0 1]      [0]
                s(x1) = [5 4] x1 + [2]
                        [2 0]      [2]
                0() = [2]
                      [2]
                n__first(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                                   [0 1]      [0 0]      [0]
                terms^#(x1) = [4 4] x1 + [0]
                              [0 2]      [0]
                c_0(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [2 2] x1 + [0]
                            [2 2]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [1]
                        [0]
                first^#(x1, x2) = [2 2] x1 + [4 0] x2 + [0]
                                  [2 2]      [2 5]      [0]
                c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [6]
                              [0 2]      [0 0]      [3]
                activate^#(x1) = [4 5] x1 + [0]
                                 [7 0]      [2]
                c_14(x1) = [1 2] x1 + [2]
                           [0 0]      [6]
                c_18(x1) = [1 0] x1 + [0]
                           [0 0]      [2]

           * Path {9,19}->{15}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}->{3}->{12}: NA
             -------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , dbl(X) -> n__dbl(X)
                , activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , first(X1, X2) -> n__first(X1, X2)}

             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 {9,19}->{15}->{1}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {2}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {s^#(X) -> c_11()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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) = {2},
                 Uargs(sqr^#) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__terms(x1) = [1 2] x1 + [2]
                               [0 1]      [2]
                s(x1) = [2 2] x1 + [4]
                        [2 2]      [0]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [2]
                terms^#(x1) = [0 0] x1 + [6]
                              [4 4]      [0]
                c_0(x1, x2) = [0 0] x1 + [4 0] x2 + [2]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s^#(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
                                  [2 2]      [4 0]      [0]
                c_8(x1, x2) = [2 2] x1 + [1 0] x2 + [4]
                              [0 0]      [0 0]      [7]
                activate^#(x1) = [3 3] x1 + [0]
                                 [3 3]      [0]
                c_11() = [0]
                         [0]
                c_14(x1) = [2 0] x1 + [0]
                           [2 0]      [0]
                c_18(x1) = [1 0] x1 + [7]
                           [0 0]      [7]

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {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]      [1]
                n__terms(x1) = [1 2] x1 + [2]
                               [0 1]      [2]
                s(x1) = [4 2] x1 + [2]
                        [0 0]      [0]
                n__first(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [2]
                terms^#(x1) = [0 2] x1 + [4]
                              [0 2]      [0]
                first^#(x1, x2) = [2 0] x1 + [3 0] x2 + [0]
                                  [0 0]      [0 2]      [6]
                c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
                              [0 0]      [0 0]      [7]
                activate^#(x1) = [3 3] x1 + [0]
                                 [2 5]      [0]
                c_9() = [1]
                        [0]
                c_14(x1) = [2 2] x1 + [3]
                           [2 2]      [2]
                c_18(x1) = [1 2] x1 + [0]
                           [0 2]      [2]

           * Path {9,19}->{16}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{6}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {add^#(0(), X) -> c_5()}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_15) = {1},
                 Uargs(c_18) = {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]      [3]
                s(x1) = [0 2] x1 + [2]
                        [2 0]      [0]
                0() = [2]
                      [2]
                n__add(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                 [0 1]      [0 0]      [0]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [0]
                add^#(x1, x2) = [0 2] x1 + [0 0] x2 + [2]
                                [2 2]      [0 0]      [2]
                c_5() = [1]
                        [0]
                first^#(x1, x2) = [2 6] x1 + [6 1] x2 + [0]
                                  [1 0]      [0 2]      [2]
                c_8(x1, x2) = [2 0] x1 + [1 0] x2 + [2]
                              [0 0]      [0 0]      [6]
                activate^#(x1) = [6 1] x1 + [0]
                                 [6 1]      [0]
                c_15(x1) = [4 2] x1 + [0]
                           [2 2]      [0]
                c_18(x1) = [1 2] x1 + [6]
                           [0 0]      [7]

           * Path {9,19}->{16}->{7}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{7}->{12}: MAYBE
             -----------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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:
                 {  add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
                  , activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
                  , s^#(X) -> c_11()
                  , activate(n__terms(X)) -> terms(X)
                  , activate(n__add(X1, X2)) -> add(X1, X2)
                  , activate(n__s(X)) -> s(X)
                  , activate(n__dbl(X)) -> dbl(X)
                  , activate(n__first(X1, X2)) -> first(X1, X2)
                  , activate(X) -> X
                  , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(n__add(activate(X), Y))
                  , first(0(), X) -> nil()
                  , first(s(X), cons(Y, Z)) ->
                    cons(Y, n__first(activate(X), activate(Z)))
                  , terms(X) -> n__terms(X)
                  , add(X1, X2) -> n__add(X1, X2)
                  , s(X) -> n__s(X)
                  , dbl(X) -> n__dbl(X)
                  , first(X1, X2) -> n__first(X1, X2)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,19}->{16}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {add^#(X1, X2) -> c_10()}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_15) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 4] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [2 4]      [0]
                n__add(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                 [0 0]      [0 0]      [0]
                n__first(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
                                   [0 0]      [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                                [0 0]      [0 0]      [0]
                first^#(x1, x2) = [0 2] x1 + [2 0] x2 + [0]
                                  [0 0]      [0 0]      [2]
                c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [1]
                activate^#(x1) = [2 0] x1 + [0]
                                 [4 0]      [0]
                c_10() = [0]
                         [0]
                c_15(x1) = [4 0] x1 + [0]
                           [0 0]      [7]
                c_18(x1) = [1 2] x1 + [0]
                           [0 2]      [2]

           * Path {9,19}->{17}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [1 1] x1 + [0]
                           [0 1]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [3 0] x1 + [0]
                          [3 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [1 3] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {activate^#(n__s(X)) -> c_16(s^#(X))}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_16) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [3 3]      [2]
                n__s(x1) = [1 2] x1 + [2]
                           [0 1]      [2]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [2]
                s^#(x1) = [0 0] x1 + [2]
                          [0 2]      [0]
                first^#(x1, x2) = [0 4] x1 + [3 0] x2 + [0]
                                  [0 4]      [6 0]      [0]
                c_8(x1, x2) = [2 2] x1 + [1 0] x2 + [1]
                              [1 2]      [2 0]      [0]
                activate^#(x1) = [3 3] x1 + [2]
                                 [3 3]      [0]
                c_16(x1) = [2 2] x1 + [1]
                           [0 0]      [6]
                c_18(x1) = [1 0] x1 + [7]
                           [0 0]      [7]

