Problem Maude 06 LengthOfFiniteLists nokinds

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LengthOfFiniteLists nokinds

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LengthOfFiniteLists nokinds

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  zeros() -> cons(0(), zeros())
     , U11(tt(), L) -> s(length(L))
     , and(tt(), X) -> X
     , isNat(0()) -> tt()
     , isNat(length(V1)) -> isNatList(V1)
     , isNat(s(V1)) -> isNat(V1)
     , isNatIList(V) -> isNatList(V)
     , isNatIList(zeros()) -> tt()
     , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
     , isNatList(nil()) -> tt()
     , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
     , length(nil()) -> 0()
     , length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)}

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: zeros^#() -> c_0(zeros^#())
              , 2: U11^#(tt(), L) -> c_1(length^#(L))
              , 3: and^#(tt(), X) -> c_2()
              , 4: isNat^#(0()) -> c_3()
              , 5: isNat^#(length(V1)) -> c_4(isNatList^#(V1))
              , 6: isNat^#(s(V1)) -> c_5(isNat^#(V1))
              , 7: isNatIList^#(V) -> c_6(isNatList^#(V))
              , 8: isNatIList^#(zeros()) -> c_7()
              , 9: isNatIList^#(cons(V1, V2)) ->
                   c_8(and^#(isNat(V1), isNatIList(V2)))
              , 10: isNatList^#(nil()) -> c_9()
              , 11: isNatList^#(cons(V1, V2)) ->
                    c_10(and^#(isNat(V1), isNatList(V2)))
              , 12: length^#(nil()) -> c_11()
              , 13: length^#(cons(N, L)) ->
                    c_12(U11^#(and(isNatList(L), isNat(N)), L))}

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

             ->{9}                                                       [         NA         ]
                |
                `->{3}                                                   [         NA         ]

             ->{8}                                                       [         NA         ]

             ->{7}                                                       [         NA         ]
                |
                |->{10}                                                  [         NA         ]
                |
                `->{11}                                                  [         NA         ]
                    |
                    `->{3}                                               [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]
                    |
                    |->{10}                                              [         NA         ]
                    |
                    `->{11}                                              [         NA         ]
                        |
                        `->{3}                                           [         NA         ]

             ->{2,13}                                                    [         NA         ]
                |
                `->{12}                                                  [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {1},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [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:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zeros^#() -> c_0(zeros^#())}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2,13}: NA
             ---------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 1] x1 + [1 2 1] x2 + [0]
                               [0 0 2]      [0 1 0]      [1]
                               [0 0 0]      [0 0 1]      [3]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 2]      [2]
                and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 2 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 1] x1 + [1]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 1 1] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [1]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(x1, x2) = [1 0 0] x1 + [2 2 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length^#(x1) = [2 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {2,13}->{12}: NA
             ---------------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 3]      [0 1 0]      [3]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [2]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 2 0]      [0]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 2] x1 + [2]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 3 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [1]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {6}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{4}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{5}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{5}->{10}: NA
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{5}->{11}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 2 0] x1 + [1 0 1] x2 + [3]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [0]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [1]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [1]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [3 0 0]      [3]
                             [0 0 0]      [0]
                and(x1, x2) = [1 0 2] x1 + [1 0 1] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 1 0] x1 + [1]
                            [0 0 0]      [0]
                            [0 0 0]      [2]
                isNatList(x1) = [3 0 0] x1 + [3]
                                [0 0 0]      [0]
                                [0 0 0]      [1]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [3 0 2] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {6}->{5}->{11}->{3}: NA
             ----------------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 1 1]      [1]
                               [0 0 1]      [0 0 0]      [3]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 1]      [0]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 0 1] x1 + [2]
                            [0 0 0]      [0]
                            [0 0 1]      [0]
                isNatList(x1) = [0 1 1] x1 + [0]
                                [0 0 0]      [0]
                                [0 1 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [1]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {7}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [1 0 1] x1 + [0]
                                  [1 0 1]      [0]
                                  [0 3 1]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNatIList^#(x1) = [3 3 3] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 1 1] x1 + [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]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {7}->{10}: NA
             ------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(x1) = [0 0 0] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {7}->{11}: NA
             ------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 1] x1 + [1 1 1] x2 + [0]
                               [0 0 2]      [0 1 1]      [3]
                               [0 0 0]      [0 0 0]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [1]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [2]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 2]      [2]
                and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 2 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 1] x1 + [2]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 1 1] x1 + [3]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(x1) = [1 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]
                isNatIList^#(x1) = [0 0 0] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {7}->{11}->{3}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 3]      [0 1 2]      [3]
                               [0 0 1]      [0 0 0]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [2]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 2]      [3]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 2 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 0 1] x1 + [1]
                            [0 0 1]      [1]
                            [0 0 2]      [0]
                isNatList(x1) = [0 1 1] x1 + [3]
                                [0 1 2]      [2]
                                [0 2 0]      [2]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(x1) = [0 0 0] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {8}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {9}: NA
             ------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 2] x1 + [1 2 0] x2 + [1]
                               [0 1 1]      [0 1 0]      [2]
                               [0 0 0]      [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 2 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 1]      [0]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 1 0]      [1]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 0 1] x1 + [2]
                            [1 0 0]      [0]
                            [2 0 0]      [0]
                isNatList(x1) = [0 1 1] x1 + [1]
                                [0 1 0]      [0]
                                [0 1 0]      [0]
                isNatIList(x1) = [2 1 1] x1 + [2]
                                 [0 1 0]      [3]
                                 [0 1 1]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(x1) = [2 0 2] 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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {9}->{3}: NA
             -----------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 1 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 1]      [0 0 1]      [3]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 1 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 2]      [2]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 0 1] x1 + [1]
                            [1 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 1] x1 + [1]
                                [0 1 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 1] x1 + [2]
                                 [0 1 0]      [0]
                                 [1 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

