Problem HirokawaMiddeldorp 04 t002

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	HirokawaMiddeldorp 04 t002

stdout:

MAYBE

Problem:
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
if(true(),x,y) -> x
if(false(),x,y) -> y
-(x,0()) -> x
-(s(x),s(y)) -> -(x,y)
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if(leq(y,x),mod(-(s(x),s(y)),s(y)),s(x))

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	HirokawaMiddeldorp 04 t002

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	HirokawaMiddeldorp 04 t002

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

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

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

             ->{10}                                                      [     inherited      ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]

             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {leq^#(s(x), s(y)) -> c_2(leq^#(x, y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                leq^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                                [0 0]      [0 0]      [4]
                c_2(x1) = [1 0] x1 + [1]
                          [0 0]      [3]

           * Path {3}->{1}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {leq^#(0(), y) -> c_0()}
               Weak Rules: {leq^#(s(x), s(y)) -> c_2(leq^#(x, y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                leq^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                                [3 3]      [0 1]      [0]
                c_0() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]

           * Path {3}->{2}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [2]
                leq^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {1},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                -^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                              [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {-^#(s(x), s(y)) -> c_6(-^#(x, y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                -^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [4]
                c_6(x1) = [1 0] x1 + [1]
                          [0 0]      [3]

           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {1},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {-^#(x, 0()) -> c_5()}
               Weak Rules: {-^#(s(x), s(y)) -> c_6(-^#(x, y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 3] x1 + [2]
                        [0 1]      [2]
                -^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
                              [0 2]      [3 2]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [7]
                          [0 0]      [7]

           * Path {8}: 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(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [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: {mod^#(0(), y) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [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: {mod^#(s(x), 0()) -> c_8()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(mod^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [2]
                mod^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                                [0 0]      [0 0]      [3]
                c_8() = [0]
                        [1]

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{4}.

           * Path {10}->{4}: NA
             ------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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 {10}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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.

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

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

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

             ->{10}                                                      [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [         NA         ]

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {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(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                mod(x1, x2) = [0] x1 + [0] x2 + [0]
                leq^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_3() = [0]
                c_4() = [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                mod^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [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: {leq^#(s(x), s(y)) -> c_2(leq^#(x, y))}
               Weak Rules: {}

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

           * Path {3}->{1}: 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(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                mod(x1, x2) = [0] x1 + [0] x2 + [0]
                leq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_3() = [0]
                c_4() = [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                mod^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [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: {leq^#(0(), y) -> c_0()}
               Weak Rules: {leq^#(s(x), s(y)) -> c_2(leq^#(x, y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                leq^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_0() = [1]
                c_2(x1) = [1] x1 + [5]

           * Path {3}->{2}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                leq^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_1() = [1]
                c_2(x1) = [1] x1 + [7]

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

             The usable rules of this path are empty.

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

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

           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                -^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_5() = [1]
                c_6(x1) = [1] x1 + [6]

           * Path {8}: 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(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                mod(x1, x2) = [0] x1 + [0] x2 + [0]
                leq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_3() = [0]
                c_4() = [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                mod^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [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: {mod^#(0(), y) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(mod^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [0] x1 + [2]
                mod^#(x1, x2) = [2] x1 + [2] x2 + [7]
                c_8() = [0]

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{4}.

           * Path {10}->{4}: MAYBE
             ---------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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:
                 {  mod^#(s(x), s(y)) ->
                    c_9(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)))
                  , if^#(true(), x, y) -> c_3()
                  , leq(0(), y) -> true()
                  , leq(s(x), 0()) -> false()
                  , leq(s(x), s(y)) -> leq(x, y)
                  , -(x, 0()) -> x
                  , -(s(x), s(y)) -> -(x, y)
                  , mod(0(), y) -> 0()
                  , mod(s(x), 0()) -> 0()
                  , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                  , if(true(), x, y) -> x
                  , if(false(), x, y) -> y}

             Proof Output:
               The input cannot be shown compatible

           * Path {10}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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.

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

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

    5) '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	HirokawaMiddeldorp 04 t002

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	HirokawaMiddeldorp 04 t002

stdout:

MAYBE

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

Proof Output:
  None of the processors succeeded.

