Problem AProVE 08 thiemann40 modified

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	AProVE 08 thiemann40 modified

stdout:

MAYBE

Problem:
nonZero(0()) -> false()
nonZero(s(x)) -> true()
p(s(0())) -> 0()
p(s(s(x))) -> s(p(s(x)))
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x,0())
rand(x,y) -> if(nonZero(x),x,y)
if(false(),x,y) -> y
if(true(),x,y) -> rand(p(x),id_inc(y))

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	AProVE 08 thiemann40 modified

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	AProVE 08 thiemann40 modified

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  nonZero(0()) -> false()
     , nonZero(s(x)) -> true()
     , p(s(0())) -> 0()
     , p(s(s(x))) -> s(p(s(x)))
     , id_inc(x) -> x
     , id_inc(x) -> s(x)
     , random(x) -> rand(x, 0())
     , rand(x, y) -> if(nonZero(x), x, y)
     , if(false(), x, y) -> y
     , if(true(), x, y) -> rand(p(x), id_inc(y))}

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: nonZero^#(0()) -> c_0()
              , 2: nonZero^#(s(x)) -> c_1()
              , 3: p^#(s(0())) -> c_2()
              , 4: p^#(s(s(x))) -> c_3(p^#(s(x)))
              , 5: id_inc^#(x) -> c_4()
              , 6: id_inc^#(x) -> c_5()
              , 7: random^#(x) -> c_6(rand^#(x, 0()))
              , 8: rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
              , 9: if^#(false(), x, y) -> c_8()
              , 10: if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))}

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

             ->{7}                                                       [     inherited      ]
                |
                `->{8,10}                                                [         NA         ]
                    |
                    `->{9}                                               [         NA         ]

             ->{6}                                                       [    YES(?,O(1))     ]

             ->{5}                                                       [    YES(?,O(1))     ]

             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8() = [0]
                        [0]
                        [0]
                c_9(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: {nonZero^#(0()) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(nonZero^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                nonZero^#(x1) = [0 2 0] x1 + [7]
                                [2 2 0]      [3]
                                [2 2 2]      [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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8() = [0]
                        [0]
                        [0]
                c_9(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: {nonZero^#(s(x)) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8() = [0]
                        [0]
                        [0]
                c_9(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: {p^#(s(s(x))) -> c_3(p^#(s(x)))}
               Weak Rules: {}

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

           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8() = [0]
                        [0]
                        [0]
                c_9(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: {p^#(s(0())) -> c_2()}
               Weak Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                      [0]
                s(x1) = [1 3 2] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                p^#(x1) = [2 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [2 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [1 0 0]      [0]

           * Path {5}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8() = [0]
                        [0]
                        [0]
                c_9(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: {id_inc^#(x) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [0 0 0] x1 + [7]
                               [0 0 0]      [7]
                               [0 0 0]      [7]
                c_4() = [0]
                        [3]
                        [3]

           * Path {6}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8() = [0]
                        [0]
                        [0]
                c_9(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: {id_inc^#(x) -> c_5()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [0 0 0] x1 + [7]
                               [0 0 0]      [7]
                               [0 0 0]      [7]
                c_5() = [0]
                        [3]
                        [3]

           * Path {7}: inherited
             -------------------

             This path is subsumed by the proof of path {7}->{8,10}.

           * Path {7}->{8,10}: NA
             --------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {7}->{8,10}->{9}: NA
             -------------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {1}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {1}, Uargs(rand^#) = {1, 2}, Uargs(c_7) = {1},
                 Uargs(if^#) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [2]
                              [0 0 0]      [3]
                              [0 0 0]      [3]
                0() = [0]
                      [0]
                      [0]
                false() = [1]
                          [1]
                          [1]
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 0]      [0]
                        [0 0 0]      [2]
                true() = [1]
                         [1]
                         [1]
                p(x1) = [2 0 1] x1 + [0]
                        [0 0 0]      [0]
                        [1 0 0]      [0]
                id_inc(x1) = [3 3 3] x1 + [3]
                             [0 3 3]      [3]
                             [0 0 3]      [3]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                rand^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                if^#(x1, x2, x3) = [3 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_8() = [0]
                        [0]
                        [0]
                c_9(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^2))