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

             {  1: zeros^#() -> c_0(zeros^#())
              , 2: U11^#(tt(), L) -> c_1(length^#(L))
              , 3: and^#(tt(), X) -> c_2()
              , 4: isNat^#(0()) -> c_3()
              , 5: isNat^#(length(V1)) -> c_4(isNatList^#(V1))
              , 6: isNat^#(s(V1)) -> c_5(isNat^#(V1))
              , 7: isNatIList^#(V) -> c_6(isNatList^#(V))
              , 8: isNatIList^#(zeros()) -> c_7()
              , 9: isNatIList^#(cons(V1, V2)) ->
                   c_8(and^#(isNat(V1), isNatIList(V2)))
              , 10: isNatList^#(nil()) -> c_9()
              , 11: isNatList^#(cons(V1, V2)) ->
                    c_10(and^#(isNat(V1), isNatList(V2)))
              , 12: length^#(nil()) -> c_11()
              , 13: length^#(cons(N, L)) ->
                    c_12(U11^#(and(isNatList(L), isNat(N)), L))}

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

             ->{9}                                                       [         NA         ]
                |
                `->{3}                                                   [         NA         ]

             ->{8}                                                       [         NA         ]

             ->{7}                                                       [         NA         ]
                |
                |->{10}                                                  [         NA         ]
                |
                `->{11}                                                  [         NA         ]
                    |
                    `->{3}                                               [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]
                    |
                    |->{10}                                              [         NA         ]
                    |
                    `->{11}                                              [         NA         ]
                        |
                        `->{3}                                           [         NA         ]