           * Path {9,19}->{17}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {1}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {s^#(X) -> c_11()}
               Weak Rules:
                 {  activate^#(n__s(X)) -> c_16(s^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_16) = {1},
                 Uargs(c_18) = {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]      [1]
                s(x1) = [4 2] x1 + [2]
                        [0 0]      [0]
                n__s(x1) = [1 2] x1 + [2]
                           [0 1]      [2]
                n__first(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [2]
                s^#(x1) = [0 2] x1 + [4]
                          [0 2]      [0]
                first^#(x1, x2) = [2 0] x1 + [3 0] x2 + [0]
                                  [0 0]      [0 2]      [6]
                c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
                              [0 0]      [0 0]      [7]
                activate^#(x1) = [3 3] x1 + [0]
                                 [2 5]      [0]
                c_11() = [1]
                         [0]
                c_16(x1) = [2 2] x1 + [3]
                           [2 2]      [2]
                c_18(x1) = [1 2] x1 + [0]
                           [0 2]      [2]

           * Path {9,19}->{18}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {dbl^#(0()) -> c_3()}
               Weak Rules:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 4] x2 + [2]
                               [0 0]      [0 1]      [0]
                s(x1) = [3 1] x1 + [2]
                        [2 4]      [0]
                0() = [2]
                      [2]
                n__dbl(x1) = [1 2] x1 + [2]
                             [0 0]      [2]
                n__first(x1, x2) = [1 3] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 0]      [2]
                dbl^#(x1) = [0 2] x1 + [2]
                            [2 2]      [0]
                c_3() = [1]
                        [0]
                first^#(x1, x2) = [2 3] x1 + [2 0] x2 + [4]
                                  [2 2]      [2 1]      [0]
                c_8(x1, x2) = [3 1] x1 + [1 0] x2 + [1]
                              [2 2]      [0 0]      [0]
                activate^#(x1) = [2 3] x1 + [2]
                                 [2 2]      [2]
                c_17(x1) = [2 0] x1 + [7]
                           [2 0]      [3]
                c_18(x1) = [1 0] x1 + [6]
                           [0 0]      [6]

           * Path {9,19}->{18}->{5}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{5}->{12}: NA
             --------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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 {9,19}->{18}->{13}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {dbl^#(X) -> c_12()}
               Weak Rules:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {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]      [1]
                s(x1) = [4 2] x1 + [2]
                        [0 0]      [0]
                n__dbl(x1) = [1 2] x1 + [2]
                             [0 1]      [2]
                n__first(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [2]
                dbl^#(x1) = [0 2] x1 + [4]
                            [0 2]      [0]
                first^#(x1, x2) = [2 0] x1 + [3 0] x2 + [0]
                                  [0 0]      [0 2]      [6]
                c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [7]
                              [0 0]      [0 0]      [7]
                activate^#(x1) = [3 3] x1 + [0]
                                 [2 5]      [0]
                c_12() = [1]
                         [0]
                c_17(x1) = [2 2] x1 + [3]
                           [2 2]      [2]
                c_18(x1) = [1 2] x1 + [0]
                           [0 2]      [2]

           * Path {9,19}->{20}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                activate^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11() = [0]
                         [0]
                c_12() = [0]
                         [0]
                c_13() = [0]
                         [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19() = [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: {activate^#(X) -> c_19()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1, 2}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
                               [0 0]      [0 1]      [2]
                s(x1) = [2 1] x1 + [2]
                        [2 3]      [2]
                n__first(x1, x2) = [1 4] x1 + [1 4] x2 + [2]
                                   [0 1]      [0 1]      [2]
                first^#(x1, x2) = [2 2] x1 + [2 2] x2 + [0]
                                  [2 2]      [4 0]      [0]
                c_8(x1, x2) = [1 2] x1 + [1 0] x2 + [4]
                              [0 2]      [2 0]      [0]
                activate^#(x1) = [2 1] x1 + [2]
                                 [2 3]      [2]
                c_18(x1) = [1 0] x1 + [7]
                           [0 0]      [7]
                c_19() = [1]
                         [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), s^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) ->
                   c_2(s^#(n__add(sqr(activate(X)), dbl(activate(X)))))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(s^#(n__s(n__dbl(activate(X)))))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
              , 10: terms^#(X) -> c_9()
              , 11: add^#(X1, X2) -> c_10()
              , 12: s^#(X) -> c_11()
              , 13: dbl^#(X) -> c_12()
              , 14: first^#(X1, X2) -> c_13()
              , 15: activate^#(n__terms(X)) -> c_14(terms^#(X))
              , 16: activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
              , 17: activate^#(n__s(X)) -> c_16(s^#(X))
              , 18: activate^#(n__dbl(X)) -> c_17(dbl^#(X))
              , 19: activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
              , 20: activate^#(X) -> c_19()}

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

             ->{9,19}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^1))    ]
                |
                |->{14}                                                  [   YES(?,O(n^1))    ]
                |
                |->{15}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^1))    ]
                |   |   |
                |   |   |->{3}                                           [     inherited      ]
                |   |   |   |
                |   |   |   `->{12}                                      [         NA         ]
                |   |   |
                |   |   `->{12}                                          [   YES(?,O(n^1))    ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^1))    ]
                |
                |->{16}                                                  [     inherited      ]
                |   |
                |   |->{6}                                               [   YES(?,O(n^1))    ]
                |   |
                |   |->{7}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [       MAYBE        ]
                |   |
                |   `->{11}                                              [   YES(?,O(n^1))    ]
                |
                |->{17}                                                  [   YES(?,O(n^1))    ]
                |   |
                |   `->{12}                                              [   YES(?,O(n^1))    ]
                |
                |->{18}                                                  [     inherited      ]
                |   |
                |   |->{4}                                               [   YES(?,O(n^1))    ]
                |   |
                |   |->{5}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [         NA         ]
                |   |
                |   `->{13}                                              [   YES(?,O(n^1))    ]
                |
                `->{20}                                                  [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {9,19}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1, 2}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                s(x1) = [2] x1 + [2]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_7() = [1]
                c_8(x1, x2) = [2] x1 + [1] x2 + [7]
                activate^#(x1) = [2] x1 + [0]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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_13()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1, 2}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                s(x1) = [4] x1 + [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                first^#(x1, x2) = [2] x1 + [2] x2 + [1]
                c_8(x1, x2) = [1] x1 + [1] x2 + [1]
                activate^#(x1) = [2] x1 + [1]
                c_13() = [0]
                c_18(x1) = [1] x1 + [0]

