Problem CSR 04 Ex5 DLMMU04

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex5 DLMMU04

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex5 DLMMU04

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  pairNs() -> cons(0(), incr(oddNs()))
     , oddNs() -> incr(pairNs())
     , incr(cons(X, XS)) -> cons(s(X), incr(XS))
     , take(0(), XS) -> nil()
     , take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
     , zip(nil(), XS) -> nil()
     , zip(X, nil()) -> nil()
     , zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
     , tail(cons(X, XS)) -> XS
     , repItems(nil()) -> nil()
     , repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))}

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: pairNs^#() -> c_0(incr^#(oddNs()))
              , 2: oddNs^#() -> c_1(incr^#(pairNs()))
              , 3: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
              , 6: zip^#(nil(), XS) -> c_5()
              , 7: zip^#(X, nil()) -> c_6()
              , 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))
              , 9: tail^#(cons(X, XS)) -> c_8()
              , 10: repItems^#(nil()) -> c_9()
              , 11: repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [   YES(?,O(n^2))    ]
                |
                `->{10}                                                  [   YES(?,O(n^2))    ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [   YES(?,O(n^3))    ]
                |
                |->{6}                                                   [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^3))    ]
             
             ->{5}                                                       [   YES(?,O(n^3))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}.
           
           * Path {1}->{3}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  oddNs() -> incr(pairNs())
                , pairNs() -> cons(0(), incr(oddNs()))
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}: inherited
             -------------------
             
             This path is subsumed by the proof of path {2}->{3}.
           
           * Path {2}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  pairNs() -> cons(0(), incr(oddNs()))
                , oddNs() -> incr(pairNs())
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  oddNs^#() -> c_1(incr^#(pairNs()))
                  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
                  , pairNs() -> cons(0(), incr(oddNs()))
                  , oddNs() -> incr(pairNs())
                  , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 1 0]      [0 1 3]      [0]
                               [0 0 0]      [0 0 1]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 2] x2 + [0]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                take^#(x1, x2) = [0 4 0] x1 + [4 1 0] x2 + [0]
                                 [0 0 0]      [2 0 0]      [0]
                                 [4 0 0]      [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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: {take^#(0(), XS) -> c_3()}
               Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 0] x2 + [0]
                               [0 0 0]      [0 0 2]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [2]
                      [0]
                      [0]
                s(x1) = [1 4 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                take^#(x1, x2) = [2 0 0] x1 + [2 2 0] x2 + [0]
                                 [2 0 0]      [4 0 0]      [0]
                                 [0 0 0]      [4 4 0]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {8}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [1 3 0] x1 + [1 3 3] x2 + [0]
                               [0 1 0]      [0 1 3]      [0]
                               [0 0 0]      [0 0 1]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zip^#(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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, 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 + [2]
                               [0 0 0]      [0 0 2]      [2]
                               [0 0 0]      [0 0 0]      [0]
                zip^#(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_7(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {8}->{6}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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: {zip^#(nil(), XS) -> c_5()}
               Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 4 2] x2 + [0]
                               [0 0 0]      [0 0 2]      [0]
                               [0 0 0]      [0 0 1]      [0]
                nil() = [2]
                        [2]
                        [2]
                zip^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                                [2 2 2]      [0 0 4]      [0]
                                [2 2 2]      [0 0 0]      [0]
                c_5() = [1]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {8}->{7}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             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: {zip^#(X, nil()) -> c_6()}
               Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [1 2 0] x2 + [2]
                               [0 0 0]      [0 1 3]      [2]
                               [0 0 0]      [0 0 1]      [2]
                nil() = [2]
                        [2]
                        [2]
                zip^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
                                [0 0 2]      [2 2 0]      [0]
                                [0 0 0]      [0 2 2]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [7]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(X, XS)) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [2]
                               [0 0 0]      [0 0 0]      [2]
                               [0 0 0]      [0 0 0]      [2]
                tail^#(x1) = [0 2 0] x1 + [7]
                             [2 2 0]      [3]
                             [2 2 2]      [3]
                c_8() = [0]
                        [1]
                        [1]
           
