Problem Transformed CSR 04 Ex2 Luc02a L

Tool CaT

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

stdout:

MAYBE

Problem:
terms(N) -> cons(recip(sqr(N)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)

Proof:
Open

Tool IRC1

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

stdout:

MAYBE

Tool IRC2

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

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)))
     , sqr(0()) -> 0()
     , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
     , dbl(0()) -> 0()
     , dbl(s(X)) -> s(s(dbl(X)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y)) -> cons(Y)}

Proof Output:
  None of the processors succeeded.

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

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y)) -> c_8()}

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

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

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

             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [       MAYBE        ]
                    |
                    `->{7}                                               [     inherited      ]
                        |
                        `->{6}                                           [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

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

           * Path {1}->{2}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(terms^#) = {}, Uargs(c_0) = {1}, Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                terms^#(x1) = [7 7 7] x1 + [7]
                              [7 7 7]      [7]
                              [7 7 7]      [7]
                c_0(x1) = [2 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
                sqr^#(x1) = [2 0 2] x1 + [0]
                            [2 0 0]      [0]
                            [2 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]

           * Path {1}->{3}: inherited
             ------------------------

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

           * Path {1}->{3}->{6}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , add^#(0(), X) -> c_5()
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{3}->{7}: inherited
             -----------------------------

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

           * Path {1}->{3}->{7}->{6}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

           * Path {8}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y)) -> c_8()}
               Weak Rules: {}

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

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

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y)) -> c_8()}

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

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

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

             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [       MAYBE        ]
                    |
                    `->{7}                                               [     inherited      ]
                        |
                        `->{6}                                           [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

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

           * Path {1}->{2}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(terms^#) = {}, Uargs(c_0) = {1}, Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                terms^#(x1) = [7 7] x1 + [7]
                              [7 7]      [7]
                c_0(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
                sqr^#(x1) = [0 0] x1 + [4]
                            [2 2]      [0]
                c_1() = [1]
                        [0]

           * Path {1}->{3}: inherited
             ------------------------

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

           * Path {1}->{3}->{6}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , add^#(0(), X) -> c_5()
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{3}->{7}: inherited
             -----------------------------

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

           * Path {1}->{3}->{7}->{6}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

           * Path {8}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]

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

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

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

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

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y)) -> c_8()}

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

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

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

             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [       MAYBE        ]
                    |
                    `->{7}                                               [     inherited      ]
                        |
                        `->{6}                                           [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

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

           * Path {1}->{2}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1) = [0] x1 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(terms^#) = {}, Uargs(c_0) = {1}, Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                terms^#(x1) = [7] x1 + [7]
                c_0(x1) = [1] x1 + [3]
                sqr^#(x1) = [7] x1 + [1]
                c_1() = [0]

           * Path {1}->{3}: inherited
             ------------------------

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

           * Path {1}->{3}->{6}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , add^#(0(), X) -> c_5()
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{3}->{7}: inherited
             -----------------------------

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

           * Path {1}->{3}->{7}->{6}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

           * Path {8}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1) = [0] x1 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_6) = {},
                 Uargs(first^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1) = [0] x1 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]

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

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

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

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

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

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

Tool RC1

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

stdout:

MAYBE

Tool RC2

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

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y)) -> c_8(Y)}

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

             ->{9}                                                       [   YES(?,O(n^3))    ]

             ->{8}                                                       [    YES(?,O(1))     ]

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

             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [       MAYBE        ]
                    |
                    `->{7}                                               [     inherited      ]
                        |
                        `->{6}                                           [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

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

           * Path {1}->{2}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(terms^#) = {}, Uargs(c_0) = {1}, Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                terms^#(x1) = [7 7 7] x1 + [7]
                              [7 7 7]      [7]
                              [7 7 7]      [7]
                c_0(x1) = [2 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
                sqr^#(x1) = [2 0 2] x1 + [0]
                            [2 0 0]      [0]
                            [2 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]

           * Path {1}->{3}: inherited
             ------------------------

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

           * Path {1}->{3}->{6}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , add^#(0(), X) -> c_5(X)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{3}->{7}: inherited
             -----------------------------

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

           * Path {1}->{3}->{7}->{6}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

           * Path {8}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y)) -> c_8(Y)}
               Weak Rules: {}

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

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

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y)) -> c_8(Y)}

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

             ->{9}                                                       [   YES(?,O(n^2))    ]

             ->{8}                                                       [    YES(?,O(1))     ]

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

             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [       MAYBE        ]
                    |
                    `->{7}                                               [     inherited      ]
                        |
                        `->{6}                                           [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

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

           * Path {1}->{2}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(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) = [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(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(terms^#) = {}, Uargs(c_0) = {1}, Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                terms^#(x1) = [7 7] x1 + [7]
                              [7 7]      [7]
                c_0(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
                sqr^#(x1) = [0 0] x1 + [4]
                            [2 2]      [0]
                c_1() = [1]
                        [0]

           * Path {1}->{3}: inherited
             ------------------------

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

           * Path {1}->{3}->{6}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , add^#(0(), X) -> c_5(X)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{3}->{7}: inherited
             -----------------------------

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

           * Path {1}->{3}->{7}->{6}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

           * Path {8}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(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) = [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(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: 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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 1]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 0]      [0]

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

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

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

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

             {  1: terms^#(N) -> c_0(sqr^#(N))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y)) -> c_8(Y)}

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

             ->{9}                                                       [   YES(?,O(n^1))    ]

             ->{8}                                                       [    YES(?,O(1))     ]

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

             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [       MAYBE        ]
                    |
                    `->{7}                                               [     inherited      ]
                        |
                        `->{6}                                           [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------

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

           * Path {1}->{2}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1) = [0] x1 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sqr^#(0()) -> c_1()}
               Weak Rules: {terms^#(N) -> c_0(sqr^#(N))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(terms^#) = {}, Uargs(c_0) = {1}, Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                terms^#(x1) = [7] x1 + [7]
                c_0(x1) = [1] x1 + [3]
                sqr^#(x1) = [7] x1 + [1]
                c_1() = [0]

           * Path {1}->{3}: inherited
             ------------------------

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

           * Path {1}->{3}->{6}: MAYBE
             -------------------------

             The usable rules for this path are:

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

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

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
                  , terms^#(N) -> c_0(sqr^#(N))
                  , add^#(0(), X) -> c_5(X)
                  , sqr(0()) -> 0()
                  , sqr(s(X)) -> s(add(sqr(X), dbl(X)))
                  , dbl(0()) -> 0()
                  , dbl(s(X)) -> s(s(dbl(X)))
                  , add(0(), X) -> X
                  , add(s(X), Y) -> s(add(X, Y))}

             Proof Output:
               The input cannot be shown compatible

           * Path {1}->{3}->{7}: inherited
             -----------------------------

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

           * Path {1}->{3}->{7}->{6}: NA
             ---------------------------

             The usable rules for this path are:

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

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

             The usable rules of this path are empty.

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

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

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

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

           * Path {8}: YES(?,O(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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1) = [0] x1 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: 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(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {},
                 Uargs(sqr^#) = {}, Uargs(c_2) = {}, Uargs(add^#) = {},
                 Uargs(dbl^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1) = [0] x1 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(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: {first^#(s(X), cons(Y)) -> c_8(Y)}
               Weak Rules: {}

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

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

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

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