           * Path {9,19}->{15}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [1] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [1] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                n__terms(x1) = [1] x1 + [0]
                s(x1) = [4] x1 + [6]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                terms^#(x1) = [2] x1 + [4]
                c_0(x1, x2) = [1] x1 + [7] x2 + [0]
                sqr^#(x1) = [2] x1 + [4]
                s^#(x1) = [0] x1 + [0]
                c_1() = [1]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2) = [2] x1 + [1] x2 + [0]
                activate^#(x1) = [2] x1 + [4]
                c_14(x1) = [1] x1 + [0]
                c_18(x1) = [1] x1 + [7]

           * Path {9,19}->{15}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}->{3}->{12}: NA
             -------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , dbl(X) -> n__dbl(X)
                , activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , first(X1, X2) -> n__first(X1, X2)}

             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 {9,19}->{15}->{1}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {2}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [1] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [1] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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: {s^#(X) -> c_11()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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) = {2},
                 Uargs(sqr^#) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                n__terms(x1) = [1] x1 + [2]
                s(x1) = [6] x1 + [4]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                terms^#(x1) = [1] x1 + [4]
                c_0(x1, x2) = [1] x1 + [4] x2 + [0]
                sqr^#(x1) = [1] x1 + [0]
                s^#(x1) = [0] x1 + [1]
                first^#(x1, x2) = [2] x1 + [3] x2 + [0]
                c_8(x1, x2) = [3] x1 + [1] x2 + [0]
                activate^#(x1) = [3] x1 + [2]
                c_11() = [0]
                c_14(x1) = [2] x1 + [0]
                c_18(x1) = [1] x1 + [0]

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [1] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                n__terms(x1) = [1] x1 + [2]
                s(x1) = [3] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                terms^#(x1) = [0] x1 + [2]
                first^#(x1, x2) = [2] x1 + [3] x2 + [4]
                c_8(x1, x2) = [2] x1 + [1] x2 + [0]
                activate^#(x1) = [3] x1 + [2]
                c_9() = [1]
                c_14(x1) = [4] x1 + [0]
                c_18(x1) = [1] x1 + [4]

           * Path {9,19}->{16}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{6}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [1] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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: {add^#(0(), X) -> c_5()}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_15) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [4] x1 + [4]
                0() = [0]
                n__add(x1, x2) = [1] x1 + [1] x2 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                add^#(x1, x2) = [0] x1 + [0] x2 + [1]
                c_5() = [0]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [7]
                activate^#(x1) = [2] x1 + [0]
                c_15(x1) = [4] x1 + [0]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{16}->{7}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{7}->{12}: MAYBE
             -----------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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:
                 {  add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
                  , activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
                  , s^#(X) -> c_11()
                  , activate(n__terms(X)) -> terms(X)
                  , activate(n__add(X1, X2)) -> add(X1, X2)
                  , activate(n__s(X)) -> s(X)
                  , activate(n__dbl(X)) -> dbl(X)
                  , activate(n__first(X1, X2)) -> first(X1, X2)
                  , activate(X) -> X
                  , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(n__add(activate(X), Y))
                  , first(0(), X) -> nil()
                  , first(s(X), cons(Y, Z)) ->
                    cons(Y, n__first(activate(X), activate(Z)))
                  , terms(X) -> n__terms(X)
                  , add(X1, X2) -> n__add(X1, X2)
                  , s(X) -> n__s(X)
                  , dbl(X) -> n__dbl(X)
                  , first(X1, X2) -> n__first(X1, X2)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,19}->{16}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [1] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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: {add^#(X1, X2) -> c_10()}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_15) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [2] x1 + [2]
                n__add(x1, x2) = [1] x1 + [1] x2 + [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [3]
                add^#(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_8(x1, x2) = [2] x1 + [1] x2 + [1]
                activate^#(x1) = [2] x1 + [2]
                c_10() = [1]
                c_15(x1) = [1] x1 + [0]
                c_18(x1) = [1] x1 + [2]

           * Path {9,19}->{17}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [1] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [3] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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^#(n__s(X)) -> c_16(s^#(X))}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_16) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [6] x1 + [4]
                n__s(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                s^#(x1) = [7] x1 + [0]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2) = [2] x1 + [1] x2 + [3]
                activate^#(x1) = [2] x1 + [0]
                c_16(x1) = [0] x1 + [3]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{17}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {1}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [1] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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: {s^#(X) -> c_11()}
               Weak Rules:
                 {  activate^#(n__s(X)) -> c_16(s^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_16) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [3] x1 + [2]
                n__s(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                s^#(x1) = [0] x1 + [2]
                first^#(x1, x2) = [2] x1 + [3] x2 + [4]
                c_8(x1, x2) = [2] x1 + [1] x2 + [0]
                activate^#(x1) = [3] x1 + [2]
                c_11() = [1]
                c_16(x1) = [4] x1 + [0]
                c_18(x1) = [1] x1 + [4]

           * Path {9,19}->{18}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [1] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                s(x1) = [2] x1 + [2]
                0() = [2]
                n__dbl(x1) = [1] x1 + [0]
                n__first(x1, x2) = [1] x1 + [1] x2 + [3]
                dbl^#(x1) = [2] x1 + [0]
                c_3() = [1]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2) = [2] x1 + [1] x2 + [1]
                activate^#(x1) = [2] x1 + [2]
                c_17(x1) = [1] x1 + [2]
                c_18(x1) = [1] x1 + [0]