           * Path {11}: YES(?,O(n^2))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [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]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [3 3 3]      [0]
                                 [3 3 3]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(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: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {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]
                repItems^#(x1) = [0 1 0] x1 + [2]
                                 [6 0 0]      [0]
                                 [2 3 0]      [2]
                c_10(x1) = [1 0 0] x1 + [1]
                           [2 0 2]      [0]
                           [0 0 0]      [0]
           
           * Path {11}->{10}: YES(?,O(n^2))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                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_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [0]
                           [0 0 1]      [0]
             
             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: {repItems^#(nil()) -> c_9()}
               Weak Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {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]
                repItems^#(x1) = [2 2 2] x1 + [0]
                                 [0 6 0]      [0]
                                 [0 0 2]      [0]
                c_9() = [1]
                        [0]
                        [0]
                c_10(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: pairNs^#() -> c_0(incr^#(oddNs()))
              , 2: oddNs^#() -> c_1(incr^#(pairNs()))
              , 3: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
              , 6: zip^#(nil(), XS) -> c_5()
              , 7: zip^#(X, nil()) -> c_6()
              , 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))
              , 9: tail^#(cons(X, XS)) -> c_8()
              , 10: repItems^#(nil()) -> c_9()
              , 11: repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [   YES(?,O(n^1))    ]
                |
                `->{10}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [   YES(?,O(n^2))    ]
                |
                |->{6}                                                   [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [         NA         ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{3}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}.
           
           * Path {1}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  oddNs() -> incr(pairNs())
                , pairNs() -> cons(0(), incr(oddNs()))
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  pairNs^#() -> c_0(incr^#(oddNs()))
                  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
                  , oddNs() -> incr(pairNs())
                  , pairNs() -> cons(0(), incr(oddNs()))
                  , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: inherited
             -------------------
             
             This path is subsumed by the proof of path {2}->{3}.
           
           * Path {2}->{3}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  pairNs() -> cons(0(), incr(oddNs()))
                , oddNs() -> incr(pairNs())
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {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]      [0]
                s(x1) = [1 4] x1 + [0]
                        [0 1]      [2]
                take^#(x1, x2) = [1 4] x1 + [0 1] x2 + [0]
                                 [0 2]      [0 0]      [2]
                c_4(x1) = [1 2] x1 + [3]
                          [0 0]      [3]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {take^#(0(), XS) -> c_3()}
               Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {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]      [0]
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [2]
                take^#(x1, x2) = [1 2] x1 + [0 2] x2 + [2]
                                 [3 3]      [0 4]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [1]
                          [2 0]      [4]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zip^#(x1, x2) = [3 3] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, 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 1]      [2]
                zip^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                [0 2]      [0 0]      [0]
                c_7(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {8}->{6}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zip^#(nil(), XS) -> c_5()}
               Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                nil() = [2]
                        [2]
                zip^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                                [4 1]      [2 0]      [0]
                c_5() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zip^#(X, nil()) -> c_6()}
               Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [0]
                nil() = [2]
                        [0]
                zip^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                                [0 0]      [4 1]      [0]
                c_6() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(X, XS)) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
                               [0 0]      [0 0]      [2]
                tail^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_8() = [0]
                        [1]
           
           * Path {11}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [3 3] x1 + [0]
                                 [3 3]      [0]
                c_9() = [0]
                        [0]
                c_10(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: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {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]
                repItems^#(x1) = [0 1] x1 + [1]
                                 [0 0]      [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 0]      [0]
           
