Problem Strategy outermost added 08 ExSec11 1 Luc02a

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExSec11 1 Luc02a

stdout:

MAYBE

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

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExSec11 1 Luc02a

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExSec11 1 Luc02a

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))
              , 10: half^#(0()) -> c_9()
              , 11: half^#(s(0())) -> c_10()
              , 12: half^#(s(s(X))) -> c_11(half^#(X))
              , 13: half^#(dbl(X)) -> c_12()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{12}                                                      [   YES(?,O(n^2))    ]
                |
                |->{10}                                                  [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [   YES(?,O(n^2))    ]
                |
                `->{13}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^3))    ]
                |
                `->{8}                                                   [   YES(?,O(n^1))    ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [         NA         ]
                    |
                    `->{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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1, 2}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12() = [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), terms^#(s(N)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
                 Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [0]
                      [0]
                terms^#(x1) = [6 0 0] x1 + [0]
                              [2 2 2]      [0]
                              [2 2 6]      [0]
                c_0(x1, x2) = [2 0 0] x1 + [2 2 2] x2 + [0]
                              [0 0 0]      [2 2 0]      [0]
                              [0 0 0]      [2 2 2]      [0]
                sqr^#(x1) = [2 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [2 0 0]      [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}: 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 {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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5() = [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]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12() = [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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5() = [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]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                dbl^#(x1) = [2 2 2] x1 + [0]
                            [0 6 0]      [0]
                            [0 0 2]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
           
           * Path {9}: 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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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]
                                  [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12() = [0]
                         [0]
                         [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                s(x1) = [1 4 4] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                first^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
                                  [2 0 2]      [2 0 0]      [0]
                                  [0 2 2]      [1 0 0]      [0]
                c_8(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]
           
           * Path {9}->{8}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12() = [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:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [0]
                               [0 0 0]      [0 0 4]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [0]
                      [0]
                first^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
                                  [2 0 0]      [2 0 0]      [4]
                                  [0 0 0]      [0 2 0]      [2]
                c_7() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [0]
                          [0 0 0]      [2]
           
           * Path {12}: YES(?,O(n^2))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0 0] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12() = [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: {half^#(s(s(X))) -> c_11(half^#(X))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [2]
                half^#(x1) = [2 0 2] x1 + [0]
                             [4 0 2]      [0]
                             [0 0 0]      [2]
                c_11(x1) = [1 0 2] x1 + [3]
                           [2 0 0]      [0]
                           [0 0 0]      [2]
           
           * Path {12}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12() = [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:    innermost DP runtime-complexity with respect to
               Strict Rules: {half^#(0()) -> c_9()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                half^#(x1) = [2 0 0] x1 + [0]
                             [0 0 2]      [4]
                             [1 2 2]      [0]
                c_9() = [1]
                        [0]
                        [0]
                c_11(x1) = [1 0 0] x1 + [3]
                           [0 0 0]      [7]
                           [0 0 0]      [6]
           
           * Path {12}->{11}: YES(?,O(n^2))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12() = [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: {half^#(s(0())) -> c_10()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                half^#(x1) = [2 3 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 5 0]      [0]
                c_10() = [1]
                         [0]
                         [0]
                c_11(x1) = [1 0 1] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
           
           * Path {12}->{13}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12() = [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:    innermost DP runtime-complexity with respect to
               Strict Rules: {half^#(dbl(X)) -> c_12()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl) = {}, Uargs(half^#) = {},
                 Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                dbl(x1) = [0 0 0] x1 + [2]
                          [0 0 0]      [2]
                          [0 0 0]      [2]
                half^#(x1) = [2 0 0] x1 + [0]
                             [0 0 2]      [4]
                             [1 2 2]      [0]
                c_11(x1) = [1 0 0] x1 + [3]
                           [0 0 0]      [7]
                           [0 0 0]      [6]
                c_12() = [1]
                         [0]
                         [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))
              , 10: half^#(0()) -> c_9()
              , 11: half^#(s(0())) -> c_10()
              , 12: half^#(s(s(X))) -> c_11(half^#(X))
              , 13: half^#(dbl(X)) -> c_12()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{12}                                                      [   YES(?,O(n^1))    ]
                |
                |->{10}                                                  [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [   YES(?,O(n^2))    ]
                |
                `->{13}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^2))    ]
                |
                `->{8}                                                   [   YES(?,O(n^2))    ]
             