           * Path {9,19}->{18}->{5}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{5}->{12}: NA
             --------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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 {9,19}->{18}->{13}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [1] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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^#(X) -> c_12()}
               Weak Rules:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [3] x1 + [2]
                n__dbl(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                dbl^#(x1) = [0] x1 + [2]
                first^#(x1, x2) = [2] x1 + [3] x2 + [4]
                c_8(x1, x2) = [2] x1 + [1] x2 + [0]
                activate^#(x1) = [3] x1 + [2]
                c_12() = [1]
                c_17(x1) = [4] x1 + [0]
                c_18(x1) = [1] x1 + [4]

           * Path {9,19}->{20}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1, 2}, Uargs(activate^#) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                activate^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11() = [0]
                c_12() = [0]
                c_13() = [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19() = [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_19()}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) -> c_8(activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {1, 2}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [2] x1 + [3]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_8(x1, x2) = [2] x1 + [1] x2 + [0]
                activate^#(x1) = [3] x1 + [3]
                c_18(x1) = [1] x1 + [2]
                c_19() = [0]

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

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

    5) '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 Ex26 Luc03b Z

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex26 Luc03b 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(n__add(sqr(activate(X)), dbl(activate(X))))
     , dbl(0()) -> 0()
     , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(n__add(activate(X), Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) ->
       cons(Y, n__first(activate(X), activate(Z)))
     , terms(X) -> n__terms(X)
     , add(X1, X2) -> n__add(X1, X2)
     , s(X) -> n__s(X)
     , dbl(X) -> n__dbl(X)
     , first(X1, X2) -> n__first(X1, X2)
     , activate(n__terms(X)) -> terms(X)
     , activate(n__add(X1, X2)) -> add(X1, X2)
     , activate(n__s(X)) -> s(X)
     , activate(n__dbl(X)) -> dbl(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), s^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) ->
                   c_2(s^#(n__add(sqr(activate(X)), dbl(activate(X)))))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(s^#(n__s(n__dbl(activate(X)))))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) ->
                   c_8(Y, activate^#(X), activate^#(Z))
              , 10: terms^#(X) -> c_9(X)
              , 11: add^#(X1, X2) -> c_10(X1, X2)
              , 12: s^#(X) -> c_11(X)
              , 13: dbl^#(X) -> c_12(X)
              , 14: first^#(X1, X2) -> c_13(X1, X2)
              , 15: activate^#(n__terms(X)) -> c_14(terms^#(X))
              , 16: activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
              , 17: activate^#(n__s(X)) -> c_16(s^#(X))
              , 18: activate^#(n__dbl(X)) -> c_17(dbl^#(X))
              , 19: activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
              , 20: activate^#(X) -> c_19(X)}

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

             ->{9,19}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^2))    ]
                |
                |->{14}                                                  [   YES(?,O(n^2))    ]
                |
                |->{15}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^1))    ]
                |   |   |
                |   |   |->{3}                                           [     inherited      ]
                |   |   |   |
                |   |   |   `->{12}                                      [         NA         ]
                |   |   |
                |   |   `->{12}                                          [   YES(?,O(n^2))    ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^2))    ]
                |
                |->{16}                                                  [     inherited      ]
                |   |
                |   |->{6}                                               [   YES(?,O(n^2))    ]
                |   |
                |   |->{7}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [       MAYBE        ]
                |   |
                |   `->{11}                                              [   YES(?,O(n^2))    ]
                |
                |->{17}                                                  [   YES(?,O(n^2))    ]
                |   |
                |   `->{12}                                              [   YES(?,O(n^2))    ]
                |
                |->{18}                                                  [     inherited      ]
                |   |
                |   |->{4}                                               [   YES(?,O(n^2))    ]
                |   |
                |   |->{5}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [         NA         ]
                |   |
                |   `->{13}                                              [   YES(?,O(n^2))    ]
                |
                `->{20}                                                  [   YES(?,O(n^2))    ]



         Sub-problems:
         -------------
           * Path {9,19}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 7] x1 + [1 4] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [3 0] x1 + [4]
                        [3 2]      [0]
                0() = [2]
                      [2]
                n__first(x1, x2) = [1 2] x1 + [1 0] x2 + [3]
                                   [0 1]      [0 1]      [2]
                first^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                                  [2 2]      [2 0]      [0]
                c_7() = [1]
                        [0]
                c_8(x1, x2, x3) = [0 0] x1 + [2 2] x2 + [1 0] x3 + [7]
                                  [0 0]      [0 0]      [0 0]      [6]
                activate^#(x1) = [2 1] x1 + [0]
                                 [3 0]      [0]
                c_18(x1) = [1 0] x1 + [7]
                           [0 0]      [7]

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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^#(X1, X2) -> c_13(X1, X2)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) ->
                    c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_13) = {}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 4] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [2 4] x1 + [2]
                        [2 1]      [2]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [0]
                first^#(x1, x2) = [2 3] x1 + [2 0] x2 + [2]
                                  [2 2]      [2 0]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [2 2] x2 + [1 0] x3 + [0]
                                  [0 0]      [2 0]      [0 0]      [3]
                activate^#(x1) = [2 5] x1 + [2]
                                 [2 0]      [2]
                c_13(x1, x2) = [0 1] x1 + [0 0] x2 + [1]
                               [0 0]      [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 0]      [2]

           * Path {9,19}->{15}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {sqr^#(0()) -> c_1()}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__terms(x1) = [1 3] x1 + [1]
                               [0 0]      [2]
                s(x1) = [2 2] x1 + [0]
                        [3 2]      [3]
                0() = [2]
                      [2]
                n__first(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
                                   [0 0]      [0 0]      [2]
                terms^#(x1) = [2 2] x1 + [2]
                              [0 0]      [2]
                c_0(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                              [0 0]      [0 0]      [1]
                sqr^#(x1) = [2 2] x1 + [0]
                            [0 2]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [1]
                        [0]
                first^#(x1, x2) = [2 2] x1 + [3 0] x2 + [0]
                                  [1 3]      [0 0]      [2]
                c_8(x1, x2, x3) = [1 0] x1 + [2 0] x2 + [1 0] x3 + [3]
                                  [0 0]      [0 0]      [0 0]      [7]
                activate^#(x1) = [3 2] x1 + [0]
                                 [4 2]      [0]
                c_14(x1) = [1 0] x1 + [1]
                           [0 2]      [3]
                c_18(x1) = [1 1] x1 + [1]
                           [1 2]      [0]