             ->{2,13}                                                    [         NA         ]
                |
                `->{12}                                                  [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {1},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [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:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zeros^#() -> c_0(zeros^#())}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2,13}: NA
             ---------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 1]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 1]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [2]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [1 0] x1 + [2 2] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [2 2] x1 + [0]
                               [3 3]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {2,13}->{12}: NA
             ---------------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [2]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [1]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 1] x1 + [0]
                             [0 3]      [2]
                and(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [2]
                            [2 0]      [0]
                isNatList(x1) = [0 3] x1 + [2]
                                [0 1]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {6}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [3 3] x1 + [0]
                              [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{4}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{5}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{5}->{10}: NA
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{5}->{11}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [3]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [1]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatList^#(x1) = [1 2] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {6}->{5}->{11}->{3}: NA
             ----------------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [3]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 2]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [1]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [1 1] x1 + [0]
                                  [3 3]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [3 3] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {7}->{10}: NA
             ------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {7}->{11}: NA
             ------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 0] x1 + [0]
                             [0 3]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 3]      [0]
                isNatList(x1) = [0 3] x1 + [0]
                                [0 2]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [3]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [2 2] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {7}->{11}->{3}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [2]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 2] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 2] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [1]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {8}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {9}: NA
             ------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 2]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [2 1] x1 + [2]
                                 [0 0]      [0]
                nil() = [0]
                        [2]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [2 1] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {9}->{3}: NA
             -----------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [1]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [1]
                and(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [1]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 0]      [1]
                isNatIList(x1) = [0 3] x1 + [2]
                                 [0 0]      [1]
                nil() = [0]
                        [1]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             {  1: zeros^#() -> c_0(zeros^#())
              , 2: U11^#(tt(), L) -> c_1(length^#(L))
              , 3: and^#(tt(), X) -> c_2()
              , 4: isNat^#(0()) -> c_3()
              , 5: isNat^#(length(V1)) -> c_4(isNatList^#(V1))
              , 6: isNat^#(s(V1)) -> c_5(isNat^#(V1))
              , 7: isNatIList^#(V) -> c_6(isNatList^#(V))
              , 8: isNatIList^#(zeros()) -> c_7()
              , 9: isNatIList^#(cons(V1, V2)) ->
                   c_8(and^#(isNat(V1), isNatIList(V2)))
              , 10: isNatList^#(nil()) -> c_9()
              , 11: isNatList^#(cons(V1, V2)) ->
                    c_10(and^#(isNat(V1), isNatList(V2)))
              , 12: length^#(nil()) -> c_11()
              , 13: length^#(cons(N, L)) ->
                    c_12(U11^#(and(isNatList(L), isNat(N)), L))}

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

             ->{9}                                                       [         NA         ]
                |
                `->{3}                                                   [         NA         ]

             ->{8}                                                       [         NA         ]

             ->{7}                                                       [         NA         ]
                |
                |->{10}                                                  [         NA         ]
                |
                `->{11}                                                  [         NA         ]
                    |
                    `->{3}                                               [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]
                    |
                    |->{10}                                              [         NA         ]
                    |
                    `->{11}                                              [         NA         ]
                        |
                        `->{3}                                           [         NA         ]

             ->{2,13}                                                    [         NA         ]
                |
                `->{12}                                                  [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {1},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [1] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zeros^#() -> c_0(zeros^#())}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2,13}: NA
             ---------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             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 {2,13}->{12}: NA
             ---------------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [3]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [1]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [2]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {6}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{4}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{5}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{5}->{10}: NA
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{5}->{11}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [1]
                s(x1) = [1] x1 + [3]
                length(x1) = [3] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [0]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [1] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                isNatList^#(x1) = [3] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {6}->{5}->{11}->{3}: NA
             ----------------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {1},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [3]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [2]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [2]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [3] x1 + [0]
                c_6(x1) = [3] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {7}->{10}: NA
             ------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {7}->{11}: NA
             ------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [1] x1 + [1] x2 + [1]
                isNat(x1) = [2] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [1] x1 + [1] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [3] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}->{11}->{3}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_10) = {1},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [1]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {8}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {9}: NA
             ------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [1]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [2]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [2]
                isNatIList(x1) = [3] x1 + [3]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [1] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [3] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {9}->{3}: NA
             -----------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(isNat^#) = {}, Uargs(c_4) = {},
                 Uargs(isNatList^#) = {}, Uargs(c_5) = {}, Uargs(isNatIList^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_10) = {},
                 Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [3]
                cons(x1, x2) = [1] x1 + [1] x2 + [3]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [1]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [2] x1 + [2]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LengthOfFiniteLists nokinds

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Maude 06 LengthOfFiniteLists nokinds

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  zeros() -> cons(0(), zeros())
     , U11(tt(), L) -> s(length(L))
     , and(tt(), X) -> X
     , isNat(0()) -> tt()
     , isNat(length(V1)) -> isNatList(V1)
     , isNat(s(V1)) -> isNat(V1)
     , isNatIList(V) -> isNatList(V)
     , isNatIList(zeros()) -> tt()
     , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
     , isNatList(nil()) -> tt()
     , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
     , length(nil()) -> 0()
     , length(cons(N, L)) -> U11(and(isNatList(L), isNat(N)), L)}

