Problem Strategy outermost added 08 muladd

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 muladd

stdout:

MAYBE

Problem:
*(X,+(Y,1())) -> +(*(X,+(Y,*(1(),0()))),X)
*(X,1()) -> X
*(X,0()) -> X
*(X,0()) -> 0()

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 muladd

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 muladd

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
     , *(X, 1()) -> X
     , *(X, 0()) -> X
     , *(X, 0()) -> 0()}

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: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
              , 2: *^#(X, 1()) -> c_1()
              , 3: *^#(X, 0()) -> c_2()
              , 4: *^#(X, 0()) -> c_3()}

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

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

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

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

             ->{1}                                                       [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: NA
             ------------

             The usable rules for this path are:

               {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                , *(X, 1()) -> X
                , *(X, 0()) -> X
                , *(X, 0()) -> 0()}

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

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

           * Path {2}: 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(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                +(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                1() = [0]
                      [0]
                      [0]
                0() = [0]
                      [0]
                      [0]
                *^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]

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

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

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                +(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                1() = [0]
                      [0]
                      [0]
                0() = [0]
                      [0]
                      [0]
                *^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]

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

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

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                +(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                1() = [0]
                      [0]
                      [0]
                0() = [0]
                      [0]
                      [0]
                *^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]

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

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

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

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

             {  1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
              , 2: *^#(X, 1()) -> c_1()
              , 3: *^#(X, 0()) -> c_2()
              , 4: *^#(X, 0()) -> c_3()}

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------

             The usable rules for this path are:

               {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                , *(X, 1()) -> X
                , *(X, 0()) -> X
                , *(X, 0()) -> 0()}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
                  , *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                  , *(X, 1()) -> X
                  , *(X, 0()) -> X
                  , *(X, 0()) -> 0()}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                +(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                1() = [0]
                      [0]
                0() = [0]
                      [0]
                *^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [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: {*^#(X, 1()) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                +(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                1() = [0]
                      [0]
                0() = [0]
                      [0]
                *^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [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: {*^#(X, 0()) -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                +(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                1() = [0]
                      [0]
                0() = [0]
                      [0]
                *^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [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: {*^#(X, 0()) -> c_3()}
               Weak Rules: {}

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

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

             {  1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
              , 2: *^#(X, 1()) -> c_1()
              , 3: *^#(X, 0()) -> c_2()
              , 4: *^#(X, 0()) -> c_3()}

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------

             The usable rules for this path are:

               {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                , *(X, 1()) -> X
                , *(X, 0()) -> X
                , *(X, 0()) -> 0()}

             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:
                 {  *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))))
                  , *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                  , *(X, 1()) -> X
                  , *(X, 0()) -> X
                  , *(X, 0()) -> 0()}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0] x1 + [0] x2 + [0]
                +(x1, x2) = [0] x1 + [0] x2 + [0]
                1() = [0]
                0() = [0]
                *^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [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: {*^#(X, 1()) -> c_1()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0] x1 + [0] x2 + [0]
                +(x1, x2) = [0] x1 + [0] x2 + [0]
                1() = [0]
                0() = [0]
                *^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [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: {*^#(X, 0()) -> c_2()}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0] x1 + [0] x2 + [0]
                +(x1, x2) = [0] x1 + [0] x2 + [0]
                1() = [0]
                0() = [0]
                *^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [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: {*^#(X, 0()) -> c_3()}
               Weak Rules: {}

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

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

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

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

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 muladd

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 muladd

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
     , *(X, 1()) -> X
     , *(X, 0()) -> X
     , *(X, 0()) -> 0()}

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: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
              , 2: *^#(X, 1()) -> c_1(X)
              , 3: *^#(X, 0()) -> c_2(X)
              , 4: *^#(X, 0()) -> c_3()}

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

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

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

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

             ->{1}                                                       [         NA         ]



         Sub-problems:
         -------------
           * Path {1}: NA
             ------------

             The usable rules for this path are:

               {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                , *(X, 1()) -> X
                , *(X, 0()) -> X
                , *(X, 0()) -> 0()}

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

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

           * Path {2}: 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(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                +(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                1() = [0]
                      [0]
                      [0]
                0() = [0]
                      [0]
                      [0]
                *^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]

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

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

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                +(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                1() = [0]
                      [0]
                      [0]
                0() = [0]
                      [0]
                      [0]
                *^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]

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

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

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                +(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                1() = [0]
                      [0]
                      [0]
                0() = [0]
                      [0]
                      [0]
                *^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3() = [0]
                        [0]
                        [0]

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

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

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

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

             {  1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
              , 2: *^#(X, 1()) -> c_1(X)
              , 3: *^#(X, 0()) -> c_2(X)
              , 4: *^#(X, 0()) -> c_3()}

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------

             The usable rules for this path are:

               {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                , *(X, 1()) -> X
                , *(X, 0()) -> X
                , *(X, 0()) -> 0()}

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

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

             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
                  , *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                  , *(X, 1()) -> X
                  , *(X, 0()) -> X
                  , *(X, 0()) -> 0()}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                +(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                1() = [0]
                      [0]
                0() = [0]
                      [0]
                *^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [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: {*^#(X, 1()) -> c_1(X)}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                +(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                1() = [0]
                      [0]
                0() = [0]
                      [0]
                *^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_3() = [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: {*^#(X, 0()) -> c_2(X)}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                +(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                1() = [0]
                      [0]
                0() = [0]
                      [0]
                *^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3() = [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: {*^#(X, 0()) -> c_3()}
               Weak Rules: {}

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

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

             {  1: *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
              , 2: *^#(X, 1()) -> c_1(X)
              , 3: *^#(X, 0()) -> c_2(X)
              , 4: *^#(X, 0()) -> c_3()}

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

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

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

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

             ->{1}                                                       [       MAYBE        ]



         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------

             The usable rules for this path are:

               {  *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                , *(X, 1()) -> X
                , *(X, 0()) -> X
                , *(X, 0()) -> 0()}

             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:
                 {  *^#(X, +(Y, 1())) -> c_0(*^#(X, +(Y, *(1(), 0()))), X)
                  , *(X, +(Y, 1())) -> +(*(X, +(Y, *(1(), 0()))), X)
                  , *(X, 1()) -> X
                  , *(X, 0()) -> X
                  , *(X, 0()) -> 0()}

             Proof Output:
               The input cannot be shown compatible

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0] x1 + [0] x2 + [0]
                +(x1, x2) = [0] x1 + [0] x2 + [0]
                1() = [0]
                0() = [0]
                *^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [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: {*^#(X, 1()) -> c_1(X)}
               Weak Rules: {}

             Proof Output:
               The following argument positions are usable:
                 Uargs(*^#) = {}, Uargs(c_1) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1() = [5]
                *^#(x1, x2) = [7] x1 + [3] x2 + [0]
                c_1(x1) = [1] x1 + [0]

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0] x1 + [0] x2 + [0]
                +(x1, x2) = [0] x1 + [0] x2 + [0]
                1() = [0]
                0() = [0]
                *^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                c_3() = [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: {*^#(X, 0()) -> c_2(X)}
               Weak Rules: {}

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

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

             The usable rules of this path are empty.

             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(*) = {}, Uargs(+) = {}, Uargs(*^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                *(x1, x2) = [0] x1 + [0] x2 + [0]
                +(x1, x2) = [0] x1 + [0] x2 + [0]
                1() = [0]
                0() = [0]
                *^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3() = [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: {*^#(X, 0()) -> c_3()}
               Weak Rules: {}

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

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

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

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