Problem Strategy outermost added 08 Ex4 7 37 Bor03

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex4 7 37 Bor03

stdout:

MAYBE

Problem:
 from(X) -> cons(X,from(s(X)))
 sel(0(),cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,XS)
 minus(X,0()) -> 0()
 minus(s(X),s(Y)) -> minus(X,Y)
 quot(0(),s(Y)) -> 0()
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 zWquot(XS,nil()) -> nil()
 zWquot(nil(),XS) -> nil()
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),zWquot(XS,YS))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex4 7 37 Bor03

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex4 7 37 Bor03

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, from(s(X)))
     , sel(0(), cons(X, XS)) -> X
     , sel(s(N), cons(X, XS)) -> sel(N, XS)
     , minus(X, 0()) -> 0()
     , minus(s(X), s(Y)) -> minus(X, Y)
     , quot(0(), s(Y)) -> 0()
     , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
     , zWquot(XS, nil()) -> nil()
     , zWquot(nil(), XS) -> nil()
     , zWquot(cons(X, XS), cons(Y, YS)) ->
       cons(quot(X, Y), zWquot(XS, YS))}

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: from^#(X) -> c_0(from^#(s(X)))
              , 2: sel^#(0(), cons(X, XS)) -> c_1()
              , 3: sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
              , 4: minus^#(X, 0()) -> c_3()
              , 5: minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))
              , 6: quot^#(0(), s(Y)) -> c_5()
              , 7: quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
              , 8: zWquot^#(XS, nil()) -> c_7()
              , 9: zWquot^#(nil(), XS) -> c_8()
              , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
                    c_9(quot^#(X, Y), zWquot^#(XS, YS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |
                |->{7}                                                   [         NA         ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                |->{8}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [   YES(?,O(n^3))    ]
                |
                `->{2}                                                   [         NA         ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {1}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [3 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_0(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [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:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(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_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [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:    innermost DP runtime-complexity with respect to
               Strict Rules: {sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {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]
                sel^#(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_2(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
             
             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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [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:    innermost DP runtime-complexity with respect to
               Strict Rules: {minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                minus^#(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_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {5}->{4}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}: NA
             -------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                 [3 0 0]      [3 0 0]      [0]
                                 [3 0 0]      [3 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [0]
                                   [3 3 3]      [3 3 3]      [0]
                                   [3 3 3]      [3 3 3]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(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]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{6}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [1]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                      [2]
                      [0]
                minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
                                [0 0 0]      [0 2 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, x2) = [1 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(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}->{6}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [2]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [1 0 0] x1 + [3 0 0] x2 + [2]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{8}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(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]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(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]
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(from^#(s(X)))
              , 2: sel^#(0(), cons(X, XS)) -> c_1()
              , 3: sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
              , 4: minus^#(X, 0()) -> c_3()
              , 5: minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))
              , 6: quot^#(0(), s(Y)) -> c_5()
              , 7: quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
              , 8: zWquot^#(XS, nil()) -> c_7()
              , 9: zWquot^#(nil(), XS) -> c_8()
              , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
                    c_9(quot^#(X, Y), zWquot^#(XS, YS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |
                |->{7}                                                   [         NA         ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                |->{8}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {1}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}->{4}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}: NA
             -------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                 [3 3]      [3 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                                   [3 3]      [3 3]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{6}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}->{6}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [2]
                      [0]
                minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
                                [0 0]      [2 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{8}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(from^#(s(X)))
              , 2: sel^#(0(), cons(X, XS)) -> c_1()
              , 3: sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
              , 4: minus^#(X, 0()) -> c_3()
              , 5: minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))
              , 6: quot^#(0(), s(Y)) -> c_5()
              , 7: quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
              , 8: zWquot^#(XS, nil()) -> c_7()
              , 9: zWquot^#(nil(), XS) -> c_8()
              , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
                    c_9(quot^#(X, Y), zWquot^#(XS, YS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |
                |->{7}                                                   [         NA         ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                |->{8}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {1}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1) = [1] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {1}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}->{4}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}: NA
             -------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [1] x1 + [1] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [3] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{6}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [1] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [3]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [2]
                minus(x1, x2) = [1] x1 + [0] x2 + [3]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [1] x1 + [1] x2 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}->{6}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {1}, Uargs(c_6) = {1}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [3]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [3]
                minus(x1, x2) = [1] x1 + [1] x2 + [3]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [1] x1 + [1] x2 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{8}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_6) = {}, Uargs(zWquot^#) = {},
                 Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1) = [0] x1 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex4 7 37 Bor03

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 Ex4 7 37 Bor03

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, from(s(X)))
     , sel(0(), cons(X, XS)) -> X
     , sel(s(N), cons(X, XS)) -> sel(N, XS)
     , minus(X, 0()) -> 0()
     , minus(s(X), s(Y)) -> minus(X, Y)
     , quot(0(), s(Y)) -> 0()
     , quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
     , zWquot(XS, nil()) -> nil()
     , zWquot(nil(), XS) -> nil()
     , zWquot(cons(X, XS), cons(Y, YS)) ->
       cons(quot(X, Y), zWquot(XS, YS))}