           * Path {9,19}->{15}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}->{3}->{12}: NA
             -------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , dbl(X) -> n__dbl(X)
                , activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , first(X1, X2) -> n__first(X1, X2)}

             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 {9,19}->{15}->{1}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {2}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s^#(x1) = [3 3] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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 1] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {s^#(X) -> c_11(X)}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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) = {2},
                 Uargs(sqr^#) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_11) = {},
                 Uargs(c_14) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 7] x1 + [1 2] x2 + [0]
                               [0 1]      [0 0]      [1]
                n__terms(x1) = [1 3] x1 + [0]
                               [0 1]      [2]
                s(x1) = [0 4] x1 + [6]
                        [6 5]      [0]
                n__first(x1, x2) = [1 1] x1 + [1 3] x2 + [0]
                                   [0 0]      [0 1]      [0]
                terms^#(x1) = [0 2] x1 + [2]
                              [0 0]      [4]
                c_0(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 0]      [3]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s^#(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                first^#(x1, x2) = [0 2] x1 + [2 0] x2 + [0]
                                  [1 0]      [0 2]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [2 2] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [7]
                activate^#(x1) = [2 3] x1 + [0]
                                 [1 2]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_14(x1) = [2 0] x1 + [2]
                           [2 0]      [0]
                c_18(x1) = [1 2] x1 + [0]
                           [0 0]      [0]

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_14) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                n__terms(x1) = [1 2] x1 + [2]
                               [0 1]      [0]
                s(x1) = [3 2] x1 + [0]
                        [3 4]      [2]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [1]
                terms^#(x1) = [0 0] x1 + [2]
                              [0 2]      [2]
                first^#(x1, x2) = [2 2] x1 + [3 0] x2 + [0]
                                  [2 2]      [2 2]      [0]
                c_8(x1, x2, x3) = [1 0] x1 + [2 2] x2 + [1 0] x3 + [0]
                                  [0 0]      [2 2]      [0 0]      [3]
                activate^#(x1) = [3 4] x1 + [2]
                                 [3 2]      [2]
                c_9(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
                c_14(x1) = [2 2] x1 + [0]
                           [2 2]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 0]      [7]

           * Path {9,19}->{16}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{6}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5(x1) = [1 1] 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {add^#(0(), X) -> c_5(X)}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(c_5) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_15) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                s(x1) = [2 2] x1 + [2]
                        [2 2]      [0]
                0() = [0]
                      [2]
                n__add(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                 [0 1]      [0 1]      [2]
                n__first(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
                                   [0 0]      [0 1]      [3]
                add^#(x1, x2) = [0 2] x1 + [0 3] x2 + [6]
                                [0 2]      [0 2]      [0]
                c_5(x1) = [0 1] x1 + [1]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [1 3]      [2 5]      [0]
                c_8(x1, x2, x3) = [1 0] x1 + [2 2] x2 + [1 0] x3 + [2]
                                  [0 0]      [0 0]      [0 0]      [2]
                activate^#(x1) = [3 3] x1 + [2]
                                 [3 3]      [2]
                c_15(x1) = [2 1] x1 + [2]
                           [0 0]      [2]
                c_18(x1) = [1 0] x1 + [6]
                           [0 0]      [7]

           * Path {9,19}->{16}->{7}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{7}->{12}: MAYBE
             -----------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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:
                 {  add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
                  , activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
                  , s^#(X) -> c_11(X)
                  , activate(n__terms(X)) -> terms(X)
                  , activate(n__add(X1, X2)) -> add(X1, X2)
                  , activate(n__s(X)) -> s(X)
                  , activate(n__dbl(X)) -> dbl(X)
                  , activate(n__first(X1, X2)) -> first(X1, X2)
                  , activate(X) -> X
                  , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(n__add(activate(X), Y))
                  , first(0(), X) -> nil()
                  , first(s(X), cons(Y, Z)) ->
                    cons(Y, n__first(activate(X), activate(Z)))
                  , terms(X) -> n__terms(X)
                  , add(X1, X2) -> n__add(X1, X2)
                  , s(X) -> n__s(X)
                  , dbl(X) -> n__dbl(X)
                  , first(X1, X2) -> n__first(X1, X2)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,19}->{16}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {add^#(X1, X2) -> c_10(X1, X2)}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_10) = {},
                 Uargs(c_15) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [0 0] x1 + [2]
                        [4 5]      [0]
                n__add(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                 [0 1]      [0 1]      [0]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [0]
                add^#(x1, x2) = [0 2] x1 + [0 2] x2 + [1]
                                [0 2]      [0 2]      [0]
                first^#(x1, x2) = [0 2] x1 + [2 0] x2 + [6]
                                  [4 3]      [4 0]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [2 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [2 0]      [2 0]      [0]
                activate^#(x1) = [2 5] x1 + [2]
                                 [2 5]      [0]
                c_10(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_15(x1) = [2 2] x1 + [1]
                           [0 2]      [3]
                c_18(x1) = [1 0] x1 + [0]
                           [0 0]      [3]

           * Path {9,19}->{17}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [1 1] x1 + [0]
                           [0 1]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [3 0] x1 + [0]
                          [3 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [0 1]      [0]
                activate^#(x1) = [1 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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {activate^#(n__s(X)) -> c_16(s^#(X))}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) ->
                    c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_16) = {},
                 Uargs(c_18) = {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]      [0]
                s(x1) = [2 3] x1 + [0]
                        [4 0]      [2]
                n__s(x1) = [1 2] x1 + [2]
                           [0 1]      [2]
                n__first(x1, x2) = [1 4] x1 + [1 2] x2 + [2]
                                   [0 0]      [0 1]      [2]
                s^#(x1) = [0 2] x1 + [2]
                          [0 2]      [2]
                first^#(x1, x2) = [3 2] x1 + [3 3] x2 + [0]
                                  [4 0]      [2 0]      [0]
                c_8(x1, x2, x3) = [1 0] x1 + [1 2] x2 + [1 0] x3 + [3]
                                  [0 0]      [2 0]      [0 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [3 3]      [0]
                c_16(x1) = [2 2] x1 + [1]
                           [0 2]      [7]
                c_18(x1) = [1 0] x1 + [6]
                           [0 0]      [7]

