Problem Strategy outermost added 08 Ex1 GL02a

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 GL02a

stdout:

MAYBE

Problem:
 eq(0(),0()) -> true()
 eq(s(X),s(Y)) -> eq(X,Y)
 eq(X,Y) -> false()
 inf(X) -> cons(X,inf(s(X)))
 take(0(),X) -> nil()
 take(s(X),cons(Y,L)) -> cons(Y,take(X,L))
 length(nil()) -> 0()
 length(cons(X,L)) -> s(length(L))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 GL02a

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 GL02a

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  eq(0(), 0()) -> true()
     , eq(s(X), s(Y)) -> eq(X, Y)
     , eq(X, Y) -> false()
     , inf(X) -> cons(X, inf(s(X)))
     , take(0(), X) -> nil()
     , take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
     , length(nil()) -> 0()
     , length(cons(X, L)) -> s(length(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: eq^#(0(), 0()) -> c_0()
              , 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
              , 3: eq^#(X, Y) -> c_2()
              , 4: inf^#(X) -> c_3(inf^#(s(X)))
              , 5: take^#(0(), X) -> c_4()
              , 6: take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))
              , 7: length^#(nil()) -> c_6()
              , 8: length^#(cons(X, L)) -> c_7(length^#(L))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{6}                                                       [   YES(?,O(n^3))    ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [       MAYBE        ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [3 3 3]      [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                eq^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                               [2 2 0]      [0 2 0]      [0]
                               [4 0 0]      [0 2 0]      [0]
                c_1(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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 {2}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {eq^#(X, Y) -> c_2()}
               Weak Rules: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                eq^#(x1, x2) = [0 0 0] x1 + [4 0 0] x2 + [1]
                               [2 0 0]      [0 2 0]      [0]
                               [0 4 0]      [2 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
           
           * Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {1},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [3 3 3] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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: {inf^#(X) -> c_3(inf^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}: YES(?,O(n^3))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [3 3 3]      [3 3 3]      [0]
                                 [3 3 3]      [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(cons) = {}, Uargs(take^#) = {},
                 Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 4] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                take^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
                                 [2 0 2]      [2 0 0]      [0]
                                 [0 2 2]      [1 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]
           
           * 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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 {8}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
                               [0 0 0]      [0 1 2]      [2]
                               [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 1 0] x1 + [2]
                               [6 0 0]      [0]
                               [2 3 0]      [2]
                c_7(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]
           
           * Path {8}->{7}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(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]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {length^#(nil()) -> c_6()}
               Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
                               [0 0 0]      [0 1 1]      [1]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [2]
                        [2]
                        [2]
                length^#(x1) = [2 2 2] x1 + [0]
                               [0 6 0]      [0]
                               [0 0 2]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: eq^#(0(), 0()) -> c_0()
              , 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
              , 3: eq^#(X, Y) -> c_2()
              , 4: inf^#(X) -> c_3(inf^#(s(X)))
              , 5: take^#(0(), X) -> c_4()
              , 6: take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))
              , 7: length^#(nil()) -> c_6()
              , 8: length^#(cons(X, L)) -> c_7(length^#(L))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [       MAYBE        ]
             
             ->{2}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {1},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [3 3] x1 + [0]
                            [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(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: {inf^#(X) -> c_3(inf^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [1]
                length^#(x1) = [0 1] x1 + [1]
                               [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {8}->{7}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {length^#(nil()) -> c_6()}
               Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
                               [0 0]      [0 0]      [3]
                nil() = [2]
                        [2]
                length^#(x1) = [1 2] x1 + [2]
                               [6 1]      [0]
                c_6() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: eq^#(0(), 0()) -> c_0()
              , 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
              , 3: eq^#(X, Y) -> c_2()
              , 4: inf^#(X) -> c_3(inf^#(s(X)))
              , 5: take^#(0(), X) -> c_4()
              , 6: take^#(s(X), cons(Y, L)) -> c_5(take^#(X, L))
              , 7: length^#(nil()) -> c_6()
              , 8: length^#(cons(X, L)) -> c_7(length^#(L))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [         NA         ]
                |
                `->{7}                                                   [         NA         ]
             
             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [       MAYBE        ]
             
             ->{2}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {1},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [3] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(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: {inf^#(X) -> c_3(inf^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {1}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                length^#(x1) = [3] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}->{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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 GL02a