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

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

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

             ->{10}                                                      [     inherited      ]
                |
                |->{4}                                                   [         NA         ]
                |
                `->{5}                                                   [         NA         ]

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]

             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]



         Sub-problems:
         -------------
           * Path {3}: YES(?,O(n^2))
             -----------------------

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {leq^#(s(x), s(y)) -> c_2(leq^#(x, y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                leq^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                                [0 0]      [0 0]      [4]
                c_2(x1) = [1 0] x1 + [1]
                          [0 0]      [3]

           * Path {3}->{1}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {leq^#(0(), y) -> c_0()}
               Weak Rules: {leq^#(s(x), s(y)) -> c_2(leq^#(x, y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                leq^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                                [3 3]      [0 1]      [0]
                c_0() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]

           * Path {3}->{2}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [2]
                leq^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                              [3 3]      [3 3]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {-^#(s(x), s(y)) -> c_6(-^#(x, y))}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                -^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [4]
                c_6(x1) = [1 0] x1 + [1]
                          [0 0]      [3]

           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_5(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {-^#(x, 0()) -> c_5(x)}
               Weak Rules: {-^#(s(x), s(y)) -> c_6(-^#(x, y))}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [2 6] x2 + [0]
                              [2 5]      [5 3]      [0]
                c_5(x1) = [0 0] x1 + [1]
                          [0 1]      [0]
                c_6(x1) = [1 0] x1 + [3]
                          [0 0]      [2]

           * Path {8}: 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(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [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: {mod^#(0(), y) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                mod(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                leq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(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_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                mod^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [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: {mod^#(s(x), 0()) -> c_8()}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(mod^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [2]
                mod^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                                [0 0]      [0 0]      [3]
                c_8() = [0]
                        [1]

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{4}.

           * Path {10}->{4}: NA
             ------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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 {10}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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.

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

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

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

             ->{10}                                                      [     inherited      ]
                |
                |->{4}                                                   [       MAYBE        ]
                |
                `->{5}                                                   [         NA         ]

             ->{9}                                                       [    YES(?,O(1))     ]

             ->{8}                                                       [    YES(?,O(1))     ]

             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]

             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]



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

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

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                leq^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_0() = [1]
                c_2(x1) = [1] x1 + [5]

           * Path {3}->{2}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                leq^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_1() = [1]
                c_2(x1) = [1] x1 + [7]

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

             The usable rules of this path are empty.

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

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

           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [1] x1 + [2]
                -^#(x1, x2) = [2] x1 + [2] x2 + [1]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [1] x1 + [0]

           * Path {8}: 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(leq) = {}, Uargs(s) = {}, Uargs(if) = {}, Uargs(-) = {},
                 Uargs(mod) = {}, Uargs(leq^#) = {}, Uargs(c_2) = {},
                 Uargs(if^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(-^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(mod^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                leq(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                mod(x1, x2) = [0] x1 + [0] x2 + [0]
                leq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                mod^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_7() = [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:    DP runtime-complexity with respect to
               Strict Rules: {mod^#(0(), y) -> c_7()}
               Weak Rules: {}

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

           * Path {9}: YES(?,O(1))
             ---------------------

             The usable rules of this path are empty.

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

             Proof Output:
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(mod^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [0] x1 + [2]
                mod^#(x1, x2) = [2] x1 + [2] x2 + [7]
                c_8() = [0]

           * Path {10}: inherited
             --------------------

             This path is subsumed by the proof of path {10}->{4}.

           * Path {10}->{4}: MAYBE
             ---------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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:
                 {  mod^#(s(x), s(y)) ->
                    c_9(if^#(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x)))
                  , if^#(true(), x, y) -> c_3(x)
                  , leq(0(), y) -> true()
                  , leq(s(x), 0()) -> false()
                  , leq(s(x), s(y)) -> leq(x, y)
                  , -(x, 0()) -> x
                  , -(s(x), s(y)) -> -(x, y)
                  , mod(0(), y) -> 0()
                  , mod(s(x), 0()) -> 0()
                  , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                  , if(true(), x, y) -> x
                  , if(false(), x, y) -> y}

             Proof Output:
               The input cannot be shown compatible

           * Path {10}->{5}: NA
             ------------------

             The usable rules for this path are:

               {  leq(0(), y) -> true()
                , leq(s(x), 0()) -> false()
                , leq(s(x), s(y)) -> leq(x, y)
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)
                , mod(0(), y) -> 0()
                , mod(s(x), 0()) -> 0()
                , mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
                , if(true(), x, y) -> x
                , if(false(), x, y) -> y}

             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.

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

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

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