           * Path {11}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8() = [0]
                        [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(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: {repItems^#(nil()) -> c_9()}
               Weak Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {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]
                repItems^#(x1) = [1 2] x1 + [2]
                                 [6 1]      [0]
                c_9() = [1]
                        [0]
                c_10(x1) = [1 0] x1 + [5]
                           [2 0]      [3]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: pairNs^#() -> c_0(incr^#(oddNs()))
              , 2: oddNs^#() -> c_1(incr^#(pairNs()))
              , 3: incr^#(cons(X, XS)) -> c_2(incr^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))
              , 6: zip^#(nil(), XS) -> c_5()
              , 7: zip^#(X, nil()) -> c_6()
              , 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))
              , 9: tail^#(cons(X, XS)) -> c_8()
              , 10: repItems^#(nil()) -> c_9()
              , 11: repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [   YES(?,O(n^1))    ]
                |
                `->{10}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [         NA         ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{3}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}.
           
           * Path {1}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  oddNs() -> incr(pairNs())
                , pairNs() -> cons(0(), incr(oddNs()))
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  pairNs^#() -> c_0(incr^#(oddNs()))
                  , incr^#(cons(X, XS)) -> c_2(incr^#(XS))
                  , oddNs() -> incr(pairNs())
                  , pairNs() -> cons(0(), incr(oddNs()))
                  , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: inherited
             -------------------
             
             This path is subsumed by the proof of path {2}->{3}.
           
           * Path {2}->{3}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  pairNs() -> cons(0(), incr(oddNs()))
                , oddNs() -> incr(pairNs())
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                s(x1) = [1] x1 + [2]
                take^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_4(x1) = [1] x1 + [3]
           
           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {1}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {take^#(0(), XS) -> c_3()}
               Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(take^#(N, XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                0() = [2]
                s(x1) = [1] x1 + [0]
                take^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_3() = [1]
                c_4(x1) = [1] x1 + [3]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                zip^#(x1, x2) = [3] x1 + [1] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                zip^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_7(x1) = [1] x1 + [7]
           
           * Path {8}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zip^#(nil(), XS) -> c_5()}
               Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                nil() = [2]
                zip^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_5() = [1]
                c_7(x1) = [1] x1 + [2]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {1}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {zip^#(X, nil()) -> c_6()}
               Weak Rules: {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [2]
                nil() = [2]
                zip^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_6() = [1]
                c_7(x1) = [1] x1 + [7]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(X, XS)) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [0] x2 + [7]
                tail^#(x1) = [1] x1 + [7]
                c_8() = [1]
           
           * Path {11}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [3] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [4]
                repItems^#(x1) = [2] x1 + [0]
                c_10(x1) = [1] x1 + [7]
           
           * Path {11}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(repItems^#) = {},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8() = [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {repItems^#(nil()) -> c_9()}
               Weak Rules: {repItems^#(cons(X, XS)) -> c_10(repItems^#(XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [0]
                nil() = [2]
                repItems^#(x1) = [2] x1 + [0]
                c_9() = [1]
                c_10(x1) = [1] x1 + [0]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex5 DLMMU04

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputCSR 04 Ex5 DLMMU04

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  pairNs() -> cons(0(), incr(oddNs()))
     , oddNs() -> incr(pairNs())
     , incr(cons(X, XS)) -> cons(s(X), incr(XS))
     , take(0(), XS) -> nil()
     , take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
     , zip(nil(), XS) -> nil()
     , zip(X, nil()) -> nil()
     , zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), zip(XS, YS))
     , tail(cons(X, XS)) -> XS
     , repItems(nil()) -> nil()
     , repItems(cons(X, XS)) -> cons(X, cons(X, repItems(XS)))}

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: pairNs^#() -> c_0(incr^#(oddNs()))
              , 2: oddNs^#() -> c_1(incr^#(pairNs()))
              , 3: incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
              , 6: zip^#(nil(), XS) -> c_5()
              , 7: zip^#(X, nil()) -> c_6()
              , 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))
              , 9: tail^#(cons(X, XS)) -> c_8(XS)
              , 10: repItems^#(nil()) -> c_9()
              , 11: repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [   YES(?,O(n^3))    ]
                |
                `->{10}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^3))    ]
             
             ->{8}                                                       [   YES(?,O(n^3))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^3))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [         NA         ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{3}                                                   [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}.
           