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: zeros^#() -> c_0(zeros^#())
              , 2: U11^#(tt(), L) -> c_1(length^#(L))
              , 3: and^#(tt(), X) -> c_2(X)
              , 4: isNat^#(0()) -> c_3()
              , 5: isNat^#(length(V1)) -> c_4(isNatList^#(V1))
              , 6: isNat^#(s(V1)) -> c_5(isNat^#(V1))
              , 7: isNatIList^#(V) -> c_6(isNatList^#(V))
              , 8: isNatIList^#(zeros()) -> c_7()
              , 9: isNatIList^#(cons(V1, V2)) ->
                   c_8(and^#(isNat(V1), isNatIList(V2)))
              , 10: isNatList^#(nil()) -> c_9()
              , 11: isNatList^#(cons(V1, V2)) ->
                    c_10(and^#(isNat(V1), isNatList(V2)))
              , 12: length^#(nil()) -> c_11()
              , 13: length^#(cons(N, L)) ->
                    c_12(U11^#(and(isNatList(L), isNat(N)), L))}

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

             ->{9}                                                       [         NA         ]
                |
                `->{3}                                                   [         NA         ]

             ->{8}                                                       [         NA         ]

             ->{7}                                                       [         NA         ]
                |
                |->{10}                                                  [         NA         ]
                |
                `->{11}                                                  [         NA         ]
                    |
                    `->{3}                                               [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]
                    |
                    |->{10}                                              [         NA         ]
                    |
                    `->{11}                                              [         NA         ]
                        |
                        `->{3}                                           [         NA         ]

             ->{2,13}                                                    [         NA         ]
                |
                `->{12}                                                  [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {1},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [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:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zeros^#() -> c_0(zeros^#())}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2,13}: NA
             ---------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 1] x1 + [1 2 1] x2 + [0]
                               [0 0 2]      [0 1 0]      [1]
                               [0 0 0]      [0 0 1]      [3]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 2]      [2]
                and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 2 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 1] x1 + [1]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 1 1] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [1]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(x1, x2) = [1 0 0] x1 + [2 2 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length^#(x1) = [2 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {2,13}->{12}: NA
             ---------------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 3]      [0 1 0]      [3]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [2]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 2 0]      [0]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 2] x1 + [2]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 3 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [1]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {6}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{4}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{5}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{5}->{10}: NA
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {6}->{5}->{11}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 2 0] x1 + [1 0 1] x2 + [3]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [0]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [1]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [1]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [3 0 0]      [3]
                             [0 0 0]      [0]
                and(x1, x2) = [1 0 2] x1 + [1 0 1] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 1 0] x1 + [1]
                            [0 0 0]      [0]
                            [0 0 0]      [2]
                isNatList(x1) = [3 0 0] x1 + [3]
                                [0 0 0]      [0]
                                [0 0 0]      [1]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(x1) = [3 0 2] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {6}->{5}->{11}->{3}: NA
             ----------------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 2]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 1]      [3]
                length(x1) = [0 2 0] x1 + [0]
                             [0 2 0]      [0]
                             [0 3 2]      [0]
                and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [2]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 1 2] x1 + [1]
                            [2 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 2 3] x1 + [0]
                                [0 2 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [2]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [3 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNat^#(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]
                isNatList^#(x1) = [0 0 0] x1 + [0]
                                  [0 0 0]      [0]
                                  [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {7}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(x1) = [1 0 1] x1 + [0]
                                  [1 0 1]      [0]
                                  [0 3 1]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNatIList^#(x1) = [3 3 3] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 1 1] x1 + [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]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {7}->{10}: NA
             ------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(x1) = [0 0 0] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {7}->{11}: NA
             ------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 1] x1 + [1 1 1] x2 + [0]
                               [0 0 2]      [0 1 1]      [3]
                               [0 0 0]      [0 0 0]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [1]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [2]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 2]      [2]
                and(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 2 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 1] x1 + [2]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 1 1] x1 + [3]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(x1) = [1 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]
                isNatIList^#(x1) = [0 0 0] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {7}->{11}->{3}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 1]      [0 0 1]      [3]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 1]      [1]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 2 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 0 1] x1 + [1]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 1] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [1]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [3 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(x1) = [0 0 0] x1 + [0]
                                   [0 0 0]      [0]
                                   [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {8}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [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]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                and(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]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                isNatList(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                isNatIList(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(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_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]