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: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: sel^#(0(), cons(X, XS)) -> c_1(X)
              , 3: sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
              , 4: minus^#(X, 0()) -> c_3()
              , 5: minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))
              , 6: quot^#(0(), s(Y)) -> c_5()
              , 7: quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
              , 8: zWquot^#(XS, nil()) -> c_7()
              , 9: zWquot^#(nil(), XS) -> c_8()
              , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
                    c_9(quot^#(X, Y), zWquot^#(XS, YS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |
                |->{7}                                                   [         NA         ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                |->{8}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [   YES(?,O(n^3))    ]
                |
                `->{2}                                                   [         NA         ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {2}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [1 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_0(x1, x2) = [0 1 0] x1 + [1 0 0] x2 + [0]
                              [0 0 0]      [0 1 0]      [0]
                              [0 0 0]      [0 0 1]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [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:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: 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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [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: {sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel^#) = {},
                 Uargs(c_2) = {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]
                sel^#(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_2(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
                               [0 1 3]      [0 0 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [3 1 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
             
             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(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {1}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [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: {minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                minus^#(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_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {5}->{4}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {1}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}: NA
             -------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                               [0 1 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                 [3 0 0]      [3 0 0]      [0]
                                 [3 0 0]      [3 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [0]
                                   [3 3 3]      [3 3 3]      [0]
                                   [3 3 3]      [3 3 3]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(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]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{6}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_6) = {1},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [1]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                      [2]
                      [0]
                minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [2]
                                [0 0 0]      [0 2 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, x2) = [1 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(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}->{6}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_6) = {1},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [2]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [1 0 0] x1 + [3 0 0] x2 + [2]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                              [0 1 0]      [0 1 0]      [0]
                              [0 0 1]      [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{8}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(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]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                sel(x1, 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]
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                quot(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                zWquot(x1, 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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                sel^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                minus^#(x1, 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]
                quot^#(x1, 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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                zWquot^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(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]
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: sel^#(0(), cons(X, XS)) -> c_1(X)
              , 3: sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
              , 4: minus^#(X, 0()) -> c_3()
              , 5: minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))
              , 6: quot^#(0(), s(Y)) -> c_5()
              , 7: quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
              , 8: zWquot^#(XS, nil()) -> c_7()
              , 9: zWquot^#(nil(), XS) -> c_8()
              , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
                    c_9(quot^#(X, Y), zWquot^#(XS, YS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |
                |->{7}                                                   [         NA         ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                |->{8}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {2}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_0(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 2] x1 + [1 1] x2 + [0]
                               [0 0]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [1 0] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {1}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}->{4}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {1}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}: NA
             -------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 1]      [0 1]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                 [3 3]      [3 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                                   [3 3]      [3 3]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{6}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_6) = {1},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
                                 [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}->{6}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_6) = {1},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [2]
                      [0]
                minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
                                [0 0]      [2 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{8}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                zWquot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                sel^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                minus^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                zWquot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(X, from^#(s(X)))
              , 2: sel^#(0(), cons(X, XS)) -> c_1(X)
              , 3: sel^#(s(N), cons(X, XS)) -> c_2(sel^#(N, XS))
              , 4: minus^#(X, 0()) -> c_3()
              , 5: minus^#(s(X), s(Y)) -> c_4(minus^#(X, Y))
              , 6: quot^#(0(), s(Y)) -> c_5()
              , 7: quot^#(s(X), s(Y)) -> c_6(quot^#(minus(X, Y), s(Y)))
              , 8: zWquot^#(XS, nil()) -> c_7()
              , 9: zWquot^#(nil(), XS) -> c_8()
              , 10: zWquot^#(cons(X, XS), cons(Y, YS)) ->
                    c_9(quot^#(X, Y), zWquot^#(XS, YS))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                |->{6}                                                   [         NA         ]
                |
                |->{7}                                                   [         NA         ]
                |   |
                |   `->{6}                                               [         NA         ]
                |
                |->{8}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{5}                                                       [         NA         ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [         NA         ]
                |
                `->{2}                                                   [         NA         ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {2}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [3] x1 + [0]
                c_0(x1, x2) = [2] x1 + [1] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_0(X, from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [1] x1 + [3] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {1}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: NA
             ------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {1}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}->{4}: NA
             -----------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {1}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}: NA
             -------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [1] x1 + [1] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [3] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{6}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [1] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_6) = {1},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [3]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [2]
                minus(x1, x2) = [1] x1 + [0] x2 + [3]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [1] x1 + [1] x2 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{7}->{6}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(X, 0()) -> 0()
                , minus(s(X), s(Y)) -> minus(X, Y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_6) = {1},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {1, 2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [3]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [3]
                minus(x1, x2) = [1] x1 + [1] x2 + [3]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [1] x1 + [1] x2 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{8}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(s) = {}, Uargs(sel) = {},
                 Uargs(minus) = {}, Uargs(quot) = {}, Uargs(zWquot) = {},
                 Uargs(from^#) = {}, Uargs(c_0) = {}, Uargs(sel^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(minus^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {}, Uargs(c_6) = {},
                 Uargs(zWquot^#) = {}, Uargs(c_9) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                sel(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                zWquot(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                sel^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                zWquot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [0]
                c_8() = [0]
                c_9(x1, x2) = [0] x1 + [1] x2 + [0]
             
             We have not generated a proof for the resulting sub-problem.
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.