           * Path {1}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  oddNs() -> incr(pairNs())
                , pairNs() -> cons(0(), incr(oddNs()))
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  pairNs^#() -> c_0(incr^#(oddNs()))
                  , incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
                  , oddNs() -> incr(pairNs())
                  , pairNs() -> cons(0(), incr(oddNs()))
                  , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: inherited
             -------------------
             
             This path is subsumed by the proof of path {2}->{3}.
           
           * Path {2}->{3}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  pairNs() -> cons(0(), incr(oddNs()))
                , oddNs() -> incr(pairNs())
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: YES(?,O(n^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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [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]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 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]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(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]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 2 2] x2 + [0]
                               [0 0 2]      [0 1 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                take^#(x1, x2) = [2 2 0] x1 + [3 3 2] x2 + [0]
                                 [2 3 0]      [2 5 4]      [0]
                                 [4 0 0]      [3 2 2]      [0]
                c_4(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [3]
                              [1 0 0]      [0 0 0]      [0]
           
           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(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]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 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: {take^#(0(), XS) -> c_3()}
               Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 6 0] x2 + [2]
                               [0 0 2]      [0 0 0]      [2]
                               [0 0 0]      [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 0 0] x1 + [2]
                        [0 1 0]      [0]
                        [0 0 1]      [0]
                take^#(x1, x2) = [2 2 2] x1 + [2 2 0] x2 + [0]
                                 [2 3 1]      [2 2 0]      [0]
                                 [2 2 0]      [2 2 0]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [4]
                              [0 0 0]      [0 0 0]      [7]
           
           * Path {8}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [1 3 3] x1 + [1 0 0] x2 + [0]
                               [0 1 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                zip^#(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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 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:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 4 0] x2 + [4]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zip^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
                                [2 0 0]      [0 0 0]      [0]
                                [2 0 0]      [0 0 0]      [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [6]
           
           * Path {8}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 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: {zip^#(nil(), XS) -> c_5()}
               Weak Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 0 0] x2 + [2]
                               [0 0 2]      [0 1 0]      [2]
                               [0 0 0]      [0 0 1]      [2]
                nil() = [2]
                        [2]
                        [2]
                zip^#(x1, x2) = [2 2 2] x1 + [0 1 0] x2 + [0]
                                [2 2 0]      [0 0 0]      [0]
                                [2 0 2]      [0 2 0]      [0]
                c_5() = [1]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [1]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [7]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 1]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zip^#(X, nil()) -> c_6()}
               Weak Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0 2] x1 + [1 2 0] x2 + [0]
                               [0 0 2]      [0 0 2]      [0]
                               [0 0 1]      [0 0 1]      [0]
                nil() = [2]
                        [2]
                        [2]
                zip^#(x1, x2) = [2 4 0] x1 + [1 2 2] x2 + [0]
                                [2 2 0]      [2 3 2]      [0]
                                [2 2 0]      [2 3 2]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
           
           * Path {9}: YES(?,O(n^3))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [1 3 3] x1 + [0 0 0] x2 + [0]
                               [0 1 1]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tail^#(x1) = [1 3 3] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 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: {tail^#(cons(X, XS)) -> c_8(XS)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {}
               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 0 2]      [2]
                               [0 0 0]      [0 0 0]      [2]
                tail^#(x1) = [2 2 2] x1 + [3]
                             [2 2 2]      [3]
                             [2 2 2]      [3]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [0 0 0]      [1]
           