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

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

             {  1: nonZero^#(0()) -> c_0()
              , 2: nonZero^#(s(x)) -> c_1()
              , 3: p^#(s(0())) -> c_2()
              , 4: p^#(s(s(x))) -> c_3(p^#(s(x)))
              , 5: id_inc^#(x) -> c_4()
              , 6: id_inc^#(x) -> c_5()
              , 7: random^#(x) -> c_6(rand^#(x, 0()))
              , 8: rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
              , 9: if^#(false(), x, y) -> c_8()
              , 10: if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))}

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

             ->{7}                                                       [     inherited      ]
                |
                `->{8,10}                                                [       MAYBE        ]
                    |
                    `->{9}                                               [         NA         ]

             ->{6}                                                       [    YES(?,O(1))     ]

             ->{5}                                                       [    YES(?,O(1))     ]

             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(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: {nonZero^#(0()) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(nonZero^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                nonZero^#(x1) = [2 0] x1 + [7]
                                [2 2]      [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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(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: {nonZero^#(s(x)) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(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: {p^#(s(s(x))) -> c_3(p^#(s(x)))}
               Weak Rules: {}

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

           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(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: {p^#(s(0())) -> c_2()}
               Weak Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                s(x1) = [0 2] x1 + [2]
                        [0 1]      [0]
                p^#(x1) = [2 0] x1 + [0]
                          [2 0]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [3]

           * Path {5}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(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: {id_inc^#(x) -> c_4()}
               Weak Rules: {}

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

           * Path {6}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(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: {id_inc^#(x) -> c_5()}
               Weak Rules: {}

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

           * Path {7}: inherited
             -------------------

             This path is subsumed by the proof of path {7}->{8,10}.

           * Path {7}->{8,10}: MAYBE
             -----------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  random^#(x) -> c_6(rand^#(x, 0()))
                  , rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
                  , if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))
                  , nonZero(0()) -> false()
                  , nonZero(s(x)) -> true()
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))
                  , id_inc(x) -> x
                  , id_inc(x) -> s(x)}

             Proof Output:
               The input cannot be shown compatible

           * Path {7}->{8,10}->{9}: NA
             -------------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {1}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {1}, Uargs(rand^#) = {1, 2}, Uargs(c_7) = {1},
                 Uargs(if^#) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [3]
                              [3 3]      [3]
                0() = [0]
                      [0]
                false() = [0]
                          [1]
                s(x1) = [1 0] x1 + [1]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [2 0] x1 + [0]
                        [2 0]      [0]
                id_inc(x1) = [3 3] x1 + [3]
                             [0 3]      [3]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                rand^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                if^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             {  1: nonZero^#(0()) -> c_0()
              , 2: nonZero^#(s(x)) -> c_1()
              , 3: p^#(s(0())) -> c_2()
              , 4: p^#(s(s(x))) -> c_3(p^#(s(x)))
              , 5: id_inc^#(x) -> c_4()
              , 6: id_inc^#(x) -> c_5()
              , 7: random^#(x) -> c_6(rand^#(x, 0()))
              , 8: rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
              , 9: if^#(false(), x, y) -> c_8()
              , 10: if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))}

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

             ->{7}                                                       [     inherited      ]
                |
                `->{8,10}                                                [       MAYBE        ]
                    |
                    `->{9}                                               [         NA         ]

             ->{6}                                                       [    YES(?,O(1))     ]

             ->{5}                                                       [    YES(?,O(1))     ]

             ->{4}                                                       [   YES(?,O(n^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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(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: {nonZero^#(0()) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(nonZero^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                nonZero^#(x1) = [1] x1 + [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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(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: {nonZero^#(s(x)) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [1] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [3] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                p^#(x1) = [1] x1 + [0]
                c_3(x1) = [1] x1 + [3]

           * Path {4}->{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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(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: {p^#(s(0())) -> c_2()}
               Weak Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [3]
                p^#(x1) = [2] x1 + [2]
                c_2() = [1]
                c_3(x1) = [1] x1 + [0]

           * Path {5}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(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: {id_inc^#(x) -> c_4()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [0] x1 + [7]
                c_4() = [0]

           * Path {6}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {}, Uargs(rand^#) = {}, Uargs(c_7) = {},
                 Uargs(if^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(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: {id_inc^#(x) -> c_5()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [0] x1 + [7]
                c_5() = [0]

           * Path {7}: inherited
             -------------------

             This path is subsumed by the proof of path {7}->{8,10}.