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

           * Path {9}: NA
             ------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 2] x1 + [1 2 0] x2 + [1]
                               [0 1 1]      [0 1 0]      [2]
                               [0 0 0]      [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 2 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 1 1]      [0]
                and(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                              [0 0 0]      [0 1 0]      [1]
                              [0 0 0]      [0 0 1]      [0]
                isNat(x1) = [0 0 1] x1 + [2]
                            [1 0 0]      [0]
                            [2 0 0]      [0]
                isNatList(x1) = [0 1 1] x1 + [1]
                                [0 1 0]      [0]
                                [0 1 0]      [0]
                isNatIList(x1) = [2 1 1] x1 + [2]
                                 [0 1 0]      [3]
                                 [0 1 1]      [0]
                nil() = [0]
                        [0]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [2 0 0] x1 + [1 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(x1) = [2 0 2] 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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))

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

           * Path {9}->{3}: NA
             -----------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                          [0]
                cons(x1, x2) = [1 0 2] x1 + [1 0 0] x2 + [1]
                               [0 0 3]      [0 1 0]      [2]
                               [0 0 1]      [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                U11(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]
                tt() = [0]
                       [1]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [1]
                length(x1) = [0 0 0] x1 + [1]
                             [0 0 0]      [0]
                             [0 1 1]      [0]
                and(x1, x2) = [2 1 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 2]      [0]
                isNat(x1) = [0 0 1] x1 + [1]
                            [0 0 0]      [1]
                            [3 0 0]      [1]
                isNatList(x1) = [0 1 1] x1 + [0]
                                [0 0 0]      [1]
                                [0 0 0]      [0]
                isNatIList(x1) = [2 2 1] x1 + [1]
                                 [0 0 0]      [1]
                                 [0 0 0]      [0]
                nil() = [0]
                        [1]
                        [0]
                zeros^#() = [0]
                            [0]
                            [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                U11^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                and^#(x1, x2) = [3 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                isNat^#(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]
                isNatList^#(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]
                isNatIList^#(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]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11() = [0]
                         [0]
                         [0]
                c_12(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

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

             {  1: zeros^#() -> c_0(zeros^#())
              , 2: U11^#(tt(), L) -> c_1(length^#(L))
              , 3: and^#(tt(), X) -> c_2(X)
              , 4: isNat^#(0()) -> c_3()
              , 5: isNat^#(length(V1)) -> c_4(isNatList^#(V1))
              , 6: isNat^#(s(V1)) -> c_5(isNat^#(V1))
              , 7: isNatIList^#(V) -> c_6(isNatList^#(V))
              , 8: isNatIList^#(zeros()) -> c_7()
              , 9: isNatIList^#(cons(V1, V2)) ->
                   c_8(and^#(isNat(V1), isNatIList(V2)))
              , 10: isNatList^#(nil()) -> c_9()
              , 11: isNatList^#(cons(V1, V2)) ->
                    c_10(and^#(isNat(V1), isNatList(V2)))
              , 12: length^#(nil()) -> c_11()
              , 13: length^#(cons(N, L)) ->
                    c_12(U11^#(and(isNatList(L), isNat(N)), L))}

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

             ->{9}                                                       [         NA         ]
                |
                `->{3}                                                   [         NA         ]

             ->{8}                                                       [         NA         ]

             ->{7}                                                       [         NA         ]
                |
                |->{10}                                                  [         NA         ]
                |
                `->{11}                                                  [         NA         ]
                    |
                    `->{3}                                               [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]
                    |
                    |->{10}                                              [         NA         ]
                    |
                    `->{11}                                              [         NA         ]
                        |
                        `->{3}                                           [         NA         ]

