Problem Strategy outermost added 08 Ex15 Luc98

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex15 Luc98

stdout:

MAYBE

Problem:
and(true(),X) -> X
and(false(),Y) -> false()
if(true(),X,Y) -> X
if(false(),X,Y) -> Y
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
from(X) -> cons(X,from(s(X)))

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex15 Luc98

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex15 Luc98

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  and(true(), X) -> X
     , and(false(), Y) -> false()
     , if(true(), X, Y) -> X
     , if(false(), X, Y) -> Y
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
     , from(X) -> cons(X, from(s(X)))}

Proof Output:
  None of the processors succeeded.

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

             {  1: and^#(true(), X) -> c_0()
              , 2: and^#(false(), Y) -> c_1()
              , 3: if^#(true(), X, Y) -> c_2()
              , 4: if^#(false(), X, Y) -> c_3()
              , 5: add^#(0(), X) -> c_4()
              , 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
              , 7: first^#(0(), X) -> c_6()
              , 8: first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))
              , 9: from^#(X) -> c_8(from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [   YES(?,O(n^3))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]

             ->{6}                                                       [   YES(?,O(n^2))    ]
                |
                `->{5}                                                   [   YES(?,O(n^2))    ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(true(), X) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                         [2]
                and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                [2 2 0]      [0 0 0]      [3]
                                [2 2 2]      [0 0 0]      [3]
                c_0() = [0]
                        [1]
                        [1]

           * Path {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(false(), Y) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                          [2]
                and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                [2 2 0]      [0 0 0]      [3]
                                [2 2 2]      [0 0 0]      [3]
                c_1() = [0]
                        [1]
                        [1]

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {if^#(true(), X, Y) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                         [2]
                if^#(x1, x2, x3) = [0 2 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
                                   [2 2 0]      [0 0 0]      [0 0 0]      [3]
                                   [2 2 2]      [0 0 0]      [0 0 0]      [3]
                c_2() = [0]
                        [1]
                        [1]

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {if^#(false(), X, Y) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                          [2]
                if^#(x1, x2, x3) = [0 2 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [7]
                                   [2 2 0]      [0 0 0]      [0 0 0]      [3]
                                   [2 2 2]      [0 0 0]      [0 0 0]      [3]
                c_3() = [0]
                        [1]
                        [1]

           * Path {6}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 0 2]      [3]
                        [0 0 1]      [2]
                add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
                                [0 2 1]      [0 0 0]      [2]
                                [4 0 2]      [0 0 4]      [0]
                c_5(x1) = [1 0 0] x1 + [1]
                          [0 0 0]      [2]
                          [2 2 0]      [3]

           * Path {6}->{5}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

             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: {add^#(0(), X) -> c_4()}
               Weak Rules: {add^#(s(X), Y) -> c_5(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 4]      [0]
                        [0 0 0]      [2]
                add^#(x1, x2) = [1 3 1] x1 + [0 0 0] x2 + [0]
                                [3 2 2]      [0 0 4]      [0]
                                [0 2 2]      [0 0 2]      [2]
                c_4() = [1]
                        [0]
                        [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [3]
                          [0 0 0]      [2]

           * Path {8}: YES(?,O(n^3))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 3 0] x1 + [0 0 0] x2 + [0]
                               [0 1 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 4] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [2]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 1]      [2]
                first^#(x1, x2) = [0 2 0] x1 + [0 0 2] x2 + [0]
                                  [2 0 2]      [2 0 0]      [0]
                                  [0 2 2]      [1 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [5]
                          [2 0 2]      [3]
                          [0 0 0]      [7]

           * Path {8}->{7}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_6()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                      [0]
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [1 4 4] x2 + [0]
                               [0 0 0]      [0 0 4]      [0]
                               [0 0 0]      [0 0 0]      [0]
                first^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
                                  [2 0 0]      [2 0 0]      [4]
                                  [0 0 0]      [0 2 0]      [2]
                c_6() = [1]
                        [0]
                        [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [0]
                          [0 0 0]      [2]