           * Path {7}->{8,10}: MAYBE
             -----------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  random^#(x) -> c_6(rand^#(x, 0()))
                  , rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
                  , if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))
                  , nonZero(0()) -> false()
                  , nonZero(s(x)) -> true()
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))
                  , id_inc(x) -> x
                  , id_inc(x) -> s(x)}

             Proof Output:
               The input cannot be shown compatible

           * Path {7}->{8,10}->{9}: NA
             -------------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {1}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(random^#) = {},
                 Uargs(c_6) = {1}, Uargs(rand^#) = {1, 2}, Uargs(c_7) = {1},
                 Uargs(if^#) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [3] x1 + [3]
                0() = [1]
                false() = [1]
                s(x1) = [1] x1 + [1]
                true() = [1]
                p(x1) = [2] x1 + [0]
                id_inc(x1) = [3] x1 + [3]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                rand^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7(x1) = [1] x1 + [0]
                if^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	AProVE 08 thiemann40 modified

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	AProVE 08 thiemann40 modified

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  nonZero(0()) -> false()
     , nonZero(s(x)) -> true()
     , p(s(0())) -> 0()
     , p(s(s(x))) -> s(p(s(x)))
     , id_inc(x) -> x
     , id_inc(x) -> s(x)
     , random(x) -> rand(x, 0())
     , rand(x, y) -> if(nonZero(x), x, y)
     , if(false(), x, y) -> y
     , if(true(), x, y) -> rand(p(x), id_inc(y))}

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: nonZero^#(0()) -> c_0()
              , 2: nonZero^#(s(x)) -> c_1()
              , 3: p^#(s(0())) -> c_2()
              , 4: p^#(s(s(x))) -> c_3(p^#(s(x)))
              , 5: id_inc^#(x) -> c_4(x)
              , 6: id_inc^#(x) -> c_5(x)
              , 7: random^#(x) -> c_6(rand^#(x, 0()))
              , 8: rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
              , 9: if^#(false(), x, y) -> c_8(y)
              , 10: if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))}

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

             ->{7}                                                       [     inherited      ]
                |
                `->{8,10}                                                [         NA         ]
                    |
                    `->{9}                                               [         NA         ]

             ->{6}                                                       [    YES(?,O(1))     ]

             ->{5}                                                       [    YES(?,O(1))     ]

             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [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]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {nonZero^#(0()) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(nonZero^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                nonZero^#(x1) = [0 2 0] x1 + [7]
                                [2 2 0]      [3]
                                [2 2 2]      [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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [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]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {nonZero^#(s(x)) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [3 3 3]      [0]
                          [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [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]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

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

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

           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [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]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]

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

             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {p^#(s(0())) -> c_2()}
               Weak Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                      [0]
                s(x1) = [1 3 2] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                p^#(x1) = [2 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [2 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [1 0 0]      [0]

           * Path {5}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [3 3 3] x1 + [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) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {id_inc^#(x) -> c_4(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [7 7 7] x1 + [7]
                               [7 7 7]      [7]
                               [7 7 7]      [7]
                c_4(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]

           * Path {6}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                p(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [3 3 3] x1 + [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) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                rand^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 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_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {id_inc^#(x) -> c_5(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [7 7 7] x1 + [7]
                               [7 7 7]      [7]
                               [7 7 7]      [7]
                c_5(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]

           * Path {7}: inherited
             -------------------

             This path is subsumed by the proof of path {7}->{8,10}.