             ->{2,13}                                                    [         NA         ]
                |
                `->{12}                                                  [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {1},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [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:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zeros^#() -> c_0(zeros^#())}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2,13}: NA
             ---------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 1]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 1]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [2]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [1 0] x1 + [2 2] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [2 2] x1 + [0]
                               [3 3]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {2,13}->{12}: NA
             ---------------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [2]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [1]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 1] x1 + [0]
                             [0 3]      [2]
                and(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [2]
                            [2 0]      [0]
                isNatList(x1) = [0 3] x1 + [2]
                                [0 1]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {6}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [3 3] x1 + [0]
                              [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{4}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{5}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{5}->{10}: NA
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {6}->{5}->{11}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [3]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [1]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatList^#(x1) = [1 2] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {6}->{5}->{11}->{3}: NA
             ----------------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [2]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [2]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 2]      [2]
                and(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 2] x1 + [1]
                            [0 1]      [2]
                isNatList(x1) = [0 3] x1 + [2]
                                [0 1]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [2]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [1 1] x1 + [0]
                                  [3 3]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [3 3] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {7}->{10}: NA
             ------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {7}->{11}: NA
             ------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 0] x1 + [0]
                             [0 3]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 3]      [0]
                isNatList(x1) = [0 3] x1 + [0]
                                [0 2]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [3]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [2 2] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {7}->{11}->{3}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 2]      [3]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 2]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 1] x1 + [3]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {8}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                isNatList(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]

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

           * Path {9}: NA
             ------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 2] x1 + [1 0] x2 + [2]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [0]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 2]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 0]      [0]
                isNatList(x1) = [0 1] x1 + [0]
                                [0 0]      [0]
                isNatIList(x1) = [2 1] x1 + [2]
                                 [0 0]      [0]
                nil() = [0]
                        [2]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [2 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [2 1] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

           * Path {9}->{3}: NA
             -----------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                          [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 1]      [0 1]      [3]
                0() = [0]
                      [0]
                U11(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                length(x1) = [0 0] x1 + [0]
                             [0 1]      [1]
                and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                              [0 0]      [0 1]      [0]
                isNat(x1) = [0 1] x1 + [1]
                            [0 2]      [0]
                isNatList(x1) = [0 1] x1 + [1]
                                [0 1]      [1]
                isNatIList(x1) = [0 1] x1 + [2]
                                 [1 1]      [1]
                nil() = [0]
                        [0]
                zeros^#() = [0]
                            [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                U11^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                and^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatList^#(x1) = [0 0] x1 + [0]
                                  [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                isNatIList^#(x1) = [0 0] x1 + [0]
                                   [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11() = [0]
                         [0]
                c_12(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             {  1: zeros^#() -> c_0(zeros^#())
              , 2: U11^#(tt(), L) -> c_1(length^#(L))
              , 3: and^#(tt(), X) -> c_2(X)
              , 4: isNat^#(0()) -> c_3()
              , 5: isNat^#(length(V1)) -> c_4(isNatList^#(V1))
              , 6: isNat^#(s(V1)) -> c_5(isNat^#(V1))
              , 7: isNatIList^#(V) -> c_6(isNatList^#(V))
              , 8: isNatIList^#(zeros()) -> c_7()
              , 9: isNatIList^#(cons(V1, V2)) ->
                   c_8(and^#(isNat(V1), isNatIList(V2)))
              , 10: isNatList^#(nil()) -> c_9()
              , 11: isNatList^#(cons(V1, V2)) ->
                    c_10(and^#(isNat(V1), isNatList(V2)))
              , 12: length^#(nil()) -> c_11()
              , 13: length^#(cons(N, L)) ->
                    c_12(U11^#(and(isNatList(L), isNat(N)), L))}

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

             ->{9}                                                       [         NA         ]
                |
                `->{3}                                                   [         NA         ]

             ->{8}                                                       [         NA         ]

             ->{7}                                                       [         NA         ]
                |
                |->{10}                                                  [         NA         ]
                |
                `->{11}                                                  [         NA         ]
                    |
                    `->{3}                                               [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]
                    |
                    |->{10}                                              [         NA         ]
                    |
                    `->{11}                                              [         NA         ]
                        |
                        `->{3}                                           [         NA         ]