           * Path {9}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 1 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                from^#(x1) = [3 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_8(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: and^#(true(), X) -> c_0()
              , 2: and^#(false(), Y) -> c_1()
              , 3: if^#(true(), X, Y) -> c_2()
              , 4: if^#(false(), X, Y) -> c_3()
              , 5: add^#(0(), X) -> c_4()
              , 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
              , 7: first^#(0(), X) -> c_6()
              , 8: first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))
              , 9: from^#(X) -> c_8(from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [         NA         ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(true(), X) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                [2 2]      [0 0]      [7]
                c_0() = [0]
                        [1]

           * Path {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(false(), Y) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                [2 2]      [0 0]      [7]
                c_1() = [0]
                        [1]

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {if^#(true(), X, Y) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
                                   [2 2]      [0 0]      [0 0]      [7]
                c_2() = [0]
                        [1]

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {if^#(false(), X, Y) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
                                   [2 2]      [0 0]      [0 0]      [7]
                c_3() = [0]
                        [1]

           * Path {6}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                                [0 0]      [0 4]      [4]
                c_5(x1) = [1 0] x1 + [0]
                          [0 0]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {8}->{7}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

           * Path {9}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                from^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [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_8(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

    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: and^#(true(), X) -> c_0()
              , 2: and^#(false(), Y) -> c_1()
              , 3: if^#(true(), X, Y) -> c_2()
              , 4: if^#(false(), X, Y) -> c_3()
              , 5: add^#(0(), X) -> c_4()
              , 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
              , 7: first^#(0(), X) -> c_6()
              , 8: first^#(s(X), cons(Y, Z)) -> c_7(first^#(X, Z))
              , 9: from^#(X) -> c_8(from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(true(), X) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [7]
                and^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_0() = [1]

           * Path {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {and^#(false(), Y) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [7]
                and^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_1() = [1]

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {if^#(true(), X, Y) -> c_2()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [7]
                if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
                c_2() = [1]

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {if^#(false(), X, Y) -> c_3()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [7]
                if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
                c_3() = [1]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {1}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {8}->{7}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {1},
                 Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1) = [0] x1 + [0]

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

           * Path {9}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(if^#) = {}, Uargs(add^#) = {},
                 Uargs(c_5) = {}, Uargs(first^#) = {}, Uargs(c_7) = {},
                 Uargs(from^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3() = [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_8(x1) = [1] x1 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_8(from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex15 Luc98

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 Ex15 Luc98

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  and(true(), X) -> X
     , and(false(), Y) -> false()
     , if(true(), X, Y) -> X
     , if(false(), X, Y) -> Y
     , add(0(), X) -> X
     , add(s(X), Y) -> s(add(X, Y))
     , first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
     , from(X) -> cons(X, from(s(X)))}

Proof Output:
  None of the processors succeeded.

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

             {  1: and^#(true(), X) -> c_0(X)
              , 2: and^#(false(), Y) -> c_1()
              , 3: if^#(true(), X, Y) -> c_2(X)
              , 4: if^#(false(), X, Y) -> c_3(Y)
              , 5: add^#(0(), X) -> c_4(X)
              , 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
              , 7: first^#(0(), X) -> c_6()
              , 8: first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))
              , 9: from^#(X) -> c_8(X, from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [   YES(?,O(n^3))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]

             ->{6}                                                       [   YES(?,O(n^2))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(true(), X) -> c_0(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [0]
                         [2]
                and^#(x1, x2) = [2 0 2] x1 + [7 7 7] x2 + [7]
                                [2 0 2]      [7 7 7]      [7]
                                [2 0 2]      [7 7 7]      [7]
                c_0(x1) = [1 3 3] x1 + [0]
                          [1 1 1]      [1]
                          [1 1 1]      [1]

           * Path {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(false(), Y) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                          [2]
                and^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                [2 2 0]      [0 0 0]      [3]
                                [2 2 2]      [0 0 0]      [3]
                c_1() = [0]
                        [1]
                        [1]