           * Path {11}: 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [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]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [3 3 3]      [0]
                                 [3 3 3]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                   [0 0 0]      [0 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^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2 2] x1 + [1 4 2] x2 + [2]
                               [0 0 2]      [0 1 1]      [2]
                               [0 0 0]      [0 0 1]      [2]
                repItems^#(x1) = [2 1 3] x1 + [0]
                                 [0 2 2]      [0]
                                 [2 2 0]      [0]
                c_10(x1, x2, x3) = [0 4 3] x1 + [2 0 3] x2 + [1 0 0] x3 + [7]
                                   [0 0 4]      [0 0 0]      [0 0 0]      [6]
                                   [2 4 3]      [0 0 4]      [0 0 0]      [0]
           
           * Path {11}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                           [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                incr(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                oddNs() = [0]
                          [0]
                          [0]
                s(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]
                nil() = [0]
                        [0]
                        [0]
                zip(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                pair(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                tail(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                repItems(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                pairNs^#() = [0]
                             [0]
                             [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                incr^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                oddNs^#() = [0]
                            [0]
                            [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1, 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_3() = [0]
                        [0]
                        [0]
                c_4(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                zip^#(x1, 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_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                tail^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                repItems^#(x1) = [0 0 0] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_9() = [0]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 1 0]      [0]
                                   [0 0 0]      [0 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^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {repItems^#(nil()) -> c_9()}
               Weak Rules: {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0 0] x1 + [1 3 0] x2 + [0]
                               [0 0 0]      [0 0 4]      [0]
                               [0 0 0]      [0 0 0]      [0]
                nil() = [2]
                        [2]
                        [2]
                repItems^#(x1) = [1 2 2] x1 + [0]
                                 [3 2 2]      [0]
                                 [3 2 2]      [0]
                c_9() = [1]
                        [0]
                        [0]
                c_10(x1, x2, x3) = [0 0 0] x1 + [1 0 0] x2 + [1 0 0] x3 + [0]
                                   [3 0 0]      [0 0 0]      [2 0 0]      [0]
                                   [1 0 0]      [2 0 0]      [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: pairNs^#() -> c_0(incr^#(oddNs()))
              , 2: oddNs^#() -> c_1(incr^#(pairNs()))
              , 3: incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
              , 6: zip^#(nil(), XS) -> c_5()
              , 7: zip^#(X, nil()) -> c_6()
              , 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))
              , 9: tail^#(cons(X, XS)) -> c_8(XS)
              , 10: repItems^#(nil()) -> c_9()
              , 11: repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [   YES(?,O(n^2))    ]
                |
                `->{10}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^2))    ]
             
             ->{8}                                                       [   YES(?,O(n^2))    ]
                |
                |->{6}                                                   [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}.
           
           * Path {1}->{3}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  oddNs() -> incr(pairNs())
                , pairNs() -> cons(0(), incr(oddNs()))
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}: inherited
             -------------------
             
             This path is subsumed by the proof of path {2}->{3}.
           
           * Path {2}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  pairNs() -> cons(0(), incr(oddNs()))
                , oddNs() -> incr(pairNs())
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  oddNs^#() -> c_1(incr^#(pairNs()))
                  , incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
                  , pairNs() -> cons(0(), incr(oddNs()))
                  , oddNs() -> incr(pairNs())
                  , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [3]
                               [0 0]      [0 1]      [2]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                take^#(x1, x2) = [0 0] x1 + [2 3] x2 + [0]
                                 [0 1]      [3 3]      [0]
                c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [5]
                              [1 0]      [0 0]      [7]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {take^#(0(), XS) -> c_3()}
               Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                               [0 0]      [0 1]      [0]
                0() = [3]
                      [2]
                s(x1) = [1 2] x1 + [3]
                        [0 1]      [0]
                take^#(x1, x2) = [2 1] x1 + [2 1] x2 + [0]
                                 [3 2]      [4 1]      [0]
                c_3() = [1]
                        [1]
                c_4(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
                              [0 0]      [0 0]      [6]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zip^#(x1, x2) = [2 1] x1 + [1 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 1] x1 + [1 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [0]
                               [0 0]      [0 0]      [2]
                zip^#(x1, x2) = [2 4] x1 + [0 0] x2 + [0]
                                [0 4]      [2 0]      [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [5]
                                  [0 0]      [0 0]      [0 0]      [6]
           