             ->{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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1, 2}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12() = [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), terms^#(s(N)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
                 Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [2]
                      [2]
                terms^#(x1) = [2 4] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [1 0] x1 + [2 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [2 2] x1 + [0]
                            [0 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), terms^#(s(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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5() = [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]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12() = [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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5() = [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]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12() = [0]
                         [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                dbl^#(x1) = [1 2] x1 + [2]
                            [6 1]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
           
           * Path {9}: 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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12() = [0]
                         [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [4]
                s(x1) = [1 2] x1 + [2]
                        [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 2] x2 + [0]
                                  [4 0]      [4 0]      [0]
                c_8(x1) = [1 0] x1 + [5]
                          [0 0]      [7]
           
           * Path {9}->{8}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12() = [0]
                         [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 6] x2 + [0]
                               [0 0]      [0 1]      [2]
                s(x1) = [1 6] x1 + [0]
                        [0 1]      [2]
                0() = [0]
                      [2]
                first^#(x1, x2) = [1 2] x1 + [2 2] x2 + [0]
                                  [0 2]      [0 3]      [0]
                c_7() = [1]
                        [0]
                c_8(x1) = [1 2] x1 + [6]
                          [0 0]      [2]
           
           * Path {12}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0] x1 + [0]
                             [3 3]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12() = [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: {half^#(s(s(X))) -> c_11(half^#(X))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                half^#(x1) = [2 2] x1 + [2]
                             [6 0]      [0]
                c_11(x1) = [1 0] x1 + [5]
                           [2 0]      [7]
           
           * Path {12}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12() = [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: {half^#(0()) -> c_9()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                half^#(x1) = [2 2] x1 + [4]
                             [6 2]      [0]
                c_9() = [1]
                        [0]
                c_11(x1) = [1 0] x1 + [3]
                           [2 0]      [3]
           
           * Path {12}->{11}: YES(?,O(n^2))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12() = [0]
                         [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {half^#(s(0())) -> c_10()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                0() = [1]
                      [0]
                half^#(x1) = [2 1] x1 + [1]
                             [0 0]      [7]
                c_10() = [1]
                         [1]
                c_11(x1) = [1 1] x1 + [3]
                           [0 0]      [3]
           
           * Path {12}->{13}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5() = [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]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12() = [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: {half^#(dbl(X)) -> c_12()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl) = {}, Uargs(half^#) = {},
                 Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [2]
                half^#(x1) = [2 2] x1 + [4]
                             [6 2]      [0]
                c_11(x1) = [1 0] x1 + [3]
                           [2 0]      [3]
                c_12() = [1]
                         [0]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5()
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))
              , 10: half^#(0()) -> c_9()
              , 11: half^#(s(0())) -> c_10()
              , 12: half^#(s(s(X))) -> c_11(half^#(X))
              , 13: half^#(dbl(X)) -> c_12()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{12}                                                      [   YES(?,O(n^1))    ]
                |
                |->{10}                                                  [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [    YES(?,O(1))     ]
                |
                `->{13}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^1))    ]
                |
                `->{8}                                                   [   YES(?,O(n^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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1, 2}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [1] x1 + [1] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12() = [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), terms^#(s(N)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
                 Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [1]
                terms^#(x1) = [6] x1 + [0]
                c_0(x1, x2) = [4] x1 + [2] x2 + [0]
                sqr^#(x1) = [1] x1 + [0]
                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), terms^#(s(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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12() = [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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12() = [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                dbl^#(x1) = [2] x1 + [0]
                c_3() = [1]
                c_4(x1) = [1] x1 + [0]
           
           * Path {9}: 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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12() = [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [2]
                first^#(x1, x2) = [3] x1 + [0] x2 + [2]
                c_8(x1) = [1] x1 + [5]
           
           * Path {9}->{8}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {1},
                 Uargs(half^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12() = [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(first^#(X, Z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                s(x1) = [1] x1 + [0]
                0() = [2]
                first^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_7() = [1]
                c_8(x1) = [1] x1 + [3]
           
           * Path {12}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [3] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12() = [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: {half^#(s(s(X))) -> c_11(half^#(X))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                half^#(x1) = [2] x1 + [0]
                c_11(x1) = [1] x1 + [7]
           