             ->{2,13}                                                    [         NA         ]
                |
                `->{12}                                                  [         NA         ]

             ->{1}                                                       [       MAYBE        ]



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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {1},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [1] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zeros^#() -> c_0(zeros^#())}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

           * Path {2,13}: NA
             ---------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             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 {2,13}->{12}: NA
             ---------------------

             The usable rules for this path are:

               {  and(tt(), X) -> X
                , isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {1}, Uargs(c_1) = {1}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [3]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [1]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [2]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {6}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [3] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{4}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{5}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{5}->{10}: NA
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {6}->{5}->{11}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [1]
                s(x1) = [1] x1 + [3]
                length(x1) = [3] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [0]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [1] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                isNatList^#(x1) = [3] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {6}->{5}->{11}->{3}: NA
             ----------------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {1}, Uargs(isNatList^#) = {}, Uargs(c_5) = {1},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [1]
                s(x1) = [1] x1 + [3]
                length(x1) = [3] x1 + [3]
                and(x1, x2) = [1] x1 + [1] x2 + [0]
                isNat(x1) = [2] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [3] x1 + [0]
                c_6(x1) = [3] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {7}->{10}: NA
             ------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {7}->{11}: NA
             ------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [1] x1 + [1] x2 + [1]
                isNat(x1) = [2] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [1] x1 + [1] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [3] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {7}->{11}->{3}: NA
             -----------------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))
                , and(tt(), X) -> X}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {1}, Uargs(c_8) = {},
                 Uargs(c_10) = {1}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [1]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [1] x1 + [1] x2 + [0]
                isNat(x1) = [2] x1 + [0]
                isNatList(x1) = [2] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {8}: NA
             ------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                length(x1) = [0] x1 + [0]
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                isNat(x1) = [0] x1 + [0]
                isNatList(x1) = [0] x1 + [0]
                isNatIList(x1) = [0] x1 + [0]
                nil() = [0]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]

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

           * Path {9}: NA
             ------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [1]
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [2]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [2]
                isNatIList(x1) = [3] x1 + [3]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [1] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [3] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

           * Path {9}->{3}: NA
             -----------------

             The usable rules for this path are:

               {  isNat(0()) -> tt()
                , isNat(length(V1)) -> isNatList(V1)
                , isNat(s(V1)) -> isNat(V1)
                , isNatIList(V) -> isNatList(V)
                , isNatIList(zeros()) -> tt()
                , isNatIList(cons(V1, V2)) -> and(isNat(V1), isNatIList(V2))
                , and(tt(), X) -> X
                , isNatList(nil()) -> tt()
                , isNatList(cons(V1, V2)) -> and(isNat(V1), isNatList(V2))}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(U11) = {}, Uargs(s) = {},
                 Uargs(length) = {}, Uargs(and) = {1, 2}, Uargs(isNat) = {},
                 Uargs(isNatList) = {}, Uargs(isNatIList) = {}, Uargs(c_0) = {},
                 Uargs(U11^#) = {}, Uargs(c_1) = {}, Uargs(length^#) = {},
                 Uargs(and^#) = {1, 2}, Uargs(c_2) = {1}, Uargs(isNat^#) = {},
                 Uargs(c_4) = {}, Uargs(isNatList^#) = {}, Uargs(c_5) = {},
                 Uargs(isNatIList^#) = {}, Uargs(c_6) = {}, Uargs(c_8) = {1},
                 Uargs(c_10) = {}, Uargs(c_12) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                zeros() = [2]
                cons(x1, x2) = [1] x1 + [1] x2 + [1]
                0() = [3]
                U11(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [2]
                s(x1) = [1] x1 + [3]
                length(x1) = [2] x1 + [3]
                and(x1, x2) = [2] x1 + [1] x2 + [0]
                isNat(x1) = [1] x1 + [0]
                isNatList(x1) = [2] x1 + [2]
                isNatIList(x1) = [2] x1 + [3]
                nil() = [3]
                zeros^#() = [0]
                c_0(x1) = [0] x1 + [0]
                U11^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                and^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2(x1) = [1] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                isNatList^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                isNatIList^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
                c_11() = [0]
                c_12(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

    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.