           * Path {7}->{8,10}: NA
             --------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

           * Path {7}->{8,10}->{9}: NA
             -------------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {1}, Uargs(s) = {1}, Uargs(p) = {1},
                 Uargs(id_inc) = {1}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {1},
                 Uargs(rand^#) = {1, 2}, Uargs(c_7) = {1}, Uargs(if^#) = {1, 2, 3},
                 Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [2 0 0] x1 + [2]
                              [0 0 0]      [3]
                              [0 0 0]      [3]
                0() = [1]
                      [0]
                      [0]
                false() = [1]
                          [1]
                          [1]
                s(x1) = [1 0 0] x1 + [1]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                true() = [1]
                         [1]
                         [1]
                p(x1) = [2 0 0] x1 + [1]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                id_inc(x1) = [3 3 3] x1 + [3]
                             [0 3 3]      [3]
                             [0 0 3]      [3]
                random(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                rand(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 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]
                nonZero^#(x1) = [0 0 0] x1 + [0]
                                [0 0 0]      [0]
                                [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                p^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                id_inc^#(x1) = [0 0 0] x1 + [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]
                random^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                rand^#(x1, x2) = [3 0 0] x1 + [3 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_7(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                if^#(x1, x2, x3) = [3 0 0] x1 + [3 3 3] x2 + [3 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_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_9(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             {  1: nonZero^#(0()) -> c_0()
              , 2: nonZero^#(s(x)) -> c_1()
              , 3: p^#(s(0())) -> c_2()
              , 4: p^#(s(s(x))) -> c_3(p^#(s(x)))
              , 5: id_inc^#(x) -> c_4(x)
              , 6: id_inc^#(x) -> c_5(x)
              , 7: random^#(x) -> c_6(rand^#(x, 0()))
              , 8: rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
              , 9: if^#(false(), x, y) -> c_8(y)
              , 10: if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))}

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

             ->{7}                                                       [     inherited      ]
                |
                `->{8,10}                                                [       MAYBE        ]
                    |
                    `->{9}                                               [         NA         ]

             ->{6}                                                       [    YES(?,O(1))     ]

             ->{5}                                                       [    YES(?,O(1))     ]

             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {nonZero^#(0()) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(nonZero^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                nonZero^#(x1) = [2 0] x1 + [7]
                                [2 2]      [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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {nonZero^#(s(x)) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}
               Weak Rules: {}

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

           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {p^#(s(0())) -> c_2()}
               Weak Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                s(x1) = [0 2] x1 + [2]
                        [0 1]      [0]
                p^#(x1) = [2 0] x1 + [0]
                          [2 0]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [3]

           * Path {5}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [3 3] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {id_inc^#(x) -> c_4(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [7 7] x1 + [7]
                               [7 7]      [7]
                c_4(x1) = [1 3] x1 + [0]
                          [3 1]      [3]

           * Path {6}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                true() = [0]
                         [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                id_inc(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [3 3] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                rand^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {id_inc^#(x) -> c_5(x)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(id_inc^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                id_inc^#(x1) = [7 7] x1 + [7]
                               [7 7]      [7]
                c_5(x1) = [1 3] x1 + [0]
                          [3 1]      [3]

           * Path {7}: inherited
             -------------------

             This path is subsumed by the proof of path {7}->{8,10}.

           * Path {7}->{8,10}: MAYBE
             -----------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  random^#(x) -> c_6(rand^#(x, 0()))
                  , rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
                  , if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))
                  , nonZero(0()) -> false()
                  , nonZero(s(x)) -> true()
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))
                  , id_inc(x) -> x
                  , id_inc(x) -> s(x)}

             Proof Output:
               The input cannot be shown compatible

           * Path {7}->{8,10}->{9}: NA
             -------------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {1}, Uargs(s) = {1}, Uargs(p) = {1},
                 Uargs(id_inc) = {1}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {1},
                 Uargs(rand^#) = {1, 2}, Uargs(c_7) = {1}, Uargs(if^#) = {1, 2, 3},
                 Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [2 0] x1 + [0]
                              [3 3]      [3]
                0() = [2]
                      [0]
                false() = [1]
                          [1]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                true() = [1]
                         [1]
                p(x1) = [2 0] x1 + [0]
                        [2 0]      [0]
                id_inc(x1) = [3 3] x1 + [3]
                             [0 3]      [3]
                random(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                rand(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                nonZero^#(x1) = [0 0] x1 + [0]
                                [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                id_inc^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                random^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                rand^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                if^#(x1, x2, x3) = [3 0] x1 + [3 3] x2 + [3 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