           * Path {12}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12() = [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: {half^#(0()) -> c_9()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                half^#(x1) = [2] x1 + [4]
                c_9() = [1]
                c_11(x1) = [1] x1 + [0]
           
           * Path {12}->{11}: 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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12() = [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: {half^#(s(0())) -> c_10()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [0]
                half^#(x1) = [0] x1 + [1]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
           
           * Path {12}->{13}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_6) = {}, Uargs(first^#) = {}, Uargs(c_8) = {},
                 Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12() = [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: {half^#(dbl(X)) -> c_12()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl) = {}, Uargs(half^#) = {},
                 Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                dbl(x1) = [0] x1 + [2]
                half^#(x1) = [2] x1 + [4]
                c_11(x1) = [1] x1 + [0]
                c_12() = [1]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExSec11 1 Luc02a

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExSec11 1 Luc02a

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
              , 10: half^#(0()) -> c_9()
              , 11: half^#(s(0())) -> c_10()
              , 12: half^#(s(s(X))) -> c_11(half^#(X))
              , 13: half^#(dbl(X)) -> c_12(X)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{12}                                                      [   YES(?,O(n^2))    ]
                |
                |->{10}                                                  [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [   YES(?,O(n^2))    ]
                |
                `->{13}                                                  [   YES(?,O(n^2))    ]
             
             ->{9}                                                       [   YES(?,O(n^3))    ]
                |
                `->{8}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [    YES(?,O(1))     ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{6}                                               [         NA         ]
                    |
                    `->{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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1, 2}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [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, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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), terms^#(s(N)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
                 Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [0]
                      [0]
                terms^#(x1) = [6 0 0] x1 + [0]
                              [2 2 2]      [0]
                              [2 2 6]      [0]
                c_0(x1, x2) = [2 0 0] x1 + [2 2 2] x2 + [0]
                              [0 0 0]      [2 2 0]      [0]
                              [0 0 0]      [2 2 2]      [0]
                sqr^#(x1) = [2 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [2 0 0]      [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}: 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 {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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                dbl^#(x1) = [2 2 2] x1 + [0]
                            [0 6 0]      [0]
                            [0 0 2]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
           
           * Path {9}: 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(half) = {}, 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) = {2}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
                               [0 1 3]      [0 1 0]      [0]
                               [0 0 1]      [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]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 0] x1 + [1 0 2] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [6]
                s(x1) = [1 1 1] x1 + [0]
                        [0 1 2]      [3]
                        [0 0 1]      [0]
                first^#(x1, x2) = [1 1 1] x1 + [2 0 1] x2 + [0]
                                  [4 2 0]      [0 0 0]      [2]
                                  [0 4 0]      [4 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [6]
                              [0 0 0]      [0 0 0]      [7]
           
           * Path {9}->{8}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {2}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0 2] x1 + [1 3 0] x2 + [1]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 2 0] x1 + [2]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                first^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                  [2 2 2]      [0 5 0]      [0]
                                  [0 0 0]      [1 1 0]      [0]
                c_7() = [1]
                        [0]
                        [0]
                c_8(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [2]
                              [1 0 0]      [0 0 0]      [1]
           
           * Path {12}: YES(?,O(n^2))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [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: {half^#(s(s(X))) -> c_11(half^#(X))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [2]
                half^#(x1) = [2 0 2] x1 + [0]
                             [4 0 2]      [0]
                             [0 0 0]      [2]
                c_11(x1) = [1 0 2] x1 + [3]
                           [2 0 0]      [0]
                           [0 0 0]      [2]
           
           * Path {12}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {half^#(0()) -> c_9()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                half^#(x1) = [2 0 0] x1 + [0]
                             [0 0 2]      [4]
                             [1 2 2]      [0]
                c_9() = [1]
                        [0]
                        [0]
                c_11(x1) = [1 0 0] x1 + [3]
                           [0 0 0]      [7]
                           [0 0 0]      [6]
           
           * Path {12}->{11}: YES(?,O(n^2))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [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: {half^#(s(0())) -> c_10()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                half^#(x1) = [2 3 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 5 0]      [0]
                c_10() = [1]
                         [0]
                         [0]
                c_11(x1) = [1 0 1] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
           