           * Path {8}->{6}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zip^#(nil(), XS) -> c_5()}
               Weak Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [2]
                nil() = [2]
                        [2]
                zip^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                                [2 2]      [2 1]      [0]
                c_5() = [1]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [6]
                                  [0 0]      [0 0]      [0 0]      [7]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                  [0 0]      [0 0]      [0 1]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zip^#(X, nil()) -> c_6()}
               Weak Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [2]
                               [0 0]      [0 1]      [2]
                nil() = [2]
                        [0]
                zip^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                                [2 2]      [2 1]      [0]
                c_6() = [1]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [7]
                                  [0 0]      [0 0]      [0 0]      [7]
           
           * Path {9}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tail^#(x1) = [3 3] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(X, XS)) -> c_8(XS)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                tail^#(x1) = [2 2] x1 + [7]
                             [2 0]      [7]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [1]
           
           * Path {11}: YES(?,O(n^2))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [1 3] x1 + [0]
                                 [3 3]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [1 3] x1 + [0 3] x2 + [1 0] x3 + [0]
                                   [0 0]      [0 0]      [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [1 2] x2 + [1]
                               [0 0]      [0 1]      [2]
                repItems^#(x1) = [2 4] x1 + [0]
                                 [2 3]      [0]
                c_10(x1, x2, x3) = [0 4] x1 + [2 0] x2 + [1 0] x3 + [1]
                                   [0 4]      [2 0]      [0 0]      [0]
           
           * Path {11}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                           [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                incr(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                oddNs() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                take(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                zip(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                pair(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                repItems(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                pairNs^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                incr^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                oddNs^#() = [0]
                            [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(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_3() = [0]
                        [0]
                c_4(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                zip^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                repItems^#(x1) = [0 0] x1 + [0]
                                 [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [0]
                                   [0 0]      [0 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: {repItems^#(nil()) -> c_9()}
               Weak Rules: {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 3] x1 + [1 1] x2 + [2]
                               [0 0]      [0 0]      [0]
                nil() = [0]
                        [2]
                repItems^#(x1) = [5 2] x1 + [0]
                                 [4 2]      [0]
                c_9() = [1]
                        [0]
                c_10(x1, x2, x3) = [0 7] x1 + [3 7] x2 + [1 0] x3 + [7]
                                   [2 7]      [1 5]      [0 0]      [6]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: pairNs^#() -> c_0(incr^#(oddNs()))
              , 2: oddNs^#() -> c_1(incr^#(pairNs()))
              , 3: incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
              , 4: take^#(0(), XS) -> c_3()
              , 5: take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))
              , 6: zip^#(nil(), XS) -> c_5()
              , 7: zip^#(X, nil()) -> c_6()
              , 8: zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))
              , 9: tail^#(cons(X, XS)) -> c_8(XS)
              , 10: repItems^#(nil()) -> c_9()
              , 11: repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [   YES(?,O(n^1))    ]
                |
                `->{10}                                                  [   YES(?,O(n^1))    ]
             
             ->{9}                                                       [   YES(?,O(n^1))    ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [     inherited      ]
                |
                `->{3}                                                   [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}.
           
           * Path {1}->{3}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  oddNs() -> incr(pairNs())
                , pairNs() -> cons(0(), incr(oddNs()))
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2}: inherited
             -------------------
             
             This path is subsumed by the proof of path {2}->{3}.
           