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [3 3 3] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {if^#(true(), X, Y) -> c_2(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [0]
                         [2]
                if^#(x1, x2, x3) = [2 0 2] x1 + [7 7 7] x2 + [0 0 0] x3 + [7]
                                   [2 0 2]      [7 7 7]      [0 0 0]      [7]
                                   [2 0 2]      [7 7 7]      [0 0 0]      [7]
                c_2(x1) = [1 3 3] x1 + [0]
                          [1 1 1]      [1]
                          [1 1 1]      [1]

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [3 3 3] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {if^#(false(), X, Y) -> c_3(Y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [0]
                          [2]
                if^#(x1, x2, x3) = [2 0 2] x1 + [0 0 0] x2 + [7 7 7] x3 + [7]
                                   [2 0 2]      [0 0 0]      [7 7 7]      [7]
                                   [2 0 2]      [0 0 0]      [7 7 7]      [7]
                c_3(x1) = [1 3 3] x1 + [0]
                          [1 1 1]      [1]
                          [1 1 1]      [1]

           * Path {6}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

             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: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 0 2]      [3]
                        [0 0 1]      [2]
                add^#(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
                                [0 2 1]      [0 0 0]      [2]
                                [4 0 2]      [0 0 4]      [0]
                c_5(x1) = [1 0 0] x1 + [1]
                          [0 0 0]      [2]
                          [2 2 0]      [3]

           * Path {6}->{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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [3 3 3] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

             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: {add^#(0(), X) -> c_4(X)}
               Weak Rules: {add^#(s(X), Y) -> c_5(add^#(X, Y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 2] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [2]
                add^#(x1, x2) = [2 2 2] x1 + [0 0 0] x2 + [0]
                                [2 2 2]      [4 4 4]      [0]
                                [2 2 2]      [4 0 4]      [0]
                c_4(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [2]

           * Path {8}: YES(?,O(n^3))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                cons(x1, x2) = [1 3 3] x1 + [1 3 0] x2 + [0]
                               [0 1 3]      [0 1 0]      [0]
                               [0 0 1]      [0 0 0]      [0]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 1] x1 + [0]
                        [0 1 2]      [3]
                        [0 0 1]      [0]
                cons(x1, x2) = [1 2 0] x1 + [1 0 2] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [6]
                first^#(x1, x2) = [1 1 1] x1 + [2 0 1] x2 + [0]
                                  [4 2 0]      [0 0 0]      [2]
                                  [0 4 0]      [4 0 0]      [0]
                c_7(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [6]
                              [0 0 0]      [0 0 0]      [7]

           * Path {8}->{7}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(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]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_8(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {first^#(0(), X) -> c_6()}
               Weak Rules: {first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(cons) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 0] x1 + [2]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                cons(x1, x2) = [1 0 2] x1 + [1 3 0] x2 + [1]
                               [0 0 0]      [0 1 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                first^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                  [2 2 2]      [0 5 0]      [0]
                                  [0 0 0]      [1 1 0]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_7(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [7]
                              [0 0 0]      [0 0 0]      [2]
                              [1 0 0]      [0 0 0]      [1]

           * Path {9}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                false() = [0]
                          [0]
                          [0]
                if(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
                add(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 1 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 1]      [0]
                first(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                nil() = [0]
                        [0]
                        [0]
                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]
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                and^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                if^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                add^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                first^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                from^#(x1) = [1 3 3] x1 + [0]
                             [3 3 3]      [0]
                             [3 3 3]      [0]
                c_8(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]

             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_8(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

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

             {  1: and^#(true(), X) -> c_0(X)
              , 2: and^#(false(), Y) -> c_1()
              , 3: if^#(true(), X, Y) -> c_2(X)
              , 4: if^#(false(), X, Y) -> c_3(Y)
              , 5: add^#(0(), X) -> c_4(X)
              , 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
              , 7: first^#(0(), X) -> c_6()
              , 8: first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))
              , 9: from^#(X) -> c_8(X, from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [         NA         ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(true(), X) -> c_0(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                and^#(x1, x2) = [2 2] x1 + [7 7] x2 + [7]
                                [2 2]      [7 7]      [3]
                c_0(x1) = [1 3] x1 + [0]
                          [1 1]      [1]

           * Path {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(false(), Y) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                and^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                [2 2]      [0 0]      [7]
                c_1() = [0]
                        [1]