             {  1: nonZero^#(0()) -> c_0()
              , 2: nonZero^#(s(x)) -> c_1()
              , 3: p^#(s(0())) -> c_2()
              , 4: p^#(s(s(x))) -> c_3(p^#(s(x)))
              , 5: id_inc^#(x) -> c_4(x)
              , 6: id_inc^#(x) -> c_5(x)
              , 7: random^#(x) -> c_6(rand^#(x, 0()))
              , 8: rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
              , 9: if^#(false(), x, y) -> c_8(y)
              , 10: if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))}

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

             ->{7}                                                       [     inherited      ]
                |
                `->{8,10}                                                [       MAYBE        ]
                    |
                    `->{9}                                               [         NA         ]

             ->{6}                                                       [    YES(?,O(1))     ]

             ->{5}                                                       [    YES(?,O(1))     ]

             ->{4}                                                       [   YES(?,O(n^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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {nonZero^#(0()) -> c_0()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(nonZero^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                nonZero^#(x1) = [1] x1 + [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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {nonZero^#(s(x)) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [1] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [3] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                p^#(x1) = [1] x1 + [0]
                c_3(x1) = [1] x1 + [3]

           * Path {4}->{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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {1}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {p^#(s(0())) -> c_2()}
               Weak Rules: {p^#(s(s(x))) -> c_3(p^#(s(x)))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [3]
                p^#(x1) = [2] x1 + [2]
                c_2() = [1]
                c_3(x1) = [1] x1 + [0]

           * Path {5}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [3] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {id_inc^#(x) -> c_4(x)}
               Weak Rules: {}

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

           * Path {6}: 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(nonZero) = {}, Uargs(s) = {}, Uargs(p) = {},
                 Uargs(id_inc) = {}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {},
                 Uargs(rand^#) = {}, Uargs(c_7) = {}, Uargs(if^#) = {},
                 Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                true() = [0]
                p(x1) = [0] x1 + [0]
                id_inc(x1) = [0] x1 + [0]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [3] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                rand^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(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:    DP runtime-complexity with respect to
               Strict Rules: {id_inc^#(x) -> c_5(x)}
               Weak Rules: {}

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

           * Path {7}: inherited
             -------------------

             This path is subsumed by the proof of path {7}->{8,10}.

           * Path {7}->{8,10}: MAYBE
             -----------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE

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

             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  random^#(x) -> c_6(rand^#(x, 0()))
                  , rand^#(x, y) -> c_7(if^#(nonZero(x), x, y))
                  , if^#(true(), x, y) -> c_9(rand^#(p(x), id_inc(y)))
                  , nonZero(0()) -> false()
                  , nonZero(s(x)) -> true()
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))
                  , id_inc(x) -> x
                  , id_inc(x) -> s(x)}

             Proof Output:
               The input cannot be shown compatible

           * Path {7}->{8,10}->{9}: NA
             -------------------------

             The usable rules for this path are:

               {  nonZero(0()) -> false()
                , nonZero(s(x)) -> true()
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , id_inc(x) -> x
                , id_inc(x) -> s(x)}

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(nonZero) = {1}, Uargs(s) = {1}, Uargs(p) = {1},
                 Uargs(id_inc) = {1}, Uargs(random) = {}, Uargs(rand) = {},
                 Uargs(if) = {}, Uargs(nonZero^#) = {}, Uargs(p^#) = {},
                 Uargs(c_3) = {}, Uargs(id_inc^#) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(random^#) = {}, Uargs(c_6) = {1},
                 Uargs(rand^#) = {1, 2}, Uargs(c_7) = {1}, Uargs(if^#) = {1, 2, 3},
                 Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nonZero(x1) = [2] x1 + [0]
                0() = [2]
                false() = [1]
                s(x1) = [1] x1 + [2]
                true() = [1]
                p(x1) = [2] x1 + [0]
                id_inc(x1) = [3] x1 + [3]
                random(x1) = [0] x1 + [0]
                rand(x1, x2) = [0] x1 + [0] x2 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                nonZero^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                p^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                id_inc^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                random^#(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]
                rand^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_7(x1) = [1] x1 + [0]
                if^#(x1, x2, x3) = [3] x1 + [3] x2 + [3] x3 + [0]
                c_8(x1) = [1] x1 + [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))

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

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

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

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