           * Path {12}->{13}: YES(?,O(n^2))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                recip(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                sqr(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                dbl(x1) = [0 3 3] x1 + [0]
                          [0 3 3]      [0]
                          [0 3 3]      [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]
                half(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                terms^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sqr^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                dbl^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                half^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10() = [0]
                         [0]
                         [0]
                c_11(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_12(x1) = [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: {half^#(dbl(X)) -> c_12(X)}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl) = {}, Uargs(half^#) = {},
                 Uargs(c_11) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                dbl(x1) = [2 2 2] x1 + [2]
                          [2 2 2]      [2]
                          [2 2 2]      [2]
                half^#(x1) = [3 2 2] x1 + [0]
                             [2 2 2]      [0]
                             [2 2 2]      [0]
                c_11(x1) = [1 0 0] x1 + [3]
                           [0 0 0]      [7]
                           [0 0 0]      [3]
                c_12(x1) = [0 0 0] x1 + [1]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
              , 10: half^#(0()) -> c_9()
              , 11: half^#(s(0())) -> c_10()
              , 12: half^#(s(s(X))) -> c_11(half^#(X))
              , 13: half^#(dbl(X)) -> c_12(X)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{12}                                                      [   YES(?,O(n^1))    ]
                |
                |->{10}                                                  [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [   YES(?,O(n^2))    ]
                |
                `->{13}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^2))    ]
                |
                `->{8}                                                   [   YES(?,O(n^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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1, 2}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [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, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             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), terms^#(s(N)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
                 Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [2]
                      [2]
                terms^#(x1) = [2 4] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [1 0] x1 + [2 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [2 2] x1 + [0]
                            [0 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), terms^#(s(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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                dbl^#(x1) = [1 2] x1 + [2]
                            [6 1]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
           
           * Path {9}: 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(half) = {}, 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) = {2}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [0]
                first^#(x1, x2) = [0 2] x1 + [2 2] x2 + [2]
                                  [2 5]      [6 1]      [0]
                c_8(x1, x2) = [0 0] x1 + [1 0] x2 + [5]
                              [0 0]      [2 0]      [6]
           
           * Path {9}->{8}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {2}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             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^#(0(), X) -> c_7()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 4] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                0() = [2]
                      [2]
                first^#(x1, x2) = [2 0] x1 + [2 0] x2 + [4]
                                  [2 2]      [1 0]      [4]
                c_7() = [1]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [1 0]      [0 0]      [2]
           
           * Path {12}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [3 3]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(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: {half^#(s(s(X))) -> c_11(half^#(X))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                half^#(x1) = [2 2] x1 + [2]
                             [6 0]      [0]
                c_11(x1) = [1 0] x1 + [5]
                           [2 0]      [7]
           
           * Path {12}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [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: {half^#(0()) -> c_9()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                half^#(x1) = [2 2] x1 + [4]
                             [6 2]      [0]
                c_9() = [1]
                        [0]
                c_11(x1) = [1 0] x1 + [3]
                           [2 0]      [3]
           
           * Path {12}->{11}: YES(?,O(n^2))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {half^#(s(0())) -> c_10()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                0() = [1]
                      [0]
                half^#(x1) = [2 1] x1 + [1]
                             [0 0]      [7]
                c_10() = [1]
                         [1]
                c_11(x1) = [1 1] x1 + [3]
                           [0 0]      [3]
           