           * Path {9,19}->{17}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {1}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [3 3] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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 1] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {s^#(X) -> c_11(X)}
               Weak Rules:
                 {  activate^#(n__s(X)) -> c_16(s^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_11) = {},
                 Uargs(c_16) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                s(x1) = [3 2] x1 + [0]
                        [3 4]      [2]
                n__s(x1) = [1 2] x1 + [2]
                           [0 1]      [0]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [1]
                s^#(x1) = [0 0] x1 + [2]
                          [0 2]      [2]
                first^#(x1, x2) = [2 2] x1 + [3 0] x2 + [0]
                                  [2 2]      [2 2]      [0]
                c_8(x1, x2, x3) = [1 0] x1 + [2 2] x2 + [1 0] x3 + [0]
                                  [0 0]      [2 2]      [0 0]      [3]
                activate^#(x1) = [3 4] x1 + [2]
                                 [3 2]      [2]
                c_11(x1) = [0 0] x1 + [1]
                           [0 0]      [0]
                c_16(x1) = [2 2] x1 + [0]
                           [2 2]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 0]      [7]

           * Path {9,19}->{18}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {dbl^#(0()) -> c_3()}
               Weak Rules:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {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]      [0]
                s(x1) = [2 0] x1 + [4]
                        [2 4]      [0]
                0() = [2]
                      [2]
                n__dbl(x1) = [1 2] x1 + [2]
                             [0 1]      [2]
                n__first(x1, x2) = [1 3] x1 + [1 0] x2 + [2]
                                   [0 1]      [0 0]      [0]
                dbl^#(x1) = [2 2] x1 + [0]
                            [2 0]      [0]
                c_3() = [1]
                        [0]
                first^#(x1, x2) = [2 2] x1 + [4 0] x2 + [0]
                                  [2 2]      [4 0]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [2 0] x2 + [1 0] x3 + [7]
                                  [0 0]      [0 0]      [0 0]      [6]
                activate^#(x1) = [4 1] x1 + [0]
                                 [2 2]      [0]
                c_17(x1) = [2 0] x1 + [7]
                           [0 0]      [7]
                c_18(x1) = [1 0] x1 + [2]
                           [0 0]      [3]

           * Path {9,19}->{18}->{5}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{5}->{12}: NA
             --------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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 {9,19}->{18}->{13}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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 1] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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: {dbl^#(X) -> c_12(X)}
               Weak Rules:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_12) = {},
                 Uargs(c_17) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                s(x1) = [3 2] x1 + [0]
                        [3 4]      [2]
                n__dbl(x1) = [1 2] x1 + [2]
                             [0 1]      [0]
                n__first(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                                   [0 1]      [0 1]      [1]
                dbl^#(x1) = [0 0] x1 + [2]
                            [0 2]      [2]
                first^#(x1, x2) = [2 2] x1 + [3 0] x2 + [0]
                                  [2 2]      [2 2]      [0]
                c_8(x1, x2, x3) = [1 0] x1 + [2 2] x2 + [1 0] x3 + [0]
                                  [0 0]      [2 2]      [0 0]      [3]
                activate^#(x1) = [3 4] x1 + [2]
                                 [3 2]      [2]
                c_12(x1) = [0 0] x1 + [1]
                           [0 0]      [0]
                c_17(x1) = [2 2] x1 + [0]
                           [2 2]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 0]      [7]

           * Path {9,19}->{20}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                n__dbl(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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]
                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]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [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]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 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 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 1]      [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) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_13(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_14(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_15(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_16(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_17(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_18(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_19(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_19(X)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) ->
                    c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 1]      [3]
                s(x1) = [3 2] x1 + [0]
                        [2 0]      [0]
                n__first(x1, x2) = [1 4] x1 + [1 4] x2 + [2]
                                   [0 0]      [0 0]      [0]
                first^#(x1, x2) = [2 1] x1 + [2 4] x2 + [0]
                                  [0 0]      [4 0]      [0]
                c_8(x1, x2, x3) = [0 0] x1 + [2 2] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                activate^#(x1) = [2 0] x1 + [4]
                                 [2 0]      [0]
                c_18(x1) = [1 0] x1 + [7]
                           [0 0]      [4]
                c_19(x1) = [0 0] x1 + [1]
                           [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), s^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) ->
                   c_2(s^#(n__add(sqr(activate(X)), dbl(activate(X)))))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(s^#(n__s(n__dbl(activate(X)))))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) ->
                   c_8(Y, activate^#(X), activate^#(Z))
              , 10: terms^#(X) -> c_9(X)
              , 11: add^#(X1, X2) -> c_10(X1, X2)
              , 12: s^#(X) -> c_11(X)
              , 13: dbl^#(X) -> c_12(X)
              , 14: first^#(X1, X2) -> c_13(X1, X2)
              , 15: activate^#(n__terms(X)) -> c_14(terms^#(X))
              , 16: activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
              , 17: activate^#(n__s(X)) -> c_16(s^#(X))
              , 18: activate^#(n__dbl(X)) -> c_17(dbl^#(X))
              , 19: activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
              , 20: activate^#(X) -> c_19(X)}