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {if^#(true(), X, Y) -> c_2(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                if^#(x1, x2, x3) = [2 2] x1 + [7 7] x2 + [0 0] x3 + [7]
                                   [2 2]      [7 7]      [0 0]      [3]
                c_2(x1) = [1 3] x1 + [0]
                          [1 1]      [1]

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {if^#(false(), X, Y) -> c_3(Y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                if^#(x1, x2, x3) = [2 2] x1 + [0 0] x2 + [7 7] x3 + [7]
                                   [2 2]      [0 0]      [7 7]      [3]
                c_3(x1) = [1 3] x1 + [0]
                          [1 1]      [1]

           * Path {6}: YES(?,O(n^1))
             -----------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {add^#(s(X), Y) -> c_5(add^#(X, Y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(add^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                add^#(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                                [0 0]      [0 4]      [4]
                c_5(x1) = [1 0] x1 + [0]
                          [0 0]      [3]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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 {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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [1 3] x1 + [1 3] x2 + [0]
                               [0 1]      [0 1]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [3 3] x1 + [1 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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 {8}->{7}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_8(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 {9}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                first(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                and^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                add^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                from^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_8(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [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_8(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               The input cannot be shown compatible

    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: and^#(true(), X) -> c_0(X)
              , 2: and^#(false(), Y) -> c_1()
              , 3: if^#(true(), X, Y) -> c_2(X)
              , 4: if^#(false(), X, Y) -> c_3(Y)
              , 5: add^#(0(), X) -> c_4(X)
              , 6: add^#(s(X), Y) -> c_5(add^#(X, Y))
              , 7: first^#(0(), X) -> c_6()
              , 8: first^#(s(X), cons(Y, Z)) -> c_7(Y, first^#(X, Z))
              , 9: from^#(X) -> c_8(X, from^#(s(X)))}

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

             ->{9}                                                       [       MAYBE        ]

             ->{8}                                                       [         NA         ]
                |
                `->{7}                                                   [         NA         ]

             ->{6}                                                       [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{4}                                                       [    YES(?,O(1))     ]

             ->{3}                                                       [    YES(?,O(1))     ]

             ->{2}                                                       [    YES(?,O(1))     ]

             ->{1}                                                       [    YES(?,O(1))     ]



         Sub-problems:
         -------------
           * Path {1}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(true(), X) -> c_0(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [5]
                and^#(x1, x2) = [3] x1 + [7] x2 + [0]
                c_0(x1) = [1] x1 + [0]

           * Path {2}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {and^#(false(), Y) -> c_1()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(and^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [7]
                and^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_1() = [1]

           * Path {3}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
                c_2(x1) = [1] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {if^#(true(), X, Y) -> c_2(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [5]
                if^#(x1, x2, x3) = [3] x1 + [7] x2 + [0] x3 + [0]
                c_2(x1) = [1] x1 + [0]

           * Path {4}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {if^#(false(), X, Y) -> c_3(Y)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [5]
                if^#(x1, x2, x3) = [3] x1 + [0] x2 + [7] x3 + [0]
                c_3(x1) = [1] x1 + [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {1}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [1] x1 + [1] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [1] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {8}->{7}: NA
             -----------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {2}, Uargs(from^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [1] x2 + [0]
                from^#(x1) = [0] x1 + [0]
                c_8(x1, x2) = [0] x1 + [0] x2 + [0]

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

           * Path {9}: 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(and) = {}, Uargs(if) = {}, Uargs(add) = {}, Uargs(s) = {},
                 Uargs(first) = {}, Uargs(cons) = {}, Uargs(from) = {},
                 Uargs(and^#) = {}, Uargs(c_0) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(add^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(first^#) = {},
                 Uargs(c_7) = {}, Uargs(from^#) = {}, Uargs(c_8) = {2}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                and(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                first(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                from(x1) = [0] x1 + [0]
                and^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                first^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
                from^#(x1) = [3] x1 + [0]
                c_8(x1, x2) = [2] x1 + [1] 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_8(X, from^#(s(X)))}
               Weak Rules: {}

             Proof Output:
               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.