           * Path {12}->{13}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                recip(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                sqr(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                dbl(x1) = [3 3] x1 + [0]
                          [3 3]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                terms^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sqr^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                dbl^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                half^#(x1) = [1 2] x1 + [0]
                             [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_12(x1) = [1 1] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {half^#(dbl(X)) -> c_12(X)}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl) = {}, Uargs(half^#) = {},
                 Uargs(c_11) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [2]
                dbl(x1) = [2 2] x1 + [2]
                          [2 2]      [2]
                half^#(x1) = [2 2] x1 + [0]
                             [2 2]      [0]
                c_11(x1) = [1 0] x1 + [7]
                           [0 0]      [6]
                c_12(x1) = [0 0] x1 + [1]
                           [0 0]      [0]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: terms^#(N) -> c_0(sqr^#(N), terms^#(s(N)))
              , 2: sqr^#(0()) -> c_1()
              , 3: sqr^#(s(X)) -> c_2(add^#(sqr(X), dbl(X)))
              , 4: dbl^#(0()) -> c_3()
              , 5: dbl^#(s(X)) -> c_4(dbl^#(X))
              , 6: add^#(0(), X) -> c_5(X)
              , 7: add^#(s(X), Y) -> c_6(add^#(X, Y))
              , 8: first^#(0(), X) -> c_7()
              , 9: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))
              , 10: half^#(0()) -> c_9()
              , 11: half^#(s(0())) -> c_10()
              , 12: half^#(s(s(X))) -> c_11(half^#(X))
              , 13: half^#(dbl(X)) -> c_12(X)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{12}                                                      [   YES(?,O(n^1))    ]
                |
                |->{10}                                                  [   YES(?,O(n^1))    ]
                |
                |->{11}                                                  [    YES(?,O(1))     ]
                |
                `->{13}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^1))    ]
                |
                `->{8}                                                   [   YES(?,O(n^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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {1, 2}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [1] x1 + [1] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(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), terms^#(s(N)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(terms^#) = {}, Uargs(c_0) = {1, 2},
                 Uargs(sqr^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [1]
                terms^#(x1) = [6] x1 + [0]
                c_0(x1, x2) = [4] x1 + [2] x2 + [0]
                sqr^#(x1) = [1] x1 + [0]
                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), terms^#(s(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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(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(half) = {}, Uargs(terms^#) = {},
                 Uargs(c_0) = {}, Uargs(sqr^#) = {}, Uargs(c_2) = {},
                 Uargs(add^#) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {dbl^#(0()) -> c_3()}
               Weak Rules: {dbl^#(s(X)) -> c_4(dbl^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                dbl^#(x1) = [2] x1 + [0]
                c_3() = [1]
                c_4(x1) = [1] x1 + [0]
           
           * Path {9}: 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(half) = {}, 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) = {2}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [1] x1 + [1] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                s(x1) = [1] x1 + [0]
                first^#(x1, x2) = [0] x1 + [2] x2 + [4]
                c_8(x1, x2) = [0] x1 + [1] x2 + [3]
           
           * Path {9}->{8}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {2}, Uargs(half^#) = {}, Uargs(c_11) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [1] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
                c_12(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_7()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(first^#) = {},
                 Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [2]
                0() = [2]
                first^#(x1, x2) = [3] x1 + [0] x2 + [2]
                c_7() = [1]
                c_8(x1, x2) = [0] x1 + [1] x2 + [5]
           
           * Path {12}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [3] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12(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: {half^#(s(s(X))) -> c_11(half^#(X))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                half^#(x1) = [2] x1 + [0]
                c_11(x1) = [1] x1 + [7]
           
           * Path {12}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12(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: {half^#(0()) -> c_9()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                half^#(x1) = [2] x1 + [4]
                c_9() = [1]
                c_11(x1) = [1] x1 + [0]
           
           * Path {12}->{11}: 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(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12(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: {half^#(s(0())) -> c_10()}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [0]
                half^#(x1) = [0] x1 + [1]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
           
           * Path {12}->{13}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(terms) = {}, Uargs(cons) = {}, Uargs(recip) = {},
                 Uargs(sqr) = {}, Uargs(s) = {}, Uargs(add) = {}, Uargs(dbl) = {},
                 Uargs(first) = {}, Uargs(half) = {}, 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) = {}, Uargs(half^#) = {}, Uargs(c_11) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                terms(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                recip(x1) = [0] x1 + [0]
                sqr(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl(x1) = [3] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                half(x1) = [0] x1 + [0]
                terms^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sqr^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                dbl^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]
                half^#(x1) = [1] x1 + [0]
                c_9() = [0]
                c_10() = [0]
                c_11(x1) = [1] x1 + [0]
                c_12(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: {half^#(dbl(X)) -> c_12(X)}
               Weak Rules: {half^#(s(s(X))) -> c_11(half^#(X))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(dbl) = {}, Uargs(half^#) = {},
                 Uargs(c_11) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                dbl(x1) = [5] x1 + [2]
                half^#(x1) = [2] x1 + [0]
                c_11(x1) = [1] x1 + [0]
                c_12(x1) = [0] x1 + [1]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.