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

             ->{9,19}                                                    [     inherited      ]
                |
                |->{8}                                                   [   YES(?,O(n^1))    ]
                |
                |->{14}                                                  [   YES(?,O(n^1))    ]
                |
                |->{15}                                                  [     inherited      ]
                |   |
                |   |->{1}                                               [     inherited      ]
                |   |   |
                |   |   |->{2}                                           [   YES(?,O(n^1))    ]
                |   |   |
                |   |   |->{3}                                           [     inherited      ]
                |   |   |   |
                |   |   |   `->{12}                                      [         NA         ]
                |   |   |
                |   |   `->{12}                                          [   YES(?,O(n^1))    ]
                |   |
                |   `->{10}                                              [   YES(?,O(n^1))    ]
                |
                |->{16}                                                  [     inherited      ]
                |   |
                |   |->{6}                                               [   YES(?,O(n^1))    ]
                |   |
                |   |->{7}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [       MAYBE        ]
                |   |
                |   `->{11}                                              [   YES(?,O(n^1))    ]
                |
                |->{17}                                                  [   YES(?,O(n^1))    ]
                |   |
                |   `->{12}                                              [   YES(?,O(n^1))    ]
                |
                |->{18}                                                  [     inherited      ]
                |   |
                |   |->{4}                                               [   YES(?,O(n^1))    ]
                |   |
                |   |->{5}                                               [     inherited      ]
                |   |   |
                |   |   `->{12}                                          [         NA         ]
                |   |
                |   `->{13}                                              [   YES(?,O(n^1))    ]
                |
                `->{20}                                                  [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {9,19}: inherited
             ----------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                s(x1) = [2] x1 + [2]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_7() = [1]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [7]
                activate^#(x1) = [2] x1 + [0]
                c_18(x1) = [1] x1 + [0]

           * Path {9,19}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [1] x1 + [1] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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_13(X1, X2)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) ->
                    c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_13) = {}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [6] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [3]
                first^#(x1, x2) = [2] x1 + [3] x2 + [4]
                c_8(x1, x2, x3) = [1] x1 + [4] x2 + [1] x3 + [3]
                activate^#(x1) = [3] x1 + [0]
                c_13(x1, x2) = [0] x1 + [1] x2 + [1]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{15}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [1] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [1] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_14) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                n__terms(x1) = [1] x1 + [2]
                s(x1) = [4] x1 + [4]
                0() = [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                terms^#(x1) = [2] x1 + [6]
                c_0(x1, x2) = [1] x1 + [1] x2 + [0]
                sqr^#(x1) = [2] x1 + [4]
                s^#(x1) = [0] x1 + [2]
                c_1() = [1]
                first^#(x1, x2) = [2] x1 + [6] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [1] x2 + [1] x3 + [7]
                activate^#(x1) = [6] x1 + [0]
                c_14(x1) = [2] x1 + [0]
                c_18(x1) = [1] x1 + [0]

           * Path {9,19}->{15}->{1}->{3}: inherited
             --------------------------------------

             This path is subsumed by the proof of path {9,19}->{15}->{1}->{3}->{12}.

           * Path {9,19}->{15}->{1}->{3}->{12}: NA
             -------------------------------------

             The usable rules for this path are:

               {  sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , dbl(X) -> n__dbl(X)
                , activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , first(X1, X2) -> n__first(X1, X2)}

             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 {9,19}->{15}->{1}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {2}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [1] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [1] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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: {s^#(X) -> c_11(X)}
               Weak Rules:
                 {  terms^#(N) -> c_0(sqr^#(N), s^#(N))
                  , activate^#(n__terms(X)) -> c_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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) = {2},
                 Uargs(sqr^#) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_11) = {},
                 Uargs(c_14) = {1}, Uargs(c_18) = {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) = [6] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                terms^#(x1) = [0] x1 + [1]
                c_0(x1, x2) = [7] x1 + [1] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [1]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [1]
                activate^#(x1) = [2] x1 + [2]
                c_11(x1) = [0] x1 + [0]
                c_14(x1) = [4] x1 + [2]
                c_18(x1) = [1] x1 + [2]

           * Path {9,19}->{15}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {1},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [1] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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_14(terms^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(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) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_9) = {},
                 Uargs(c_14) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                n__terms(x1) = [1] x1 + [2]
                s(x1) = [2] x1 + [6]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                terms^#(x1) = [1] x1 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [0]
                activate^#(x1) = [2] x1 + [4]
                c_9(x1) = [1] x1 + [1]
                c_14(x1) = [2] x1 + [4]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{16}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{6}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_5(x1) = [1] 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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [1] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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: {add^#(0(), X) -> c_5(X)}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(c_5) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_15) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [3] x1 + [3]
                0() = [0]
                n__add(x1, x2) = [1] x1 + [1] x2 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [1]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [2] x1 + [3] x2 + [2]
                c_8(x1, x2, x3) = [1] x1 + [2] x2 + [1] x3 + [1]
                activate^#(x1) = [3] x1 + [2]
                c_15(x1) = [4] x1 + [3]
                c_18(x1) = [1] x1 + [0]

           * Path {9,19}->{16}->{7}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{16}->{7}->{12}.

           * Path {9,19}->{16}->{7}->{12}: MAYBE
             -----------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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:
                 {  add^#(s(X), Y) -> c_6(s^#(n__add(activate(X), Y)))
                  , activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))
                  , s^#(X) -> c_11(X)
                  , activate(n__terms(X)) -> terms(X)
                  , activate(n__add(X1, X2)) -> add(X1, X2)
                  , activate(n__s(X)) -> s(X)
                  , activate(n__dbl(X)) -> dbl(X)
                  , activate(n__first(X1, X2)) -> first(X1, X2)
                  , activate(X) -> X
                  , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(n__add(activate(X), Y))
                  , first(0(), X) -> nil()
                  , first(s(X), cons(Y, Z)) ->
                    cons(Y, n__first(activate(X), activate(Z)))
                  , terms(X) -> n__terms(X)
                  , add(X1, X2) -> n__add(X1, X2)
                  , s(X) -> n__s(X)
                  , dbl(X) -> n__dbl(X)
                  , first(X1, X2) -> n__first(X1, X2)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             Proof Output:
               The input cannot be shown compatible