           * Path {2}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  pairNs() -> cons(0(), incr(oddNs()))
                , oddNs() -> incr(pairNs())
                , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  oddNs^#() -> c_1(incr^#(pairNs()))
                  , incr^#(cons(X, XS)) -> c_2(X, incr^#(XS))
                  , pairNs() -> cons(0(), incr(oddNs()))
                  , oddNs() -> incr(pairNs())
                  , incr(cons(X, XS)) -> cons(s(X), incr(XS))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [1] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [1] x1 + [1] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                s(x1) = [1] x1 + [2]
                take^#(x1, x2) = [2] x1 + [3] x2 + [0]
                c_4(x1, x2) = [1] x1 + [1] x2 + [1]
           
           * Path {5}->{4}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {2}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [1] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {take^#(0(), XS) -> c_3()}
               Weak Rules: {take^#(s(N), cons(X, XS)) -> c_4(X, take^#(N, XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(take^#) = {},
                 Uargs(c_4) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                0() = [2]
                s(x1) = [1] x1 + [2]
                take^#(x1, x2) = [3] x1 + [2] x2 + [2]
                c_3() = [1]
                c_4(x1, x2) = [0] x1 + [1] x2 + [5]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zip^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                zip^#(x1, x2) = [4] x1 + [2] x2 + [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
           
           * Path {8}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zip^#(nil(), XS) -> c_5()}
               Weak Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                nil() = [2]
                zip^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_5() = [1]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
           
           * 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(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {3}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {zip^#(X, nil()) -> c_6()}
               Weak Rules:
                 {zip^#(cons(X, XS), cons(Y, YS)) -> c_7(X, Y, zip^#(XS, YS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(zip^#) = {}, Uargs(c_7) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [2]
                nil() = [2]
                zip^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_6() = [1]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [7]
           
           * Path {9}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tail^#(x1) = [3] x1 + [0]
                c_8(x1) = [1] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(X, XS)) -> c_8(XS)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [7]
                tail^#(x1) = [1] x1 + [7]
                c_8(x1) = [1] x1 + [1]
           
           * Path {11}: YES(?,O(n^1))
             ------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [3] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [2] x1 + [1] x2 + [1] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [4]
                repItems^#(x1) = [2] x1 + [0]
                c_10(x1, x2, x3) = [2] x1 + [0] x2 + [1] x3 + [7]
           
           * Path {11}->{10}: YES(?,O(n^1))
             ------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(incr) = {}, Uargs(s) = {},
                 Uargs(take) = {}, Uargs(zip) = {}, Uargs(pair) = {},
                 Uargs(tail) = {}, Uargs(repItems) = {}, Uargs(c_0) = {},
                 Uargs(incr^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(take^#) = {}, Uargs(c_4) = {}, Uargs(zip^#) = {},
                 Uargs(c_7) = {}, Uargs(tail^#) = {}, Uargs(c_8) = {},
                 Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                pairNs() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                incr(x1) = [0] x1 + [0]
                oddNs() = [0]
                s(x1) = [0] x1 + [0]
                take(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                zip(x1, x2) = [0] x1 + [0] x2 + [0]
                pair(x1, x2) = [0] x1 + [0] x2 + [0]
                tail(x1) = [0] x1 + [0]
                repItems(x1) = [0] x1 + [0]
                pairNs^#() = [0]
                c_0(x1) = [0] x1 + [0]
                incr^#(x1) = [0] x1 + [0]
                oddNs^#() = [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1, x2) = [0] x1 + [0] x2 + [0]
                take^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1, x2) = [0] x1 + [0] x2 + [0]
                zip^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]
                repItems^#(x1) = [0] x1 + [0]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {repItems^#(nil()) -> c_9()}
               Weak Rules: {repItems^#(cons(X, XS)) -> c_10(X, X, repItems^#(XS))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(repItems^#) = {}, Uargs(c_10) = {3}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                nil() = [0]
                repItems^#(x1) = [0] x1 + [1]
                c_9() = [0]
                c_10(x1, x2, x3) = [0] x1 + [0] x2 + [1] x3 + [0]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.