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex1 GL02a

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  eq(0(), 0()) -> true()
     , eq(s(X), s(Y)) -> eq(X, Y)
     , eq(X, Y) -> false()
     , inf(X) -> cons(X, inf(s(X)))
     , take(0(), X) -> nil()
     , take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
     , length(nil()) -> 0()
     , length(cons(X, L)) -> s(length(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: eq^#(0(), 0()) -> c_0()
              , 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
              , 3: eq^#(X, Y) -> c_2()
              , 4: inf^#(X) -> c_3(X, inf^#(s(X)))
              , 5: take^#(0(), X) -> c_4()
              , 6: take^#(s(X), cons(Y, L)) -> c_5(Y, take^#(X, L))
              , 7: length^#(nil()) -> c_6()
              , 8: length^#(cons(X, L)) -> c_7(length^#(L))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [       MAYBE        ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [3 3 3]      [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                eq^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                               [2 2 0]      [0 2 0]      [0]
                               [4 0 0]      [0 2 0]      [0]
                c_1(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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 {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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 {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {2},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [1 3 3] x1 + [0]
                            [3 3 3]      [0]
                            [3 3 3]      [0]
                c_3(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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: {inf^#(X) -> c_3(X, inf^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
                               [0 1 3]      [0 1 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [3 3 3]      [3 3 3]      [0]
                                 [3 3 3]      [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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 {8}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [2]
                               [0 0 0]      [0 1 2]      [2]
                               [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 1 0] x1 + [2]
                               [6 0 0]      [0]
                               [2 3 0]      [2]
                c_7(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]
           
           * Path {8}->{7}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, 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]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                false() = [0]
                          [0]
                          [0]
                inf(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                take(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                length(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                eq^#(x1, 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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                inf^#(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                c_3(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                take^#(x1, 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_4() = [0]
                        [0]
                        [0]
                c_5(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                length^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {length^#(nil()) -> c_6()}
               Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 1 0] x2 + [0]
                               [0 0 0]      [0 1 1]      [1]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [2]
                        [2]
                        [2]
                length^#(x1) = [2 2 2] x1 + [0]
                               [0 6 0]      [0]
                               [0 0 2]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: eq^#(0(), 0()) -> c_0()
              , 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
              , 3: eq^#(X, Y) -> c_2()
              , 4: inf^#(X) -> c_3(X, inf^#(s(X)))
              , 5: take^#(0(), X) -> c_4()
              , 6: take^#(s(X), cons(Y, L)) -> c_5(Y, take^#(X, L))
              , 7: length^#(nil()) -> c_6()
              , 8: length^#(cons(X, L)) -> c_7(length^#(L))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [       MAYBE        ]
             
             ->{2}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {2},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [1 3] x1 + [0]
                            [3 3]      [0]
                c_3(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(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: {inf^#(X) -> c_3(X, inf^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                length^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [1]
                length^#(x1) = [0 1] x1 + [1]
                               [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {8}->{7}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                inf(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                length(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                inf^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_3(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                length^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {length^#(nil()) -> c_6()}
               Weak Rules: {length^#(cons(X, L)) -> c_7(length^#(L))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(length^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [1]
                               [0 0]      [0 0]      [3]
                nil() = [2]
                        [2]
                length^#(x1) = [1 2] x1 + [2]
                               [6 1]      [0]
                c_6() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: eq^#(0(), 0()) -> c_0()
              , 2: eq^#(s(X), s(Y)) -> c_1(eq^#(X, Y))
              , 3: eq^#(X, Y) -> c_2()
              , 4: inf^#(X) -> c_3(X, inf^#(s(X)))
              , 5: take^#(0(), X) -> c_4()
              , 6: take^#(s(X), cons(Y, L)) -> c_5(Y, take^#(X, L))
              , 7: length^#(nil()) -> c_6()
              , 8: length^#(cons(X, L)) -> c_7(length^#(L))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [         NA         ]
                |
                `->{7}                                                   [         NA         ]
             
             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [       MAYBE        ]
             
             ->{2}                                                       [         NA         ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{1}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}->{3}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {1}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {4}: 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {2},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [3] x1 + [0]
                c_3(x1, x2) = [2] x1 + [1] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(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: {inf^#(X) -> c_3(X, inf^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * 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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [1] x1 + [1] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {2}, Uargs(length^#) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [1] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
             
             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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [3] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}->{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(eq) = {}, Uargs(s) = {}, Uargs(inf) = {}, Uargs(cons) = {},
                 Uargs(take) = {}, Uargs(length) = {}, Uargs(eq^#) = {},
                 Uargs(c_1) = {}, Uargs(inf^#) = {}, Uargs(c_3) = {},
                 Uargs(take^#) = {}, Uargs(c_5) = {}, Uargs(length^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                inf(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                length(x1) = [0] x1 + [0]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                inf^#(x1) = [0] x1 + [0]
                c_3(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1, x2) = [0] x1 + [0] x2 + [0]
                length^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
    
    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.