           * Path {9,19}->{16}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {1}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [3] x1 + [3] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [1] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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: {add^#(X1, X2) -> c_10(X1, X2)}
               Weak Rules:
                 {  activate^#(n__add(X1, X2)) -> c_15(add^#(X1, X2))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__add) = {},
                 Uargs(n__first) = {}, Uargs(add^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_10) = {},
                 Uargs(c_15) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [6] x1 + [4]
                n__add(x1, x2) = [1] x1 + [1] x2 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                add^#(x1, x2) = [0] x1 + [0] x2 + [1]
                first^#(x1, x2) = [2] x1 + [3] x2 + [0]
                c_8(x1, x2, x3) = [1] x1 + [2] x2 + [1] x3 + [1]
                activate^#(x1) = [3] x1 + [2]
                c_10(x1, x2) = [0] x1 + [0] x2 + [0]
                c_15(x1) = [4] x1 + [3]
                c_18(x1) = [1] x1 + [6]

           * Path {9,19}->{17}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [1] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [1] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [3] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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: {activate^#(n__s(X)) -> c_16(s^#(X))}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) ->
                    c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_16) = {},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [6] x1 + [6]
                n__s(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                s^#(x1) = [1] x1 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [0]
                activate^#(x1) = [2] x1 + [4]
                c_16(x1) = [2] x1 + [3]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{17}->{12}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {1}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [1] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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: {s^#(X) -> c_11(X)}
               Weak Rules:
                 {  activate^#(n__s(X)) -> c_16(s^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__s) = {},
                 Uargs(n__first) = {}, Uargs(s^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_11) = {},
                 Uargs(c_16) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [2] x1 + [6]
                n__s(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                s^#(x1) = [1] x1 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [0]
                activate^#(x1) = [2] x1 + [4]
                c_11(x1) = [1] x1 + [1]
                c_16(x1) = [2] x1 + [4]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{18}: inherited
             ----------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [1] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [4]
                s(x1) = [2] x1 + [0]
                0() = [2]
                n__dbl(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [2]
                dbl^#(x1) = [2] x1 + [4]
                c_3() = [1]
                first^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [0]
                activate^#(x1) = [2] x1 + [4]
                c_17(x1) = [1] x1 + [2]
                c_18(x1) = [1] x1 + [2]

           * Path {9,19}->{18}->{5}: inherited
             ---------------------------------

             This path is subsumed by the proof of path {9,19}->{18}->{5}->{12}.

           * Path {9,19}->{18}->{5}->{12}: NA
             --------------------------------

             The usable rules for this path are:

               {  activate(n__terms(X)) -> terms(X)
                , activate(n__add(X1, X2)) -> add(X1, X2)
                , activate(n__s(X)) -> s(X)
                , activate(n__dbl(X)) -> dbl(X)
                , activate(n__first(X1, X2)) -> first(X1, X2)
                , activate(X) -> X
                , terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
                , dbl(0()) -> 0()
                , dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
                , add(0(), X) -> X
                , add(s(X), Y) -> s(n__add(activate(X), Y))
                , first(0(), X) -> nil()
                , first(s(X), cons(Y, Z)) ->
                  cons(Y, n__first(activate(X), activate(Z)))
                , terms(X) -> n__terms(X)
                , add(X1, X2) -> n__add(X1, X2)
                , s(X) -> n__s(X)
                , dbl(X) -> n__dbl(X)
                , first(X1, X2) -> n__first(X1, X2)
                , sqr(0()) -> 0()
                , sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))}

             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 {9,19}->{18}->{13}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {1},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [1] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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^#(X) -> c_12(X)}
               Weak Rules:
                 {  activate^#(n__dbl(X)) -> c_17(dbl^#(X))
                  , first^#(s(X), cons(Y, Z)) -> c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__dbl) = {},
                 Uargs(n__first) = {}, Uargs(dbl^#) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2, 3}, Uargs(activate^#) = {}, Uargs(c_12) = {},
                 Uargs(c_17) = {1}, Uargs(c_18) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [2] x1 + [6]
                n__dbl(x1) = [1] x1 + [2]
                n__first(x1, x2) = [1] x1 + [1] x2 + [0]
                dbl^#(x1) = [1] x1 + [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [0]
                activate^#(x1) = [2] x1 + [4]
                c_12(x1) = [1] x1 + [1]
                c_17(x1) = [2] x1 + [4]
                c_18(x1) = [1] x1 + [3]

           * Path {9,19}->{20}: 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(n__add) = {}, Uargs(activate) = {}, Uargs(dbl) = {},
                 Uargs(n__s) = {}, Uargs(n__dbl) = {}, Uargs(add) = {},
                 Uargs(first) = {}, Uargs(n__first) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(s^#) = {},
                 Uargs(c_2) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(add^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}, Uargs(c_13) = {}, Uargs(c_14) = {},
                 Uargs(c_15) = {}, Uargs(c_16) = {}, Uargs(c_17) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               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]
                n__add(x1, x2) = [0] x1 + [0] x2 + [0]
                activate(x1) = [0] x1 + [0]
                dbl(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                n__dbl(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                n__first(x1, x2) = [0] x1 + [0] x2 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [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 + [1] 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) = [0] x1 + [0]
                c_13(x1, x2) = [0] x1 + [0] x2 + [0]
                c_14(x1) = [0] x1 + [0]
                c_15(x1) = [0] x1 + [0]
                c_16(x1) = [0] x1 + [0]
                c_17(x1) = [0] x1 + [0]
                c_18(x1) = [1] x1 + [0]
                c_19(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_19(X)}
               Weak Rules:
                 {  first^#(s(X), cons(Y, Z)) ->
                    c_8(Y, activate^#(X), activate^#(Z))
                  , activate^#(n__first(X1, X2)) -> c_18(first^#(X1, X2))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(n__first) = {},
                 Uargs(first^#) = {}, Uargs(c_8) = {2, 3}, Uargs(activate^#) = {},
                 Uargs(c_18) = {1}, Uargs(c_19) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [2] x1 + [4]
                n__first(x1, x2) = [1] x1 + [1] x2 + [4]
                first^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1, x2, x3) = [0] x1 + [2] x2 + [1] x3 + [4]
                activate^#(x1) = [2] x1 + [1]
                c_18(x1) = [1] x1 + [7]
                c_19(x1) = [0] x1 + [0]

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

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

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